Electrochemical Methods: Fundamentals and Applications [3 ed.] 1119334063, 9781119334064

Table of contents :
Cover
Title Page
Copyright
Contents
Preface
Major Symbols and Abbreviations
About the Companion Website
Chapter 1 Overview of Electrode Processes
1.1 Basic Ideas
1.1.1 Electrochemical Cells and Reactions
1.1.2 Interfacial Potential Differences and Cell Potential
1.1.3 Reference Electrodes and Control of Potential at a Working Electrode
1.1.4 Potential as an Expression of Electron Energy
1.1.5 Current as an Expression of Reaction Rate
1.1.6 Magnitudes in Electrochemical Systems
1.1.7 Current–Potential Curves
1.1.8 Control of Current vs. Control of Potential
1.1.9 Faradaic and Nonfaradaic Processes
1.2 Faradaic Processes and Factors Affecting Rates of Electrode Reactions
1.2.1 Electrochemical Cells—Types and Definitions
1.2.2 The Electrochemical Experiment and Variables in Electrochemical Cells
1.2.3 Factors Affecting Electrode Reaction Rate and Current
1.3 Mass‐Transfer‐Controlled Reactions
1.3.1 Modes of Mass Transfer
1.3.2 Semiempirical Treatment of Steady‐State Mass Transfer
1.4 Semiempirical Treatment of Nernstian Reactions with Coupled Chemical Reactions
1.4.1 Coupled Reversible Reactions
1.4.2 Coupled Irreversible Chemical Reactions
1.5 Cell Resistance and the Measurement of Potential
1.5.1 Components of the Applied Voltage When Current Flows
1.5.2 Two‐Electrode Cells
1.5.3 Three‐Electrode Cells
1.5.4 Uncompensated Resistance
1.6 The Electrode/Solution Interface and Charging Current
1.6.1 The Ideally Polarizable Electrode
1.6.2 Capacitance and Charge at an Electrode
1.6.3 Brief Description of the Electrical Double Layer
1.6.4 Double‐Layer Capacitance and Charging Current
1.7 Organization of this Book
1.8 The Literature of Electrochemistry
1.8.1 Reference Sources
1.8.2 Sources on Laboratory Techniques
1.8.3 Review Series
1.9 Lab Note: Potentiostats and Cell Behavior
1.9.1 Potentiostats
1.9.2 Background Processes in Actual Cells
1.9.3 Further Work with Simple RC Networks
1.10 References
1.11 Problems
Chapter 2 Potentials and Thermodynamics of Cells
2.1 Basic Electrochemical Thermodynamics
2.1.1 Reversibility
2.1.2 Reversibility and Gibbs Free Energy
2.1.3 Free Energy and Cell emf
2.1.4 Half‐Reactions and Standard Electrode Potentials
2.1.5 Standard States and Activity
2.1.6 emf and Concentration
2.1.7 Formal Potentials
2.1.8 Reference Electrodes
2.1.9 Potential–pH Diagrams and Thermodynamic Predictions
2.2 A More Detailed View of Interfacial Potential Differences
2.2.1 The Physics of Phase Potentials
2.2.2 Interactions Between Conducting Phases
2.2.3 Measurement of Potential Differences
2.2.4 Electrochemical Potentials
2.2.5 Fermi Energy and Absolute Potential
2.3 Liquid Junction Potentials
2.3.1 Potential Differences at an Electrolyte–Electrolyte Boundary
2.3.2 Types of Liquid Junctions
2.3.3 Conductance, Transference Numbers, and Mobility
2.3.4 Calculation of Liquid Junction Potentials
2.3.5 Minimizing Liquid Junction Potentials
2.3.6 Junctions of Two Immiscible Liquids
2.4 Ion‐Selective Electrodes
2.4.1 Selective Interfaces
2.4.2 Glass Electrodes
2.4.3 Other Ion‐Selective Electrodes
2.4.4 Gas‐Sensing ISEs
2.5 Lab Note: Practical Use of Reference Electrodes
2.5.1 Leakage at the Reference Tip
2.5.2 Quasireference Electrodes
2.6 References
2.7 Problems
Chapter 3 Basic Kinetics of Electrode Reactions
3.1 Review of Homogeneous Kinetics
3.1.1 Dynamic Equilibrium
3.1.2 The Arrhenius Equation and Potential Energy Surfaces
3.1.3 Transition State Theory
3.2 Essentials of Electrode Reactions
3.3 Butler–Volmer Model of Electrode Kinetics
3.3.1 Effects of Potential on Energy Barriers
3.3.2 One‐Step, One‐Electron Process
3.3.3 The Standard Rate Constant
3.3.4 The Transfer Coefficient
3.4 Implications of the Butler–Volmer Model for the One‐Step, One‐Electron Process
3.4.1 Equilibrium Conditions and the Exchange Current
3.4.2 The Current–Overpotential Equation
3.4.3 Approximate Forms of the i–η Equation
3.4.4 Exchange Current Plots
3.4.5 Very Facile Kinetics and Reversible Behavior
3.4.6 Effects of Mass Transfer
3.4.7 Limits of Basic Butler–Volmer Equations
3.5 Microscopic Theories of Charge Transfer
3.5.1 Inner‐Sphere and Outer‐Sphere Electrode Reactions
3.5.2 Extended Charge Transfer and Adiabaticity
3.5.3 The Marcus Microscopic Model
3.5.4 Implications of the Marcus Theory
3.5.5 A Model Based on Distributions of Energy States
3.6 Open‐Circuit Potential and Multiple Half‐Reactions at an Electrode
3.6.1 Open‐Circuit Potential in Multicomponent Systems
3.6.2 Establishment or Loss of Nernstian Behavior at an Electrode
3.6.3 Multiple Half‐Reaction Currents in i–E Curves
3.7 Multistep Mechanisms
3.7.1 The Primacy of One‐Electron Transfers
3.7.2 Rate‐Determining, Outer‐Sphere Electron Transfer
3.7.3 Multistep Processes at Equilibrium
3.7.4 Nernstian Multistep Processes
3.7.5 Quasireversible and Irreversible Multistep Processes
3.8 References
3.9 Problems
Chapter 4 Mass Transfer by Migration and Diffusion
4.1 General Mass‐Transfer Equations
4.2 Migration in Bulk Solution
4.3 Mixed Migration and Diffusion Near an Active Electrode
4.3.1 Balance Sheets for Mass Transfer During Electrolysis
4.3.2 Utility of a Supporting Electrolyte
4.4 Diffusion
4.4.1 A Microscopic View
4.4.2 Fick's Laws of Diffusion
4.4.3 Flux of an Electroreactant at an Electrode Surface
4.5 Formulation and Solution of Mass‐Transfer Problems
4.5.1 Initial and Boundary Conditions in Electrochemical Problems
4.5.2 General Formulation of a Linear Diffusion Problem
4.5.3 Systems Involving Migration or Convection
4.5.4 Practical Means for Reaching Solutions
4.6 References
4.7 Problems
Chapter 5 Steady‐State Voltammetry at Ultramicroelectrodes
5.1 Steady‐State Voltammetry at a Spherical UME
5.1.1 Steady‐State Diffusion
5.1.2 Steady‐State Current
5.1.3 Convergence on the Steady State
5.1.4 Steady‐State Voltammetry
5.2 Shapes and Properties of Ultramicroelectrodes
5.2.1 Spherical or Hemispherical UME
5.2.2 Disk UME
5.2.3 Cylindrical UME
5.2.4 Band UME
5.2.5 Summary of Steady‐State Behavior at UMEs
5.3 Reversible Electrode Reactions
5.3.1 Shape of the Wave
5.3.2 Applications of Reversible i–E Curves
5.4 Quasireversible and Irreversible Electrode Reactions
5.4.1 Effect of Electrode Kinetics on Steady‐State Responses
5.4.2 Total Irreversibility
5.4.3 Kinetic Regimes
5.4.4 Influence of Electrode Shape
5.4.5 Applications of Irreversible i–E Curves
5.4.6 Evaluation of Kinetic Parameters by Varying Mass‐Transfer Rates
5.5 Multicomponent Systems and Multistep Charge Transfers
5.6 Additional Attributes of Ultramicroelectrodes
5.6.1 Uncompensated Resistance at a UME
5.6.2 Effects of Conductivity on Voltammetry at a UME
5.6.3 Applications Based on Spatial Resolution
5.7 Migration in Steady‐State Voltammetry
5.7.1 Mathematical Approach to Problems Involving Migration
5.7.2 Concentration Profiles in the Diffusion–Migration Layer
5.7.3 Wave Shape at Low Electrolyte Concentration
5.7.4 Effects of Migration on Wave Height in SSV
5.8 Analysis at High Analyte Concentrations
5.9 Lab Note: Preparation of Ultramicroelectrodes
5.9.1 Preparation and Characterization of UMEs
5.9.2 Testing the Integrity of a UME
5.9.3 Estimating the Size of a UME
5.10 References
5.11 Problems
Chapter 6 Transient Methods Based on Potential Steps
6.1 Chronoamperometry Under Diffusion Control
6.1.1 Linear Diffusion at a Plane
6.1.2 Response at a Spherical Electrode
6.1.3 Transients at Other Ultramicroelectrodes
6.1.4 Information from Chronoamperometric Results
6.1.5 Microscopic and Geometric Areas
6.2 Sampled‐Transient Voltammetry for Reversible Electrode Reactions
6.2.1 A Step to an Arbitrary Potential
6.2.2 Shape of the Voltammogram
6.2.3 Concentration Profiles When R Is Initially Absent
6.2.4 Simplified Current–Concentration Relationships
6.2.5 Applications of Reversible i–E Curves
6.3 Sampled‐Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions
6.3.1 Effect of Electrode Kinetics on Transient Behavior
6.3.2 Sampled‐Transient Voltammetry for Reduction of O
6.3.3 Sampled Transient Voltammetry for Oxidation of R
6.3.4 Totally Irreversible Reactions
6.3.5 Kinetic Regimes
6.3.6 Applications of Irreversible i–E Curves
6.4 Multicomponent Systems and Multistep Charge Transfers
6.5 Chronoamperometric Reversal Techniques
6.5.1 Approaches to the Problem
6.5.2 Current–Time Responses
6.6 Chronocoulometry
6.6.1 Large‐Amplitude Potential Step
6.6.2 Reversal Experiments Under Diffusion Control
6.6.3 Effects of Heterogeneous Kinetics
6.7 Cell Time Constants at Microelectrodes
6.8 Lab Note: Practical Concerns with Potential Step Methods
6.8.1 Preparation of the Electrode Surface at a Microelectrode
6.8.2 Interference from Charging Current
6.9 References
6.10 Problems
Chapter 7 Linear Sweep and Cyclic Voltammetry
7.1 Transient Responses to a Potential Sweep
7.2 Nernstian (Reversible) Systems
7.2.1 Linear Sweep Voltammetry
7.2.2 Cyclic Voltammetry
7.3 Quasireversible Systems
7.3.1 Linear Sweep Voltammetry
7.3.2 Cyclic Voltammetry
7.4 Totally Irreversible Systems
7.4.1 Linear Sweep Voltammetry
7.4.2 Cyclic Voltammetry
7.5 Multicomponent Systems and Multistep Charge Transfers
7.5.1 Multicomponent Systems
7.5.2 Multistep Charge Transfers
7.6 Fast Cyclic Voltammetry
7.7 Convolutive Transformation
7.8 Voltammetry at Liquid–Liquid Interfaces
7.8.1 Experimental Approach to Voltammetry
7.8.2 Effect of Interfacial Potential on Composition
7.8.3 Voltammetric Behavior
7.9 Lab Note: Practical Aspects of Cyclic Voltammetry
7.9.1 Basic Experimental Conditions
7.9.2 Choice of Initial and Final Potentials
7.9.3 Deaeration
7.10 References
7.11 Problems
Chapter 8 Polarography, Pulse Voltammetry, and Square‐Wave Voltammetry
8.1 Polarography
8.1.1 The Dropping Mercury Electrode
8.1.2 The Ilkovič Equation
8.1.3 Polarographic Waves
8.1.4 Practical Advantages of the DME
8.1.5 Polarographic Analysis
8.1.6 Residual Current and Detection Limits
8.2 Normal Pulse Voltammetry
8.2.1 Implementation
8.2.2 Renewal at Stationary Electrodes
8.2.3 Normal Pulse Polarography
8.2.4 Practical Application
8.3 Reverse Pulse Voltammetry
8.4 Differential Pulse Voltammetry
8.4.1 Concept of the Method
8.4.2 Theory
8.4.3 Renewal vs. Pre‐Electrolysis
8.4.4 Residual Currents
8.4.5 Differential Pulse Polarography
8.5 Square‐Wave Voltammetry
8.5.1 Experimental Concept and Practice
8.5.2 Theoretical Prediction of Response
8.5.3 Background Currents
8.5.4 Applications
8.6 Analysis by Pulse Voltammetry
8.7 References
8.8 Problems
Chapter 9 Controlled‐Current Techniques
9.1 Introduction to Chronopotentiometry
9.2 Theory of Controlled‐Current Methods
9.2.1 General Treatment for Linear Diffusion
9.2.2 Constant‐Current Electrolysis—The Sand Equation
9.2.3 Programmed Current
Chronopotentiometry
9.3 Potential–Time Curves in Constant‐Current Electrolysis
9.3.1 Reversible (Nernstian) Waves
9.3.2 Totally Irreversible Waves
9.3.3 Quasireversible Waves
9.3.4 Practical Issues in the Measurement of Transition Time
9.4 Reversal Techniques
9.4.1 Response Function Principle
9.4.2 Current Reversal
9.5 Multicomponent Systems and Multistep Reactions
9.6 The Galvanostatic Double Pulse Method
9.7 Charge Step (Coulostatic) Methods
9.7.1 Small Excursions
9.7.2 Large Excursions
9.7.3 Coulostatic Perturbation by Temperature Jump
9.8 References
9.9 Problems
Chapter 10 Methods Involving Forced Convection—Hydrodynamic Methods
10.1 Theory of Convective Systems
10.1.1 The Convective‐Diffusion Equation
10.1.2 Determination of the Velocity Profile
10.2 Rotating Disk Electrode
10.2.1 The Velocity Profile at a Rotating Disk
10.2.2 Solution of the Convective‐Diffusion Equation
10.2.3 Concentration Profile
10.2.4 General i–E Curves at the RDE
10.2.5 The Koutecký–Levich Method
10.2.6 Current Distribution at the RDE
10.2.7 Practical Considerations for Application of the RDE
10.3 Rotating Ring and Ring‐Disk Electrodes
10.3.1 Rotating Ring Electrode
10.3.2 The Rotating Ring‐Disk Electrode
10.4 Transient Currents
10.4.1 Transients at the RDE
10.4.2 Transients at the RRDE
10.5 Modulation of the RDE
10.6 Electrohydrodynamic Phenomena
10.7 References
10.8 Problems
Chapter 11 Electrochemical Impedance Spectroscopy and ac Voltammetry
11.1 A Simple Measurement of Cell Impedance
11.2 Brief Review of ac Circuits
11.3 Equivalent Circuits of a Cell
11.3.1 The Randles Equivalent Circuit
11.3.2 Interpretation of the Faradaic Impedance
11.3.3 Behavior and Uses of the Faradaic Impedance
11.4 Electrochemical Impedance Spectroscopy
11.4.1 Conditions of Measurement
11.4.2 A System with Simple Faradaic Kinetics
11.4.3 Measurement of Resistance and Capacitance
11.4.4 A Confined Electroactive Domain
11.4.5 Other Applications
11.5 ac Voltammetry
11.5.1 Reversible Systems
11.5.2 Quasireversible and Irreversible Systems
11.5.3 Cyclic ac Voltammetry
11.6 Nonlinear Responses
11.6.1 Second Harmonic ac Voltammetry
11.6.2 Large Amplitude ac Voltammetry
11.7 Chemical Analysis by ac Voltammetry
11.8 Instrumentation for Electrochemical Impedance Methods
11.8.1 Frequency‐Domain Instruments
11.8.2 Time‐Domain Instruments
11.9 Analysis of Data in the Laplace Plane
11.10 References
11.11 Problems
Chapter 12 Bulk Electrolysis
12.1 General Considerations
12.1.1 Completeness of an Electrode Process
12.1.2 Current Efficiency
12.1.3 Experimental Concerns
12.2 Controlled‐Potential Methods
12.2.1 Current–Time Behavior
12.2.2 Practical Aspects
12.2.3 Coulometry
12.2.4 Electrogravimetry
12.2.5 Electroseparations
12.3 Controlled‐Current Methods
12.3.1 Characteristics of Controlled‐Current Electrolysis
12.3.2 Coulometric Titrations
12.3.3 Practical Aspects of Constant‐Current Electrolysis
12.4 Electrometric End‐Point Detection
12.4.1 Current–Potential Curves During Titration
12.4.2 Potentiometric Methods
12.4.3 Amperometric Methods
12.5 Flow Electrolysis
12.5.1 Mathematical Treatment
12.5.2 Dual‐Electrode Flow Cells
12.5.3 Microfluidic Flow Cells
12.6 Thin‐Layer Electrochemistry
12.6.1 Chronoamperometry and Coulometry
12.6.2 Potential Sweep in a Nernstian System
12.6.3 Dual‐Electrode Thin‐Layer Cells
12.6.4 Applications of the Thin‐Layer Concept
12.7 Stripping Analysis
12.7.1 Introduction
12.7.2 Principles and Theory
12.7.3 Applications and Variations
12.8 References
12.9 Problems
Chapter 13 Electrode Reactions with Coupled Homogeneous Chemical Reactions
13.1 Classification of Reactions
13.1.1 Reactions with One E‐Step
13.1.2 Reactions with Two or More E‐Steps
13.2 Impact of Coupled Reactions on Cyclic Voltammetry
13.2.1 Diagnostic Criteria
13.2.2 Characteristic Times
13.2.3 An Example
13.2.4 Including Kinetics in Theory
13.2.5 Comparative Simulation
13.3 Survey of Behavior
13.3.1 Following Reaction—Case ErCi
13.3.2 Effect of Electrode Kinetics in ECi Systems
13.3.3 Bidirectional Following Reaction
13.3.4 Catalytic Reaction—Case ErCi′
13.3.5 Preceding Reaction—Case CrEr
13.3.6 Multistep Electron Transfers
13.3.7 ECE/DISP Reactions
13.3.8 Concerted vs. Stepwise Reaction
13.3.9 Elaboration of Reaction Schemes
13.4 Behavior with Other Electrochemical Methods
13.5 References
13.6 Problems
Chapter 14 Double‐Layer Structure and Adsorption
14.1 Thermodynamics of the Double Layer
14.1.1 The Gibbs Adsorption Isotherm
14.1.2 The Electrocapillary Equation
14.1.3 Relative Surface Excesses
14.2 Experimental Evaluations
14.2.1 Electrocapillarity
14.2.2 Excess Charge and Capacitance
14.2.3 Relative Surface Excesses
14.3 Models for Double‐Layer Structure
14.3.1 The Helmholtz Model
14.3.2 The Gouy–Chapman Theory
14.3.3 Stern's Modification
14.3.4 Specific Adsorption
14.4 Studies at Solid Electrodes
14.4.1 Well‐Defined Single‐Crystal Electrode Surfaces
14.4.2 The Double Layer at Solids
14.5 Extent and Rate of Specific Adsorption
14.5.1 Nature and Extent of Specific Adsorption
14.5.2 Electrosorption Valency
14.5.3 Adsorption Isotherms
14.5.4 Rate of Adsorption
14.6 Practical Aspects of Adsorption
14.7 Double‐Layer Effects on Electrode Reaction Rates
14.7.1 Introduction and Principles
14.7.2 Double‐Layer Effects Without Specific Adsorption of Electrolyte
14.7.3 Double‐Layer Effects with Specific Adsorption
14.7.4 Diffuse Double‐Layer Effects on Mass Transport
14.8 References
14.9 Problems
Chapter 15 Inner‐Sphere Electrode Reactions and Electrocatalysis
15.1 Inner‐Sphere Heterogenous Electron‐Transfer Reactions
15.1.1 The Role of the Electrode Surface
15.1.2 Energetics of 1e Electron‐Transfer Reactions
15.1.3 Adsorption Energies
15.2 Electrocatalytic Reaction Mechanisms
15.2.1 Hydrogen Evolution Reaction
15.2.2 Tafel Plot Analysis of HER Kinetics
15.3 Additional Examples of Inner‐Sphere Reactions
15.3.1 Oxygen Reduction Reaction
15.3.2 Chlorine Evolution
15.3.3 Methanol Oxidation
15.3.4 CO2 Reduction
15.3.5 Oxidation of NH3 to N2
15.3.6 Organic Halide Reduction
15.3.7 Hydrogen Peroxide Oxidation and Reduction
15.4 Computational Analyses of Inner‐Sphere Electron‐Transfer Reactions
15.4.1 Density Functional Theory Analysis of Electrocatalytic Reactions
15.4.2 Hydrogen Evolution Reaction
15.4.3 Oxygen Reduction Reaction
15.5 Electrocatalytic Correlations
15.6 Electrochemical Phase Transformations
15.6.1 Nucleation and Growth of a New Phase
15.6.2 Classical Nucleation Theory
15.6.3 Electrodeposition
15.6.4 Gas Evolution
15.7 References
15.8 Problems
Chapter 16 Electrochemical Instrumentation
16.1 Operational Amplifiers
16.1.1 Ideal Properties
16.1.2 Nonidealities
16.2 Current Feedback
16.2.1 Current Follower
16.2.2 Scaler/Inverter
16.2.3 Adders
16.2.4 Integrators
16.3 Voltage Feedback
16.3.1 Voltage Follower
16.3.2 Control Functions
16.4 Potentiostats
16.4.1 Basic Considerations
16.4.2 The Adder Potentiostat
16.4.3 Refinements to the Adder Potentiostat
16.4.4 Bipotentiostats
16.4.5 Four‐Electrode Potentiostats
16.5 Galvanostats
16.6 Integrated Electrochemical Instrumentation
16.7 Difficulties with Potential Control
16.7.1 Types of Control Problems
16.7.2 Cell Properties and Electrode Placement
16.7.3 Electronic Compensation of Resistance
16.8 Measurement of Low Currents
16.8.1 Fundamental Limits
16.8.2 Practical Considerations
16.8.3 Current Amplifier
16.8.4 Simplified Instruments and Cells
16.9 Instruments for Short Time Scales
16.10 Lab Note: Practical Use of Electrochemical Instruments
16.10.1 Caution Regarding Electrochemical Workstations
16.10.2 Troubleshooting Electrochemical Systems
16.11 References
16.12 Problems
Chapter 17 Electroactive Layers and Modified Electrodes
17.1 Monolayers and Submonolayers on Electrodes
17.2 Cyclic Voltammetry of Adsorbed Layers
17.2.1 Fundamentals
17.2.2 Reversible Adsorbate Couples
17.2.3 Irreversible Adsorbate Couples
17.2.4 Nernstian Processes Involving Adsorbates and Solutes
17.2.5 More Complex Systems
17.2.6 Electric‐Field‐Driven Acid–Base Chemistry in Adsorbate Layers
17.3 Other Useful Methods for Adsorbed Monolayers
17.3.1 Chronocoulometry
17.3.2 Coulometry in Thin‐Layer Cells
17.3.3 Impedance Measurements
17.3.4 Chronopotentiometry
17.4 Thick Modification Layers on Electrodes
17.5 Dynamics in Modification Layers
17.5.1 Steady State at a Rotating Disk
17.5.2 Principal Dynamic Processes in Modifying Films
17.5.3 Interplay of Dynamical Elements
17.6 Blocking Layers
17.6.1 Permeation Through Pores and Pinholes
17.6.2 Tunneling Through Blocking Films
17.7 Other Methods for Characterizing Layers on Electrodes
17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification
17.8.1 Electrocatalysis
17.8.2 Bioelectrocatalysis Based on Enzyme‐Modified Electrodes
17.8.3 Electrochemical Sensors
17.8.4 Faradaic Electrochemical Measurements in vivo
17.9 References
17.10 Problems
Chapter 18 Scanning Electrochemical Microscopy
18.1 Principles
18.2 Approach Curves
18.3 Imaging Surface Topography and Reactivity
18.3.1 Imaging Based on Conductivity of the Substrate
18.3.2 Imaging Based on Heterogeneous Electron‐Transfer Reactivity
18.3.3 Simultaneous Imaging of Topography and Reactivity
18.4 Measurements of Kinetics
18.4.1 Heterogeneous Electron‐Transfer Reactions
18.4.2 Homogeneous Reactions
18.5 Surface Interrogation
18.6 Potentiometric Tips
18.7 Other Applications
18.7.1 Detection of Species Released from Surfaces, Films, or Pores
18.7.2 Biological Systems
18.7.3 Probing the Interior of a Layer on a Substrate
18.8 Scanning Electrochemical Cell Microscopy
18.9 References
18.10 Problems
Chapter 19 Single‐Particle Electrochemistry
19.1 General Considerations in Single‐Particle Electrochemistry
19.2 Particle Collision Experiments
19.3 Particle Collision Rate at a Disk‐Shaped UME
19.3.1 Collision Frequency
19.3.2 Variance in the Number of Particle Collisions
19.3.3 Time of First Arrival
19.4 Nanoparticle Collision Behavior
19.4.1 Blocking Collisions
19.4.2 Electrocatalytic Amplification Collisions
19.4.3 Electrolysis Collisions
19.5 Electrochemistry at Single Atoms and Atomic Clusters
19.6 Single‐Molecule Electrochemistry
19.7 References
19.8 Problems
Chapter 20 Photoelectrochemistry and Electrogenerated Chemiluminescence
20.1 Solid Materials
20.1.1 The Band Model
20.1.2 Categories of Pure Crystalline Solids
20.1.3 Doped Semiconductors
20.1.4 Fermi Energy
20.1.5 Highly Conducting Oxides
20.2 Semiconductor Electrodes
20.2.1 Interface at a Semiconducting Electrode in the Dark
20.2.2 Current–Potential Curves at Semiconductor Electrodes
20.2.3 Conducting Polymer Electrodes
20.3 Photoelectrochemistry at Semiconductors
20.3.1 Photoeffects at Semiconductor Electrodes
20.3.2 Photoelectrochemical Systems
20.3.3 Dye Sensitization
20.3.4 Surface Photocatalytic Processes at Semiconductor Particles
20.4 Radiolytic Products in Solution
20.4.1 Photoemission of Electrons from an Electrode
20.4.2 Detection and Use of Radiolytic Products in Solution
20.4.3 Photogalvanic Cells
20.5 Electrogenerated Chemiluminescence
20.5.1 Chemical Fundamentals
20.5.2 Fundamental Studies of Radical‐Ion Annihilation
20.5.3 Single‐Potential Generation Based on a Coreactant
20.5.4 ECL Based on Quantum Dots
20.5.5 Analytical Applications of ECL
20.5.6 ECL Beyond the Solution Phase
20.6 References
20.7 Problems
Chapter 21 In situ Characterization of Electrochemical Systems
21.1 Microscopy
21.1.1 Scanning Tunneling Microscopy
21.1.2 Atomic Force Microscopy
21.1.3 Optical Microscopy
21.1.4 Transmission Electron Microscopy
21.2 Quartz Crystal Microbalance
21.2.1 Basic Method
21.2.2 QCM with Dissipation Monitoring
21.3 UV–Visible Spectrometry
21.3.1 Absorption Spectroscopy with Thin‐Layer Cells
21.3.2 Ellipsometry
21.3.3 Surface Plasmon Resonance
21.4 Vibrational Spectroscopy
21.4.1 Infrared Spectroscopy
21.4.2 Raman Spectroscopy
21.5 X‐Ray Methods
21.6 Mass Spectrometry
21.7 Magnetic Resonance Spectroscopy
21.7.1 ESR
21.7.2 NMR
21.8 Ex‐situ Techniques
21.8.1 Electron Microscopy
21.8.2 Electron and Ion Spectrometry
21.9 References
A Mathematical Methods
A.1 Solving Differential Equations by the Laplace Transform Technique
A.1.1 Partial Differential Equations
A.1.2 Introduction to the Laplace Transformation
A.1.3 Fundamental Properties of the Transform
A.1.4 Solving Ordinary Differential Equations by Laplace Transformation
A.1.5 Simultaneous Linear Ordinary Differential Equations
A.1.6 Mass‐Transfer Problems Based on Partial Differential Equations
A.1.7 The Zero‐Shift Theorem
A.2 Taylor Expansions
A.2.1 Expansion of a Function of Several Variables
A.2.2 Expansion of a Function of a Single Variable
A.2.3 Maclaurin Series
A.3 The Error Function and the Gaussian Distribution
A.4 Leibnitz Rule
A.5 Complex Notation
A.6 Fourier Series and Fourier Transformation
A.7 References
A.8 Problems
B Basic Concepts of Simulation
B.1 Setting Up the Model
B.1.1 A Discrete System
B.1.2 Diffusion
B.1.3 Dimensionless Parameters
B.1.4 Time
B.1.5 Distance
B.1.6 Current
B.1.7 Thickness of the Diffusion Layer
B.1.8 Diffusion Coefficients
B.2 An Example
B.2.1 Organization of the Spreadsheet
B.2.2 Concentration Arrays
B.2.3 Results and Error Detection
B.2.4 Performance
B.3 Incorporating Homogeneous Kinetics
B.3.1 Unimolecular Reactions
B.3.2 Bimolecular Reactions
B.4 Boundary Conditions for Various Techniques
B.4.1 Potential Steps in Nernstian Systems
B.4.2 Heterogeneous Kinetics
B.4.3 Potential Sweeps
B.4.4 Controlled Current
B.5 More Complex Systems
B.6 References
B.7 Problems
C Reference Tables
References
Index

Citation preview

ISTUDY

Electrochemical Methods

Electrochemical Methods Fundamentals and Applications

Third Edition

Allen J. Bard Department of Chemistry and Center for Electrochemistry University of Texas at Austin

Larry R. Faulkner Department of Chemistry and Center for Electrochemistry University of Texas at Austin

Henry S. White Department of Chemistry University of Utah

This edition first published 2022 © 2022 John Wiley & Sons, Ltd Edition History John Wiley & Sons Ltd (1e, 1980; 2e, 2001) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/ permissions. The right of Allen J. Bard, Larry R. Faulkner and Henry S. White to be identified as the authors of this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office 9600 Garsington Road, Oxford, OX4 2DQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress Cataloging-in-Publication Data applied for: ISBN: 9781119334064 Cover Design: Wiley Cover Image: Courtesy of the Authors Set in 10/12pt Warnock by Straive, Chennai, India

10 9 8 7 6 5 4 3 2 1

v

Contents Preface xxi Major Symbols and Abbreviations xxv About the Companion Website liii 1

1.1

1.2

1.3

1.4

1.5

1.6

Overview of Electrode Processes 1 Basic Ideas 2 1.1.1 Electrochemical Cells and Reactions 2 1.1.2 Interfacial Potential Differences and Cell Potential 4 1.1.3 Reference Electrodes and Control of Potential at a Working Electrode 1.1.4 Potential as an Expression of Electron Energy 6 1.1.5 Current as an Expression of Reaction Rate 6 1.1.6 Magnitudes in Electrochemical Systems 8 1.1.7 Current–Potential Curves 9 1.1.8 Control of Current vs. Control of Potential 16 1.1.9 Faradaic and Nonfaradaic Processes 17 Faradaic Processes and Factors Affecting Rates of Electrode Reactions 17 1.2.1 Electrochemical Cells—Types and Definitions 17 1.2.2 The Electrochemical Experiment and Variables in Electrochemical Cells 18 1.2.3 Factors Affecting Electrode Reaction Rate and Current 21 Mass-Transfer-Controlled Reactions 23 1.3.1 Modes of Mass Transfer 24 1.3.2 Semiempirical Treatment of Steady-State Mass Transfer 25 Semiempirical Treatment of Nernstian Reactions with Coupled Chemical Reactions 31 1.4.1 Coupled Reversible Reactions 31 1.4.2 Coupled Irreversible Chemical Reactions 32 Cell Resistance and the Measurement of Potential 34 1.5.1 Components of the Applied Voltage When Current Flows 35 1.5.2 Two-Electrode Cells 37 1.5.3 Three-Electrode Cells 37 1.5.4 Uncompensated Resistance 38 The Electrode/Solution Interface and Charging Current 41 1.6.1 The Ideally Polarizable Electrode 41 1.6.2 Capacitance and Charge at an Electrode 41 1.6.3 Brief Description of the Electrical Double Layer 42 1.6.4 Double-Layer Capacitance and Charging Current 44

5

vi

Contents

1.7 1.8

1.9

1.10 1.11 2

2.1

2.2

2.3

2.4

2.5

2.6 2.7

Organization of this Book 51 The Literature of Electrochemistry 52 1.8.1 Reference Sources 52 1.8.2 Sources on Laboratory Techniques 53 1.8.3 Review Series 53 Lab Note: Potentiostats and Cell Behavior 54 1.9.1 Potentiostats 54 1.9.2 Background Processes in Actual Cells 55 1.9.3 Further Work with Simple RC Networks 56 References 57 Problems 57 Potentials and Thermodynamics of Cells 61 Basic Electrochemical Thermodynamics 61 2.1.1 Reversibility 61 2.1.2 Reversibility and Gibbs Free Energy 64 2.1.3 Free Energy and Cell emf 64 2.1.4 Half-Reactions and Standard Electrode Potentials 66 2.1.5 Standard States and Activity 67 2.1.6 emf and Concentration 69 2.1.7 Formal Potentials 71 2.1.8 Reference Electrodes 72 2.1.9 Potential–pH Diagrams and Thermodynamic Predictions 76 A More Detailed View of Interfacial Potential Differences 80 2.2.1 The Physics of Phase Potentials 80 2.2.2 Interactions Between Conducting Phases 82 2.2.3 Measurement of Potential Differences 84 2.2.4 Electrochemical Potentials 85 2.2.5 Fermi Energy and Absolute Potential 88 Liquid Junction Potentials 91 2.3.1 Potential Differences at an Electrolyte–Electrolyte Boundary 91 2.3.2 Types of Liquid Junctions 91 2.3.3 Conductance, Transference Numbers, and Mobility 92 2.3.4 Calculation of Liquid Junction Potentials 96 2.3.5 Minimizing Liquid Junction Potentials 100 2.3.6 Junctions of Two Immiscible Liquids 101 Ion-Selective Electrodes 101 2.4.1 Selective Interfaces 101 2.4.2 Glass Electrodes 102 2.4.3 Other Ion-Selective Electrodes 106 2.4.4 Gas-Sensing ISEs 111 Lab Note: Practical Use of Reference Electrodes 112 2.5.1 Leakage at the Reference Tip 112 2.5.2 Quasireference Electrodes 112 References 113 Problems 116

Contents

3

3.1

3.2 3.3

3.4

3.5

3.6

3.7

3.8 3.9 4

4.1 4.2 4.3

4.4

Basic Kinetics of Electrode Reactions 121 Review of Homogeneous Kinetics 121 3.1.1 Dynamic Equilibrium 121 3.1.2 The Arrhenius Equation and Potential Energy Surfaces 122 3.1.3 Transition State Theory 123 Essentials of Electrode Reactions 125 Butler–Volmer Model of Electrode Kinetics 126 3.3.1 Effects of Potential on Energy Barriers 127 3.3.2 One-Step, One-Electron Process 127 3.3.3 The Standard Rate Constant 130 3.3.4 The Transfer Coefficient 131 Implications of the Butler–Volmer Model for the One-Step, One-Electron Process 132 3.4.1 Equilibrium Conditions and the Exchange Current 133 3.4.2 The Current–Overpotential Equation 133 3.4.3 Approximate Forms of the i–𝜂 Equation 135 3.4.4 Exchange Current Plots 139 3.4.5 Very Facile Kinetics and Reversible Behavior 139 3.4.6 Effects of Mass Transfer 140 3.4.7 Limits of Basic Butler–Volmer Equations 141 Microscopic Theories of Charge Transfer 142 3.5.1 Inner-Sphere and Outer-Sphere Electrode Reactions 142 3.5.2 Extended Charge Transfer and Adiabaticity 143 3.5.3 The Marcus Microscopic Model 146 3.5.4 Implications of the Marcus Theory 152 3.5.5 A Model Based on Distributions of Energy States 162 Open-Circuit Potential and Multiple Half-Reactions at an Electrode 168 3.6.1 Open-Circuit Potential in Multicomponent Systems 169 3.6.2 Establishment or Loss of Nernstian Behavior at an Electrode 170 3.6.3 Multiple Half-Reaction Currents in i–E Curves 171 Multistep Mechanisms 171 3.7.1 The Primacy of One-Electron Transfers 172 3.7.2 Rate-Determining, Outer-Sphere Electron Transfer 173 3.7.3 Multistep Processes at Equilibrium 173 3.7.4 Nernstian Multistep Processes 174 3.7.5 Quasireversible and Irreversible Multistep Processes 174 References 177 Problems 180 Mass Transfer by Migration and Diffusion 183 General Mass-Transfer Equations 183 Migration in Bulk Solution 186 Mixed Migration and Diffusion Near an Active Electrode 187 4.3.1 Balance Sheets for Mass Transfer During Electrolysis 188 4.3.2 Utility of a Supporting Electrolyte 192 Diffusion 193

vii

viii

Contents

4.5

4.6 4.7 5

5.1

5.2

5.3

5.4

5.5 5.6

5.7

5.8 5.9

5.10 5.11

4.4.1 A Microscopic View 193 4.4.2 Fick’s Laws of Diffusion 196 4.4.3 Flux of an Electroreactant at an Electrode Surface 199 Formulation and Solution of Mass-Transfer Problems 199 4.5.1 Initial and Boundary Conditions in Electrochemical Problems 4.5.2 General Formulation of a Linear Diffusion Problem 201 4.5.3 Systems Involving Migration or Convection 202 4.5.4 Practical Means for Reaching Solutions 202 References 204 Problems 205

200

Steady-State Voltammetry at Ultramicroelectrodes 207 Steady-State Voltammetry at a Spherical UME 207 5.1.1 Steady-State Diffusion 208 5.1.2 Steady-State Current 211 5.1.3 Convergence on the Steady State 211 5.1.4 Steady-State Voltammetry 212 Shapes and Properties of Ultramicroelectrodes 214 5.2.1 Spherical or Hemispherical UME 215 5.2.2 Disk UME 215 5.2.3 Cylindrical UME 221 5.2.4 Band UME 221 5.2.5 Summary of Steady-State Behavior at UMEs 222 Reversible Electrode Reactions 224 5.3.1 Shape of the Wave 224 5.3.2 Applications of Reversible i–E Curves 226 Quasireversible and Irreversible Electrode Reactions 230 5.4.1 Effect of Electrode Kinetics on Steady-State Responses 230 5.4.2 Total Irreversibility 232 5.4.3 Kinetic Regimes 234 5.4.4 Influence of Electrode Shape 234 5.4.5 Applications of Irreversible i–E Curves 235 5.4.6 Evaluation of Kinetic Parameters by Varying Mass-Transfer Rates Multicomponent Systems and Multistep Charge Transfers 239 Additional Attributes of Ultramicroelectrodes 241 5.6.1 Uncompensated Resistance at a UME 241 5.6.2 Effects of Conductivity on Voltammetry at a UME 242 5.6.3 Applications Based on Spatial Resolution 243 Migration in Steady-State Voltammetry 245 5.7.1 Mathematical Approach to Problems Involving Migration 245 5.7.2 Concentration Profiles in the Diffusion–Migration Layer 246 5.7.3 Wave Shape at Low Electrolyte Concentration 248 5.7.4 Effects of Migration on Wave Height in SSV 248 Analysis at High Analyte Concentrations 251 Lab Note: Preparation of Ultramicroelectrodes 253 5.9.1 Preparation and Characterization of UMEs 254 5.9.2 Testing the Integrity of a UME 254 5.9.3 Estimating the Size of a UME 256 References 257 Problems 258

237

Contents

6

6.1

6.2

6.3

6.4 6.5

6.6

6.7 6.8

6.9 6.10 7

7.1 7.2

7.3

7.4

7.5

Transient Methods Based on Potential Steps 261 Chronoamperometry Under Diffusion Control 261 6.1.1 Linear Diffusion at a Plane 262 6.1.2 Response at a Spherical Electrode 265 6.1.3 Transients at Other Ultramicroelectrodes 267 6.1.4 Information from Chronoamperometric Results 270 6.1.5 Microscopic and Geometric Areas 271 Sampled-Transient Voltammetry for Reversible Electrode Reactions 275 6.2.1 A Step to an Arbitrary Potential 276 6.2.2 Shape of the Voltammogram 277 6.2.3 Concentration Profiles When R Is Initially Absent 278 6.2.4 Simplified Current–Concentration Relationships 279 6.2.5 Applications of Reversible i–E Curves 279 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions 279 6.3.1 Effect of Electrode Kinetics on Transient Behavior 280 6.3.2 Sampled-Transient Voltammetry for Reduction of O 282 6.3.3 Sampled Transient Voltammetry for Oxidation of R 284 6.3.4 Totally Irreversible Reactions 285 6.3.5 Kinetic Regimes 287 6.3.6 Applications of Irreversible i–E Curves 287 Multicomponent Systems and Multistep Charge Transfers 289 Chronoamperometric Reversal Techniques 290 6.5.1 Approaches to the Problem 292 6.5.2 Current–Time Responses 293 Chronocoulometry 294 6.6.1 Large-Amplitude Potential Step 295 6.6.2 Reversal Experiments Under Diffusion Control 296 6.6.3 Effects of Heterogeneous Kinetics 299 Cell Time Constants at Microelectrodes 300 Lab Note: Practical Concerns with Potential Step Methods 303 6.8.1 Preparation of the Electrode Surface at a Microelectrode 303 6.8.2 Interference from Charging Current 305 References 306 Problems 307

311 Transient Responses to a Potential Sweep 311 Nernstian (Reversible) Systems 313 7.2.1 Linear Sweep Voltammetry 313 7.2.2 Cyclic Voltammetry 321 Quasireversible Systems 325 7.3.1 Linear Sweep Voltammetry 326 7.3.2 Cyclic Voltammetry 326 Totally Irreversible Systems 329 7.4.1 Linear Sweep Voltammetry 329 7.4.2 Cyclic Voltammetry 332 Multicomponent Systems and Multistep Charge Transfers 7.5.1 Multicomponent Systems 332 7.5.2 Multistep Charge Transfers 333 Linear Sweep and Cyclic Voltammetry

332

ix

x

Contents

7.6 7.7 7.8

7.10 7.11

Fast Cyclic Voltammetry 334 Convolutive Transformation 336 Voltammetry at Liquid–Liquid Interfaces 339 7.8.1 Experimental Approach to Voltammetry 340 7.8.2 Effect of Interfacial Potential on Composition 341 7.8.3 Voltammetric Behavior 341 Lab Note: Practical Aspects of Cyclic Voltammetry 344 7.9.1 Basic Experimental Conditions 344 7.9.2 Choice of Initial and Final Potentials 345 7.9.3 Deaeration 347 References 347 Problems 349

8

Polarography, Pulse Voltammetry, and Square-Wave Voltammetry 355

8.1

Polarography 355 8.1.1 The Dropping Mercury Electrode 355 8.1.2 The Ilkoviˇc Equation 356 8.1.3 Polarographic Waves 357 8.1.4 Practical Advantages of the DME 358 8.1.5 Polarographic Analysis 358 8.1.6 Residual Current and Detection Limits 359 Normal Pulse Voltammetry 361 8.2.1 Implementation 362 8.2.2 Renewal at Stationary Electrodes 363 8.2.3 Normal Pulse Polarography 364 8.2.4 Practical Application 366 Reverse Pulse Voltammetry 367 Differential Pulse Voltammetry 369 8.4.1 Concept of the Method 370 8.4.2 Theory 371 8.4.3 Renewal vs. Pre-Electrolysis 374 8.4.4 Residual Currents 375 8.4.5 Differential Pulse Polarography 375 Square-Wave Voltammetry 376 8.5.1 Experimental Concept and Practice 376 8.5.2 Theoretical Prediction of Response 377 8.5.3 Background Currents 380 8.5.4 Applications 381 Analysis by Pulse Voltammetry 383 References 385 Problems 386

7.9

8.2

8.3 8.4

8.5

8.6 8.7 8.8 9

9.1 9.2

Controlled-Current Techniques 389 Introduction to Chronopotentiometry 389 Theory of Controlled-Current Methods 391 9.2.1 General Treatment for Linear Diffusion 391 9.2.2 Constant-Current Electrolysis—The Sand Equation 392 9.2.3 Programmed Current Chronopotentiometry 394

Contents

9.3

9.4

9.5 9.6 9.7

9.8 9.9 10

10.1

10.2

10.3

10.4

10.5 10.6 10.7 10.8 11

11.1 11.2 11.3

11.4

Potential–Time Curves in Constant-Current Electrolysis 394 9.3.1 Reversible (Nernstian) Waves 394 9.3.2 Totally Irreversible Waves 394 9.3.3 Quasireversible Waves 395 9.3.4 Practical Issues in the Measurement of Transition Time Reversal Techniques 398 9.4.1 Response Function Principle 398 9.4.2 Current Reversal 398 Multicomponent Systems and Multistep Reactions 400 The Galvanostatic Double Pulse Method 401 Charge Step (Coulostatic) Methods 403 9.7.1 Small Excursions 404 9.7.2 Large Excursions 405 9.7.3 Coulostatic Perturbation by Temperature Jump 405 References 406 Problems 407

396

Methods Involving Forced Convection—Hydrodynamic Methods 411 Theory of Convective Systems 411 10.1.1 The Convective-Diffusion Equation 412 10.1.2 Determination of the Velocity Profile 412 Rotating Disk Electrode 414 10.2.1 The Velocity Profile at a Rotating Disk 414 10.2.2 Solution of the Convective-Diffusion Equation 416 10.2.3 Concentration Profile 418 10.2.4 General i–E Curves at the RDE 419 10.2.5 The Koutecký–Levich Method 420 10.2.6 Current Distribution at the RDE 423 10.2.7 Practical Considerations for Application of the RDE 426 Rotating Ring and Ring-Disk Electrodes 426 10.3.1 Rotating Ring Electrode 427 10.3.2 The Rotating Ring-Disk Electrode 428 Transient Currents 432 10.4.1 Transients at the RDE 432 10.4.2 Transients at the RRDE 433 Modulation of the RDE 435 Electrohydrodynamic Phenomena 436 References 439 Problems 440 Electrochemical Impedance Spectroscopy and ac Voltammetry 443 A Simple Measurement of Cell Impedance 444 Brief Review of ac Circuits 446 Equivalent Circuits of a Cell 450 11.3.1 The Randles Equivalent Circuit 451 11.3.2 Interpretation of the Faradaic Impedance 452 11.3.3 Behavior and Uses of the Faradaic Impedance 455 Electrochemical Impedance Spectroscopy 458

xi

xii

Contents

11.5

11.6

11.7 11.8

11.9 11.10 11.11 12

12.1

12.2

12.3

12.4

12.5

12.6

11.4.1 Conditions of Measurement 458 11.4.2 A System with Simple Faradaic Kinetics 460 11.4.3 Measurement of Resistance and Capacitance 465 11.4.4 A Confined Electroactive Domain 466 11.4.5 Other Applications 470 ac Voltammetry 470 11.5.1 Reversible Systems 470 11.5.2 Quasireversible and Irreversible Systems 473 11.5.3 Cyclic ac Voltammetry 477 Nonlinear Responses 477 11.6.1 Second Harmonic ac Voltammetry 478 11.6.2 Large Amplitude ac Voltammetry 479 Chemical Analysis by ac Voltammetry 481 Instrumentation for Electrochemical Impedance Methods 482 11.8.1 Frequency-Domain Instruments 482 11.8.2 Time-Domain Instruments 483 Analysis of Data in the Laplace Plane 485 References 485 Problems 487 Bulk Electrolysis 489 General Considerations 490 12.1.1 Completeness of an Electrode Process 490 12.1.2 Current Efficiency 491 12.1.3 Experimental Concerns 491 Controlled-Potential Methods 495 12.2.1 Current–Time Behavior 495 12.2.2 Practical Aspects 497 12.2.3 Coulometry 498 12.2.4 Electrogravimetry 500 12.2.5 Electroseparations 501 Controlled-Current Methods 501 12.3.1 Characteristics of Controlled-Current Electrolysis 501 12.3.2 Coulometric Titrations 503 12.3.3 Practical Aspects of Constant-Current Electrolysis 506 Electrometric End-Point Detection 507 12.4.1 Current–Potential Curves During Titration 507 12.4.2 Potentiometric Methods 508 12.4.3 Amperometric Methods 509 Flow Electrolysis 510 12.5.1 Mathematical Treatment 510 12.5.2 Dual-Electrode Flow Cells 515 12.5.3 Microfluidic Flow Cells 516 Thin-Layer Electrochemistry 521 12.6.1 Chronoamperometry and Coulometry 521 12.6.2 Potential Sweep in a Nernstian System 524 12.6.3 Dual-Electrode Thin-Layer Cells 526 12.6.4 Applications of the Thin-Layer Concept 526

Contents

12.7

12.8 12.9

Stripping Analysis 527 12.7.1 Introduction 527 12.7.2 Principles and Theory 528 12.7.3 Applications and Variations 529 References 531 Problems 534

13

Electrode Reactions with Coupled Homogeneous Chemical Reactions

13.1

Classification of Reactions 539 13.1.1 Reactions with One E-Step 541 13.1.2 Reactions with Two or More E-Steps 542 Impact of Coupled Reactions on Cyclic Voltammetry 545 13.2.1 Diagnostic Criteria 545 13.2.2 Characteristic Times 547 13.2.3 An Example 547 13.2.4 Including Kinetics in Theory 548 13.2.5 Comparative Simulation 551 Survey of Behavior 552 13.3.1 Following Reaction—Case Er Ci 552 13.3.2 Effect of Electrode Kinetics in ECi Systems 556 13.3.3 Bidirectional Following Reaction 558 13.3.4 Catalytic Reaction—Case Er Ci ′ 561 13.3.5 Preceding Reaction—Case Cr Er 564 13.3.6 Multistep Electron Transfers 569 13.3.7 ECE/DISP Reactions 576 13.3.8 Concerted vs. Stepwise Reaction 584 13.3.9 Elaboration of Reaction Schemes 590 Behavior with Other Electrochemical Methods 591 References 593 Problems 595

13.2

13.3

13.4 13.5 13.6 14

14.1

14.2

14.3

14.4

Double-Layer Structure and Adsorption 599 Thermodynamics of the Double Layer 599 14.1.1 The Gibbs Adsorption Isotherm 599 14.1.2 The Electrocapillary Equation 601 14.1.3 Relative Surface Excesses 601 Experimental Evaluations 602 14.2.1 Electrocapillarity 602 14.2.2 Excess Charge and Capacitance 603 14.2.3 Relative Surface Excesses 606 Models for Double-Layer Structure 606 14.3.1 The Helmholtz Model 607 14.3.2 The Gouy–Chapman Theory 609 14.3.3 Stern’s Modification 614 14.3.4 Specific Adsorption 617 Studies at Solid Electrodes 619 14.4.1 Well-Defined Single-Crystal Electrode Surfaces 14.4.2 The Double Layer at Solids 623

620

539

xiii

xiv

Contents

14.5

14.6 14.7

14.8 14.9 15

15.1

15.2

15.3

15.4

15.5 15.6

15.7 15.8 16

16.1

16.2

Extent and Rate of Specific Adsorption 627 14.5.1 Nature and Extent of Specific Adsorption 628 14.5.2 Electrosorption Valency 629 14.5.3 Adsorption Isotherms 630 14.5.4 Rate of Adsorption 633 Practical Aspects of Adsorption 634 Double-Layer Effects on Electrode Reaction Rates 636 14.7.1 Introduction and Principles 636 14.7.2 Double-Layer Effects Without Specific Adsorption of Electrolyte 638 14.7.3 Double-Layer Effects with Specific Adsorption 639 14.7.4 Diffuse Double-Layer Effects on Mass Transport 640 References 645 Problems 648 653 Inner-Sphere Heterogenous Electron-Transfer Reactions 653 15.1.1 The Role of the Electrode Surface 653 15.1.2 Energetics of 1e Electron-Transfer Reactions 654 15.1.3 Adsorption Energies 657 Electrocatalytic Reaction Mechanisms 657 15.2.1 Hydrogen Evolution Reaction 657 15.2.2 Tafel Plot Analysis of HER Kinetics 660 Additional Examples of Inner-Sphere Reactions 667 15.3.1 Oxygen Reduction Reaction 667 15.3.2 Chlorine Evolution 670 15.3.3 Methanol Oxidation 670 15.3.4 CO2 Reduction 673 15.3.5 Oxidation of NH3 to N2 674 15.3.6 Organic Halide Reduction 676 15.3.7 Hydrogen Peroxide Oxidation and Reduction 677 Computational Analyses of Inner-Sphere Electron-Transfer Reactions 678 15.4.1 Density Functional Theory Analysis of Electrocatalytic Reactions 679 15.4.2 Hydrogen Evolution Reaction 679 15.4.3 Oxygen Reduction Reaction 681 Electrocatalytic Correlations 684 Electrochemical Phase Transformations 688 15.6.1 Nucleation and Growth of a New Phase 688 15.6.2 Classical Nucleation Theory 689 15.6.3 Electrodeposition 699 15.6.4 Gas Evolution 707 References 713 Problems 718 Inner-Sphere Electrode Reactions and Electrocatalysis

Electrochemical Instrumentation 721 Operational Amplifiers 721 16.1.1 Ideal Properties 721 16.1.2 Nonidealities 723 Current Feedback 725 16.2.1 Current Follower 725

Contents

16.3

16.4

16.5 16.6 16.7

16.8

16.9 16.10

16.11 16.12 17

17.1 17.2

17.3

17.4 17.5

16.2.2 Scaler/Inverter 726 16.2.3 Adders 726 16.2.4 Integrators 727 Voltage Feedback 728 16.3.1 Voltage Follower 728 16.3.2 Control Functions 729 Potentiostats 730 16.4.1 Basic Considerations 730 16.4.2 The Adder Potentiostat 731 16.4.3 Refinements to the Adder Potentiostat 732 16.4.4 Bipotentiostats 733 16.4.5 Four-Electrode Potentiostats 734 Galvanostats 734 Integrated Electrochemical Instrumentation 736 Difficulties with Potential Control 737 16.7.1 Types of Control Problems 737 16.7.2 Cell Properties and Electrode Placement 740 16.7.3 Electronic Compensation of Resistance 740 Measurement of Low Currents 744 16.8.1 Fundamental Limits 744 16.8.2 Practical Considerations 746 16.8.3 Current Amplifier 746 16.8.4 Simplified Instruments and Cells 746 Instruments for Short Time Scales 748 Lab Note: Practical Use of Electrochemical Instruments 749 16.10.1 Caution Regarding Electrochemical Workstations 749 16.10.2 Troubleshooting Electrochemical Systems 749 References 751 Problems 752 755 Monolayers and Submonolayers on Electrodes 756 Cyclic Voltammetry of Adsorbed Layers 757 17.2.1 Fundamentals 757 17.2.2 Reversible Adsorbate Couples 758 17.2.3 Irreversible Adsorbate Couples 763 17.2.4 Nernstian Processes Involving Adsorbates and Solutes 766 17.2.5 More Complex Systems 770 17.2.6 Electric-Field-Driven Acid–Base Chemistry in Adsorbate Layers Other Useful Methods for Adsorbed Monolayers 775 17.3.1 Chronocoulometry 775 17.3.2 Coulometry in Thin-Layer Cells 777 17.3.3 Impedance Measurements 778 17.3.4 Chronopotentiometry 779 Thick Modification Layers on Electrodes 780 Dynamics in Modification Layers 782 17.5.1 Steady State at a Rotating Disk 783 17.5.2 Principal Dynamic Processes in Modifying Films 784 17.5.3 Interplay of Dynamical Elements 789 Electroactive Layers and Modified Electrodes

771

xv

xvi

Contents

17.6

17.7 17.8

17.9 17.10 18

18.1 18.2 18.3

18.4

18.5 18.6 18.7

18.8 18.9 18.10 19

19.1 19.2 19.3

19.4

19.5 19.6 19.7 19.8

Blocking Layers 791 17.6.1 Permeation Through Pores and Pinholes 792 17.6.2 Tunneling Through Blocking Films 796 Other Methods for Characterizing Layers on Electrodes 798 Electrochemical Methods Based on Electroactive Layers or Electrode Modification 798 17.8.1 Electrocatalysis 799 17.8.2 Bioelectrocatalysis Based on Enzyme-Modified Electrodes 799 17.8.3 Electrochemical Sensors 803 17.8.4 Faradaic Electrochemical Measurements in vivo 809 References 812 Problems 817 819 Principles 819 Approach Curves 821 Imaging Surface Topography and Reactivity 825 18.3.1 Imaging Based on Conductivity of the Substrate 825 18.3.2 Imaging Based on Heterogeneous Electron-Transfer Reactivity 826 18.3.3 Simultaneous Imaging of Topography and Reactivity 827 Measurements of Kinetics 828 18.4.1 Heterogeneous Electron-Transfer Reactions 828 18.4.2 Homogeneous Reactions 831 Surface Interrogation 835 Potentiometric Tips 839 Other Applications 839 18.7.1 Detection of Species Released from Surfaces, Films, or Pores 839 18.7.2 Biological Systems 840 18.7.3 Probing the Interior of a Layer on a Substrate 841 Scanning Electrochemical Cell Microscopy 841 References 846 Problems 849 Scanning Electrochemical Microscopy

Single-Particle Electrochemistry 851 General Considerations in Single-Particle Electrochemistry 851 Particle Collision Experiments 852 Particle Collision Rate at a Disk-Shaped UME 854 19.3.1 Collision Frequency 854 19.3.2 Variance in the Number of Particle Collisions 855 19.3.3 Time of First Arrival 856 Nanoparticle Collision Behavior 857 19.4.1 Blocking Collisions 857 19.4.2 Electrocatalytic Amplification Collisions 861 19.4.3 Electrolysis Collisions 864 Electrochemistry at Single Atoms and Atomic Clusters 870 Single-Molecule Electrochemistry 875 References 879 Problems 881

Contents

20.6 20.7

885 Solid Materials 885 20.1.1 The Band Model 885 20.1.2 Categories of Pure Crystalline Solids 886 20.1.3 Doped Semiconductors 889 20.1.4 Fermi Energy 890 20.1.5 Highly Conducting Oxides 891 Semiconductor Electrodes 892 20.2.1 Interface at a Semiconducting Electrode in the Dark 892 20.2.2 Current–Potential Curves at Semiconductor Electrodes 896 20.2.3 Conducting Polymer Electrodes 899 Photoelectrochemistry at Semiconductors 901 20.3.1 Photoeffects at Semiconductor Electrodes 901 20.3.2 Photoelectrochemical Systems 903 20.3.3 Dye Sensitization 905 20.3.4 Surface Photocatalytic Processes at Semiconductor Particles 906 Radiolytic Products in Solution 908 20.4.1 Photoemission of Electrons from an Electrode 908 20.4.2 Detection and Use of Radiolytic Products in Solution 909 20.4.3 Photogalvanic Cells 909 Electrogenerated Chemiluminescence 910 20.5.1 Chemical Fundamentals 910 20.5.2 Fundamental Studies of Radical-Ion Annihilation 912 20.5.3 Single-Potential Generation Based on a Coreactant 916 20.5.4 ECL Based on Quantum Dots 917 20.5.5 Analytical Applications of ECL 918 20.5.6 ECL Beyond the Solution Phase 922 References 922 Problems 927

21

In situ Characterization of Electrochemical Systems 931

21.1

Microscopy 931 21.1.1 Scanning Tunneling Microscopy 932 21.1.2 Atomic Force Microscopy 934 21.1.3 Optical Microscopy 937 21.1.4 Transmission Electron Microscopy 938 Quartz Crystal Microbalance 940 21.2.1 Basic Method 940 21.2.2 QCM with Dissipation Monitoring 942 UV–Visible Spectrometry 942 21.3.1 Absorption Spectroscopy with Thin-Layer Cells 21.3.2 Ellipsometry 945 21.3.3 Surface Plasmon Resonance 946 Vibrational Spectroscopy 947 21.4.1 Infrared Spectroscopy 947 21.4.2 Raman Spectroscopy 950 X-Ray Methods 953 Mass Spectrometry 954 Magnetic Resonance Spectroscopy 955

20

20.1

20.2

20.3

20.4

20.5

21.2

21.3

21.4

21.5 21.6 21.7

Photoelectrochemistry and Electrogenerated Chemiluminescence

942

xvii

xviii

Contents

21.8

21.9

21.7.1 ESR 955 21.7.2 NMR 956 Ex-situ Techniques 957 21.8.1 Electron Microscopy 957 21.8.2 Electron and Ion Spectrometry References 960

958

Mathematical Methods 967 Solving Differential Equations by the Laplace Transform Technique 967 A.1.1 Partial Differential Equations 967 A.1.2 Introduction to the Laplace Transformation 968 A.1.3 Fundamental Properties of the Transform 969 A.1.4 Solving Ordinary Differential Equations by Laplace Transformation 970 A.1.5 Simultaneous Linear Ordinary Differential Equations 972 A.1.6 Mass-Transfer Problems Based on Partial Differential Equations 973 A.1.7 The Zero-Shift Theorem 975 Taylor Expansions 976 A.2.1 Expansion of a Function of Several Variables 976 A.2.2 Expansion of a Function of a Single Variable 977 A.2.3 Maclaurin Series 977 The Error Function and the Gaussian Distribution 977 Leibnitz Rule 979 Complex Notation 979 Fourier Series and Fourier Transformation 981 References 982 Problems 983

Appendix A

A.1

A.2

A.3 A.4 A.5 A.6 A.7 A.8

Basic Concepts of Simulation 985 Setting Up the Model 985 B.1.1 A Discrete System 985 B.1.2 Diffusion 986 B.1.3 Dimensionless Parameters 987 B.1.4 Time 990 B.1.5 Distance 990 B.1.6 Current 991 B.1.7 Thickness of the Diffusion Layer 992 B.1.8 Diffusion Coefficients 993 An Example 993 B.2.1 Organization of the Spreadsheet 993 B.2.2 Concentration Arrays 996 B.2.3 Results and Error Detection 996 B.2.4 Performance 997 Incorporating Homogeneous Kinetics 999 B.3.1 Unimolecular Reactions 999 B.3.2 Bimolecular Reactions 1000 Boundary Conditions for Various Techniques 1001 B.4.1 Potential Steps in Nernstian Systems 1001 B.4.2 Heterogeneous Kinetics 1002

Appendix B

B.1

B.2

B.3

B.4

Contents

B.5 B.6 B.7

B.4.3 Potential Sweeps 1003 B.4.4 Controlled Current 1003 More Complex Systems 1004 References 1005 Problems 1005

Reference Tables 1007 References 1013 Appendix C

Index 1015

xix

xxi

Preface Since the appearance of our 1980 and 2001 editions, electrochemistry has developed remarkably. Phenomena are better understood; experimental tools have become more sophisticated; and new methods have emerged. With this new edition, we have striven to accommodate an evolved, enlarged field, while extending this book’s value as a general introduction. Our overall goal is to provide an authoritative resource for students and new practitioners, covering the core of what they now must know to be successful in research. Accordingly, the emphasis has shifted in this edition to methods that are extensively practiced and to phenomenological questions of current concern. The reconception has led to changes in scope and organization as outlined below. Moreover, we now address a much broader audience. Electrochemistry’s clear relevance to energy and environment has attracted scientists and engineers with educational backgrounds outside of chemistry and chemical engineering. The prior editions were written principally for graduate students in chemistry and for practicing researchers in electrochemistry, for whom a formal preparation through physical chemistry could be assumed. In this edition, we teach, instead, from a foundation of basic university courses in general chemistry, physics, and mathematics. We have sought to make the book self-contained by developing almost all key ideas from fundamental principles of chemistry and physics. This volume includes numerous problems and chemical examples; illustrations are used to clarify presentations; and the style is pedagogical throughout. The book can be used in formal courses at the senior undergraduate and graduate levels, but we have also tried to write in a way that enables self-study by interested individuals in mid-career. Because we stress foundations and limits of application, the book continues to present the mathematical theory underlying methodology; however, the key ideas are consistently discussed apart from the mathematical basis. The end-of-chapter problems have been devised as teaching tools, often extending concepts introduced in the text or showing how experimental data are reduced to fundamental results. The cited literature is extensive, but mainly includes only seminal papers and reviews. Major changes are found throughout: • An entirely new Chapter 5 (“Steady-State Voltammetry at Ultramicroelectrodes”) has been created in support of the authors’ view that steady-state voltammetry is now the best starting point for the methodological sequence. Potential step methods—formerly providing the starting point—are now treated in a redesigned Chapter 6 (“Transient Methods Based on Potential Steps”). • Also completely new is Chapter 15 (“Inner-Sphere Electrode Reactions and Electrocatalysis”), giving a substantial introduction to the electrode kinetics of important complex reactions.

xxii

Preface

• A third new unit is Chapter 19 (“Single-Particle Electrochemistry”), which explores a frontier where individual elementary events come into focus. • Chapter 1 (“Overview of Electrode Processes”) has been reorganized and revised to improve its effectiveness for new readers coming into electrochemistry from diverse backgrounds. • Chapter 3 (“Basic Kinetics of Electrode Reactions”) now includes an extensive treatment of Marcus kinetics as applied to electrode reactions. • Chapter 4 (“Mass Transfer by Migration and Diffusion”) includes a fuller introduction to migration, which is now regularly encountered in the chapters on methodology. • Chapter 11 (“Electrochemical Impedance Spectroscopy and ac Voltammetry”) has a stronger focus on EIS with an expanded presentation. • Chapter 13 (“Electrode Reactions with Coupled Homogeneous Chemical Reactions”) has been streamlined using cyclic voltammetry as a consistent methodological context and including more experimental examples. • Chapter 18 (“Scanning Electrochemical Microscopy”) is now wholly dedicated to SECM and SECCM. This domain has greatly expanded since the second edition appeared in 2001. • Chapter 21 (“In situ Characterization of Electrochemical Systems”) has been reconceived to emphasize methods with in situ or operando capabilities. • All other chapters have been edited toward clear, efficient presentation of current knowledge and practice. • Finally, we have added Lab Notes to many chapters to help newcomers with the transition from concept to actual practice in the laboratory. A goal has been to keep this book as close in size as possible to the 2001 edition. Naturally, we have deleted or abbreviated topics to make room for more current matters. In such cases, references have been provided to the corresponding passages in earlier editions, so that interested readers can still find coverage of a deleted or attenuated topic. In general, we have excluded or reduced the coverage of techniques that, while functional, are not widely practiced. Apart from what we include in the Lab Notes, laboratory procedures remain outside our intended scope. Just after this Preface, the reader will find a convenient unit, “Major Symbols and Abbreviations,” offering definitions, units, and section references. Indeed, Tables 1–5 contained in that unit comprise a functional alternative index. Table 5 identifies abbreviated chemical substances by names recognized by CAS and with references to chemical structures, most displayed in Figure 1, following Table 5. Our uses of symbols usually adhere to the recommendations of the IUPAC Commission on Electrochemistry [R. Parsons et al., Pure Appl. Chem., 37, 503 (1974)]. Exceptions have been made where customary usage or clarity of notation seemed compelling. As with both prior editions, we owe thanks to others. Commissioning Editor Sarah Higginbotham and Managing Editor Stefanie Volk of John Wiley & Sons provided calm support and excellent judgment from start to finish. Sundaramoorthy Balasubramani was invaluable to us during the process of production. Once again, Cynthia Zoski and Johna Leddy generously agreed to prepare the “Instructor’s Solutions Manual” and have often commented helpfully as we developed the book manuscript. Valuable comments or answers to queries were also provided by C. Amatore, A. Bond, F. Dalton, B. Dunn, B. Feldman, W. Geiger, P. He, W. Heineman, A. Heller, P. Kissinger, S. Lin, S. Minteer, M. Mirkin, B. Mullins, M. Neurock, D. Pletcher, H. Ren, M. Robert, L. Sombers, P. Unwin, J. Wadhawan, and F. Zamborini. The late Jean-Michel Savéant, an invaluable colleague for decades, was a helpful commentator on all three editions. He remains in our memories as a beacon of science, representing the finest of insight and quality. We thank all we have named and our many other colleagues throughout

Preface

the electrochemical community, who have taught us patiently over the years. Once more, we thank our families for affording us the time and freedom required for this large project. 24 February 2022

Allen J. Bard Department of Chemistry and Center for Electrochemistry University of Texas at Austin Larry R. Faulkner Department of Chemistry and Center for Electrochemistry University of Texas at Austin Henry S. White Department of Chemistry University of Utah

xxiii

xxv

Major Symbols and Abbreviations In five tables below are symbols and abbreviations used in several chapters or in large portions of a chapter. Similar symbols may have different local meanings. Usage normally follows the recommendations of the IUPAC Commission on Electrochemistry [R. Parsons et al., Pure Appl. Chem., 37, 503 (1974).]; however, there are exceptions. A bar over a concentration or a current [e.g., C O (x, s)] indicates the Laplace transform of the variable. The exception is when i indicates an average current in dc polarography. Table 1 Standard Subscripts and Superscripts. 0

standard (superscript)

dl

double layer

O

for species O in O + e ⇄ R

a

anodic

eq

equilibrium

p

(a) peak

ads

adsorbed

f

(a) forward

c

(a) cathodic

(b) faradaic

(b) p-type carrier R

(a) for species R in O + e ⇄ R

(b) charging

l

limiting

(b) ring

D

disk

M

metal (superscript)

r

reverse

d

diffusion

n

n-type carrier

S

solution (superscript)

Table 2 Roman Symbols.

Symbol

Meaning

Usual Units

Section

A

(a) area

cm2

1.1.5

(b) cross-sectional area of a porous electrode

cm2

12.5.1

(c) frequency factor in a rate constant

depends on nature and order

3.1.2

(d) open-loop gain of an amplifier

none

16.1.1

dc open-loop gain of an amplifier

none

16.1.2(a)

geometric area of an electrode

cm2

6.1.5

microscopic area of an electrode

cm2

6.1.5

Adc Ag Am

(Continued)

xxvi

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

a

(a) activity

none

2.1.5

(b) internal area of a porous electrode

cm2

12.5.1

(c) radius of a disk-shaped tip in SECM

μm, nm

18.2

(a) activity of substance j

none

2.1.5

aj

(b) interaction parameter among adsorbates

none

17.2.2

a𝛼j

activity of substance j in a phase 𝛼

none

2.1.5

ap

in flow electrolysis, total open area on the face of a porous electrode

cm2

12.5.1

b

𝛼Fv/RT = 𝛼fv

s−1

7.4.1(a)

bj

for adsorption of species j, 𝛽 j Γj,s

mol/cm2

14.5.4

C

capacitance

F

1.6.2, 11.2

c

speed of light in vacuo

m/s

CB

series equivalent capacitance of a cell

F

11.1, 11.3

CD

predicted differential capacitance of a diffuse layer

F, F/cm2

14.3.2(c), 14.3.3

Cd

differential capacitance of the double layer

F, F/cm2

1.6.2, 14.2.2

C GCS

in the Gouy–Chapman–Stern model, predicted differential capacitance of the double layer

F, F/cm2

14.3.3

CH

predicted differential capacitance of a Helmholtz layer

F, F/cm2

14.3.1, 14.3.3

Ci

integral capacitance of the double layer

F, F/cm2

14.2.2

Cj

(a) concentration of species j

M, mol/cm3

(b) capacitance of an electrical element j

F, F/cm2

bulk concentration of species j

M, mol/cm3

1.3.2, 4.5.1

standard-state concentration of species j

M

2.1.5

C j (0, t)

concentration of species j at the electrode surface at time t (linear system)

M, mol/cm3

4.5.1(c)

C j (0, t)m

in impedance theory, mean concentration of species j at the electrode surface at time t (linear system where t ≫ 1/𝜔)

M, mol/cm3

11.4.1

C j (r)

concentration of species j at radius r (radial system)

M, mol/cm3

5.1.1

C j (r = r0 )

concentration of species j at the electrode surface (radial system)

M, mol/cm3

5.1.1

C j (r, t)

concentration of species j at radius r at time t (radial system)

M, mol/cm3

4.4.2

C j (r0 , t)

concentration of species j at the electrode surface at time t (radial system)

M, mol/cm3

6.1.2

Cj∗ Cj0

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

C j (surface)

concentration of species j at the electrode surface (a general symbol encompassing multiple geometries and methods)

M, mol/cm3

5.2.5

C j (x)

concentration of species j at distance x (linear system)

M, mol/cm3

1.3.1, 4.1

C j (x = 0)

concentration of species j at the electrode surface (linear system)

M, mol/cm3

1.3.2

C j (x, t)

concentration of species j at distance x at time t (linear system)

M, mol/cm3

4.4.2

C j (y)

concentration of species j at distance y away from a rotating electrode

M, mol/cm3

10.2.2

C j (y = 0)

surface concentration of species j at a rotating electrode

M, mol/cm3

10.2.4

C j (z = 0)

surface concentration of species j at a disk UME

M, mol/cm3

5.2.2

Cs

series capacitive component of Zf

F

11.3.1, 11.3.2(c)

CgS

in CNT, supersaturation concentration of a species in solution required to nucleate a gas bubble

M, mol/cm3

15.6.4

C SC

space charge capacitance

F/cm2

20.2.1(d)

diffusion coefficient (of species j)

cm2 /s

1.3.1, 4.4

DE

diffusion coefficient for electrons in a modifying layer on an electrode

cm2 /s

17.5.2(c)

De

effective diffusion coefficient for a redox couple engaged in redox cycling

cm2 /s

19.6

Dj (𝜆, E)

concentration density of states for species j

cm−3 eV−1

3.5.5(a)

DM

in simulation, model diffusion coefficient

none

B.1.3, B.1.8

DS

in a modifying layer on an electrode, diffusion coefficient for the primary reactant (substrate)

cm2 /s

17.5.2(b)

d

(a) in AFM, SECM, or STM, distance of the tip from the substrate

μm, nm

18.1, 21.1.1

(b) in a nanoscale redox-cycling cell, electrode separation

cm, μm, nm

19.6

dj

density of phase j

kg/L, g/cm3

E

(a) potential of an electrode vs. a reference

V

1.1.2, 2.1.4

(b) emf of a reaction

V

2.1.3

(c) amplitude of an ac voltage

V

11.2

D, Dj

ΔE

(a) in DPV, pulse height

mV

8.4.2

(b) in ac voltammetry, amplitude of ac excitation (1/2 peak-to-peak)

mV

11.5.1 (Continued)

xxvii

xxviii

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

E

(a) energy

J, eV

Section

(b) electron energy

eV

2.2.5, 3.5.5(a)

E E

electric field strength

V/cm

2.2.1

electric field strength vector

V/cm

2.2.1

Ė

voltage or potential phasor

V

11.2

(a) standard potential of an electrode or a couple (with subscript or parentheses, of the O/R couple)

V

2.1.4

(b) standard emf of a half-reaction (with subscript, of the O/R half-reaction)

V

2.1.4

difference in standard potentials for two couples, E20 − E10

V

13.3.6(a)

formal potential of an electrode or a couple (with subscript or parentheses, of the O/R couple)

V

2.1.7

difference in formal potentials for two couples, ′ ′ E20 − E10

V

7.5.2, 13.3.6

electron energy corresponding to the formal potential of a couple

eV

3.5.5(a)

0a E0a , EO∕R

standard potential of an electrode or a couple on the absolute scale (with subscript, of the O/R couple)

V

2.2.5(a)

E1/2

(a) in voltammetry, measured or expected half-wave potential

V

1.3.2, 5.3–5.4, 6.2–6.3

(b) in derivations for diffusing systems, the “reversible” half-wave potential, ′ 1∕2 1∕2 E0 + (RT∕nF) ln(DR ∕DO )

V

6.2.2

E1/4

potential where i = id,c /4 (or il /4)

V

5.3.1(a)

E3/4

potential where i = 3id,c /4 (or 3il /4)

V

5.3.1(a)

0 E0 , EO∕R 0 E (O/R)

ΔE0 ′

′

0 E0 , EO∕R ′

E0 (O∕R) ΔE0 E0

′

′

EA

activation energy of a reaction

kJ/mol

3.1.2

EA

in a doped semiconductor, acceptor level

eV

20.1.3

Eac

ac component of potential

mV

11, 11.4.1

Eappl

voltage applied at the working electrode vs. the reference electrode

V

1.2.2, 1.5.1, 1.6.4(d)

Eb

base potential in NPV and RPV

V

8.2.1, 8.3

EC

in a solid, electron energy at the bottom of the CB (CB edge)

eV

20.2.1(a)

ED

potential of a rotating disk electrode

V

10.3.2

ED

in a doped semiconductor, donor level

eV

20.1.3

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

Ed

in stripping analysis, deposition potential

V

12.7.1

Edc

dc component of potential

V

11.4.1

Eeq

equilibrium potential of an electrode

V

1.1.7(a), 3.2, 3.4.1

EF , E𝛼F

Fermi energy (with superscript, in phase 𝛼)

eV

2.2.5(b), 3.5.5(a), 20.1.4

Ef

forward step potential

V

6.5

Efb

flat-band potential

V

20.2.1(b)

Eg

band gap of a material

eV

20.1.1

Ei

initial potential

V

6.5, 7.1

Ej

junction potential

mV

2.3.4

Em

membrane potential

mV

2.4.1

En

nucleation potential

V

15.6.2(a)

Ep

peak potential

V

7.2.1(b)

ΔEp

(a) in CV, peak separation, Epa − Epc or |Epf − Epr |

mV

7.2.2(c), 7.2.2(h)

(b) in SWV, pulse height

mV

8.5.1

ΔEp, 1/2

in voltammetry, peak width at half-height

mV

17.2.2

Ep/2

in LSV or CV, potential preceding Epf where i = ipf /2

V

7.2.1(b), 7.2.2(h)

Epa

anodic peak potential

V

7.2.2(a)

Epc

cathodic peak potential

V

7.2.1(b)

Epf

forward peak potential

V

7.2.2(h)

Epr

reverse peak potential

V

7.2.2(h)

ER

potential of a rotating ring electrode

V

10.3.2

Er

reverse step potential

V

6.5

ES

in SECM or EC-STM, potential of the substrate

V

18.2, 21.1.1

ET

in SECM or EC-STM, potential of the tip

V

18.2, 21.1.1

ΔEs

in SWV, staircase step height

mV

8.5.1

EV

in a solid, electron energy at the top of the VB (VB edge)

eV

20.2.1(a)

Ez

potential of zero charge

V

1.6.4(a), 14.2.2

E𝜆

in CV or CSWV, switching (reversal) potential

V

1.6.4(b), 7.2.2

E𝜏/4

in chronopotentiometry, quarter-wave potential

V

9.3.1 (Continued)

xxix

xxx

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

e

(a) electronic charge

C

(b) voltage in an electric circuit

V

Section

11.2, 16.1

ei

input voltage of a circuit

V

16.2.2

eo

output voltage of an amplifier or a circuit

V

16.1.1

es

voltage across the input terminals of an amplifier

μV

16.1.1

EA

electron affinity

eV

15.1.2, 20.1.4

erf(x)

error function of x

none

A.3

erfc(x)

error function complement of x

none

A.3

F

Faraday constant; charge per mole of electrons

C/mol

1.1.5

F 1 (𝜆)

in chronoamperometry and STV, a general kinetic function

none

6.3.2

f

(a) F/RT

V−1

(b) frequency of a sinusoidal oscillation

s−1

11.2

(c) frequency of rotation

r/s

10.2

(d) in SWV, frequency, f = 1/2t p

s−1

8.5.1

(e) fraction titrated

none

12.4.1 19.3.1

(f ) collision frequency

s−1

f (E)

Fermi function

none

3.5.5(a)

f m (j, k)

in simulation, fractional concentration of species m in box j after iteration k

none

B.1.3

G

(a) Gibbs free energy

kJ, kJ/mol

2.1.2

(b) electrical conductance

S = Ω−1

2.3.3

(c) in SECM, geometric factor relating to shape and extent of the diffusion field at the tip

none

18.1

ΔG

in a chemical process, Gibbs free energy change

kJ, kJ/mol

2.1.2, 2.1.3

G

electrochemical free energy

kJ, kJ/mol

2.2.4

ΔG‡ , ΔGj‡

standard Gibbs free energy of activation (of process j)

kJ/mol

3.1.2

G0

standard Gibbs free energy

kJ/mol

2.1.2

ΔG0

in a chemical process, standard Gibbs free energy change

kJ/mol

2.1.2, 2.1.3

ΔGtransfer, j

standard free energy of transfer for species j from phase 𝛼 into phase 𝛽

kJ/mol

2.3.6

g

(a) gravitational acceleration

m/s2 , cm/s2

(b) in adsorption isotherms, interaction parameter

J cm2 mol−1 or none

14.5.3(b), 17.2.2

in a nucleation process, the free-energy change per unit of volume created

J/m3

15.6.2(b)

0𝛼→𝛽

ΔGV

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

H

(a) enthalpy 1∕2 (b) kf ∕DO

1∕2 + kb ∕DR

Usual Units

Section

kJ, kJ/mol

2.1.2

s−1/2

6.3.1(a)

ΔH

in a chemical process, enthalpy change

kJ, kJ/mol

2.1.2

ΔH ‡

standard enthalpy of activation

kJ/mol

3.1.2

ΔH 0

in a chemical process, standard enthalpy change

kJ/mol

2.1.2

H ab

electronic coupling matrix element

eV

3.5.2(b)

h

Planck constant

J-s

I

amplitude of an ac current (1/2 peak-to-peak)

A

11.2

İ

current phasor

A

11.2

I

in dc polarography, diffusion current constant for average current

μA s1/2 mg−2/3 mM−1

8.1.5

I(t)

convolutive transform (semi-integral) of i(t)

C/s1/2

7.7

(I)max

in dc polarography, diffusion current constant for maximum current

μA s1/2 mg−2/3 mM−1

8.1.5

IE

ionization energy

eV

15.1.2

Il

in LSV or CV, limiting value of I(t)

C/s1/2

7.7

Ip

peak value of ac current amplitude

A

11.5.1

IT

in SECM, dimensionless tip current, iT /iT,∞

none

18.2

Im(w)

imaginary part of complex function w

i

current

A.5 A

1.1.5

Δi

in SWV, difference current, if − ir

A

8.5.1

𝛿i

in DPV, difference current, i(𝜏) − i(𝜏 ′ )

A

8.4.1

(𝛿i)max

in DPV, peak height

A

8.4.2

(i)RP

in RPV, current sample during the pulse

A

8.3

i(0)

in bulk electrolysis, initial current

A

12.2.1

i0

exchange current

A

3.4.1

i0,t

true (corrected) exchange current

A

14.7.1

iA

in treatment of dynamics at modified electrodes, characteristic current proportional to the flux of primary reactant to the outer film boundary

A

17.5.1

ia

anodic component current

A

3.2

iac

ac component of current

A

11

ic

(a) charging current

A

7.2.1(d)

(b) cathodic component current

A

3.2 (Continued)

xxxi

xxxii

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

iD

current at a rotating disk electrode

A

10.3.1

id , id, j

(a) diffusion-limited current (of species j)

A

5.1.2

(b) current due to diffusive flux (of species j)

A

4.1

id

at a DME, average diffusion-limited current over a drop lifetime

A

8.1.3

(id )max

at a DME, diffusion-limited current at t max (maximum current)

A

8.1.3

id,a

diffusion-limited anodic current

A

5.2.5

(id,a )RP

in RPV, diffusion-limited anodic current sample when the forward process is reduction

A

8.3

id,c

diffusion-limited cathodic current

A

5.1.2, 5.2.5

(id,c )DC

in dc polarography or RPV, diffusion-limited cathodic current sample when reduction is the forward process

A

8.2.3(a), 8.3

(id,c )NP

in NPV, diffusion-limited cathodic current

A

8.2.1

iss d,c

steady-state asymptote of the diffusion-limited current transient at a UME

A

6.1.3(a)

iE

in treatment of dynamics at modified electrodes, characteristic current describing the maximum rate of electron diffusion across the modification layer

A

17.5.2(c)

iF

in treatment of dynamics at modified electrodes, characteristic current describing the maximum rate of substrate conversion.

A

17.5.1

if

(a) faradaic current

A

7.2.1(f )

(b) current during a forward step or sweep

A

6.5.2

(c) in SWV, forward current sample

A

8.5.1

(d) feedback current

A

16.2

ij

current due to species j

A

4.1

iK

in the KL method, kinetically limited current

A

10.2.5

ik

in treatment of dynamics at modified electrodes, characteristic current describing the maximum rate of cross-reaction in the modification layer

A

17.5.2(d)

il

limiting current

A

1.3.2

il,a

limiting anodic current

A

1.3.2(b)

il,c

limiting cathodic current

A

1.3.2(b)

im , im, j

current due to a migration flux (of species j)

A

4.1

iP

in treatment of dynamics at modified electrodes, characteristic current describing the maximum rate of permeation of the primary reactant into the layer

A

17.5.2(b)

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

ip

peak current

A

7.2.1(b)

ipa

anodic peak current

A

7.2.2(a)

ipc

cathodic peak current

A

7.2.1(b)

ipf

forward peak current

A

7.2.2(h)

ipf,u

forward peak current for a process unperturbed by coupled kinetics

A

13.3.7(a)

iph

photocurrent in a photoelectrochemical cell

A

20.3.1

ipr

reversal peak current

A

7.2.2(h)

|ipr /ipf |

in CV, reversal criterion

none

7.2.2(h)

iR

current at a rotating ring electrode

A

10.3.1

ir

(a) current during a reversal step or sweep

A

6.5.2

(b) in SWV, reverse current sample

A

8.5.1

|ir (2𝜏)/if (𝜏)|

in chronoamperometry, reversal criterion

none

6.5.2

iS

(a) in treatment of dynamics at modified electrodes, characteristic current describing the maximum rate of substrate (primary reactant) diffusion across the modification layer

A

17.5.2(b)

(b) in SECM and SECCM, substrate current

A

18.1, 18.8

|iS /iT |

In SECM, collection efficiency in TG/SC mode

none

18.4.2

iT

in SECM, tip current

A

18.1

|iT /iS |

In SECM, collection efficiency in SG/TC mode

none

18.4.2

iT,∞

in SECM, tip current far from the substrate

A

18.1

Jj

flux vector for species j

mol cm−2 s−1

4.1

J j (0, t)

flux of species j at the electrode surface at time t (linear system)

mol cm−2 s−1

4.4.3

J j (r0 , t)

flux of species j at the electrode surface at time t (radial system)

mol cm−2 s−1

5.1.1, 6.1.2

J j (x)

flux of species j at location x (linear system)

mol cm−2 s−1

1.3.1, 4.1

J j (x = 0)

flux of species j at the electrode surface (linear system)

mol cm−2 s−1

1.3.1

J j (x, t)

flux of species j at location x at time t (linear system)

mol cm−2 s−1

4.4.2

Jn

nucleation rate

s−1

15.6.2(b)

J n,0

pre-exponential factor for nucleation rate

s−1

15.6.2(b)

j

(a) current density

A/cm2

1.1.5

(b) (−1)1/2

none

A.5

j0

exchange current density

A/cm2

3.4.1

jtot

in simulation, highest box number required

none

B.1.7 (Continued)

xxxiii

xxxiv

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

K, K j

equilibrium constant (of reaction j)

none

KH

Henry’s law constant

M/bar

15.6.4

K P, j

in Marcus theory, precursor state equilibrium constant for reactants or products (identified by j)

depends on mode

3.5.3(a)

k, k j

rate constant for a reaction (of process j)

depends on nature and order

k

Boltzmann constant

J/K

k 0 , kj0

standard heterogeneous rate constant (of reaction j)

cm/s

3.3.2, 3.3.3

kb

(a) heterogeneous rate constant for oxidation

cm/s

3.2

(b) rate constant for backward reaction

depends on nature and order

3.1

kd

in the ECE mechanism, rate constant for disproportionation

M−1 s−1

13.3.7

kf

(a) heterogeneous rate constant for reduction

cm/s

3.2

(b) rate constant for forward reaction

depends on nature and order

3.1

ki,j

potentiometric selectivity coefficient of interferent j toward a measurement of species i

none

2.4.2

k max

in simulation, final iteration number

none

B.1.7

kt0

true standard heterogeneous rate constant

cm/s

14.7.1

L

(a) length of a porous electrode

cm

12.5.1

(b) in SECM, dimensionless tip-to-substrate distance, d/a

none

18.2

pot

Section

L{f (t)}

Laplace transform of f (t) = f (s)

L−1 {f (s)}

inverse Laplace transform of f (s)

l

(a) length

cm

(b) thickness of a layer on an electrode

cm

11.4.4, 17.5

(c) in a thin-layer cell, thickness of solution

cm

12.6

A.1.2 A.1.4

(d) electrosorption valency

none

14.5.2

l

in simulation, number of iterations corresponding to t k

none

B.1.4

m

(a) mass

g, kg

(b) at a DME, mass flow rate of mercury

mg/s

8.1.2

mass-transfer coefficient of species j

cm/s

1.3.2

mj

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

N, N j

(a) number of items (of kind j)

none

(b) at an RRDE, collection efficiency (subscript, if used, distinguishes measurement condition)

none

10.3.2(a)

number of NP collisions in time interval Δt

none

19.3.1

(a) Avogadro constant

mol−1

(b) in a doped semiconductor, acceptor density

cm−3

20.1.3

ND

in a doped semiconductor, donor density

cm−3

20.1.3

n

(a) for an electrode reaction, stoichiometric number of electrons

none

1.1.5

(b) in a doped semiconductor, electron density in the CB

cm−3

20.1.3

N(Δt) NA

Section

(c) refractive index

none

21.3.2

n0

number concentration of each ion in a z : z electrolyte

cm−3

14.3.2

napp

apparent number of electrons passed

none

13.3.7(a)

ni

in an intrinsic semiconductor, electron density in the CB

cm−3

20.1.2

nj

(a) number of moles of species j in a volume element or a phase

mol

2.2.4, 14.1.1

(b) number concentration of ion j in an electrolyte

cm−3

14.3.2

n0j

number concentration of ion j in the bulk electrolyte

cm−3

14.3.2

O

oxidized form of O + e ⇄ R

P

pressure

Pa, bar

Pe

in CNT, hydrostatic gas pressure on the surrounding liquid

Pa, bar

15.6.2(e)

Pg

in CNT, partial pressure in equilibrium with a dissolved gas at concentration C g

Pa, bar

15.6.4

PgS

in CNT, partial pressure of a gas in equilibrium with the supersaturation concentration, CgS

Pa, bar

15.6.4

Pi

in CNT, gas pressure inside a bubble

Pa, bar

15.6.2(e)

Pj

partial pressure of species j

Pa, bar

2.1.5

Pj0

standard-state pressure of species j

Pa, bar

2.1.5

PL

in CNT, the Laplace pressure of a bubble

Pa, bar

15.6.4

(a) in a doped semiconductor, hole density in the VB

cm−3

20.1.3

(b) in bulk electrolysis, mj A/V

s−1

12.2.1

(c) in flow electrolysis, mj s/𝜀

s−1

12.5.1

(d) dimensionless kinetic parameter for ECE/DISP systems

none

13.3.7(c)

p

(Continued)

xxxv

xxxvi

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

pi

in an intrinsic semiconductor, hole density in the VB

cm−3

20.1.2

Q

charge passed in electrolysis

C

1.1.5, 6.6.1

Q0

charge required for complete electrolysis of a component by Faraday’s law

C

12.2.3

Qd

in chronocoulometry, charge from a diffusing electroreactant

C

6.6.1

none

6.6.2

Qd (2𝜏)/Qd (𝜏) in chronocoulometry, reversal criterion Qdl

charge devoted to double-layer capacitance

C

6.6.1

Qr

in chronocoulometry, charge withdrawn in reversal

C

6.6.2

q𝛼

excess charge on phase 𝛼

C, μC

1.6.2, 2.2.2

R

reduced form of O + e ⇄ R

R

(a) gas constant

J mol−1 K−1

(b) resistance

Ω

(c) in flow electrolysis, fraction of substance electrolyzed

none

12.5.1

RB

series equivalent resistance of a cell

Ω

11.1, 11.3

Rc

compensated resistance

Ω

1.5.4, 16.7.1(a)

Rct

charge-transfer resistance

Ω

1.2.3, 3.4.3(b), 3.7.5(b)

Rf

feedback resistance

Ω

16.2

Rmt

mass-transfer resistance

Ω

1.3.2(c), 3.4.6

Rs

(a) solution resistance

Ω

1.5

(b) series resistive component of Zf

Ω

11.3.1, 11.3.2(c)

Ru

uncompensated resistance

Ω

1.5.4, 16.7.1(a)

Re

Reynolds number

none

10.1.2

Re(w)

real part of complex function w

RG

in SECM, a geometric factor, rg /a

r

(a) radial distance from the center of an electrode

cm

4.4.2, 5.1, 5.2

(b) in CNT, radius of a cluster or nucleus

m, nm

15.6.2(b)

r0

radius of an electrode

cm

4.4.2, 5.1, 5.2

r1

radius of the disk in an RDE or RRDE

cm

10.3

r2

inner radius of a ring electrode

cm

10.3

r3

outer radius of a ring electrode

cm

10.3

rc

in CNT, critical radius of a cluster or nucleus

m, nm

15.6.2(b)

rd

radius of a catalytic NP deposited on an electrode surface

cm

19.5

A.5 none

18.2

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

re

contact radius of a NP with an electrode surface

cm

19.4.3

rg

in SECM, radius of the tip shroud

μm, nm

18.2

rp

radius of a particle

cm

19.4.1

S

(a) entropy

kJ/K, kJ mol−1 K−1

2.1.2

(b) in CNT, supersaturation ratio

none

15.6.4

ΔS

in a chemical process, entropy change

kJ/K, kJ mol−1 K−1

2.1.2

ΔS‡

standard entropy of activation

kJ mol−1 K−1

3.1.2

ΔS0

in a chemical process, standard entropy change

kJ mol−1 K−1

2.1.2

S𝜅 (t)

unit step function rising at t = 𝜅

none

A.1.7

s

(a) Laplace variable, usually complementary to t

A.1.2

(b) specific area of a porous electrode

cm−1

T

absolute temperature

K

t

time

s

t1

in chronopotentiometry, first current reversal time

s

9.4.2

t 2d

in redox cycling, average round-trip transit time

s

19.6

td

in stripping analysis, deposition time

s

12.7.1

tf

current sampling time in a forward step

s

6.5.2

tj

transference number of species j

none

2.3.3, 4.2

tk

in simulation, known characteristic time

s

B.1.4

t max

at a DME, drop lifetime

s

8.1.3

tp

in SWV, pulse width

s

8.5.1

tr

in chronoamperometry or chronocoulometry, current sampling time in a reversal step

s

6.5.2

U

linear flow velocity of a fluid

cm/s

12.5.1

uj

mobility of ion or charge carrier j

cm2

2.3.3, 4.1

V

volume

L, cm3

Va

in CNT, atomic volume of an electrodeposited species

m3 , nm3

15.6.2(d)

v

velocity vector

cm/s

4.1

v

linear potential scan rate

V/s

1.6.4(b), 7.2.1(a), 7.2.2

v, vj

(a) velocity (of component j or in the j direction)

cm/s

(b) homogeneous reaction rate (of process j)

mol cm−3 s−1

(c) heterogeneous reaction rate (of process j)

mol cm−2 s−1

1.1.5, 3.2

(d) in flow electrolysis, volumetric flow rate (unsubscripted)

cm3 /s

12.5.1

V−1 s−1

12.5.1

(Continued)

xxxvii

xxxviii

Major Symbols and Abbreviations

Table 2 (Continued)

Symbol

Meaning

Usual Units

Section

vmt

rate of mass transfer to a surface

mol cm−2 s−1

1.3.1

W j (𝜆, E)

probability density function for species j

eV−1

3.5.5(a)

w

width of a band electrode

cm

5.2.4

Δw

in Marcus theory, wR − wO

eV

3.5.3(c)

wj

in Marcus theory, work term in electron transfer (subscript denotes O-side or R-side of the reaction)

eV

3.5.3(c)

XC

capacitive reactance

Ω

11.2

Xj

mole fraction of species j

none

2.1.5, 14.1.3

x

(a) distance, often from a planar electrode

cm

(b) in a scanning-probe method, horizontal coordinate

μm, nm

18.3, 21.1.1, 21.1.2

x1

distance of the IHP from the electrode surface

cm

1.6.3, 14.3.4

x2

distance of the OHP from the electrode surface

cm

1.6.3, 14.3.3

admittance

S, Ω−1

11.2

Y

admittance vector

S, Ω−1

11.2

y

(a) distance from an RDE or RRDE

cm

10.2.1

(b) in a scanning probe method, horizontal coordinate

μm, nm

18.3, 21.1.1, 21.1.2

yh

hydrodynamic boundary layer thickness at an RDE

cm

10.2.1

Z

(a) impedance

Ω

11.2

(b) in simulation, dimensionless current parameter

none

B.1.6

Y

Z

impedance vector

Ω

11.2

Z(k)

in simulation, dimensionless current in iteration k

none

B.1.6

Z(𝜔)

impedance spectrum

Ω

11, 11.4

Zf

faradaic impedance

Ω

11.3.1

Z Im

imaginary part of impedance

Ω

11.2

Z Re

real part of impedance

Ω

11.2

ZW

Warburg impedance

Ω

11.3.1

z

(a) distance normal to the surface of a disk electrode or along a cylindrical electrode

cm

5.2.2(a), 6.1.3(b)

(b) in a scanning probe method, vertical coordinate

μm, nm, Å

18.3, 21.1.1, 21.1.2

(c) charge magnitude of each ion in a z : z electrolyte

none

14.3.2

signed charge on species j

none

2.3.3

zj

Major Symbols and Abbreviations

Table 3 Greek Symbols.

Symbol

Meaning

Usual Units

Section

𝛼

transfer coefficient

none

3.3.2, 3.3.4

(a) distance parameter for electron tunneling

nm−1

3.5.2(a)

none

10.3.1

𝛽

(b) RRDE geometric parameter, (r3 /r1

)3 − (r

2

/r1

)3

(c) occasionally, 1 − 𝛼

none

(a) in impedance theory, 𝜕E/𝜕C j (0, t)

V cm3 mol−1

11.3.2(b)

(b) in an adsorption isotherm, equilibrium parameter for species j

none

14.5.3

Γ∗

in a system where only adsorbed O and R are electroactive, the total surface concentration for O and R

mol/cm2

17.2.2

Γj

surface excess concentration of species j

mol/cm2

6.6.1, 14.1.1

Γj(r)

relative surface excess of species j with respect to component r

mol/cm2

14.1.3

Γs , Γj,s

surface excess at saturation (of species j)

mol/cm2

14.5.3(a)

(a) surface tension

J/m2 , dyn/cm

14.1.1

(b) ratio of supporting electrolyte concentration to electroreactant concentration

none

5.7.2

𝛾 gs

in CNT, gas–solid surface tension

J/m2 , dyn/cm

15.6.2(c)

𝛾j

activity coefficient for species j

none

2.1.5

𝛾 ls

in CNT, liquid–solid surface tension

J/m2 , dyn/cm

15.6.2(c)

𝛿j

at an electrode fed by steady-state mass transfer, Nernst layer thickness for species j

cm

1.3.2, 10.2.2

𝜀

(a) dielectric constant

none

14.3.1

(b) porosity of an electrode

none

vacuum electric permittivity

C2

𝜀j (E)

in Marcus–Gerischer theory, proportionality function (subscript ox for oxidation, red for reduction)

cm3

𝜁

zeta potential

mV

10.6

𝜂

(a) overpotential, E − Eeq

V

1.2.2, 3.4.2

(b) viscosity (of fluid j, if subscripted)

g cm−1 s−1 = poise

2.3.3, 10.1.2

𝜂 ct

charge-transfer overpotential

V

1.2.3

𝜂 HER

for Tafel plots, overpotential for the HER defined vs. Eeq for the system

V

15.2.2(a)

𝜂 mt

mass-transfer overpotential

V

1.2.3

𝜂n

nucleation overpotential

V

15.6.2(a)

𝛽j

𝛾

𝜀0

N−1

12.5.1 m−2

eV

2.2.1, 14.3.1 3.5.5(a)

(Continued)

xxxix

xl

Major Symbols and Abbreviations

Table 3 (Continued)

Symbol

Meaning

Usual Units

Section

𝜃

0 (a) enf (E−E )

none

5.4.1; 6.2.1

(b) 𝜏 1/2 + (t − 𝜏)1/2 − t 1/2

s1/2

6.6.2

(c) fractional coverage of an interface (species not denoted by subscript)

none

14.5.3(a)

′

(d) contact angle

deg

15.6.2(c)

𝜃j

fractional coverage of an interface by species j

none

14.5.3(a)

𝜃m

in impedance theory, enf (Edc

none

11.4.1

(a) conductivity

S/cm = Ω−1 cm−1

2.3.3, 4.2

(b) transmission coefficient for a reaction

none

3.1.3

(c) double-layer thickness parameter; inverse of the Debye length

cm−1

14.3.2(a)

(d) in treatment of dynamics at modified electrodes, partition coefficient for the primary reactant

none

17.5.2(b)

𝜅 el

electronic transmission coefficient

none

3.5.2(b)

Λ

in LSV and CV, dimensionless heterogeneous kinetic parameter, k 0 /(Dfv)1/2

none

7.3.1, 13.3.2

Λ0

in SSV, dimensionless heterogeneous kinetic parameter, k 0 /mO

none

5.4.1(b)

Λb

in SSV, dimensionless heterogeneous kinetic parameter, k b /mR

none

5.4.1(a)

Λf

in SSV, dimensionless heterogeneous kinetic parameter, k f /mO

none

5.4.1(a)

Λ𝛼

equivalent conductivity of a solution 𝛼

cm2 Ω−1 equiv−1

2.3.3

𝜅

𝜆

′ −E0 )

(a) reorganization energy for electron transfer

eV

3.5.3(c,d)

(b) in chronoamperometry and STV, a dimensionless kinetic parameter, −1∕2 kf t 1∕2 DO (1 + 𝜉𝜃), (t = 𝜏 in STV)

none

6.3.2

(c) dimensionless homogeneous kinetic parameter, specific to a method and mechanism, generally 𝜏 obs /𝜏 rxn

none

13.3.1(a), 13.4

(d) in CV, switching (reversal) time

s

1.6.4(b), 7.2.2

(e) wavelength

nm

𝜆0

in STV, dimensionless kinetic parameter, 1∕2 1∕2 (1 + 𝜉)k 0 𝜏 1∕2 ∕DO ≈ 2k 0 𝜏 1∕2 ∕DO

none

6.3.5

𝜆i

inner component of the reorganization energy

eV

3.5.3(d)

𝜆j

equivalent ionic conductivity for ion j

cm2 Ω−1 equiv−1

2.3.3

𝜆0j

equivalent ionic conductivity for ion j extrapolated to infinite dilution

cm2 Ω−1 equiv−1

2.3.3

𝜆o

outer component of the reorganization energy

eV

3.5.3(d)

Major Symbols and Abbreviations

Table 3 (Continued)

Symbol

Meaning

Usual Units

Section

𝜇

reaction layer thickness

cm

1.4.2

𝜇𝛼e 𝜇j𝛼 𝜇𝛼j 𝜇j0𝛼

electrochemical potential of electrons in phase 𝛼

kJ/mol

2.2.4, 2.2.5

chemical potential of species j in phase 𝛼

kJ/mol

2.2.4

electrochemical potential of species j in phase 𝛼

kJ/mol

2.2.4

standard chemical potential of species j in phase 𝛼

kJ/mol

2.2.4

𝜈

(a) in hydrodynamics, kinematic viscosity

cm2 /s

10.1.2

(b) frequency of light

s−1

𝜈j

in a chemical process, stoichiometric coefficient for species j

none

2.1.6

𝜈n

in Marcus theory, nuclear frequency factor

s−1

3.5.3(b)

generally, mO /mR ; for semi-infinite diffusion,

none

5.4.1(b), 6.2.5

(a) resistivity (of phase 𝛼)

Ω cm

4.2

(b) roughness factor (of surface 𝛼)

none

6.1.5

(c) three-dimensional charge density (in phase 𝛼)

C/cm3

10.6

electronic density of states

cm−2

3.5.5(a)

(a) in theory of LSV and CV, nFv/RT = nfv

s−1

7.2.1(a)

(b) in impedance 1∕2 1∕2 theory,(nFA21∕2 )−1 (𝛽O ∕DO − 𝛽R ∕DR )

Ω s1/2

11.3.2(b)

𝜉

1∕2

1∕2

DO ∕DR ; for SSV, DO /DR 𝜌, 𝜌𝛼

𝜌(E) 𝜎

eV−1

(c) standard deviation

A.3

(d) surface area per adsorbed molecule

nm2

17.3.2

𝜎R

parameter describing potential dependence of adsorption energy

none

17.2.4(a)

𝜎𝛼

excess surface charge density on phase 𝛼

C/cm2

1.6.2, 14.2.2

𝜏

(a) in chronopotentiometry, transition time

s

9.1, 9.2.2

(b) in STV, current-sampling time

s

6.2, 8.2.1

(c) in a double-step experiment, forward step duration

s

6.5

(d) a dimensionless parameter, 4DO t∕r02 , describing diffusion-controlled current at a UME

none

6.1.3(a)

(e) generally, a characteristic time defined by the properties of an experiment

s

(f ) an RC time constant, sometimes Ru C d

s

1.6.4(a)

𝜏′

in pulse voltammetry, start of the potential pulse, measured from beginning of each cycle

s

8.2.1

𝜏2

in chronopotentiometry, reversal transition time

s

9.4.2 (Continued)

xli

xlii

Major Symbols and Abbreviations

Table 3 (Continued)

Symbol

Meaning

Usual Units

Section

𝜏 2 /t 1

in chronopotentiometry, reversal criterion

none

9.4.2

𝜏 obs

for an electrochemical method, characteristic time of observation

s

7.6, 13.2.2, 13.4

𝜏 rev

in CV, time between E1/2 and E𝜆

s

13.3.1(b)

𝜏 rxn

for a homogeneous reaction, characteristic time of reaction

s

13.2.2

𝜏 ss

steady-state renewal time at a UME

s

5.1.3

Φ(𝜃)

in CNT, a geometric factor based on contact angle, [(2 − cos 𝜃)(1 + cos 𝜃)2 ]/4

none

15.6.2(c)

Φ𝛼

work function of a phase 𝛼

eV

2.2.5(d)

𝜙

(a) electric potential

V

2.2.1

(b) phase angle between two sinusoidal signals

deg, rad

11.2

(c) in impedance methods, phase angle between İ ac and Ė

deg, rad

11.2, 11.3.3(d)

(a) electric potential difference between two points or phases

V

2.2.1, 2.2.3

(b) in a semiconductor, potential drop in the space charge region

V

20.2.1

𝜙0

total potential drop across the solution side of the double layer

V

14.3.2(a)

𝜙2

potential at the OHP with respect to bulk solution

V

1.6.3, 14.3.3

𝜙PAD

potential at the PAD with respect to bulk solution

V

17.2.6

𝜙PET

potential at the PET with respect to bulk solution

V

17.2.2(b)

𝜙S

electric potential in bulk solution

V

1.6.3

𝜙𝛼

inner (Galvani) potential of conducting phase 𝛼

V

2.2.1

Δ𝛼𝛽 𝜙

junction potential at a liquid/liquid interface between phases 𝛼 and 𝛽

V

7.8

Δ𝛼𝛽 𝜙0j

standard Galvani potential of ion transfer for species j from phase 𝛼 to phase 𝛽

V

7.8

ac

Δ𝜙

𝜒(j)

in simulation, dimensionless distance of box j

none

B.1.5

𝜒(bt)

in LSV and CV, dimensionless current for a totally irreversible system

none

7.4.1(a)

𝜒(𝜎t)

in LSV and CV, dimensionless current for a reversible system

none

7.2.1(a)

𝜒f

in treatment of dynamics at modified electrodes, rate constant for permeation of the primary reactant

cm/s

17.5.2(b)

𝜓

in CV, dimensionless quasireversibility parameter

none

7.3.2

(a) angular velocity of rotation; 2𝜋 × rotation rate

s−1

10.2

(b) angular frequency of a sinusoidal oscillation, 2𝜋f

s−1

11.2

𝜔

Major Symbols and Abbreviations

Table 4 Standard Abbreviations. Abbreviation

Meaning

1e, 2e, …ne

one-electron, two-electron, …n-electron

Section

1D, 2D, 3D

one-dimensional, two-dimensional, three-dimensional

ADC

analog-to-digital converter

16.6

AES

Auger electron spectrometry

21.8.2

AFM

atomic force microscopy (microscope)

21.1.2

Ag/AgCl

Ag/AgCl/KCl (sat’d) reference electrode

1.1.3, 2.1.8(c)

ASV

anodic stripping voltammetry

12.7.1

ATR

attenuated total reflection (reflectance)

21.4.1

A/V

area-to-volume

5, 12

BDD

boron-doped diamond

14.4.1(b)

BiFE

bismuth film electrode

12.7.2

BV

Butler–Volmer

3.3

CACV

cyclic ac voltammetry

11.5.3

CB

conduction band

20.1.1

CE

(a) homogeneous chemical process preceding heterogeneous electron transfer(a)

13.1

(b) capillary electrophoresis

12.5.3

(a) classical nucleation theory

15.6.2

(b) carbon nanotubes

14.4.1(b)

CSWV

cyclic square wave voltammetry

8.5.4

CV

cyclic voltammetry

7.1, 7.2.2

DAC

digital-to-analog converter

16.6

DEMS

differential electrochemical mass spectrometry

21.6

CNT

DESI-MS

desorption electrospray ionization mass spectrometry

21.6

DFT

density functional theory

15.4.1

DISP

ECE system in which the second electron is transferred predominantly by homogeneous disproportionation

13.3.7(c)

DME

dropping mercury electrode

8.1.1

DMFC

direct methanol fuel cell

15.3.3

DPP

differential pulse polarography

8.4.5

DPV

differential pulse voltammetry

8.4

DSA

dimensionally stable anode

20.1.5(a)

(E)n

stepwise heterogeneous electron transfers making up an n-electron sequence (EE is preferred notation for two steps)(a)

13.3.6(b)

EA

electron affinity

20.1.4

EC

heterogeneous electron transfer followed by homogeneous chemical reaction(a)

13.1 (Continued)

xliii

xliv

Major Symbols and Abbreviations

Table 4 (Continued) Abbreviation

Meaning

Section

EC′

catalytic regeneration of the electroactive species in a following homogeneous reaction(a)

13.1

EC2

heterogeneous electron transfer followed by homogeneous dimerization(a)

13.1

ECE

heterogeneous electron transfer, homogeneous chemical reaction, and heterogeneous electron transfer, in sequence(a)

13.1

ECE/DISP

ECE system in which homogeneous disproportionation is important

13.3.7

ECEC

heterogeneous electron transfer, homogeneous chemical reaction, heterogeneous electron transfer, homogeneous chemical reaction, in sequence(a)

13.1

ECL

electrogenerated chemiluminescence

20.5

ECM

electrocapillary maximum

14.2.2

EC-STM

electrochemical scanning tunneling microscopy

21.1.1

EDS

energy dispersive X-ray spectroscopy

21.8.1

EE

stepwise heterogeneous electron transfers making up a 2-electron sequence(a)

13.1

EELS

electron energy loss spectrometry

21.8.2

EIS

electrochemical impedance spectroscopy

11, 11.4

emf

electromotive force

2.1.3

EMIRS

electrochemically modulated infrared reflectance spectroscopy

21.4.1

ESR

electron spin resonance

21.7.1

ETM

electron-transport material

20.2.3

EXAFS

extended X-ray absorption fine structure

21.5

FCC

face-centered cubic

14.4.1(a)

FFT

fast Fourier transform

A.6

FI

flow injection

12.5.3

FRA

frequency response analyzer

11.8, 11.8.1

FSCV

fast-scan cyclic voltammetry

17.8.4(b)

FT

Fourier transformation

A.6

FTAC

Fourier transform ac voltammetry, usually referring to the large-amplitude approach

11.6.2

FTIR

Fourier transform infrared (spectrometer)

21.4.1

FTO

fluorine-doped tin oxide

20.1.5(c)

GBP

gain–bandwidth product

16.1.2(b)

GC

glassy carbon

1.9.2

GCS

Gouy–Chapman–Stern

14.3.3

GDP

galvanostatic double pulse

9.6

HCP

hexagonal close-packed

14.4.1(b)

Major Symbols and Abbreviations

Table 4 (Continued) Abbreviation

Meaning

Section

HER

hydrogen evolution reaction

2.1.9(a), 15.2.1

HMDE

hanging mercury drop electrode

7.2.1(c), 8.2.3(b)

HOMO

highest occupied molecular orbital

HOPG

highly oriented pyrolytic graphite

14.4.1(b)

HREELS

high-resolution electron energy loss spectrometry

21.8.2

HTM

hole-transport material

20.2.3

ICR

ionic current rectification

10.6

IE

ionization energy

15.1.2

IHP

inner Helmholtz plane

1.6.3, 14.3.4

INE

ideally nonpolarizable electrode

1.2.2

IPE

ideally polarizable electrode

1.2.2, 1.6.1

IR

infrared

IRRAS

infrared reflection-absorption spectroscopy

21.4.1

ISE

ion-selective electrode

2.4

ITIES

interface between two immiscible electrolyte solutions

7.8

ITO

indium-tin oxide

20.1.5(c)

KL

Koutecký–Levich

10.2.5, 17.5.1

LB

Langmuir–Blodgett

17.1

LCEC

liquid chromatography with electrochemical detection

12.5.3

LEED

low-energy electron diffraction

21.8.2

LSV

linear sweep voltammetry

7, 7.2.1

LUMO

lowest unoccupied molecular orbital

MFE

mercury film electrode

12.7.2

MMO

mixed metal oxide

20.1.5(a)

MO

molecular orbital

NCE

normal calomel electrode, Hg/Hg2 Cl2 /KCl(1.0 M)

2.1.8(c)

NHE

normal hydrogen electrode (=SHE)

1.1.3, 2.1.4

NMR

nuclear magnetic resonance

21.7.2

NP

nanoparticle

19.2

NPP

normal pulse polarography

8.2.3

NPV

normal pulse voltammetry

8.2.1

NSOM

near-field scanning optical microscopy (microscope)

21.1.3

OCP

open-circuit potential

3.6

ODE

ordinary differential equation

A.1.1

OEMS

online electrochemistry-mass spectrometry

21.6

OER

oxygen evolution reaction

2.1.9(a)

OHP

outer Helmholtz plane

1.6.3, 14.3.3

ORR

oxygen reduction reaction

2.1.9(a), 15.3.1 (Continued)

xlv

xlvi

Major Symbols and Abbreviations

Table 4 (Continued) Abbreviation

Meaning

Section

OTE

optically transparent electrode

21.3.1

OTTLE

optically transparent thin-layer electrode

21.3.1

PAD

(a) pulsed amperometric detection

12.5.3(a)

(b) plane of acid dissociation

17.2.6

PCET

proton-coupled electron transfer

13.3.8(e)

PDE

partial differential equation

A.1.1

PDF

probability density function

19.6

PET

plane of electron transfer

17.2.2(b)

PNP

Poisson–Nernst–Planck

14.7.4

PZC

potential of zero charge

1.6.4(a), 14.2.2

QCM

quartz crystal microbalance

21.2

QCM-D

quartz crystal microbalance with dissipation monitoring

21.2.2

QD

quantum dot

20.1.2

QRE

quasireference electrode

2.5.2

RC-SECM

redox-competition SECM

18.4.1

RDE

rotating disk electrode

1.3.2, 10.2

RDS

rate-determining step

3.7.2

RGO

reduced graphene oxide

15.2.2(e)

RHE

reversible hydrogen electrode

2.1.8(d)

RPP

reverse pulse polarography

8.3

RPV

reverse pulse voltammetry

8.3

RRDE

rotating ring-disk electrode

10.3

RVC

reticulated vitreous carbon

12.1.3(a)

SAM

(a) self-assembled monolayer

17.1

(b) scanning Auger microprobe

21.8.2

SCE

saturated calomel electrode

1.1.3, 2.1.8(c)

SECCM

scanning electrochemical cell microscopy (microscope)

18.8

SECM

scanning electrochemical microscopy (microscope)

18

SEIRAS

surface-enhanced infrared absorption spectroscopy

21.4.1

SEM

scanning electron microscopy (microscope)

21.1.4, 21.8.1

SERS

surface enhanced Raman spectroscopy

21.4.2

SG/TC

substrate generation/tip collection, a mode of SECM

18.4.2

SHE

standard hydrogen electrode (=NHE)

1.1.3, 2.1.4

SIMS

secondary-ion mass spectrometry

21.8.2

SI-SECM

surface interrogation scanning electrochemical microscopy

18.5

SMD

single-molecule detection

19.6

SMDE

static mercury drop electrode

8.2.3(b)

SNIFTIRS

subtractively normalized interfacial Fourier transform infrared spectroscopy

21.4.1

Major Symbols and Abbreviations

Table 4 (Continued) Abbreviation

Meaning

Section

SPR

surface plasmon resonance

21.3.3

SSCE

sodium-saturated calomel electrode, Hg/Hg2 Cl2 /NaCl (sat’d), −5 mV vs. SCE

Table C.2

SSV

steady-state voltammetry

5.1.4

STEM

scanning transmission electron microscopy (microscope)

21.1.4, 21.8.1

STM

scanning tunneling microscopy (microscope)

21.1.1

STV

sampled transient voltammetry

6.2

SWV

square wave voltammetry

8.5.1

TEM

transmission electron microscopy (microscope)

21.1.4, 21.8.1

TERS

tip-enhanced Raman spectroscopy

21.4.2

TFA

time of first arrival

19.3.3

TG/SC

tip generation/substrate collection, a mode of SECM

18.4.2

UHV

ultrahigh vacuum

21.8.2

UME

ultramicroelectrode

1.3.2, 5.2

UPD

underpotential deposition

15.6.3

UV

ultraviolet

VB

valence band

20.1.1

XAS

X-ray absorption spectroscopy

21.5

XAFS

X-ray absorption fine structure

21.5

XANES

X-ray absorption near-edge structure

21.5

XPS

X-ray photoelectron spectrometry

21.8.2

XRD

X-ray diffraction

21.5

(a) Letters may be subscripted i, q, or r to indicate irreversible, quasi-reversible, or reversible reactions.

Table 5 Abbreviated Chemical Species(a) . Abbreviation

Meaning

Structure

AB

azobenzene

Figure 1

ADN

adiponitrile

Equation 18.4.7

AN

acrylonitrile

Equation 18.4.5

An

anthracene

Figure 1

2,6-AQDS

anthraquinone-2,6-disulfonate

Figure 1

AzT

azotoluene

Figure 1

B[ghi]FA

benzo[ghi]fluoranthene

Figure 1

BP

benzophenone

Figure 1

bpy

2,2′ -bipyridine

Figure 1 (Continued)

xlvii

xlviii

Major Symbols and Abbreviations

Table 5 (Continued) Abbreviation

Meaning

Structure

BQ

p-benzoquinone

Figure 1

Ch

chrysene

Figure 1

COD

cyclooctadiene

Figure 1

CP[cd]Py

cyclopenta[cd]pyrene

Figure 1

DA

dopamine

Figure 1

DCB

1,4-dicyanobenzene

Figure 1

DCE

1,2-dichloroethane

2,6-DHADS

9,10-dihydroxyanthracene-2,6-disulfonate

Figure 1

DMA

9,10-dimethylanthracene

Figure 1

DMF

N, N-dimethylformamide

DMSO

dimethylsulfoxide

DOPAC

3,4-dihydroxyphenylacetic acid

Figure 1

DPA

9, 10-diphenylanthracene

Figure 1

EDTA

ethylenediaminetetraacetate

EPI

epinephrine

FA

fluoranthene

Figure 1

Fc

ferrocene

Figure 1

FcA−

ferrocenecarboxylate

Figure 1

Figure 1

FcMeOH

ferrocenylmethanol

Figure 1

FcTMA+

ferrocenylmethyltrimethylammonium

Figure 1

FePc

iron(II) phthalocyanine

Figure 1 (MPc)

FePP

iron(II) protoporphyrin IX

Figure 1 (MPP)

GOx

glucose oxidase

5-HIAA

5-hydroxyindole-3-acetic acid

Figure 1

HQ

hydroquinone

Figure 1

5-HT

5-hydroxytryptamine (serotonin)

Figure 1

HVA

homovanillic acid

Figure 1

Me10 Fc

decamethylferrocene

Figure 1

10-MP

10-methylphenothiazine

Figure 1

MPc

metal(II) phthalocyanine

Figure 1

MPP

metal(II) protoporphyrin IX

Figure 1

MTPP

metal(II) tetraphenylporphine

Figure 1

MV2+

methyl viologen

Figure 1

Naf

NafionTM

Figure 17.4.1

NB

nitrobenzene

Figure 1

NE

norepinephrine

Figure 1

NP

naphthalene

Figure 1

Major Symbols and Abbreviations

Table 5 (Continued) Abbreviation

Meaning

Structure

pChl

p-chloranil

Figure 1

P3HT

poly(3-n-hexylthiophene)

Figure 20.2.5

PEDOT

poly(3,4-ethylenedioxythiophene)

Figure 20.2.5

PP

protoporphyrin IX

Figure 1

PPD

2,5-diphenyl-1,3,4-oxadiazole

Figure 1

PPy

polypyrrole

Figure 20.2.5

PS

polystyrene

Figure 17.4.1

PSS

poly(styrenesulfonate)

Figure 17.4.1

PT

polythiophene

Figure 20.2.5

PVFc

poly(vinylferrocene)

Figure 17.4.1

PVOS

polymerized viologen organosilane

Figure 17.4.1

PVP

poly(4-vinylpyridine)

Figure 17.4.1

PXDOT

poly(3,4-ortho-xylenedioxythiophene)

Figure 20.2.6

PXV

poly(xylylviologen)

Figure 17.4.1

Py

pyrene

Figure 1

QPVP

poly(4-vinyl-N-methylpyridinium)

Figure 17.4.1

R

rubrene

Figure 1

TBABF4

tetra-n-butylammonium tetrafluoroborate

TBAI

tetra-n-butylammonium iodide

TBAN3

tetra-n-butylammonium azide

TBAP

tetra-n-butylammonium perchlorate

TBAPF6

tetra-n-butylammonium hexafluorophosphate

TBATPB

tetra-n-butylammonium tetraphenyborate

TCNQ

7,7,8,8-tetracyanoquinodimethane

TEABF4

tetraethylammonium tetrafluoroborate

Figure 1

TEAP

tetraethylammonium perchlorate

TEMPO∙

2,2,6,6-tetramethylpiperidine-1-oxyl

Figure 1

TH

thianthrene

Figure 1

THF

tetrahydrofuran

TMPD

N, N, N ′ , N ′ -tetramethyl-p-phenylenediamine

TPAsTPB

tetraphenylarsonium tetraphenylborate

TPP

tetraphenyporphyrin

Figure 1

TPrA

tri-n-propylamine

Figure 1

TPTA

tri-p-tolylamine

Figure 1

ttBP

1,3,5-tri-tert-butylpentalene

Figure 1

TTF

tetrathiafulvalene

Figure 1

(a) Standard chemical abbreviations, such as EtOH, MeCN, and PhBr, are not included.

Figure 1

xlix

l

Major Symbols and Abbreviations

Figure 1 Structures of molecules and ions used with abbreviated names. Chemical names are identified in Table 5 (immediately preceding). Polymers used for electrode modification are depicted in Figure 17.4.1. Conducting polymers are shown in Figure 20.2.5.

Major Symbols and Abbreviations

Figure 1 (Continued)

li

liii

About the Companion Website This book is accompanied by a companion website www.wiley.com/go/BardElectrochemical3e

The website features: • Figure power point slides.

1

1 Overview of Electrode Processes Electrochemistry allows the controlled addition or removal of electrons, often one by one, at molecules or ions arriving very near a metallic or semiconducting electrode. Electrochemical systems provide access to fundamental chemical events and valuable practical applications; however, the science also entails complexity. Key events often take place in a tiny fraction of the total volume, typically at or very near metallic surfaces, or perhaps only at rare active sites on surfaces. The reacting molecules or ions must be transported to the locales of reaction, and the processes of transport influence reaction rates. Once a reaction is initiated at an electrode, it can proceed through a mechanism involving almost any kind of chemical step–proton transfer, a change in ligation, elimination, rearrangement, or even subsequent electron transfer–either at the electrode or with other molecules or ions. Reactions at electrodes involve the orbital structures of ions and molecules and the band structures of metals, semiconductors, and insulators. They depend, too, on electrostatics and thermodynamics. Among the domains of chemistry, electrochemistry offers some of the greatest challenges to effective theory and experimental examination; yet, over two centuries of genuine science, a strong base of theory and experimental methodology has been assembled. This book presents the most essential aspects. Electrochemistry connects electrical and chemical effects. Much of it deals with either chemical changes caused by passage of an electric current or the production of electrical energy by chemical reactions. Yet, the field extends broadly to encompass diverse phenomena (e.g., electrophoresis and corrosion), devices (such as electrochromic displays, electroanalytical sensors, and fuel cells), and technologies (including portable power for electronic devices or automobiles, large-scale energy storage for load management on the electric power grid, electroplating of metals, and large-scale production of materials, such as aluminum and chlorine). While the basic principles of electrochemistry discussed in this text apply to all such topics, the main emphasis here is on the application of electrochemical methods to the study of chemical systems. Scientists make electrochemical measurements for a variety of reasons. They may want to understand the kinetics of a reaction at an electrode, perhaps to speed it up, to inhibit it, or to optimize product yield. They may want to generate an unstable intermediate, such as a radical ion, and study its rate of decay or its spectroscopic properties. They may seek to analyze a solution for trace species. They may be interested in obtaining thermodynamic data about a reaction. In these examples, electrochemical methods are employed as tools in the study of chemical systems, much in the manner of spectroscopic methods. Many electrochemical methods have been devised. Their application requires an understanding of the fundamental principles of reactions at electrodes and the electrical properties of electrode/solution interfaces. This chapter sets a stage. It introduces the terms and concepts used to describe electrochemical systems. In the process, it covers interfaces and cells; potential and current; reference electrodes and the measurement of potential; concepts of electrochemical experimentation; Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

2

1 Overview of Electrode Processes

the dynamics behind current–potential curves; the extraction of chemical information from electrochemical responses; faradaic and nonfaradaic processes; double-layer structure and charging currents. The tour of ideas is designed to help the reader to establish an elementary working knowledge of electrochemical concepts and systems. If the larger goal is to learn how to employ electrochemical methods reliably, far more is needed. The journey just begins in Chapter 1. The ideas and treatments introduced here are developed much more fully and rigorously in later chapters.

1.1 Basic Ideas 1.1.1

Electrochemical Cells and Reactions

In electrochemical systems, one is concerned with the processes and factors affecting the transport of charge across interfaces between adjacent chemical phases, most commonly between an electronic conductor (an electrode) and an ionic conductor (an electrolyte). Throughout this book, we will focus on the electrode/electrolyte interface and the events that occur there when an electric potential is applied and current passes. Charge is transported through an electrode by the movement of electrons (and sometimes by “holes” in semiconductor electrodes). In an electrolyte, charge is carried by the movement of ions.1 At the interface between an electrode and an electrolyte, the passage of charge requires that these distinct conduction modes be linked through an electrode reaction, causing electrons to be consumed or produced at the electrode and ions to be produced or consumed in the electrolyte. Examples include: a) Plating of copper from an aqueous solution, Cu2+ + 2e → Cu

(1.1.1)

b) Production of hydrogen gas from alkaline water at a platinum electrode, 2H2 O + 2e → H2 + 2OH−

(1.1.2)

c) Generation of the nitrobenzene radical anion in acetonitrile at a gold electrode, PhNO2 + e → PhNO2 ∙

(1.1.3)

d) De-intercalation of lithium ion from a lithium cobalt oxide electrode in a lithium-ion battery, leaving an excess electron on the electrode to be used in an external circuit, LiCoO2 → Li1−x CoO2 + xLi+ + xe

(1.1.4)

e) Oxidation of stable ruthenium(II) complexes in water solution at a gold electrode,2 Ru(bpy)2+ → Ru(bpy)3+ +e 3 3

(1.1.5)

Typical electrode materials include solid metals (e.g., Pt, Au), liquid metals (e.g., Hg, amalgams), carbon (e.g., graphite, graphene, glassy carbon), and semiconductors (indium–tin oxide, Si, GaAs, CdS). The most common electrolytes are liquid solutions containing ionic species, 1 Questions are sometimes asked about conduction in solutions by “free” electrons. Such species are very short lived and contribute negligibly to conductivity. “Solvated” electrons can exist in exceptional circumstances (Section 20.4), but are generally very reactive. Overwhelmingly, the mobile charges in solutions are ions, and electrons exchanged at an electrode are carried in solution by atoms, molecules, and ions. 2 bpy = 2,2′ -bipyridine (Figure 1).

1.1 Basic Ideas

such as, H+ , Na+ , tetraalkylammonium, Cl− , or BF− in water or a nonaqueous solvent. To be 4 useful in an electrochemical cell, the solvent–electrolyte system must be of sufficiently low resistance (i.e., sufficiently conductive) for the electrochemical experiment envisioned. Less conventional electrolytes include fused salts (e.g., molten NaCl–KCl eutectic) and ionically conductive polymers (e.g., Nafion, polyethylene oxide–LiClO4 ). Solid electrolytes also exist (e.g., sodium β-alumina, in which charge is carried by mobile sodium ions moving between aluminum oxide sheets). It is natural to think about events at a single interface, but we will find that one cannot deal experimentally with an isolated boundary. Instead, one must study collections of interfaces called electrochemical cells, which are defined most generally as two electrodes separated by at least one electrolyte phase. There is a shorthand notation for expressing the structures of cells. For example, that pictured in Figure 1.1.1a is written compactly as Zn∕Zn2+ , Cl− ∕AgCl∕Ag

(1.1.6)

In this notation, a slash represents a phase boundary, and a comma separates two components in the same phase.3 When a gaseous phase is involved, it is written adjacent to its corresponding conducting element. For example, the cell in Figure 1.1.1b is written schematically as Pt∕H2 ∕H+ , Cl− ∕AgCl∕Ag

(1.1.7)

Generally, the electrodes in a cell are not at the same electrical potential. If the two electrodes are connected by a wire, electrons can move from the more negative electrode to the more positive, so that a chemical reaction can be sustained in the cell. Electrons produced on the negative electrode by an electrode reaction proceed to the positive electrode, where they feed a different electrode reaction, consuming the electrons. The overall chemical reaction taking place in a cell is made up of the independent half-reactions describing the separate chemical changes at the two electrodes. For example, the bidirectional half-reactions for the cell in (1.1.6) are Zn2+ + 2e ⇌ Zn and AgCl + e ⇌ Ag + Cl− . When the electrodes are connected, the reaction Pt H2 Zn

Ag

Ag

Cl–

Cl–

Zn2+

H+ Excess AgCl

(a)

Excess AgCl (b)

Figure 1.1.1 Typical electrochemical cells. (a) Zn metal and Ag wire covered with AgCl immersed in an aqueous ZnCl2 solution. (b) Pt wire in a stream of H2 and Ag wire covered with AgCl in HCl solution. 3 A double slash, not yet used here, represents a phase boundary whose interfacial potential difference is regarded as a negligible component of the overall cell potential.

3

4

1 Overview of Electrode Processes

at zinc proceeds in the backward direction, while that at silver goes forward. Zinc dissolves and AgCl is converted to Ag + Cl− . The net result is Zn + 2AgCl → Zn2+ + 2Ag + 2Cl−

(1.1.8)

Processes 1.1.1–1.1.5 are all half-reactions. None can happen alone. Each must be coupled in a cell that includes a second interface with another half-reaction proceeding in the opposite direction. 1.1.2

Interfacial Potential Differences and Cell Potential

We will find in Chapter 2 that a transition in electric potential can generally be expected in crossing from one conducting phase to another. Moreover, it usually occurs in a narrow zone right at the interface. Thus, the electric potential changes in a stairstep fashion as one proceeds from one electrode, through the intervening conducting phases, to the other electrode. The situation is as depicted in Figure 1.1.2. A high-impedance voltmeter enables one to measure a difference in electrical potential between the electrodes in a complete cell without perturbing the cell by drawing an appreciable current from it. This cell potential [in volts (V), where 1 V = 1 joule/coulomb (J/C)] expresses the energy available to drive charge externally between the electrodes and to do electrical work outside the cell. It is the sum of the individual interfacial potential differences encountered on the electrical path inside the cell from one electrode to another (shown as E in Figure 1.1.2). The potential difference between the electrodes is not merely to be observed in the manner just discussed. It can also be deliberately altered by connecting a power supply across the cell. As the applied voltage is adjusted, one can generally expect changes in the individual interfacial potential differences, especially at the electrode/electrolyte boundaries. Normally, the imposed voltage will differ from the open-circuit cell potential, and the cell will respond by drawing current from the power supply. By changing the imposed voltage, it is practical to drive the

Ag Zn2+, Cl− electrolyte

Cuʹ

Zn ϕ

ϕCuʹ−ϕCu = E

Cu

Distance across cell ′

Figure 1.1.2 Profile of electric potential, 𝜙, across Cu/Zn/Zn2+ , Cl− /AgCl/Ag/Cu at equilibrium (open-circuit, zero current). The Cu and Cu′ represent contacts made to the cell in Figure 1.1.1a using copper wire. A cell potential can be measured only between two like phases. The AgCl is not in the electrical path because it is porous, and the electrolyte has direct access to the Ag. The AgCl is, however, a critical component of the half reaction, so it is included in the representation of the cell.

1.1 Basic Ideas

current in either direction. We will shortly be discussing the magnitude and chemical effects of the current. A sharp transition in potential at a phase boundary implies that a high electric field exists there. One can expect it to influence the behavior of charge carriers (electrons or ions) in the interfacial region. More important, a potential difference at an interface affects the relative energies of electrons on opposite sides of the boundary; hence, it can determine the direction and the rate of electron transfer. Accordingly, the measurement and control of potential differences in a cell are among the most important aspects of experimental electrochemistry. 1.1.3

Reference Electrodes and Control of Potential at a Working Electrode

Even though it is always necessary to employ whole cells in electrochemical investigations, one’s interest is usually in the behavior at only one of the electrodes, called the working electrode.4 To focus on it, one can standardize the other half of the cell by using an electrode (called a reference electrode) that exhibits a reproducible, invariant potential difference at its electrode/electrolyte boundary. If one can build a cell in which a working electrode is paired with a well-behaved reference electrode, then any change in the cell potential must occur at the working electrode. The key to such a reference electrode is to maintain a known, constant composition. If all participants in a half-reaction are present at an electrode/electrolyte interface [including both redox forms (e.g., both AgCl and Ag or both H+ and H2 )], the interfacial potential difference turns out to be linked to the activities of those species near the electrode.5 The Nernst equation, familiar from introductory chemistry, describes the behavior quantitatively (Section 2.1.6). By constructing a reference electrode of reproducible composition, one obtains a reproducible potential difference at the reference interface. It is also important that the interfacial potential difference at the reference electrode not change when current passes through the cell and that the reference side of the cell not suffer contamination from the working side. We will soon see how these issues can be addressed. The design, construction, and use of reference electrodes make up a substantial topic of their own (Sections 2.1.8 and 2.5), but it should not divert us now. The present goal is just to introduce the concept of a reference electrode and to identify some common types. The internationally accepted primary reference is the standard hydrogen electrode (SHE), or normal hydrogen electrode (NHE), which has all components at unit activity at 25 ∘ C:6 Pt∕H2 (a = 1)∕H+ (a = 1, aqueous)

(1.1.9)

Since the NHE is not convenient from an experimental standpoint,7

potentials are measured and quoted with respect to other reference electrodes. An important historic reference is the saturated calomel electrode (SCE), which is Hg∕Hg2 Cl2 ∕KCl (saturated in water) Its potential at 25 ∘ C is 0.244 V vs. NHE.

(1.1.10)

4 Sometimes called the indicator electrode. 5 Activity is a concept used extensively in thermodynamics to take account of the effects of concentrations, partial pressures, or mole fractions. See Section 2.1.5 for detail. For now, just think of it as a representation of concentration, pressure, or mole fraction, in which “unit activity” would correspond roughly to a 1 M solution, to a partial pressure of 105 Pa (1 bar), or to a pure solid. 6 Thus, the NHE would involve something like 1 M H+ and gaseous H2 at close to 1 bar (1.01325 × 105 Pa). 7 The NHE is not actually realizable except by extrapolation of the behavior of real electrodes. For now, it is not necessary for us to go into how this is done. We just need to recognize that the NHE is not an everyday reference electrode.

5

6

1 Overview of Electrode Processes

A more common reference, is the silver/silver chloride electrode, Ag∕AgCl∕KCl (saturated in water) (1.1.11) ∘ with a potential at 25 C of 0.197 V vs. NHE. It is common to see potentials identified in the literature as “vs. Ag/AgCl” when this electrode is used. When a cell potential is simply observed using a high-impedance voltmeter, no significant current is allowed to pass in the external circuit between the electrodes. In that situation, the working electrode potential would change only if the composition at the working electrode were to be altered, perhaps because of a pH change or because new species were added to the surrounding solution. As mentioned above, it is also possible to shift the potential of the working electrode vs. the reference by connecting a power supply to these two electrodes, causing the cell potential to adopt the power supply’s output voltage. Ideally, any change in the cell potential from that observed without connection to the power supply would show up at the working electrode, because the reference electrode is built to maintain a constant interfacial potential difference. 8 In this way, we might achieve the ability to control the working electrode’s potential arbitrarily. In sections below, we will revisit that topic in detail. 1.1.4

Potential as an Expression of Electron Energy

We say in electrochemistry that we observe or control the potential of the working electrode with respect to the reference, which is equivalent to observing or controlling the energy of the transferable electrons on the working electrode, relative to electronic states in the electrolyte (1, 2) (Section 2.2.5). By driving the working electrode to more negative potentials, the energies of its electrons are raised. Very significant changes in energy can be achieved, because the potential can often be shifted over a range of several volts. For each volt, the energy of each electron on the working electrode changes by one electron–volt (1 eV),9 which is 96.5 kJ per mole of electrons. This is a sizable change in energy—comparable to the activation energies or overall energy changes involved in many chemical reactions—comparable, as well, to the spacings between orbitals involved in chemical bonding. The electrons on the electrode can reach a level high enough to transfer into vacant electronic states on species in the electrolyte. In that case, a flow of electrons from electrode to solution (a reduction current) occurs (Figure 1.1.3a). Similarly, the energy of the electrons can be lowered by imposing a more positive potential, and at some point, electrons on molecules or ions in the electrolyte will find a more favorable energy on the electrode and can transfer there. Their flow, from solution to electrode, is an oxidation current (Figure 1.1.3b). The critical potentials at which these processes occur are often related to the standard potentials, E0 , for the specific half-reactions in the system. We will have much more to say about standard potentials, especially in Chapter 2. 1.1.5

Current as an Expression of Reaction Rate

When the potential of the working electrode vs. the reference is varied by means of an external power supply, a current can flow in the external circuit, because electrons cross the 8 And because other interfacial potential differences in the system, such as those at metal–metal contacts (Figure 1.1.2) also remain constant. 9 The eV is a convenient unit of energy in electrochemical studies. It is the work required to move a positive test charge equal in magnitude to the electronic charge, e, across a barrier of 1 V. In general, ΔE = qΔ𝜙, where ΔE is the change in energy, q is the charge being moved, and Δ𝜙 is the electric potential difference across which the charge is moved. When q is expressed in units of e and Δ𝜙 is in V, ΔE is in eV.

1.1 Basic Ideas

Electrode

Solution

Electrode

Solution e

Vacant MO

–

Potential

Energy level of electrons

+

Occupied MO A + e → A– (a) Electrode

Solution

Electrode

Solution

Vacant MO

– Energy level of electrons Potential

e +

Occupied MO A + e → A+ (b)

Figure 1.1.3 Representation of (a) reduction and (b) oxidation of a species, A, in solution. The molecular orbitals (MO) shown for species A are the highest occupied MO and the lowest vacant MO. For simple one-electron reactions, these correspond in an approximate way to the E 0 values of the A/A− and A+ /A couples, respectively. The illustrated system could represent an aromatic hydrocarbon like 9,10-diphenylanthracene in an aprotic solvent like acetonitrile at a platinum electrode.

electrode/solution interfaces as reactions occur. The number of electrons is related stoichiometrically to the amounts of reactant consumed and product generated. If an electrode reaction consumes or produces n electrons for each reactant (e.g., 2e for the oxidation of Zn), then n moles of electrons must flow in the external circuit for every mole of electroreactant transformed. The total charge, Q, on those electrons is Q (coulombs) = nF (coulombs∕mol electrolyzed) × N (mol electrolyzed)

(1.1.12)

where F (the Faraday constant) is the charge on a mole of electrons, 96,485.3 C/mol. Equation (1.1.12) is known as Faraday’s law. The passage of 96,485.3 C causes consumption of 1 mole of reactant and production of 1 mole of product in a simple one-electron reaction.10 10 The number of moles of electrons passed is sometimes called the number of equivalents.

7

8

1 Overview of Electrode Processes

The current, expressed in amperes (A), is the rate at which the total charge is collected, i(amperes) =

dQ dN (coulombs∕s) = nF dt dt

Rate (mol∕s) =

dN i = dt nF

(1.1.13) (1.1.14)

Thus, the current is a direct measure of the rate of the reaction at the working electrode. Interpreting the rate of an electrode reaction is often more complex than for reactions occurring in solution or in the gas phase. The latter are called homogeneous reactions because they occur everywhere within the medium at a uniform rate. In contrast, electrode processes are heterogeneous reactions, occurring only at the electrode/electrolyte interface. Their rates depend on mass transfer to the electrode and various surface effects, in addition to the usual kinetic variables. Reaction rates, v, for heterogeneous reactions are usually given in units of mol/s per unit area of electrode surface, A; i.e., Rate(mol s−1 cm−2 ) = v =

j i = nFA nF

(1.1.15)

where j is the current density (A/cm2 ). 1.1.6

Magnitudes in Electrochemical Systems

It is common to characterize the scale of an electrochemical system according to the size of the working electrode, the magnitude of the current, the duration over which a current can be delivered, the timescale of an experimental perturbation, or the mass of reactant consumed or product created. Systems can be huge or tiny by any of these measures. Working electrodes can be as large as several square meters or as small as a square nanometer (10−18 m2 ). Current may be in the hundreds of thousands of amperes or in the picoamperes. A current might be required to last for nanoseconds or for years, and the timescales of experimental perturbations can extend from the indefinite to nanoseconds. The use of a system may involve no consumption of material, only a few atoms, or massive amounts. In the upper range, for example, the industrial production of metallic aluminum is wholly accomplished in electrochemical cells. The global economy produces about 50 million metric tons annually. The US share of this production (but less than 5% of the global total) still requires several percent of all US electric power production. While this book is general in its presentation of electrochemical fundamentals, the real focus is on the experimental methodology used to measure properties of chemical systems or to investigate chemical behavior. We will be thinking about “lab-sized” cells convenient to bench-top work. They generally will not be larger than a few hundred mL, but they can be much, much smaller. Cells containing microliters are common. Indeed, cells have been made with volumes in the attoliter (10−18 L) range. In most experiments, we will seek only to probe the system, not to change its overall composition appreciably. A small working electrode is usually employed. It might be a disk (made by sealing a wire in glass and polishing a cross-section), an exposed length of wire, or a droplet. It might be as large as a few centimeters in diameter or length, but usually it is smaller, and the surface area is less than 0.1 cm2 . Much work has been done with very much smaller electrodes, e.g., with disks having diameters in the range of a few μm or even down to about 10 nm. The equipment used to control and to observe electrochemical cells can sometimes deliver 1–10 A, but mostly it has less capacity. “Large” currents in lab-sized cells are 10–1000 mA. Currents at tiny working electrodes can be picoamperes (10−12 A), or even smaller.

1.1 Basic Ideas

In the pictures we have drawn so far, the geometry of the electrodes has not been critical, and they might have been made roughly, perhaps of exposed wire or foil, without much regard for the surface area. As we become more sophisticated in our discussion, we will find that geometry is often very important in electrochemical systems. We will have to pay close attention to the shapes and sizes of electrodes and to their relative placement in the system. 1.1.7

Current–Potential Curves

A plot of the current at the working electrode vs. the potential of that electrode—a current–potential ( i − E) curve—can be quite informative about reactions occurring at the working interface. Much of this book deals with how one obtains and interprets i − E curves. Let us consider the cell in Figure 1.1.4 and discuss in a qualitative way the current–potential curve that might be obtained from it. In Section 1.3 and in later chapters, we will be more quantitative. The first step is to consider the voltage measured by the high-impedance voltmeter when the switch is left open, so that no current passes through the cell. This voltage is called the open-circuit potential of the cell.11 (a) Open-Circuit Potential

For some electrochemical cells, like those in Figure 1.1.1, it is possible to calculate the open-circuit potential from thermodynamic data, i.e., via the Nernst equation from the Adjustable power supply Electron flow

i

S

Cu Negligible current V

Pt

Ag

AgBr

Pt 1 M HBr

Figure 1.1.4 Electrochemical cell Pt/H+ (1 M), Br− (1 M)/AgBr/Ag attached to apparatus for obtaining a current–potential curve. The power supply is continuously tunable from positive to negative voltage. Switch, S, allows it to be connected or disconnected from the cell. In the depicted situation, the power supply is set to draw electrons, when the switch is closed, from the Ag/AgBr electrode (where Ag is converted to AgBr) and send them to the Pt electrode (where H+ is converted to H2 ). Arrows show the path of electron flow, which passes through ammeter, i, measuring the current. The high-impedance voltmeter, V, measures the potential difference between the two electrodes, but requires only a tiny current. External wiring is of Cu (dashed segments), so there are junctions between the Cu and the electrode materials, signified by the dots just below the voltmeter. Atmospheric O2 has been removed from the cell, because it represents an interference to the processes of interest. Since the electroreduction of O2 commonly interferes, most electrochemical cells are purged of O2 by bubbling with nitrogen or other means. 11 Also called the zero-current potential or the rest potential. The term rest potential has been blurred in recent years by inconsistent usage. In this book, we will use only open-circuit potential or zero-current potential to avoid confusion.

9

10

1 Overview of Electrode Processes

standard potentials of the half-reactions involved at both electrodes (Section 2.1.6). The key requirement is that true equilibria must be established on both sides of the cell, because each electrode engages a pair of redox forms linked by a given half-reaction (i.e., a redox couple). In Figure 1.1.1b, for example, we have H+ and H2 at one electrode and Ag and AgCl at the other.12 The cell in Figure 1.1.4 is different, because an overall equilibrium cannot be established. At the Ag/AgBr electrode, a couple is present, and the half-reaction is AgBr + e ⇌ Ag + Br− E0 = 0.0711 V vs. NHE

(1.1.16)

Since both AgBr and Ag are pure solids, their activities are unity. The activity of Br− can be found from the concentration in solution; hence, the potential of this electrode (with respect to NHE) can be calculated from the Nernst equation. This electrode is at equilibrium. However, we cannot calculate a thermodynamic potential for the Pt/H+ , Br− electrode, because we cannot identify a pair of redox forms coupled by a given half-reaction. The controlling pair clearly is not the H+ /H2 couple, since no H2 has been introduced into the cell. Consequently, the Pt electrode, and the cell as a whole, are not at equilibrium. An equilibrium cell potential does not exist. Even though the equilibrium potential of the cell is not available from thermodynamic data, the cell does have a potential at open circuit, and we can place it within a range, as shown below. The actual value of the open-circuit potential in this case turns out to be governed by kinetics, as discussed in Chapter 3. (b) Background i–E Curve

Let us now consider what happens when an adjustable power supply (even a battery bridged by a potentiometer) and a microammeter are connected across the cell by closing the switch shown in Figure 1.1.4, and the potential of the Pt electrode is gradually made more negative with respect to the Ag/AgBr electrode. The latter is a reference electrode because it is at equilibrium and has a fixed potential; thus, changes in the cell potential are manifested at the Pt electrode, which we can regard as the working electrode.13 The first electrode reaction that occurs at the Pt is the reduction of protons, 2H+ + 2e → H2

(1.1.17)

The direction of electron flow is from the electrode to protons in solution, as in Figure 1.1.3a, so a reduction (cathodic) current flows. This result is shown graphically on the right side of Figure 1.1.5, our first example of a current–potential curve. In the convention used in this book, cathodic currents are taken as positive, and negative potentials are plotted to the right.14 The current corresponding to 12 When a redox couple is present at each electrode and there are no contributions from liquid junctions (yet to be discussed), the open-circuit potential is also the equilibrium potential. This is the situation for both cells in Figure 1.1.1. 13 In this case, one of the two electrodes is a genuine reference electrode, but this is not true in all two-electrode cells. 14 The convention of taking i positive for a cathodic current stems from early work at Hg electrodes, where reduction reactions were usually studied. By making this choice and plotting negative potentials toward the right, it is possible to represent data in first-quadrant graphs. This convention has the drawback of having increasingly positive currents corresponding to increasingly negative potentials; therefore, it sometimes seems peculiar. Yet, it has continued among many practitioners, even though oxidation reactions are now studied quite commonly. The authors considered a change to the opposite convention for this edition, but have elected to retain the historical practice for two main reasons: (a) The literature was dominated for a long time by this convention, and most figures that we have taken from the literature follow it. (b) There is a pedagogical advantage in consistently writing half-cell reactions in the reductive direction, as we will see in Chapter 2. Even so, many practitioners prefer to take anodic current as positive and to plot positive potentials toward the right. When looking over a derivation in the literature or examining a published i − E curve, it is important to recognize which convention is being used (i.e., “Which way is up?”). Given history and contemporary practice, one has no choice but to become versatile about these things.

1.1 Basic Ideas

the reduction of H+ rises sharply as the potential of the Pt electrode approaches E0 for the H+ /H2 reaction (0 V vs. NHE or −0.07 V vs. the Ag/AgBr electrode). At any moment, that same current passes at the Ag/AgBr reference electrode, where it causes the oxidation of Ag in the presence of Br− in solution to form AgBr. The conservation of charge requires that the rate of oxidation at the Ag electrode be equal to the rate of reduction at the Pt electrode.15 When the potential of the Pt electrode is made sufficiently positive, electrons are drawn from the solution phase into the electrode, as Br− is oxidized to Br2 (and Br3− ). The corresponding oxidation (anodic) current increases sharply (in the negative direction) as EPt approaches E0 for the half-reaction, Br2 + 2e ⇌ 2Br−

(1.1.18)

which is +1.09 V vs. NHE or +1.02 V vs. Ag/AgBr. As this reaction occurs (right-to-left) at the Pt electrode, AgBr in the reference electrode is reduced to Ag, and Br− dissolves into solution. Figure 1.1.5 is an example of a background i − E curve, depicting the behavior of a working electrode immersed in a solution containing only an electrolyte added to decrease the solution resistance (a supporting electrolyte). Background curves generally feature sharply rising currents on either side, because the corresponding reactants are present at high concentrations (e.g., H+ on the negative side and Br− on the positive side, in the case of Figure 1.1.5). At some extreme potential (relative to the zero-current potential), a background current becomes so great that it obscures any smaller currents from parallel electrode reactions (perhaps from species of interest added to the blank electrolyte for investigation). A background limit is a potential beyond

Current at the working electrode

Pt/H+,Br−(1 M)/AgBr/Ag

Cathodic

Onset of H+ reduction on Pt

1.0

0.5 Onset of Br− oxidation on Pt

0.0

–0.5

Anodic Working electrode potential/V = EPt vs. Ag/AgBr/V

Figure 1.1.5 Current–potential curve for the Pt working electrode in Pt/H+ (1 M), Br− (1 M)/AgBr/Ag, showing proton reduction and bromide oxidation. Since E Ag/AgBr = 0.07 V vs. NHE, the potential axis could be converted to E Pt vs. NHE by adding 0.07 V to each value of potential. The reader may expect the rise in reductive current to be closer to 0.0 V vs. NHE (−0.07 V on the scale used here). Because the proton concentration is so large, the hydrogen discharge wave is enormous. On the observed current scale, one sees only the very “foot” of that wave, which is significantly positive of 0.0 V vs. NHE. This effect applies to many processes determining background limits. 15 Although the flow of current in the cell will cause chemical conversion of some Ag + Br− into AgBr or vice versa, the amounts of reaction are typically very small and will not appreciably change the bulk concentration of Br− . Also, the reference electrode is often made significantly bigger than the working electrode and is sealed off from the working electrode with an ionically conducting barrier, such as a frit or a fiber. These precautions give the reference electrode an ability to accept the passage of modest current with negligible overall compositional change and to avoid contamination from the working side of the cell.

11

1 Overview of Electrode Processes

which useful information usually cannot be obtained. The working range (or potential window) is the region between the positive and negative background limits, where reactions other than the background processes can be studied, given the electrode material, solvent, and supporting electrolyte. In the system of Figure 1.1.5, the working range is roughly from +0.9 to +0.1 V. The open-circuit potential (i.e., the zero-current potential) is not well defined in Figure 1.1.5. One can say only that it lies somewhere between the background limits. The value found experimentally will depend upon trace impurities in the solution (e.g., oxygen) and the previous history of the Pt electrode. (c) A Change of Working Electrode

Let us now consider the same cell, but with the Pt replaced with a mercury working electrode: Hg∕H+ (1 M), Br− (1 M)∕AgBr∕Ag

(1.1.19)

We still cannot calculate an open-circuit potential for the cell, because we cannot define a redox couple for the Hg electrode. Upon examining the behavior of this cell with an applied external potential, we find that the electrode reactions and the observed current–potential behavior are very different from the earlier case. The results are depicted in Figure 1.1.6, showing the current at the Hg working electrode vs. the potential of that electrode, measured with respect to the reference and then shifted to the NHE scale. When the potential of the Hg is made negative, there is essentially no cathodic current in the region near 0.0 V, where thermodynamics predict that H2 evolution should occur. Indeed, the potential must be brought to considerably more negative values, as shown in Figure 1.1.6, before this reaction takes place. The thermodynamics have not changed, for the equilibrium potential of half-reaction 1.1.7 is independent of the metal electrode [Section 2.2.4(e)]. However, when mercury serves as the electrode for the hydrogen evolution reaction, the reaction rate (characterized by a heterogeneous rate constant) is much lower than at Pt. Under these circumstances, the reaction does not occur at values suggested by thermodynamics. Considerably higher electron energies (corresponding to more negative potentials) must be applied to make the reaction occur at a measurable rate. The rate constant for a heterogeneous electron-transfer reaction is a function of applied potential. Any additional potential (beyond the thermodynamic requirement) needed to drive Cathodic Current at the working electrode

12

Hg/H+,Br−(1 M)/AgBr/Ag

Onset of H+ reduction on Hg

0.0

–0.5

–1.0

–1.5

Onset of Hg oxidation

Anodic EHg (V vs. Ag/AgBr) + 0.07 V = EHg (V vs. NHE)

Figure 1.1.6 Current–potential curve for the Hg electrode in the cell Hg/H+ (1 M), Br− (1 M)/AgBr/Ag, showing the limiting processes: proton reduction with a large negative overpotential and mercury oxidation. The potential axis is defined through the process outlined in the caption to Figure 1.1.5.

1.1 Basic Ideas

a reaction at a certain rate is called an overpotential; thus, it is said that mercury shows “a high overpotential for the hydrogen evolution reaction.” Chapter 15 covers this reaction in detail and provides an explanation for the behavior at mercury. As we will see in Chapter 3, an overpotential has the effect of lowering the activation barrier for an electrode reaction. When the mercury is brought to more positive potentials, both the anodic reaction and the potential range for flow of current differ from those observed when Pt is used as the electrode. The positive background limit occurs when Hg is oxidized to Hg2 Br2 at a potential near 0.14 V vs. NHE (0.07 V vs. Ag/AgBr), characteristic of the half-reaction Hg2 Br2 + 2e ⇌ 2Hg + 2Br−

(1.1.20)

As we can surmise from these two examples, background limits vary from system to system and depend generally upon the electrode material, the solvent, and the supporting electrolyte on the working side of the cell. (d) Addition of an Electroactive Solute

Let us now move beyond background curves by considering the previous cell with the addition of a small amount of Cd2+ to the solution, Hg∕H+ (1 M), Br− (1 M), Cd2+ (10−3 M)∕AgBr∕Ag

(1.1.21)

This case exemplifies the addition of a species of interest to a blank electrolyte for electrochemical investigation. The current–potential curve is shown in Figure 1.1.7. Note the appearance of a reduction wave at about −0.4 V vs. NHE arising from the reduction reaction Hg

− CdBr2− 4 + 2e −−−→ Cd(Hg) + 4Br

(1.1.22)

where Cd(Hg) denotes cadmium amalgam (i.e., with Cd atoms dissolved in the Hg). The shape and height of such a wave will be covered in Section 1.3.2.16

Current at the working electrode

Cathodic

Hg/H+,Br−(1 M), Cd2+ (1 mM)/AgBr/Ag

Onset of Cd(II) reduction

0.0

–0.5

Onset of H+ reduction on Hg –1.0

–1.5

Onset of Hg oxidation

Anodic

EHg(V vs. Ag/AgBr) + 0.07 V = EHg (V vs. NHE)

Figure 1.1.7 Current–potential curve for the Hg working electrode in the cell Hg/H+ (1 M), Br− (1 M), Cd2+ (10−3 M)/AgBr/Ag, showing a reduction wave for Cd(II) between −0.3 and −0.5 V. Potentials have been converted to the NHE scale in the manner identified in the caption to Figure 1.1.5. 16 If Cd2+ were added to the cell in Figure 1.1.4 and a current–potential curve taken, the result would be essentially the same as in Figure 1.1.5, found in the absence of Cd2+ . At a Pt electrode, proton reduction occurs at less negative potentials than are required for the reduction of Cd(II), so the cathodic background limit occurs in 1 M HBr far before the cadmium reduction wave can be seen.

13

14

1 Overview of Electrode Processes

(e) Precedence of Electrode Reactions

When the potential of an electrode is moved from its open-circuit value toward more negative potentials, the substance that will be reduced first (assuming all possible electrode reactions have fast kinetics) is the oxidant in the couple with the least negative (or most positive) E0 . For example, consider Figure 1.1.8a, which relates to a platinum electrode immersed in an aqueous solution containing 0.01 M each of Fe3+ , Sn4+ , and Ni2+ in 1 M HCl. The first substance reduced will be Fe3+ , since the E0 of this couple is most positive. Possible oxidation reactions

Possible reduction reactions

– E0 (V vs. NHE)

–0.25

Ni2+ + 2e → Ni

0

2H+ + 2e → H2

0

Sn4+ + 2e ← Sn2+

+0.15

I2 + 2e ← 2I–

+0.54

Fe3+ + e ← Fe2+

+0.77

O2 + 4H+ + 4e ← 2H2O

+1.23

Au3+ + 3e ← Au

+1.50

Fe3+ + e → Fe2+

+0.77 (Pt)

E0 (V vs. NHE)

Approximate potential for zero current

Sn4+ + 2e → Sn2+

+0.15

–

(Au)

Approximate potential for zero current

+

(a)

– Zn2+ + 2e → Zn

–0.76

+

(b) Cr3+ + e → Cr2+

–0.41

2H+ + 2e → H2

0

(Kinetically slow)

(Hg) Approximate potential for zero current E0 (V vs. NHE) +

(c)

Figure 1.1.8 (a) Potentials for possible reductions at a platinum electrode, initially at ∼1 V vs. NHE in a solution of 0.01 M each of Fe3+ , Sn4+ , and Ni2+ in 1 M HCl. (b) Potentials for possible oxidation reactions at a gold electrode, initially at ∼0.1 V vs. NHE in a solution of 0.01 M each of Sn2+ and Fe2+ in 1 M HI. (c) Potentials for possible reductions at a mercury electrode in 0.01 M Cr3+ and Zn2+ in 1 M HCl. The arrows indicate the directions of potential change discussed in the text. Potentials are plotted with the negative direction upward, because this direction corresponds with increasing energy of the available electrons on the electrode (Sections 1.1.4 and 2.2.5). Plotting potentials in i − E curves with the negative direction to the right is consistent with this consideration.

1.1 Basic Ideas

Likewise, when the potential of the electrode is moved from its open-circuit value toward more positive potentials, the substance that will be oxidized first is the reductant in the couple of least positive (or most negative) E0 . Thus, for a gold electrode in an aqueous solution containing 0.01 M each of Sn2+ and Fe2+ in 1 M HI (Figure 1.1.8b), the Sn2+ will be first oxidized, since the E0 of this couple is least positive. One must remember, however, that these predictions are based on thermodynamic considerations (i.e., reaction energetics). Slow kinetics might prevent an electrode reaction from occurring at a significant rate in a potential region where the standard potential suggests that the reaction is possible. For a mercury electrode immersed in a solution of 0.01 M each of Cr3+ and Zn2+ in 1 M HCl (Figure 1.1.8c), the first reduction process predicted is the evolution of H2 from H+ . As discussed earlier, this reaction is very slow on mercury, so the first process actually observed is the reduction of Cr3+ . None of the cases in Figure 1.1.8 involves a redox couple for which both redox forms are present; consequently, there is no equilibrium defining the open-circuit potential of the electrode. In every electrochemical system, the open-circuit potential lies between the easiest oxidation and the easiest reduction. If, apart from species defining the background limits, the solution contains only oxidized forms eligible for reduction (as in Figure 1.1.8a,c), the open-circuit potential must lie between the positive background limit and the standard potential for the first electroreduction. If the solution contains only reduced forms eligible for oxidation (as in Figure 1.1.8b), the open-circuit potential must lie between the negative background limit and the standard potential for the first electrooxidation. (f) Both Redox Forms of a Couple as Solutes

In some systems, both redox forms of a couple are present as solutes, and the behavior differs notably from the cases we have been discussing. Suppose, for example, that we make three changes in the cell of Figure 1.1.4: • Employing a silver electrode covered with AgCl, rather than one covered with AgBr. • Adopting a vessel having two compartments, one for each electrode, separated internally by a frit. • Placing 1 M HCl in each compartment, but also including 2 mM Fe(II) and 4 mM Fe(III) in the solution at the Pt electrode. Thus, the cell becomes Pt∕H+ (1 M), Cl− (1 M), Fe(III) (4 mM), Fe(II) (2 mM)∕∕H+ (1 M), Cl− (1 M)∕AgCl∕Ag (1.1.23) The compartments are employed to keep Fe(III) species away from the silver electrode, where they would otherwise react by oxidizing the metal. The frit prevents bulk mixing of the solutions in the two compartments, but allows them to remain in ionic electrical contact. In this system, the silver electrode becomes an Ag/AgCl reference,17 and the cell potential is the potential of the Pt working electrode against that reference, which can be re-expressed on the NHE scale, essentially as was done for Figures 1.1.6 and 1.1.7. The iron species exist in 1 M HCl as chloro complexes, so we simply write the relevant couple as18 Fe(III) + e ⇌ Fe(II)

′

E0 = +0.70 V vs. NHE (1 M HCl)

(1.1.24)

17 Although with 1 M HCl, rather than the usual case of saturated KCl. ′ 18 In this case, we use a medium-specific formal potential, E0 rather than the standard potential, E0 , because of the complexation that takes place. Chapter 2 explains formal potentials. For now, just regard the formal potential as functionally the same as the standard potential.

15

1 Overview of Electrode Processes

Pt/H+,Cl− (1 M), Fe(III) (4 mM), Fe(II) (2 mM)/H+,Cl− (1 M)/AgCl/Ag

Current at the working electrode

16

Cathodic

Fe(III) + e →Fe(II) Onset of H+ reduction on Pt

Open-circuit potential = equilibrium potential 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

–0.2

Fe(II) → Fe(III) + e E0΄ [Fe(III), Fe(II)] Onset of H2O oxidation on Pt

Anodic EPt /V vs. NHE

Figure 1.1.9 Current–potential curve for a Pt working electrode in a system in which both Fe(III) and Fe(II) are present in the working solution. Cell configuration identified at top. A composite wave exists in which Fe(III) is reduced on the negative side of the open-circuit potential and Fe(II) is oxidized on the positive side.

Because the Pt electrode is in contact with both redox forms, the working electrode is ′ poised by the Fe(III)/Fe(II) and shows a true equilibrium potential near E0 for (1.1.24). The current–potential curve is depicted in Figure 1.1.9. As the potential moves negatively from the open-circuit potential, reduction of Fe(III) is ′ immediately possible, because the open-circuit potential is already in the range of E0 for the couple. At open circuit (zero current), there is a dynamic equilibrium. The redox process proceeds in both directions [Fe(III) being reduced and Fe(II) being oxidized], but the rates are balanced and there is no net current. Negative movement of the potential away from the equilibrium value unbalances the rates to give a net cathodic current. Likewise, as the potential moves positively from the open-circuit value, net oxidation of Fe(II) is immediately possible. The overall result is a composite wave for the Fe(III)/Fe(II) couple having both anodic and ′ cathodic components, spanning E0 . It is important to understand the essential continuity of oxidation and reduction for a given electrode process. We will emphasize it repeatedly in Section 1.3.2(b), in Chapters 3 and 5, and elsewhere. Through these simple examples, we have seen that current–potential curves have many features in common with spectra arising from various forms of spectroscopy. In the most general terms, spectra represent the probability of a transition vs. the energy of a probing photon, and they are marked by features that can be exploited for qualitative identification, quantitative determination, or diagnosis of behavior. Current–potential curves represent the rate and direction of reaction at a working electrode vs. the energy of transferable electrons on that electrode, and they are also marked by features that can be exploited toward the same ends. Like spectra, current–potential curves are reflective of the electronic structures of participants; but unlike most spectra, they are essentially kinetic in basis. One must understand the contributing dynamics in the relevant electrode reactions to understand the curves fully. Current–potential curves arise from many different electrochemical methods, and we will be working constantly with them. 1.1.8

Control of Current vs. Control of Potential

Our work with i − E curves is already substantial enough to suggest that there is a functional relationship between current and potential in electrochemical systems. Dictating one of these

1.2 Faradaic Processes and Factors Affecting Rates of Electrode Reactions

variables determines the other, if experimental conditions otherwise remain constant. If one chooses to control the potential of the working electrode, for example, then the reaction energetics become defined by that election. The reaction rate and its time dependence (and, thus, the current and its time dependence) follow as consequences. Alternatively, one can choose to control the current, defining the reaction rate at the working electrode. Then, the energetics follow, as does the potential–time function. There is an analogy with the behavior of homogeneous reactions, for which one can vary rate with temperature. However, one cannot control both reaction rate and reaction temperature under otherwise constant conditions. If one wants a given rate, one must accept the required temperature vs. time. Or if one wants to operate at a given temperature, one must accept the reaction rate vs. time. Newcomers to electrochemistry often miss the point that current and potential cannot be controlled simultaneously in experimental systems, unless some other important variable, such as temperature, can change at the same time. 1.1.9

Faradaic and Nonfaradaic Processes

Two distinct classes of processes occur at electrodes. The more familiar group comprises reactions like those we have been discussing, in which electrons are transferred, oxidation or reduction happens, and charge moves across the electrode/solution interface. These events are stoichiometric and are governed by Faraday’s law; thus, they are called faradaic processes. Electrodes at which faradaic processes occur are sometimes called charge-transfer electrodes. We have also seen that a given electrode/solution interface can show a range of potentials where no charge-transfer reactions occur, because all such reactions are thermodynamically or kinetically unfavorable (e.g., the region in Figure 1.1.6 between −0.2 V and −0.8 V vs. NHE). However, the distribution of ions at the electrode/solution interface does change with changing potential or solution composition. Events at the electrode surface that do not involve charge transfer are called nonfaradaic. Even though charge does not cross the interface, nonfaradaic processes can cause external currents to flow, at least transiently, when the potential, electrode area, or solution composition changes. In studies focused on the nature of the electrode/solution interface, nonfaradaic processes can become the primary concern. For now, we will continue to focus on faradaic processes, but in Section 1.6, we will discuss systems in which only nonfaradaic processes occur.

1.2 Faradaic Processes and Factors Affecting Rates of Electrode Reactions 1.2.1

Electrochemical Cells—Types and Definitions

Electrochemical cells in which faradaic currents are flowing are classified as either galvanic or electrolytic cells. A galvanic cell is one in which reactions occur spontaneously at the electrodes when they are connected externally by a conductor (Figure 1.2.1a). These cells are often employed in converting chemical energy into electrical energy. Galvanic cells of commercial importance include primary (nonrechargeable) cells (e.g., the “alkaline” Zn − MnO2 cell), secondary (rechargeable) cells (e.g., a charged Pb − PbO2 storage battery or lithium-ion battery), and fuel cells (e.g., an H2 − O2 cell). An electrolytic cell is one in which electrode reactions that would not occur spontaneously are driven by the imposition of an external voltage greater than the open-circuit potential of

17

18

1 Overview of Electrode Processes Galvanic cell

Electrolytic cell e –

e

e

Zn/Zn2+//Cu2+/Cu

–

Cu/Cu2+, H2SO4/Pt

+

–

(Anode)

(Cathode)

(Cathode)

Zn → Zn2+ + 2e

Cu2+ + 2e → Cu

Cu2+ + 2e → Cu

(a)

+

e

Power supply

+

(Anode) H2O → 1 O2 + 2H+ + 2e 2

(b)

Figure 1.2.1 (a) Galvanic and (b) electrolytic cells, both causing the plating of copper.

the cell (Figure 1.2.1b). These cells are frequently employed to carry out desired chemical reactions by expending electrical energy. Commercial processes involving electrolytic cells include electrolytic syntheses (e.g., the production of chlorine and aluminum), electrorefining (e.g., copper), and electroplating (e.g., silver and gold). The lead–acid storage cell (or any other secondary cell) is an electrolytic cell when it is being recharged, but is a galvanic cell when it is delivering power. Although it is sometimes convenient to make a distinction between galvanic and electrolytic cells, we will most often be concerned with reactions occurring at only one of the electrodes. Treatment is simplified by concentrating our attention on only one-half of the cell at a time. If necessary, the behavior of a whole cell (i.e., whether galvanic or electrolytic) can be ascertained later by combining the characteristics of the individual half-cells. The behavior of a single electrode and the fundamental nature of its reactions are independent of whether the electrode is part of a galvanic or electrolytic cell. For example, consider the cells in Figure 1.2.1. The reaction Cu2+ + 2e → Cu is the same in both cells. If one desires to plate copper, one could accomplish this either in a galvanic cell (using a counter half-cell with a more negative potential than that of Cu2+ /Cu) or in an electrolytic cell (using any counter half-cell and driving electrons to the copper electrode with an external power supply). Electrolysis is a term that we define broadly to include chemical changes accompanying faradaic reactions at electrodes in contact with electrolytes. In discussing cells, one calls the electrode at which reductions occur the cathode, and the electrode at which oxidations occur the anode. A current in which electrons cross the interface from the electrode to a species in solution is a cathodic current, while electron flow from a solution species into the electrode is an anodic current. In an electrolytic cell, the cathode is negative with respect to the anode; but in a galvanic cell, the cathode is positive with respect to the anode.19 1.2.2

The Electrochemical Experiment and Variables in Electrochemical Cells

An experimental investigation of electrochemical behavior consists of controlling certain variables of a cell and observing how other variables (usually current, potential, or concentration) vary with time or with the controlled variables. The parameters of importance in

19 Because a cathodic current and a cathodic reaction can occur at an electrode that is either positive or negative with respect to another electrode (e.g., a reference electrode), it is poor usage to associate the term “cathodic” or “anodic” with potentials of a particular sign. For example, one should not say, “The potential shifted in a cathodic direction,” when what is meant is, “The potential shifted in a negative direction.” The terms anodic and cathodic refer to electron flow or current direction, not to potential.

1.2 Faradaic Processes and Factors Affecting Rates of Electrode Reactions

External variables Electrode variables

Temperature (T) Pressure (P) Time (t)

Material Surface area (A) Geometry Surface condition

Electrical variables Potential (E) Current (i) Quantity of electricity (Q)

Mass transfer variables Mode (diffusion, convection, . . .) Surface concentrations Adsorption Solution variables Bulk concentration of electroactive species (CO, CR ) Concentrations of other species (electrolyte, pH, . . .) Solvent

Figure 1.2.2 Variables affecting the rate of an electrode reaction.

electrochemical cells are shown in Figure 1.2.2. Methods fall into broad categories; the following illustrate some of the differences in approach: • In potentiometry, i = 0, and E is determined as a function of concentration. Since no current flows in this experiment, no net faradaic reaction occurs, and the potential is frequently (but not always) governed by the thermodynamic properties of the system. Many of the variables (electrode area, mass transfer, electrode geometry) do not affect the potential directly. • In voltammetry, the potential is controlled (normally following a particular function of time), and the resulting current is measured as a function of potential. • In galvanostatic experiments, the current is controlled (usually as a defined function of time) and the potential is followed vs. time. • In coulometry, the potential is held constant at a value where an electrode reaction occurs, and one measures the total charge passed, usually by integrating the current. Most electrochemical experiments can be described in terms of the way in which the system responds to a perturbation. A certain excitation function (e.g., a potential step) is applied, and a certain response function (e.g., the resulting variation of current with time) is measured, with all other system variables held constant (Figure 1.2.3). If the excitation is time dependent, the response typically will also be a function of time. Such approaches are called transient methods. Alternatively, the excitation might be constant or a very slowly changing function of time, and the response might be steady with time. Approaches based on this concept are called steady-state methods. The aim of the experiment is to obtain information (thermodynamic, kinetic, analytical, etc.) from observations of excitation and response functions and a knowledge of appropriate models for the system. This same basic idea is used in many other types of investigation, such as circuit testing or spectrophotometric analysis. In spectrophotometry, the excitation function is light of different wavelengths; the response function is the fraction of light transmitted by the system at these wavelengths; the system model is Beer’s law or a molecular model; and the

19

20

1 Overview of Electrode Processes

(a) General concept Excitation

System

Response

(b) Spectrophotometric experiment I

𝒜 Lamp-Monochromator

λ

Phototube

Optical cell with sample

λ

(c) Electrochemical experiment i

E

Power supply

i

t

t

Figure 1.2.3 (a) General principle of studying a system by application of an excitation (or perturbation) and observation of response. (b) In a spectrophotometric experiment, the excitation is light of different wavelengths (𝜆), and the response is the absorbance (A) curve. (c) In an electrochemical potential step experiment, the excitation is the application of a potential step, and the response is the observed i − t curve.

information content includes the concentrations of absorbing species, their absorptivities, or their transition energies. Before developing some models for electrochemical systems, let us examine more closely the nature of the current and potential in an electrochemical cell. Consider Figure 1.2.4, in which a cadmium electrode immersed in 1 M Cd(NO3 )2 is coupled to an SCE. The open-circuit potential of the cell is 0.64 V, with the copper wire attached to the cadmium electrode being −

+ Power supply

Eappl i

− Cu/Cd/Cd(NO3)2 (1 M)//KCl (saturated)/Hg2Cl2/Hg/Pt/Cu′ + Cd2+ + 2e = Cd Hg2Cl2 + 2e = 2Hg + 2Cl−

E0 = –0.40 V vs. NHE ESCE = 0.24 V vs. NHE

Figure 1.2.4 Schematic cell connected to an external power supply. The resistor with an adjustable intermediate contact is called a potentiometer. The double slash in the cell notation indicates an interface between two different electrolytes (a liquid junction) with a negligible potential difference. To avoid contamination of the mercury pool in the SCE by dissolved copper, Pt is used as intermediate contact.

1.2 Faradaic Processes and Factors Affecting Rates of Electrode Reactions

negative with respect to that attached to the mercury electrode.20 When the voltage applied by the external power supply, Eappl , is 0.64 V, it exactly cancels the cell’s open-circuit potential. There is no net driving force for the movement of charge, so i = 0. When Eappl is made larger (i.e., Eappl > 0.64 V, such that the cadmium electrode is made even more negative with respect to the SCE), the power supply can drive electrons into the cadmium and withdraw them at the mercury. At the cadmium electrode, the reaction Cd2+ + 2e → Cd occurs, while at the SCE, mercury is oxidized to Hg2 Cl2 . The cell is electrolytic; the power supply is overcoming the spontaneous driving force of the cell. A question of interest might be: “If Eappl = 0.80 V (i.e., if the potential of the cadmium electrode is made −0.80 V vs. the SCE), what current will flow?” As we learned in Section 1.1.5, the question “What is i?” is, for this system, essentially the same as “What is the rate of the reaction, Cd2+ + 2e → Cd?” Information about an electrode reaction is often gained by determining the current at the working electrode as a function of potential, i.e., by obtaining i − E curves. Certain terms are sometimes associated with features of the curves.21 If a working electrode has a defined equilibrium potential, Eeq , that potential is an important reference point [Section 1.1.7(a)]. The departure of the electrode potential from the equilibrium value upon passage of faradaic current is termed polarization, which is measured by the overpotential, 𝜂,22 𝜂 = E − Eeq

(1.2.1)

Current–potential curves, particularly those obtained under steady-state conditions, are sometimes called polarization curves. An ideally nonpolarizable electrode (INE) is an electrode whose potential does not change upon passage of current, i.e., an electrode of fixed potential.23,24 Ideal nonpolarizability is characterized by a vertical region on an i − E curve (Figure 1.2.5a). An SCE constructed with a sizable mercury pool would approach ideal nonpolarizability for small currents. In contrast, an ideally polarizable electrode (IPE) shows a very large change in potential upon the passage of an infinitesimal current; thus, ideal polarizability is characterized by a horizontal, zero-current i − E curve (Figure 1.2.5b and Section 1.6.1). 1.2.3

Factors Affecting Electrode Reaction Rate and Current

Consider an overall electrode reaction, O + ne ⇌ R, composed of a series of steps that cause the conversion of the dissolved oxidized species, O, to a reduced form, R, also in solution

20 This value is calculated from the Nernst equation using the information in Figure 1.2.4. The experimental value would also include the effects of activity coefficients and the liquid junction potential, which are neglected here. See Chapter 2. 21 These terms are carryovers from history and do not always represent the best possible terminology. However, their use is so ingrained in electrochemical jargon that it seems wisest to keep them and to define them as precisely as possible. 22 In the electrochemical literature, the term polarization describes the departure of an electrode from some reference point, but not always from an equilibrium potential. Overpotential is the measure of polarization relative to the chosen reference point. 23 An INE is also known as an ideally depolarized electrode (IDE). A substance is called a depolarizer if it causes an electrode to operate at a less extreme potential (i.e., to be nearer to its zero-current value) by virtue of being oxidized or reduced. Thus, ideal depolarization would yield the i − E curve in Figure 1.2.5a. 24 The term depolarizer is also frequently used to denote a substance that is preferentially oxidized or reduced, to prevent an undesirable electrode reaction. Sometimes, it is simply another name for an electroactive substance.

21

22

1 Overview of Electrode Processes

i

i

E

E

Ideally nonpolarizable electrode

Ideally polarizable electrode

(a)

(b)

Figure 1.2.5 Current–potential curves for (a) INE and (b) IPE. Dashed lines show behavior of actual electrodes that approach ideal behavior over limited ranges of current or potential. Departures from ideality in (a) reflect limitations in the kinetics of the electrode reaction. In (b), they arise from the onset of new electrode reactions.

(Figure 1.2.6). In general, the current (or electrode reaction rate) is governed by the rates of processes such as (1, 2): • Mass transfer (e.g., of O from the bulk solution25 to the electrode surface). • Electron transfer at the electrode surface. Electrode surface region

Bulk solution

Electrode Chemical reactions ion

rpt

so Ad

O′ads ne

sor

De

O′

Mass transfer Osurf

Obulk

Rsurf

Rbulk

tion

p

Electron transfer R′ads

De

sor ptio n Ad sor ptio R′ n

Chemical reactions

Figure 1.2.6 Pathway of a general electrode reaction. 25 The bulk solution (often just called the bulk) is the body of solution sufficiently far from an electrode to have uniform composition at all points and, therefore, no gradient in concentration driven by processes at the electrode. Near the electrode, electroreactants are consumed and electroproducts are generated, so concentration gradients exist and are important. On a lab scale, the bulk is not very far from an electrode, typically less than a few hundred micrometers away. In many experiments, the bulk remains essentially unaffected by the electrode process; however, one cannot assume that it is always unaffected. Cells with large electrodes and efficient mass transfer can have the purpose of transforming the whole system, especially if the goal is to synthesize a product electrolytically. Systems built for “bulk electrolysis” are discussed in Chapter 12.

1.3 Mass-Transfer-Controlled Reactions

• Chemical reactions preceding or following the electron transfer. These might be homogeneous processes (e.g., protonation or dimerization) or heterogeneous reactions on the electrode surface (e.g., catalytic decomposition). • Other surface reactions, such as adsorption, desorption, or crystallization. The rate constants for some of these processes (e.g., electron transfer at the electrode surface or adsorption) depend upon the potential. The simplest reactions involve only mass transfer of a reactant to the electrode, heterogeneous electron transfer involving nonadsorbed species, and mass transfer of the product to the bulk solution. A representative reaction of this sort is the reduction of the aromatic hydrocarbon 9,10-diphenylanthracene (DPA; Figure 1) to the radical anion (DPA−∙ ) in an aprotic solvent [e.g., N,N-dimethylformamide (DMF)]. More complex reaction sequences are quite common. They may involve almost any kind of chemistry, including a series of electron transfers, coupling reactions in solution, protonations, branching mechanisms, parallel paths, or modifications of the electrode surface. It is possible in several ways to obtain steady-state currents, and we will concentrate on such situations in Sections 1.3 and 1.4. When there is a steady current, the rates of all reaction steps in a mechanistic series are the same. The magnitude of this current is often limited by the inherent sluggishness of one or more processes called rate-determining steps. The more facile steps are held back from their maximum rates by the slowness with which a rate-determining step disposes of their products or delivers their reactants. Each value of current is driven by a certain overpotential, 𝜂, which can be considered as a sum of terms associated with the different reaction steps: 𝜂 mt (the mass-transfer overpotential), 𝜂 ct (the charge-transfer overpotential), 𝜂 rxn (the overpotential associated with a preceding reaction), etc. The overpotential of any step is an expression of how the applied electrical energy activates that step to operate at the rate required to deliver the given current density. Since −𝜂/i has units of resistance, the electrode reaction can be represented by an overall resistance, R, composed of a series of resistances representing the various steps: Rmt , Rct , etc. (Figure 1.2.7). A kinetically facile, easily driven reaction step is characterized by a small resistance, while a sluggish step is represented by a high resistance.26 Figure 1.2.7 Processes in an electrode reaction represented as resistances.

i

ηmt

ηct

ηrxn

Rmt

Rct

Rrxn

1.3 Mass-Transfer-Controlled Reactions Let us now become more quantitative about the sizes and shapes of current–potential curves. We must be able to describe the rate of reaction at the electrode surface if we are to understand i. In the simplest electrode reactions, the participants are chemically stable, and the rates of all associated chemical reactions are very rapid compared to those of the mass-transfer processes.

26 More accurately, what we have labeled as resistances should be considered as impedances. However, these impedances are functions (often very strong functions) of E or i, unlike the analogous ideal electrical elements. Typically, linear models like Figure 1.2.7 apply quantitatively only for very small perturbations about the equilibrium point, i.e., only when a small signal is applied.

23

24

1 Overview of Electrode Processes

We will find in Chapters 3 and 5 that if an electrode reaction involves only fast heterogeneous charge-transfer kinetics and mobile, reversible homogeneous reactions, then • The homogeneous reactions can be regarded as being at equilibrium. • The surface concentrations of species involved in the faradaic process are related to the electrode potential by an equation of the Nernst form. Such electrode reactions are called reversible or nernstian, because the principal species obey thermodynamic relationships at the electrode surface. In nernstian systems, the net rate of the electrode reaction is governed totally by the rate at which the electroreactant is brought to the surface by mass transfer, vmt (mol cm−2 s−1 ). From (1.1.15),27 vmt = ±i∕nFA

(+ for a reduction, − for an oxidation)

(1.3.1)

Since mass transfer plays a big role in electrochemical dynamics, we will next review its three modes and then consider mathematical methods for treating them. 1.3.1

Modes of Mass Transfer

Mass transfer—the movement of material from one location in solution to another—arises from differences in electrical or chemical potential at the two locations or from the physical movement of volume elements of solution. The modes of mass transfer, amplified in Chapter 4, are: • Migration. Movement of a charged body under the influence of an electric field (a gradient of electrical potential).28 • Diffusion. Movement of a species under the influence of a gradient of chemical potential (i.e., a concentration gradient). • Convection. Stirring or hydrodynamic transport. Fluid flow occurs because of natural convection (convection caused by density gradients) and forced convection, and is characterized by stagnant regions, laminar flow, and turbulent flow. Mass transfer to an electrode surface is governed by the Nernst–Planck equation, written in the following way for one-dimensional mass transfer along the x-axis, normal to the surface, Jj (x) = −Dj

𝜕Cj (x) 𝜕x

−

zj F RT

Dj Cj (x)

𝜕𝜙(x) + Cj (x) v(x) 𝜕x

(1.3.2)

where J j (x) is called the flux of species j (mol s−1 cm−2 ) at distance x from the surface. This is the net rate at which the species crosses a unit area at x. A positive flux describes net movement toward greater x. In this equation, Dj is the diffusion coefficient (cm2 /s); 𝜕C j (x)/𝜕x is the concentration gradient at distance x; 𝜕𝜙/𝜕x is the potential gradient; zj and C j are the charge (dimensionless) and concentration (mol/cm3 ) of species j, respectively; and v(x) is the velocity (cm/s) with which a solution volume element at x translates along the axis. The Nernst–Planck equation is derived and discussed in Chapter 4. The three terms on the right side represent the contributions of diffusion, migration, and convection to the flux. The flux of species j at the electrode surface, J j (x = 0), has a special interpretation. If it is negative, then it describes a flux into the surface and must be understood chemically as the rate 27 vmt is an unsigned quantity. The explicit signs in (1.3.1) are in recognition that the current is positive for a reduction and negative for an oxidation. 28 Sometimes called drift.

1.3 Mass-Transfer-Controlled Reactions

at which species j is consumed by electrolysis. If positive, it describes a flux out of the surface and, therefore, the rate at which species j is produced by the electrode reaction. If it is zero, then species j is neither a reactant nor a product of the electrode reaction. Suppose species k is the only electroreactant. We can identify the flux magnitude |J k (x = 0)| as vmt in (1.3.1), from which we obtain, |i| = |Jk (x = 0)| (1.3.3) nFA In later chapters, we will use the Nernst–Planck equation to calculate mass-transfer-limited currents for many different experimental circumstances. Computing the flux is complex when all three forms of mass transfer contribute; hence, electrochemical systems are often designed to render one or two of the contributions negligible. For example, the migrational component can be reduced to unimportance by adding an inert electrolyte (a supporting electrolyte) at a concentration much larger than that of the electroactive species (Section 4.3.2). Convection can be avoided by preventing stirring in the electrochemical cell. Alternatively, convection can be made dominant in a well-defined way by using a rotating disk electrode or a flow cell. In the rest of this section, we present an approximate treatment of steady-state mass transfer, which will give insight into electrochemical reactions without elaborate mathematical detail. 1.3.2

Semiempirical Treatment of Steady-State Mass Transfer

Consider the reduction of a species O at a cathode, O + ne ⇌ R. In an actual case, the oxi, and the reduced form, R, might be Fe(CN)4− , but with dized form, O, might be Fe(CN)3− 6 6 3− only Fe(CN)6 initially present at the millimolar level in a solution of 0.1 M K2 SO4 . We envision a cell with a platinum working electrode and an SCE reference electrode, and we focus on conditions that produce steady-state mass transfer to the electrode surface. An effective approach is to make the working electrode in the form of a disk embedded in an insulator and to rotate the assembly at a known rate about its central axis; this is a rotating disk electrode (RDE) and is discussed in Section 10.2. The rotation causes steady fluid flow upward along the axis of rotation, then radially outward from the face of the electrode because of centrifugal effects. It is characteristic of the RDE to deliver a steady current when the electrode is held at a potential where an electrochemical reaction can occur. If the potential is changed, the system will establish a new steady state after a brief transition. Steady-state behavior can be obtained in other systems, too. The most important cases involve very small working electrodes shaped as spheres, hemispheres, or disks. These have radii from 25 μm to 10 nm and are known as ultramicroelectrodes (UME). Chapter 5 is wholly dedicated to steady-state experiments at UMEs. The basis for attaining a steady state at a UME is not as intuitive as at an RDE, so we concentrate now on the latter; however, the results derived here apply broadly to steady-state systems. Once electrolysis begins at an RDE, species O is consumed at the electrode surface, so ∗. its surface concentration, C O (x = 0), becomes smaller than the value in bulk solution, CO A concentration profile develops from the surface to the bulk, as shown by either of the solid curves in Figure 1.3.1. Stirring is ineffective near the electrode surface, so a stagnant layer (sometimes called the Nernst diffusion layer) extends outward for a thickness, 𝛿 O . Beyond ∗ (dashed lines in Figure 1.3.1). x = 𝛿 O , convection is assumed to maintain the concentration at CO Since there is an excess of supporting electrolyte, migration is not important anywhere. Inside the diffusion layer (x < 𝛿 O ), where neither convection nor migration are operative, the flux of O is given by the first (diffusive) term in (1.3.2). At the electrode surface, JO (x = 0) = −DO (dC O ∕dx)x=0

(1.3.4)

25

26

1 Overview of Electrode Processes

↑ CO

Figure 1.3.1 Concentration profiles (solid lines) and diffusion-layer approximation (dashed lines). x = 0 corresponds to the electrode surface, and 𝛿 O is the diffusion layer thickness. Concentration profiles are shown at two different electrode potentials: (1) where ∗ ∕2, (2) where C (x = 0) ≈ 0 and C O (x = 0) is about CO O i = il . When a system achieves steady state, the concentration profile remains steady, and the flux of O is the same at any plane.

C*O 1

CO(x = 0) 2

0

δO

x→

If one further assumes a linear concentration gradient within the diffusion layer, then, from equation (1.3.4) ∗ − C (x = 0)]∕𝛿 JO (x = 0) = −DO [CO O O

(1.3.5)

Since 𝛿 O is often unknown, it is convenient to combine it with the diffusion coefficient as mO = DO /𝛿 O , and to write equation (1.3.4) as ∗ − C (x = 0)] JO (x = 0) = −mO [CO O

(1.3.6)

where the mass-transfer coefficient, mO , has units of cm/s.29 From equations (1.3.3) and (1.3.6), with recognition that i is positive for the reduction occurring at the electrode, i ∗ − C (x = 0)] = mO [CO O nFA

(1.3.7)

At the same time, R is produced at the electrode surface, so that CR (x = 0) > CR∗ (where CR∗ is the bulk concentration of R). Therefore, i = mR [CR (x = 0) − CR∗ ] nFA

(1.3.8)

For the case when CR∗ = 0 (no R in the bulk solution), i = mR CR (x = 0) nFA

(1.3.9)

The values of C O (x = 0) and C R (x = 0) depend on the electrode potential, E. The largest rate of mass transfer of O occurs when C O (x = 0) is driven to zero—or more precisely, when ∗ , so that C ∗ − C (x = 0) ≈ C ∗ . The value of the current under these condiCO (x = 0) |Eλ − E1 |∕v

(1.6.25)

The steady-state current changes from vC d during the forward scan to −vC d during the reverse scan. The result for a system with constant C d is shown in Figure 1.6.9 for the case of a negative-going initial scan. (c) Charging Current in Electrochemical Measurements

We have taken care with charging current here, because a great fraction of electrochemical experimentation involves it. Most strategies are based on changes, over time, of potential, interfacial structure, or even electrode area; consequently, a charging current generally exists

15

–1.5 E2 –1.0

10 Eappl /V –0.5

ic / μ A vCd

5 0.0 E1 0.5 0.0

0.5

1.0

1.5

2.0

0 0.0

0.5

1.0

Time/s

Time/s

(a)

(b)

1.5

2.0

Figure 1.6.8 Behavior resulting from a linear potential sweep applied to a working electrode at which only double-layer charging can occur. (a) E appl vs. time (working vs. reference). (b) Charging current vs. time. Ru = 5 kΩ, C d = 10 μF, Ru C d = 50 ms, E 1 = 0.3 V, E 2 = − 1.3 V, v = 1 V/s.

49

1 Overview of Electrode Processes –1.5

15 Eλ

10 = e

ic /μA

=

Eappl/V –0.5

vCd

5

e op Sl

–v

–1.0 Sl op

50

v

0 −vCd

–5 0.0 –10

E1

0.5 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

–15 0.0

0.4

0.8

1.2

1.6

Time/s

Time/s

(a)

(b)

2.0

2.4

2.8

3.2

15 Forward Scan 10 vCd

5 ic /μA

0 −vCd

–5 –10 –15 0.5

Reverse Scan

E1 0.0

–0.5

Eλ –1.0

–1.5

Eappl/V

(c)

Figure 1.6.9 Behavior resulting from a cyclic linear potential sweep applied to a working electrode at which only double-layer charging can occur. (a) E appl vs. time (working vs. reference). (b) Charging current vs. time. (c) Charging current vs. E appl . Cell parameters are as for Figure 1.6.8. E 1 = 0.3 V, E λ = − 1.3 V, v = 1 V/s.

and contributes to a background against which faradaic currents are observed.55 The charging current frequently interferes with measurements of faradaic currents. When the electroactive species are present at low concentrations (e.g., as in trace analysis), the charging current can be much larger than any faradaic current of interest and commonly defines detection limits. In later chapters, we will return repeatedly to this issue and to strategies for dealing with it. Still, one must not regard charging current as simply a background concern. It is integral to the control of potential in electrochemical systems, so it is essential to experimental design and execution. Moreover, it provides access to the surface charge on the electrode and to the interfacial capacitance, which are informative about interfacial structure. In some work, the charging current is the main concern, providing access to the information sought about a system. (d) A Last Word about Eappl

In Sections 1.5 and 1.6, we have focused on the distinction between the potential of the working electrode, E, and the voltage applied externally to the working electrode, Eappl , vs. the reference. These two quantities differ by the uncompensated ohmic drop, iRu . 55 The idea of a background current was introduced in Section 1.1.7(b). In most experiments, it is the sum of a charging current plus faradaic currents from the solvent, supporting electrolyte, electrode material, or impurities. Near the background limits, faradaic currents from primary components of the system predominate, but between the background limits, there may be a region where the background current is mainly charging current.

1.7 Organization of this Book

In most electrochemical experiments, one desires to control the potential of the working electrode, and Eappl is the vehicle of control. At every moment, it is the target for E. If Eappl is programmed as a step or a sweep, for example, one desires that the working electrode follow faithfully enough for theoretical treatments of the experiment to be valid. We have seen that E does not follow Eappl strictly. How well it follows depends on the cell time constant and the magnitude of the uncompensated ohmic drop, so one must be attentive to these matters. Fortunately, it is often practical to establish experimental conditions under which E does follow Eappl well enough. This will be true when • The cell time constant can be made short compared to the time scale56 of the experiment. • The uncompensated ohmic drop is small. Both of these conditions can usually be satisfied if the solution is sufficiently conductive and the working electrode area is small enough. Depending on the system, it may be necessary to use a three-electrode cell and employ electronic compensation of resistance (Section 16.7.3). Because E is practically the same as Eappl in most experimental work, experienced practitioners rarely distinguish these quantities as they speak and write. They simply say, for example, that they have taken E (rather than Eappl ) through a step program or a triangular wave. They depict the step or the sweep ideally, even though they well understand the nonideality at a step edge or at the turning point of a sweep. In published figures, such as voltammograms, they will show the potential axis as E, even though the instruments that produce the figures actually use Eappl . In the rest of this book, we will follow the same practice. Further reference will rarely be made to Eappl . Unless a cautionary point is being made, we will just assume that we can control E ideally enough, even as we take it through complex programs. We will just have to remain on the lookout for effects of uncompensated ohmic drop in recorded results.

1.7 Organization of this Book This book has been structured to support options for teachers and students as they follow their own paths into the subject. Figure 1.7.1 illustrates possibilities. The left column comprehends the essential core, presented in the first seven chapters. It is best to cover this material in sequence, because core ideas and treatments are generally constructed on preceding concepts. In the right column, the teacher and student gain extensive flexibility to tailor further study according to interests and available time. The dots by the individual chapters and the dashed arrows signal that the chapters are, for the most part, independently selectable after prior study through Chapter 7. Even though the book is built as a complete story, with all chapters designed to be addressed in sequence (signified by the solid arrows), a reader can be effective by covering a few chapters in the right column. The final group also allows for selective study; however, Chapters 17–21 are generally built on knowledge from the Advanced Fundamentals group (Chapters 13–16). There are also appendices providing support for students as they work through this text. The mathematical techniques in Appendix A are extensively employed in Chapters 5–13. It will ease 56 The time scale of an experiment is an important concept that we will encounter repeatedly. It refers generally to the time period over which the most important data are recorded. For example, it might be taken as the duration of measurements after a potential step or the time required to sweep across a voltammetric wave. If the working electrode potential is to accurately follow time variations in Eappl , the cell time constant must be significantly shorter than the experimental time scale.

51

52

1 Overview of Electrode Processes INTRODUCTION Ch. 1

Overview of Electrode Processes

CORE FUNDAMENTALS Ch. 2: Potentials, Thermodynamics of Cells Ch. 3: Basic Kinetics of Electrode Reactions Ch. 4: Mass Transfer by Migration and Diffusion

CORE METHODS Ch. 5: Ch. 6: Ch. 7:

Steady-State Voltammetry Potential-Step Methods Linear Sweep, Cyclic Voltammetry (CV)

ADDITIONAL METHODS Ch. 8: Ch. 9: Ch. 10: Ch. 11: Ch. 12:

Polarography, Pulse Voltammetry, SWV Controlled-Current Techniques Hydrodynamic Methods Impedance: EIS and ac Voltammetry Bulk Electrolysis

ADVANCED FUNDAMENTALS Ch. 13: Coupled Homogeneous Chemical Reactions Ch. 14: Double-Layer Structure and Adsorption Ch. 15: Inner-Sphere Electrode Reactions and Electrocatalysis Ch. 16: Electrochemical Instrumentation

ADVANCED SYSTEMS AND METHODS SUPPORT AND REFERENCE Ap. A: Mathematical Methods Ap. B: Basic Concepts of Simulation Ap. C: Reference Tables

Ch. 17: Ch. 18: Ch. 19: Ch. 20:

Electroactive Layers and Modified Electrodes Scanning Electrochemical Microscopy Single-Particle Electrochemistry Photoelectrochemistry, Electrogenerated Chemiluminescence Ch. 21: In situ Characterization of Electrochem. Systems

Figure 1.7.1 Major groups of chapters.

the reader’s progress to have worked through, or at least to have reviewed, Appendix A before starting Chapter 5. Appendix B is an introduction to digital simulation methods, which are widely used to address complex problems of contemporary interest. The need for simulation will come up throughout the book, beginning in Chapter 5. Appendix C is a set of reference tables useful for the end-of-chapter problems and for daily activity.

1.8 The Literature of Electrochemistry Under this heading in the second edition, there was an extensive summary of the English-language literature of electrochemistry as it existed in 2001. Prominent books and monographs were listed in several categories. Review series and key journals were identified. The present reader might find that summary useful, even now. In this edition, an equivalent summary is not provided. Mainly, the authors desire to save the space for other matters, but also relevant is that publication patterns have changed greatly since 2001, especially for articles on original research. A summary is, in many respects, no longer practical. However, it seems still worthwhile to list general reference sources (including compilations of data), sources on laboratory techniques, and the main review series, which give rise to many references in this text. 1.8.1

Reference Sources

1) Bard, A. J., G. Inzelt, and F. Scholz, Eds., “Electrochemical Dictionary,” 2nd ed., Springer, Heidelberg, 2012. 2) Bard, A. J., and H. Lund, Eds., “Encyclopedia of the Electrochemistry of the Elements” (18 volumes), Marcel Dekker, New York, 1973–1986.

1.8 The Literature of Electrochemistry

3) Bard, A. J., R. Parsons, and J. Jordan, Eds., “Standard Potentials in Aqueous Solutions,” Marcel Dekker, New York, 1985. 4) Conway, B. E., “Electrochemical Data,” Elsevier, Amsterdam, 1952. 5) Horvath, A. L., “Handbook of Aqueous Electrolyte Solutions: Physical Properties, Estimation, and Correlation Methods,” Ellis Horwood, Chichester, UK, 1985. 6) Janz, G. J., and R. P. T. Tomkins, “Nonaqueous Electrolytes Handbook” (2 volumes), Academic, New York, 1972. 7) Meites, L., and P. Zuman, “Electrochemical Data,” Wiley, New York, 1974. 8) Meites, L., and P. Zuman et al., “CRC Handbook Series in Organic Electrochemistry” (6 volumes), CRC, Boca Raton, FL, 1977–1983. 9) Meites, L., and P. Zuman et al., “CRC Handbook Series in Inorganic Electrochemistry” (8 volumes), CRC, Boca Raton, FL, 1980–1988. 10) Parsons, R., “Handbook of Electrochemical Data,” Butterworths, London, 1959. 11) Zemaitis, J. F., D. M. Clark, M. Rafal, and N. C. Scrivner, “Handbook of Aqueous Electrolyte Thermodynamics: Theory and Applications,” Design Institute for Physical Property Data (for the American Institute of Chemical Engineers), New York, 1986. 12) Zoski, C. G., Ed., “Handbook of Electrochemistry,” Elsevier, Amsterdam, 2007. 1.8.2

Sources on Laboratory Techniques

In this book, general and specialized sources on electrochemical phenomena and theory are cited in subsequent chapters; however, books on laboratory techniques generally are not. The following list is offered now as an aid to the reader: 1) Adams, R. N., “Electrochemistry at Solid Electrodes,” Marcel Dekker, New York, 1969. 2) Gileadi, E., E. Kirowa-Eisner, and J. Penciner, “Interfacial Electrochemistry—An Experimental Approach,” Addison-Wesley, Reading, MA, 1975. 3) Kissinger, P. T., and W. R. Heineman, Eds., “Laboratory Techniques in Electroanalytical Chemistry,” 2nd ed., Marcel Dekker, New York, 1996. 4) Sawyer, D. T., A. Sobkowiak, and J. L. Roberts, Jr., “Electrochemistry for Chemists,” 2nd ed., Wiley, New York, 1995. 5) Scholz, F., Ed., “Electroanalytical Methods,” 2nd ed., Springer, Berlin, 2010. 6) Zoski, C. G., Ed., “Handbook of Electrochemistry,” Elsevier, Amsterdam, 2007. 1.8.3

Review Series

A number of review series dealing with electrochemistry and related areas exist. Volumes are published from time to time and contain chapters written by authorities.57, 58 1) “Advances in Electrochemical Science and Engineering” (18 volumes), H. Gerischer, C. W. Tobias, R. C. Alkire, D. M. Kolb, J. Lipkowski, P. N. Ross, L. A. Kibler, P. N. Bartlett, and M. T. Koper, Eds., Wiley-VCH, Weinheim, Germany, 1990–2018. 57 Most of these series have had long lives and evolving editorships and publishers. They are cited below with all editors who have ever been engaged, in order of appearance, and with the publisher of the most recent volume. Individual volumes may have involved a subset of the listed editors or a different publisher. 58 Articles in the series numbered 1, 2, 4, and 7 are cited in this book, and often elsewhere in the literature, in journal reference format with the abbreviations Adv. Electrochem. Sci. Engr., Adv. Electrochem. Electrochem. Engr., Electroanal. Chem., and Mod. Asp. Electrochem., respectively. Series 4 should not be confused with J. Electroanal. Chem.

53

54

1 Overview of Electrode Processes

2) “Advances in Electrochemistry and Electrochemical Engineering” (13 volumes), P. Delahay, C. W. Tobias, and H. Gerischer, Wiley, New York, 1961–1984. 3) “Comprehensive Treatise of Electrochemistry” (10 volumes), E. Yeager, J. O’M. Bockris, B. E. Conway, S. Sarangapani, R. E. White, S. Srinivasan, and Yu. A. Chizmadzhev, Eds., Plenum, New York, 1984. 4) “Electroanalytical Chemistry” (27 volumes), A. J. Bard, I. Rubinstein, and C. G. Zoski, Eds., Taylor and Francis, Boca Raton, FL, 1966–2017. 5) “Electrochemistry (Specialist Periodical Reports)” (10 volumes), G. J. Hills (Vols. 1–3), H. R. Thirsk (Vols. 4–7), and D. Pletcher (Vols. 8–10), Senior Reporters, The Chemical Society, London, 1971–1985. 6) “Encyclopedia of Electrochemistry” (11 volumes), A. J. Bard and M. Stratmann, Series Eds., Wiley-VCH, Wenheim, Germany, 2002–2007. 7) “Modern Aspects of Electrochemistry” (60 volumes), J. O’M. Bockris, B. E. Conway, R. E. White, C. G. Vayenas, Series Eds., Springer, New York, 1954–2022. 8) “Techniques of Electrochemistry” (3 volumes), E. Yeager and A. J. Salkind, Eds., Wiley-Interscience, New York, 1972–1978.

1.9 Lab Note: Potentiostats and Cell Behavior This note is intended to help the reader to gain practical familiarity with a basic electrochemical instrument, with electrical responses of cells, and with some instrumental limitations. 1.9.1

Potentiostats

To carry out the suggested work, you will need access to a potentiostat – the central tool in most electrochemical laboratories. This is a device for controlling a three-electrode cell and for providing information about its behavior. The potentiostat essentially manifests the functions surrounding the cell in Figure 1.5.5. It normally connects to the working, reference, and counter electrodes through a cable with clip leads.59 A potentiostat uses feedback circuitry to automatically control the power delivered between the working and counter electrodes, so that Eappl , the voltage applied at the working electrode vs. the reference, remains continuously equal to a programmed value, even if that value changes with time (see also Figure 5.1.1 and related discussion). The voltage across the current-carrying path (working electrode to counter electrode) and the current in that path are functions of Eappl and its time dependence. Within practical limits, the potentiostat will deliver whatever is needed to fulfill the control condition. Common potentiostats can place ±13–14 V across the cell and can furnish a current of ±100 mA. “Powerful” instruments can deliver ±100 V and ±1 A. Chapter 16 provides extensive discussion of these systems. As discussed in Section 1.6.4(d), Eappl is a continuous target for the actual working electrode potential, E, vs. the reference. One generally selects conditions such that E tracks Eappl faithfully enough for valid application of theory. Normally, one desires that Eappl (and, therefore, also E) be taken through a waveform (such as a step or cyclic sweep). This is achieved by varying inputs of the potentiostat with time according to the desired waveform. Chapter 16 covers the details. 59 A potentiostat can also be used with a two-electrode cell. In that case, the leads for both the counter and reference electrodes are connected to the reference.

1.9 Lab Note: Potentiostats and Cell Behavior

Many modern potentiostats, including practically all routine instruments now found in electrochemical laboratories, are fully integrated and automated. Computers oversee all operations, including management of the cell, waveform generation, data recording, and presentation of results. You should use an instrument of this kind for the experiments suggested below. 1.9.2

Background Processes in Actual Cells

In our zeal to investigate the properties of a particular species, we sometimes neglect thorough study of the properties of the blank electrolyte (the identical solution with the species of interest omitted). This is usually just the solvent and supporting electrolyte, e.g., deaerated aqueous 1 M KCl or buffer solution. As described in Sections 1.1.7(b) and 1.6.4(c), the current measured in this blank electrolyte solution informs the experimenter of the anodic and cathodic potential limits and the magnitude of the charging current. This measurement is also critically important in identifying unexpected faradaic currents resulting from impurities in the supporting electrolyte and solvent. If not identified (and eliminated, if possible), these currents may be mistaken for signatures of the redox process of interest, leading to erroneous conclusions. One can study background processes for many different electrode/electrolyte combinations. A simple experiment is to use an inexpensive commercial glassy carbon (GC) disk electrode with a diameter of 1–3 mm as the working electrode. The GC electrode should be placed into a deaerated 0.1 M KCl solution together with a Pt counter electrode and a commercial Ag/AgCl reference electrode. Deaeration is easily achieved by bubbling the solution with nitrogen for a few minutes and then maintaining a nitrogen atmosphere over the solution in the cell. The counter electrode should have an area significantly larger than that of the working electrode. Typically, Pt foil or mesh is used, but commercial Pt counter electrodes are also readily available. For homemade Pt counter electrodes, it is important to make sure that any non-Pt electrical connections to the electrode are not exposed to the solution, as these may readily oxidize and contaminate the solution (e.g., Cu wire oxidation). After constructing the cell and connecting the potentiostat leads, perform a cyclic voltammetric experiment in the blank electrolyte by scanning the potential at 0.1 V/s over a potential range where the electrode behaves nearly as an IPE. A safe range is between −0.5 and 0.5 V vs. Ag/AgCl. Plot the cyclic voltammogram and determine the value of C d for the GC electrode/0.1 M KCl interface from the steady-state charging current. Do not expect your voltammograms in blank solutions to be as ideal looking as Figure 1.6.9c; most common electrode/electrolyte interfaces do not behave as an IPE over large potential ranges. Explore how far positive and negative the GC electrode potential can be scanned before significant currents arise, marking the potential window for the system. Carry out a potential step experiment to determine Ru and C d in your blank solution, but be sure to keep E in the IPE-like zone. Plot ln i vs. t, where t is measured from the step edge. Your instrument might be able to make this plot for you, but if not, just export the data and make the plot yourself with a spreadsheet. You should be able to obtain the cell time constant from the slope and Ru from the intercept. From these measurements, estimate how fast one can charge the GC/0.1 M KCl interface. Investigate the effects of order-of-magnitude changes of electrolyte concentration on Ru and C d . If you have additional working electrodes of much different area, consider using them to study the effect of area on Ru and C d . While the above experiments may appear elementary, they are important for evaluating the available potential range over which useful electrochemical measurements can be made, as well as the time constant of the cell under study. Experienced electrochemists will always make these background measurements when beginning to study a new system to ensure that background

55

56

1 Overview of Electrode Processes

processes and cell properties do not interfere with or limit the intended measurement with the species of interest. 1.9.3

Further Work with Simple RC Networks

To explore a wider range of time scales, you can investigate the responses of resistor-capacitor networks connected to the potentiostat in place of a real cell. The network in Figure 1.9.1 mimics an IPE, because the working electrode exhibits only the double-layer capacitance, C d . The solution resistance is divided into compensated and uncompensated values, Rc and Ru , as discussed in Section 1.5.4. RC networks that are built as substitutes for real cells are typically called dummy cells. The three-element dummy cell in Figure 1.9.1 provides a good model for double-layer charging and is useful both as a teaching tool and for testing the transient characteristics of potentiostats. Construct a network like that in Figure 1.9.1 using an electronics breadboard to readily connect the two resistors and a capacitor in series. A useful first set of components is C d = 10 μF, Ru = 1 kΩ, Rc = 10 kΩ. This combination gives a cell time constant of 10 ms, so transient double-layer charging should take place in less than 50 ms. The leads from the potentiostat should be connected to the circuit as noted in the diagram. Using the automated user interface for the potentiostat, set the instrument to perform a potential step experiment from E1 = − 0.5 V to E2 = − 1.5 V.60 Use a step duration of 10Ru C d . As in the case with the real cell, plot ln i vs. t (with t measured from the step edge) and obtain the cell time constant from the slope and Ru from the intercept. Test the agreement between your experimental values of Ru and C d against those used to construct the dummy cell. Investigate the effects of varying E1 , ΔE = E2 − E1 , Rc , Ru , and C d , including changes over orders of magnitude for the latter three parameters. When using smaller values for Ru and C d , you may find that the ln i vs. t plot becomes nonlinear at very short times due to the internal filters and output limitations of the potentiostat that you are using. Estimate the time required for the potentiostat to establish Eappl between the working and reference electrodes. Perform a cyclic voltammetry (triangular wave) experiment between +0.5 and −0.5 V on the original three-element network to see if the response resembles that in Figure 1.6.9c. Using the steady-state charging current, determine the value of C d and see how closely it matches the value of the capacitor in your circuit. Next, vary the scan rate over at least an order of magnitude and observe the dependence of the charging current on v. You should observe that the steady-state charging current increases linearly with scan rate. This behavior is generally seen in real electrochemical systems. Electrochemical research groups often find dummy cells helpful for testing experimental equipment and arrangements. A common application is to calibrate the i/E response of the potentiostat and to check for nonzero instrumentation offsets in either E or i.

Cd

Working

Ru

Rc Reference

Figure 1.9.1 A three-element network responding much like an IPE. Connection points for the potentiostat leads are noted. Counter

60 The instrument might be set up only to perform double-step experiments, such as E1 to E2 , then back to E1 . You can use this mode. Just ignore the data for the second step.

1.11 Problems

1.10 References 1 L. R. Faulkner, J. Chem. Educ., 60, 262 (1983). 2 L. R. Faulkner in “Physical Methods in Modern Chemical Analysis,” Vol. 3, T. Kuwana, Ed.,

Academic, New York, 1983, pp. 137–248. 3 P. Delahay, “New Instrumental Methods in Electrochemistry,” Wiley-Interscience, New York,

1954, p. 92. 4 See, for example, G. J. Hoytink, J. Van Schooten, E. de Boer, and W. Aalbersberg, Rec. Trav.

Chim., 73, 355 (1954), for an application to the study of reactions coupled to the reduction of aromatic hydrocarbons. 5 A. Demortier and A. J. Bard, J. Am. Chem. Soc., 95, 3495 (1973).

1.11 Problems 1.1

Consider each of the following electrode/solution interfaces. Write the equations for the electrode reactions that occur when the potential is moved progressively (1) negatively and (2) positively from the open-circuit potential, until the background limit is reached. Next to each reaction write the approximate potential for the reaction in V vs. SCE (assuming the reaction is reversible). Sketch the entire i − E curve for the system. a) Pt/Cu2+ (0.01 M), Cd2+ (0.01 M), H2 SO4 (1 M) b) Pt/Sn2+ (0.01 M), Sn4+ (0.01 M), HCl(1 M) c) Hg/Cd2+ (0.01 M), Zn2+ (0.01 M), HCl(1 M)

1.2

A rotating disk electrode of area 0.30 cm2 is used for the reduction of 0.010 M Fe3+ to Fe2+ in 1 M H2 SO4 . Given DO for Fe3+ at 5.2 × 10–6 cm2 /s, 𝜈 = 0.010 cm2 /s, and the expression for mO in footnote 29, calculate the limiting current for the reduction for a disk rotation rate of 10 r/s. Include units on variables during calculation and give units of current in the answer.

1.3

A solution of volume 50 cm3 contains 2.0 × 10–3 M Fe3+ and 1.0 × 10–3 M Sn4+ in 1 M HCl. This solution is examined by voltammetry at a rotating platinum disk electrode of area 0.30 cm2 . At the rotation rate employed, both Fe3+ and Sn4+ have mass-transfer coefficients, m, of 10–2 cm/s. (a) Calculate the limiting current for the reduction of Fe3+ under these conditions. (b) A current–potential scan is taken from +1.3 to −0.40 V vs. NHE. Make a labeled, quantitatively correct, sketch of the i − E curve that would be obtained. Assume that no changes in the bulk concentrations of Fe3+ and Sn4+ occur during this scan and that all electrode reactions are nernstian.

1.4

The conductivity of a 0.1 M KCl solution is 0.013 Ω–1 cm–1 at 25 ∘ C. (a) Calculate the solution resistance between two parallel planar platinum electrodes of 0.1 cm2 area placed 3 cm apart in this solution. (b) A reference electrode with a Luggin capillary is placed the following distances from a planar platinum working electrode (A = 0.1 cm2 ) in 0.1 M KCl: 0.05, 0.1, 0.5, 1.0 cm. What is Ru in each case? (c) Repeat the calculations in part (b) for a spherical working electrode of the same area. [In (b) and (c), assume that a large counter electrode is employed.]

57

58

1 Overview of Electrode Processes

1.5

A 0.1-cm2 electrode with C d = 20 μF/cm2 is subjected to a potential step under conditions where Ru is 1, 10, or 100 Ω. In each case, what is the cell time constant, and what is the time required for the double-layer charging to be 95% complete?

1.6

Suppose the potential step in Problem 1.5 is from E1 = − 0.1 V to E2 = − 0.6 V vs. Ag/AgCl and the PZC is −0.3 V vs. Ag/AgCl. (a) Calculate the charge on the working electrode at E1 and E2 . Do the same for a PZC at (b) +0.1 V and (c) −0.1 V vs. SCE.

1.7

For the electrode in Problem 1.5, what nonfaradaic current will flow (neglecting any transients) when the electrode is subjected to linear sweeps at 0.02, 1, 20 V/s?

1.8

Consider the nernstian half-reaction: A3+ + 2e ⇌ A+

′

E0 3+ A

∕A+

= −0.500 V vs.NHE

The i–E curve for a solution at 25 ∘ C containing 2.00 mM A3+ and 1.00 mM A+ in excess electrolyte shows il,c = 4.00 μA and il,a = − 2.40 μA. (a) What is E1/2 (V vs. NHE)? (b) Sketch the expected i − E curve for this system. (c) Sketch the “log plot” (analogous to Figure 1.3.2b) for the system. 1.9

Consider the system in Problem 1.8 under the conditions that a complexing agent, L− , which reacts with A3+ according to the reaction A3+ + 4L− ⇌ AL4−

K = 1016

is added to the system. For a solution at 25 ∘ C containing only 2.0 mM A3+ and 0.1 M L− in excess inert electrolyte, answer parts (a), (b), and (c) in Problem 1.8. (Assume mO is the same for A3+ and AL4− ) 1.10 Derive the current–potential relationship under the conditions of Section 1.3.2 for a sys∗ = 0. Consider both O and tem where R is initially present at a concentration CR∗ and CO R soluble. Sketch the expected i − E curve. 1.11 Suppose a mercury pool of 1 cm2 area is immersed in a 0.1 M sodium perchlorate solution. How much charge (order of magnitude) would be required to change its potential by 1 mV? How would this be affected by a change in the electrolyte concentration to 10–2 M? Why? 1.12 Rearrangement of equation (1.3.17) yields the following expression for i as a function of E, which is convenient for calculating i − E curves for nernstian reactions: i∕il = {1 + exp[(nF∕RT)(E − E1∕2 )]}−1

(1.11.1)

a) Derive this expression. b) Consider the half-reaction Ru(NH3 )3+ + e ⇌ Ru(NH3 )2+ . The E0 for this reaction is 6 6 given in Appendix C. A steady-state i − E curve is obtained with a solution containing 10 mM Ru(NH3 )3+ and 1 M KCl (as supporting electrolyte). The working electrode is 6 a Pt disk of area 0.10 cm2 operating under conditions where m = 10−3 cm/s for both Ru species. Use a spreadsheet to calculate and plot the expected i − E curve.

1.11 Problems

1.13 a) Derive an expression for i as a function of E, analogous to that in Problem 1.12, from (1.3.21), using (1.3.16) as the definition of E1/2 , for use in solutions that contain both components of a redox couple. b) Consider the same system as in Problem 1.12, but for a solution containing 10 mM Ru(NH3 )3+ and 5.0 mM Ru(NH3 )2+ in 1 M KCl. Use a spreadsheet to calculate the 6 6 i–E curve and plot the results. c) What is 𝜂 mt at a cathodic current density of 0.48 mA/cm2 ? d) Estimate Rmt .

59

61

2 Potentials and Thermodynamics of Cells In Chapter 1, we focused on potential as an electrochemical variable. Here we look more closely at the physical meaning of potential, the origins of potential differences, and access, through potential, to chemical information. Initially, we will approach these matters through thermodynamics, which will teach us that potential differences manifest free energy changes. That linkage will open a path to chemical information through electrochemical measurements. Subsequently, we will explore the physical mechanism by which potential differences are established, and we will gain insights relevant to the measurement and control of potential.

2.1 Basic Electrochemical Thermodynamics 2.1.1

Reversibility

Thermodynamics strictly applies only to systems at equilibrium, which involves the idea that a process can move readily in opposite directions from the equilibrium position. The adjective reversible relates to this core idea; however, the word carries several different, but related, meanings. We distinguish three of them now. (a) Chemical Reversibility

Consider the electrochemical cell shown in Figure 1.1.1b: Pt∕H2 ∕H+ , Cl− ∕AgCl∕Ag

(2.1.1)

Experimentally, one finds that the difference in potential between the silver wire and the platinum wire is 0.222 V when all substances are in their standard states.1 The platinum wire is the negative electrode. When the two electrodes are shorted together, electrons flow from the Pt electrode through the external circuit to the Ag electrode, and the following reaction takes place: H2 + 2AgCl → 2Ag + 2H+ + 2Cl−

(2.1.2)

If one overcomes the cell voltage by opposing it with the output of a battery or other direct current (dc) source, the current flow through the cell reverses, and the new cell reaction is 2Ag + 2H+ + 2Cl− → H2 + 2AgCl

(2.1.3)

1 The discussion in this section rests on a basic understanding of thermodynamic quantities, including enthalpy, H, entropy, S, and Gibbs free energy, G (often just called free energy), which are typically introduced in general chemistry. The authors assume preparation at that level. There is also mention in this section of standard states, activities, and standard thermodynamic quantities. Section 2.1.5 introduces these concepts. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

62

2 Potentials and Thermodynamics of Cells

Reversing the cell current reverses the cell reaction. No new reactions appear; accordingly, the cell is termed chemically reversible. In contrast, the system Zn∕H+ , SO2− 4 ∕Pt

(2.1.4)

is not chemically reversible. The zinc electrode is negative with respect to platinum, and discharging the cell causes dissolution of the zinc: Zn → Zn2+ + 2e

(2.1.5)

At the platinum electrode, hydrogen evolves: 2H+ + 2e → H2

(2.1.6)

Thus, the net cell reaction is2 Zn + 2H+ → H2 + Zn2+

(2.1.7)

By applying an opposing voltage larger than the cell voltage, one can reverse the current, but the reactions observed are 2H+ + 2e → H2

(Zn electrode)

(2.1.8)

2H2 O → O2 + 4H+ + 4e

(Pt electrode)

(2.1.9)

2H2 O → 2H2 + O2

(Net)

(2.1.10)

Upon current reversal, one has different electrode reactions, as well as a different net process; hence, this cell is said to be chemically irreversible. One can similarly characterize half-reactions by their chemical reversibility. The reduction of nitrobenzene in oxygen-free, dry acetonitrile produces a stable radical anion in a chemically reversible, one-electron process: PhNO2 + e ⇌ PhNO2 −∙

(2.1.11)

The reduction of an aromatic halide, ArX, under similar conditions is normally chemically irreversible, because the radical anion resulting from the electron-transfer reaction rapidly decomposes: ArX + e → Ar−∙ + X−

(2.1.12)

Whether a half-reaction exhibits chemical reversibility depends upon solution conditions and the time scale of the experiment. For example, if the reduction of nitrobenzene is carried out in an acidic acetonitrile solution, the reaction is chemically irreversible, because PhNO2 −∙ reacts with protons. Alternatively, if the reduction of ArX is studied by a technique that takes only a very short time, the reaction ArX + e ⇌ ArX−∙

(2.1.13)

can appear chemically reversible, because the decay of the radical anion takes longer than the observational time scale.

2 This net reaction will also occur without a flow of electrons in the external circuit, because H+ in solution will attack the zinc. This “side reaction,” which happens to be identical with the electrochemical process, is slow if dilute acid is used.

2.1 Basic Electrochemical Thermodynamics

(b) Thermodynamic Reversibility

A process is thermodynamically reversible if an infinitesimal reversal in a driving force causes it to reverse direction. This can be true only if the system feels an infinitesimal driving force at any time; hence, it must be always at equilibrium. A reversible path between two states of the system is, therefore, one that connects a continuum of equilibrium states. Traversing it would require an infinite length of time. A cell that is chemically irreversible cannot behave reversibly in a thermodynamic sense. A chemically reversible cell may or may not operate in a manner approaching thermodynamic reversibility. (c) Practical Reversibility

Since all real processes occur at finite rates, none can proceed with strict thermodynamic reversibility. However, a process may be carried out in such a manner that thermodynamic equations apply to a desired accuracy. For the investigator’s practical purposes, he or she might then call the process “reversible.” Practical reversibility is not an absolute term; it includes attitudes and expectations that an observer has toward the process. A useful analogy involves the removal of a large weight from a spring balance. Carrying out this process strictly reversibly requires continuous equilibrium; the “thermodynamic” equation that applies is kx = mg

(2.1.14)

where k is the force constant, x is the distance the spring is stretched when mass m is added, and g is the earth’s gravitational acceleration. In the reversible process, the spring never contracts more than an infinitesimal distance, because the large weight is removed progressively in infinitesimal portions. If the same final state is reached by removing the weight all at once, then (2.1.14) applies at no time during the process, which is characterized by severe disequilibrium and is grossly irreversible. Alternatively, one could remove the weight as pieces. If there were enough pieces, then (2.1.14) would begin to apply a large fraction of the time. One might not be able to distinguish the real (but slightly irreversible) process from the strictly reversible path. In that case, one could label the real transformation as “practically reversible.” In electrochemistry, one frequently relies on the Nernst equation: ′ RT CO ln (2.1.15) E = E0 + nF CR to provide a linkage between electrode potential, E, and the concentrations of participants in the electrode process: O + ne ⇌ R

(2.1.16)

If a system follows the Nernst equation or an equation derived from it, the electrode reaction is often said to be thermodynamically or electrochemically reversible (or nernstian). Whether a process appears reversible depends on one’s ability to detect the signs of disequilibrium. In turn, that ability depends on the time domain of the possible measurements, the rate of change of the force driving the observed process, and the speed with which the system can reestablish equilibrium. If the perturbation applied to the system is small enough, or if the system can attain equilibrium rapidly enough compared to the measuring time, thermodynamic relations will apply. A given system may behave reversibly in one experiment and irreversibly in another, even of the same genre, if the experimental conditions have a wide latitude. This point will be met again and again throughout this book.

63

64

2 Potentials and Thermodynamics of Cells

2.1.2

Reversibility and Gibbs Free Energy

Consider three different methods (1) of carrying out the reaction Zn + 2AgCl → Zn2+ + 2Ag + 2Cl– : 1) Suppose zinc and silver chloride are mixed directly in a calorimeter at constant atmospheric pressure and at 25 ∘ C. Assume also that the extent of reaction is so small that the activities of all species remain unchanged during the experiment. The amount of heat liberated when all substances are in their standard states is found to be 233 kJ/mol of Zn reacted. Thus, the standard enthalpy of reaction, ΔH 0 , is −233 kJ.3 2) Suppose we now construct the cell of Figure 1.1.1a, that is, Zn∕Zn2+ (a = 1), Cl− (a = 1)∕AgCl∕Ag

(2.1.17)

and discharge it through a resistance R. Again, we assume that the extent of reaction is small enough to keep the activities essentially unchanged. During the discharge, heat evolves from both the resistor and the cell, and we can measure the total heat change by placing the entire apparatus inside a calorimeter. The heat evolved is 233 kJ/mol of Zn, independent of R. That is, ΔH 0 = − 233 kJ, regardless of the rate of cell discharge. 3) Let us now repeat the experiment with the cell and the resistor in separate calorimeters. Assume that the wires connecting them have no resistance and do not conduct any heat between the calorimeters. If we take QC as the heat change in the cell and QR as that in the resistor, we find that QC + QR = − 233 kJ/mol of Zn reacted, independent of R. However, the balance between these quantities does depend on the rate of discharge. As R increases, |QC | decreases and |QR | increases. In the limit of infinite R, QC approaches −43 kJ per mole of zinc, and QR tends toward −190 kJ. In this example, the energy QR was dissipated as heat, but it was obtained as electrical energy, so it might have been converted to light or to mechanical work. In contrast, QC is an energy change that is inevitably thermal. Discharge through R → ∞ corresponds to a thermodynamically reversible process; therefore, we can identify lim QC as the energy, Qrev , that must appear R→∞

as heat in traversing a reversible path. The entropy change, ΔS, is defined in thermodynamics as Qrev /T (2); hence, for our example, where all species are in their standard states, TΔS0 = lim QC = −43 kJ R→∞

(2.1.18)

Because ΔG0 = ΔH 0 − TΔS0 , ΔG0 = −190 kJ = lim QR R→∞

(2.1.19)

We have now identified −ΔG with the maximum net work obtainable from the cell, where net work is defined as work other than PV work (2). For any finite R, |QR | (and the net work) is less than the limiting value. 2.1.3

Free Energy and Cell emf

Just above, we recognized that if the electrochemical cell (2.1.17) were discharged through an infinite load resistance, the discharge would be reversible. The potential difference would always be the equilibrium (open-circuit) value. Since the extent of reaction is assumed to be small

3 In the thermodynamic convention, absorbed quantities are positive and evolved quantities are negative.

2.1 Basic Electrochemical Thermodynamics

enough that all activities remain constant, the potential also remains constant. Then, the energy dissipated in R is given by |ΔG| = charge passed × reversible potential difference

(2.1.20)

|ΔG| = nF|E|

(2.1.21)

where n is the number of electrons passed per atom of zinc reacted (or the number of moles of electrons per mole of Zn reacted), and F is the charge on a mole of electrons (96,485 C). We also recognize that the free energy change has a sign associated with the direction of the net cell reaction. The sign reverses with a change of direction. Yet, only an infinitesimal change in the potential difference is required for the reversal; consequently, E is essentially constant and independent of the direction of a (reversible) transformation. We have a quandary: We want to relate a direction-sensitive quantity ( ΔG) to a direction-insensitive observable ( E). This desire is the origin of the confusion that commonly exists over signs in electrochemical systems. In fact, the meaning of the signs − and + differs for free energy and potential. For free energy, − and + signify energy lost or gained from the system, a convention of thermodynamics. For potential, − and + signify an excess or deficiency of electronic charge, an electrostatic convention proposed by Benjamin Franklin even before the discovery of the electron. In most scientific discussions, this difference in meaning is not important, since the context, thermodynamic or electrostatic, is clear. But when one considers electrochemical cells, where both thermodynamic and electrostatic concepts are needed, it is necessary to distinguish clearly between these distinct usages. We rationalize this difficulty by inventing a thermodynamic construct called the emf of the cell reaction. This quantity is assigned to a reaction (not to the physical cell); hence, it has a directional aspect. In a formal way, we also associate each cell schematic with a particular cell reaction, such that the right electrode in the schematic always corresponds to reduction, and the left, to oxidation. In the case of schematic (2.1.17), for example, the corresponding cell reaction is Zn + 2AgCl → Zn2+ + 2Ag + 2Cl−

(2.1.22)

The reverse cell reaction, Zn2+ + 2Ag + 2Cl− → Zn + 2AgCl

(2.1.23)

is associated with the opposite schematic, Ag∕AgCl∕Zn2+ (a = 1), Cl− (a = 1)∕Zn

(2.1.24)

The cell reaction emf , Erxn , is defined as the electrostatic potential of the electrode written on the right in the cell schematic with respect to that on the left. For example, in cell (2.1.17), the measured potential difference is 0.985 V, and the zinc electrode is negative; thus, the emf of reaction 2.1.22, the spontaneous process, is +0.985 V. Likewise, the emf corresponding to (2.1.23), the reverse of (2.1.22), is −0.985 V. By adopting this convention, we have managed to rationalize an (observable) electrostatic quantity (the cell potential difference), which is not sensitive to the direction of the cell’s operation, with a (defined) thermodynamic quantity (the Gibbs free energy), which is sensitive to that direction. One can completely avoid confusion over sign conventions of cell potentials if one understands this formal relationship between electrostatic measurements and thermodynamic concepts (3, 4). Because our convention implies a positive emf when a reaction is spontaneous, ΔG = −nFErxn

(2.1.25)

65

66

2 Potentials and Thermodynamics of Cells

or as above, when all substances are at unit activity [i.e., in their standard states (Section 2.1.5)], ΔG0 = −nFE0rxn

(2.1.26)

0 is the standard emf of the cell reaction. where Erxn Other thermodynamic quantities can be derived from electrochemical measurements now that we have linked the potential difference across the cell to the free energy. For example, the entropy change in the cell reaction is given by the temperature dependence of ΔG: ( ) ) ( 𝜕Erxn 𝜕ΔG ΔS = − = nF (2.1.27) 𝜕T P 𝜕T P

and

[ ( ) ] 𝜕Erxn ΔH = ΔG + TΔS = nF T − Erxn 𝜕T P

(2.1.28)

The equilibrium constant of the reaction is given by RT ln Krxn = −ΔG0 = nFE0rxn

(2.1.29)

These relations allow computation of electrochemical properties from thermochemical data. Several problems following this chapter illustrate the usefulness of that approach. Large tabulations of thermodynamic quantities exist (5–8). 2.1.4

Half-Reactions and Standard Electrode Potentials

Since the overall cell reaction comprises two independent half-reactions, one might think it reasonable that the cell potential could be broken into two individual electrode potentials. Indeed, a self-consistent set of half-reaction emfs and half-cell potentials can be devised, but the process is complicated by an experimental constraint. Experimentally, one can measure a potential difference only between two electronic conductors of the same composition (Section 2.2.3). In electrochemistry, this limitation requires that one always work with whole cells of two electrodes, connected to the measuring device (e.g., a high-impedance voltmeter) via contacts having the same composition. Despite this constraint, a useful scale for individual half-cells can be constructed, if one refers electrode potentials and half-reaction emfs to a standard reference electrode featuring a standard half-reaction. The primary reference, chosen by convention, is the normal hydrogen electrode (NHE), also called the standard hydrogen electrode (SHE): Pt∕H2 (a = 1)∕H+ (a = 1)

(2.1.30)

Its potential is taken as zero at all temperatures. The standard emfs of the half-reactions, 2H+ + 2e ⇌ H2

(2.1.31)

have also been assigned values of zero at all temperatures. One can record half-cell potentials by measuring them in whole cells against the NHE [Section 2.1.8(a)]. For example, in the system Pt∕H2 (a = 1)∕H+ (a = 1)∕∕Ag+ (a = 1)∕Ag

(2.1.32)

the cell potential is 0.799 V, and silver is positive. Thus, we say that the standard potential of the Ag+ /Ag couple is +0.799 V vs. NHE. Moreover, the standard emf of the Ag+ reduction is also +0.799 V vs. NHE, but that of the Ag oxidation is −0.799 V vs. NHE. Another valid expression

2.1 Basic Electrochemical Thermodynamics

is that the standard electrode potential of Ag+ /Ag is +0.799 V vs. NHE. To sum all of this up, we write:4 Ag+ + e ⇌ Ag

E0

Ag+ ∕Ag

= +0.799 V vs. NHE

(2.1.33)

For the general system, (2.1.16), the electrostatic potential of the O/R electrode (with respect to NHE) and the emf for the reduction of O always coincide. Therefore, one can condense the electrostatic and thermodynamic information into one list by tabulating electrode potentials and writing the half-reactions as reductions. Appendix C provides some frequently encountered potentials. Reference (5) is an authoritative general source for aqueous systems. Such tables are extremely useful, because they present much chemical and electrical information very compactly. A few electrode potentials can characterize many cells and reactions. Since the potentials are indices of free energies, they are also ready means for evaluating equilibrium constants, complexation constants, and solubility products. Also, they can be taken in linear combinations to supply electrochemical information about additional half-reactions. One can tell from a glance at an ordered list of potentials whether a given redox process will proceed spontaneously. It is important to grasp that electrostatic potential (not emf ) is the measurable experimental variable. When a half-reaction is chemically reversible, the potential of its electrode will usually remain in the same neighborhood and retain the same sign, whether the reaction proceeds as an oxidation or a reduction (9) [Sections 1.1.7(f ) and 1.3.2(b)]. 2.1.5

Standard States and Activity

Standard thermodynamic quantities (e.g., standard free energies of formation for various substances) correspond to situations in which all species of interest, including all participants in any reaction, are in their standard states. Likewise, the standard potential of a cell or half-reaction is defined for conditions where all species are in their standard states (10). This provision is necessary because many thermodynamic quantities, such as the free energy of a solute, depend on concentration. To provide meaningful tabulations, one must specify the concentrations. It is helpful to adopt a standard practice—that is, to define standard states systematically. Standard thermodynamic quantities are marked with a superscript zero, e.g., ΔG0 (the standard free energy change for a reaction) or E0 (the standard potential for an electrode). The definitions of standard state are straightforward, but idealized:5 • For a pure solid or liquid, the standard state is the substance under the standard pressure of 105 Pa (1 bar). For solids, it is necessary to specify the allotrope or crystalline form. • For a pure gas, the standard state is the gas under standard pressure, but behaving as an ideal gas. • For a solute, the standard state is a solution of standard concentration under standard pressure, but behaving as though each solute molecule is infinitely diluted. The standard concentration is often 1 M or 1 m (molal), if either such measure is normal for the system; 4 In some of the older literature, the standard emfs of reduction and oxidation are called the “reduction potential” and the “oxidation potential,” respectively. These terms are intrinsically confusing and should be avoided altogether, because they conflate the chemical concept of reaction direction with the physical concept of electrical potential. 5 In 1982, the IUPAC redefined the standard pressure to 105 Pa (1 bar) from 1 atm (1.01325 × 105 Pa). Standard states were based before 1982 on the old standard pressure, and tabulations of standard data compiled before 1982 presumed those standard states. The change in the pressure standard is not consequential for most work in electrochemistry.

67

68

2 Potentials and Thermodynamics of Cells

alternatively, it could be a partial pressure of 105 Pa for a gaseous solute or unit mole fraction for a component in an alloy or any other system well described by mole fraction. The idealization of the standard states for gases and solutes is meant to produce a conceptual condition in which each gas molecule or condensed-phase solute is independent, apart from its interaction with the diluent (solvent in the case of a solution, free space for a gas). In reality, species in a gas at 105 Pa or solutes in a system with molar-level concentration are affected by neighboring entities (like themselves or different), because neighbors are close enough to exert forces on each other and to mutually affect environments. When such interactions are significant, the properties of a standard state must be determined by extrapolation based on progressive dilution to remove such effects. Of course, one usually wants to work under conditions where concentrations or partial pressures differ from the standard. Thermodynamic relationships can accommodate such situations if one expresses each concentration or partial pressure in terms of an activity, aj , defined for a dissolved or gaseous species j as aj = 𝛾j Cj ∕Cj0

aj = 𝛾j Pj ∕Pj0

(2.1.34a,b)

where C j is the concentration (or Pj is the partial pressure) of species j, Cj0 is the standard-state

concentration (or Pj0 is the standard-state partial pressure) of that species, and 𝛾 j is the activity coefficient. For an ideal system, in which each entity is independent, the activity coefficient is unity, and aj is just the concentration or partial pressure scaled to that of the standard state. For example, a solute in an ideal 1 mM solution would have an activity of 10−3 , given the common standard-state concentration of 1 M. Activity is unitless; C j or Pj must always be expressed in the units used to define the standard state. The activity coefficient reflects the nonidealities of a real system by accounting for the way in which the free energy of a solute or gas molecule is environmentally altered. As solutions become more dilute (or as pressures are reduced in gaseous systems), the systems become more ideal, and activity coefficients approach unity.6 Theories exist to predict the values of activity coefficients. The best known is the Debye–Hückel theory, which applies to dilute solutions of fully dissociated electrolytes (2, 11). It predicts that the activity coefficient of a given ionic species depends on its charge and size, the charges and concentrations of all other ions, and the dielectric constant of the solvent. The activity coefficient is unity in the limit of infinite dilution, but becomes progressively smaller as the ionic population rises from zero. For relatively dilute solutions, the Debye–Hückel theory predicts that, log 𝛾j =

−Az2j I 1∕2 1 + Bdj I 1∕2

(2.1.35)

where zj is the charge on species j, dj is its effective solvated diameter, and I is the volume ionic strength, 1∑ 2 I= z C (2.1.36) 2 m m m 6 Sometimes one sees activity described as an “idealized concentration.” Actually, it is the product of two things: (a) a normalized concentration relative to the standard state, and (b) a free-energy adjustment (expressed in the activity coefficient). Activity is needed for accuracy in equations based on free energies. It is, however, not the same as concentration. If a species is present at a given concentration in a solution for which the ionic strength is made to vary, its activity will change, but the number of molecules of that species per unit volume will not.

2.1 Basic Electrochemical Thermodynamics

Table 2.1.1 Activity Coefficients Calculated from the Debye–Hückel Theory(a). Volume Ionic Strength (M) Ion

dj

H+ Na+ Ag+

(nm)(b)

0.001

0.01

0.1

0.9

0.967

0.914

0.826

0.4

0.965

0.902

0.770

0.25

0.965

0.897

0.745

OH−

0.35

0.965

0.900

0.762

NO− 3

0.3

0.965

0.899

0.754

Mg2+

0.8

0.872

0.690

0.445

Fe2+

0.6

0.870

0.676

0.401

Fe3+

0.9

0.737

0.443

0.179

Fe(CN)3− 6

0.4

0.726

0.394

0.095

0.5

0.569

0.200

0.020

Fe(CN)4− 6

(a) In water at 25 ∘ C. From (2.1.35). (b) From Klotz (12).

In (2.1.36), the sum runs over all ionic species in the system, and the C m are molar concentrations. The constants A and B in (2.1.35) depend on the dielectric constant of the solvent and the temperature. For aqueous solutions at 298 K, A = 0.509 M−1/2 and B = 3.29 nm−1 M−1/2 . Table 2.1.1 provides some predictions of the theory, illustrating the effects of various factors. Activity coefficients can depart substantially from the ideal (unity) for species that are highly charged or physically large, or if the ionic strength is high. They also become progressively less ideal in solvents of lower dielectric constant. Neutral species remain nearly ideal under the conditions being discussed here. The Debye–Hückel theory is quantitatively valid only for fully dissociated ionic species at ionic strengths below 0.1 M. In more concentrated media, the behavior of activity coefficients can be complex, as shown in Figure 2.1.1.7 The direction of change with ionic strength can reverse, and the activity coefficients can even become much larger than unity. The behavior is discussed much more fully in the specialized thermodynamic literature (2). For solutes in practical electrochemical media (e.g., 0.5 M aqueous KCl or 0.1 M TBABF4 in CH3 CN), activity coefficients are often unknown and are not readily obtained. In electrochemical research, one often must either neglect activity effects or accommodate their effect in some general way (as we will soon see with formal potentials in Section 2.1.7). 2.1.6

emf and Concentration

Consider a general cell in which the half-reaction at the right-hand electrode is 𝜈O O + ne ⇌ 𝜈R R

(2.1.37)

where the 𝜈 j are stoichiometric coefficients. If the left side of the cell is an NHE, the cell reaction is then 𝜈H H2 + 𝜈O O → 𝜈R R + 𝜈H+ H+ 2

(2.1.38)

7 Experimental means for determining activity coefficients do not yield values for single ions, but generally a mean ionic activity coefficient, 𝛾 ± . For a 1:1 electrolyte, such as HCl, 𝛾 ± = (𝛾 + 𝛾 − )1/2 , where 𝛾 + and 𝛾 − are the single-ion values for the cation and anion, respectively.

69

70

2 Potentials and Thermodynamics of Cells

1.8 1.6

Debye–Hückel region

1.4

HCl LiCl

𝛾± 1.2 Ideal behavior

1.0

NaCl

0.8 0.6 0.0

1.0

2.0

3.0

4.0

Concentration/m

Figure 2.1.1 Experimentally determined mean ionic activity coefficients vs. concentration [in molal units (mol per kg of solvent)]. Data are for aqueous solution; molar concentrations below 1 M are similar to molal values. Debye–Hückel theory applies only in the shaded area to the left. In that region, the values of 𝛾 ± are close to the figures presented for the cations in Table 2.1.1. [Data from Moore (13).]

and its free energy change is given from basic thermodynamics (2) by ΔG =

ΔG0

+ RT ln

𝜈

𝜈 +

𝜈

𝜈H

aRR aHH+

(2.1.39)

aOO aH 2 2

where aj is the activity of species j 8 Since ΔG = − nFE and ΔG0 = − nFE0 , 𝜈

𝜈 +

R H RT aR aH+ 0 E=E − ln 𝜈 𝜈H nF aOO aH 2

(2.1.40)

2

but since aH+ = aH = 1, 2

𝜈

O RT aO E=E + ln 𝜈 nF aRR

0

(2.1.41)

This relation, the Nernst equation, furnishes the potential of the O/R electrode vs. NHE as a function of the activities of O and R. In addition, it defines the activity dependence of the emf for reaction 2.1.37. The emf of any cell reaction, in terms of the electrode potentials of the two half-reactions, is Erxn = Eright − Eleft

(2.1.42)

where Eright and Eleft refer to the cell schematic and are given by the appropriate Nernst equations. The cell potential is the magnitude of this value. The functional form of the Nernst equation varies to suit the reaction. For example, if a half-reaction involves participants other than O and R, such as ligands or protons, its Nernst 8 A consequence of the 1982 change in standard pressure5 is that the potential of the NHE now differs from that used historically. The “new NHE” is +0.169 mV vs. the “old NHE” (based on a standard state of 1 atm). This difference is rarely significant, and is never so in this book. Most tabulated standard potentials, including those in Table C.1, are referred to the old NHE. See reference (14).

2.1 Basic Electrochemical Thermodynamics

equation must reflect the added complexity. The general form of the Nernst equation for an electrode and for the corresponding reductive half-reaction is O∏ side 𝜈 j j RT E = E0 + ln R∏ side nF j

aj

𝜈j

(2.1.43)

aj

in which the top and bottom products run, respectively, over all participants on the O and R sides of the half reaction. Since activities are unitless, the argument of the logarithm remains unitless, regardless of the number and powers of activities it comprehends. 2.1.7

Formal Potentials

It is usually inconvenient to deal with activities in evaluations of half-cell potentials, because activity coefficients are almost always unknown. A device for avoiding them is the formal poten′ tial, E0 , which is the measured potential of a half-cell (vs. NHE) when • Species O and R are present at concentrations such that [O]𝜈O ∕[R]𝜈R is numerically unity, where [O] and [R] represent the total unitless molar concentrations of species O and R in all equilibrated chemical forms (e.g., solvated, protonated to various degrees, or complexed in various degrees). • Other substances, such as, components of the medium, are present at designated concentrations. For a reaction like (2.1.37), this definition allows us to write, 𝜈

E=

E0

O 𝜈 ′ RT aO RT [O] O + ln 𝜈 = E0 + ln nF nF [R]𝜈R aRR

(2.1.44)

At the least, the formal potential incorporates the standard potential and some activity coefficients; however, it also frequently includes chemical factors, such as equilibrium constants and concentrations of ligands. To understand more fully, let us rewrite (2.1.41) so that the activity coefficients are factored out of the second term: 𝜈

E=

E0

O 0 𝜈 RT 𝛾O RT (CO ∕C ) O + ln 𝜈 + ln nF nF (CR ∕C 0 )𝜈R 𝛾RR

(2.1.45)

If the system is simple, so that solvated O and solvated R are the only forms present, then ′ C O /C 0 = [O] and C R /C 0 = [R],9 and by comparison with (2.1.44), we find that E0 is just the first two terms of (2.1.45), 𝜈

E

0′

O RT 𝛾O 0 =E + ln 𝜈 nF 𝛾RR

(2.1.46)

This is a common situation. 9 Informally, and even in the literature, one commonly finds the practice of treating aj for a solute as though it could be written 𝛾 j C j . The activity coefficient is sometimes factored out, leaving the concentration. This cannot be correct, because activities and activity coefficients are unitless. If the activity coefficient is factored out, what is left behind must also be unitless. In reality, it is aj ∕𝛾j = Cj ∕Cj0 , not C j . If C j is a molar concentration, then the standard state is 1 M, so Cj ∕Cj0 has the numeric value of C j without the units. Thus, the common practice is numerically accurate, but not dimensionally so.

71

72

2 Potentials and Thermodynamics of Cells

For example, consider Fe3+ + e ⇌ Fe2+

(2.1.47)

in 1 M HClO4 . In the rightmost form of (2.1.44), the Nernst equation becomes simply RT [Fe3+ ] ln nF [Fe2+ ] and the formal potential is ′ RT 𝛾Fe3+ E0 = E0 + ln nF 𝛾Fe2+ ′

E = E0 +

(2.1.48)

(2.1.49)

In the case of Cu2+ + 2e ⇌ Cu

(2.1.50)

The Nernst equation based on the formal potential is ′ RT [ 2+ ] E = E0 + ln Cu (2.1.51) nF and ′ RT E0 = E0 + ln 𝛾Cu2+ (2.1.52) nF There is no factor for metallic Cu in either (2.1.51) or (2.1.52), because both its activity and activity coefficient are unity. ′ Ionic strength affects activity coefficients; consequently, E0 varies from medium to medium. Table C.2 lists formal potentials for the Fe(III)/Fe(II) couple in 1 M HCl, 10 M HCl, 1 M HClO4 , 1 M H2 SO4 , and 2 M H3 PO4 . Standard potentials for half-reactions and cells are, in fact, determined by measuring formal potentials at different ionic strengths and extrapolating to zero ionic strength. ′ Often E0 also contains factors related to complexation or ion pairing, as it does in fact for the Fe(III)/Fe(II) couple in HCl, H2 SO4 , and H3 PO4 solutions. Both iron species are complexed in these media; hence, (2.1.47) does not accurately describe the half-cell reaction, which is better expressed in the form of (1.1.24). The bracketed concentrations, [Fe(III)] and [Fe(II)], can be readily understood and defined, but are made up of ferric and ferrous species complexed in varying, generally unknown, degrees. By relying on the empirical formal potentials, one can sidestep the need for detailed description of the related equilibria. ′ In other situations, one can define E0 explicitly and can exploit it to obtain information about homogeneous equilibria associated with an electrode reaction. An example is found in Problem 7.12. 2.1.8

Reference Electrodes

(a) The NHE and Classic Hydrogen Electrodes

So far in this chapter, we have used the NHE exclusively as the reference for potential, because it is the thermodynamic standard. However, an NHE cannot actually be constructed, because it requires that aH = aH+ = 1, implying ideal behavior in the gas and the 2 solution. A real hydrogen electrode can approximate an NHE, and measurements against the NHE can be achieved by extrapolating results using the real electrode to infinite dilution of the electrolyte. The overall process has been clearly described (11). Such painstaking measurements have been essential to the history of electrochemical thermodynamics, including the establishment of the potential scale. A real hydrogen electrode can be constructed as illustrated on the left side of Figure 1.1.1b. A Pt electrode is immersed in an aqueous electrolyte having a known concentration of H+ ,

2.1 Basic Electrochemical Thermodynamics

then H2 is bubbled through the solution near the Pt. Normally, the Pt electrode has been “platinized,” i.e., electrodeposited with Pt by reduction in chloroplatinic acid solution. Platinization greatly increases the surface area of the electrode and improves the kinetics of the heterogeneous reaction, 2H+ + 2e ⇌ H2 . Classic hydrogen electrodes like these are well behaved and reproducible; however, they are also cumbersome, so they are rarely used for general laboratory work. More convenient reference electrodes are employed for daily measurements. The remainder of this section covers the critical properties of reference electrodes and the most widely used options. The Lab Note in this chapter (Section 2.5) covers some practicalities. Much greater detail is available in authoritative reviews (15–19). (b) Attributes of Practical Reference Electrodes

A reference electrode for routine use in the laboratory must: 1) Adopt a predictable potential with precision and hold that potential reliably over time. 2) Be packaged conveniently for insertion into a cell, for removal, and for storage, so that it can be reused easily over a long lifetime. 3) Make satisfactory ionic contact with the working solution. 4) Not contaminate the solution at the working electrode or become contaminated by that solution. 5) Accept any current required by the role of the reference electrode in the cell without significant change of potential. Reference electrodes that satisfy the Attribute 1 are well poised, meaning that they are based on an electrochemical equilibrium between an electrode and both redox forms of a kinetically facile redox couple. Such an equilibrium is the basis of predictability, precision, and stability. Many reference electrodes also embody a secondary equilibrium (commonly a solubility equilibrium) that can stabilize the activities of participants in the electrode reaction other than the redox species themselves. Section 2.1.8(c) provides examples. Packaging often resembles that shown for the Ag/AgCl reference in Figure 1.5.4a. The active electrode material and the internal electrolyte are typically contained in a glass or plastic envelope having a plastic cap bearing a sturdy conductor to which external connection can be made repeatedly by clip-lead. Reference electrodes are often fabricated in the laboratory for local use, but many are also purchased commercially. Of course, any packaging must be inert and insoluble in the working solutions to be used. Contact between the internal electrolyte of the reference electrode and the working solution is established through an ionically conducting element at the tip of the reference electrode assembly. In most cases, porous glass or porous ceramic is employed; however, an ionically conducting polymer, a porous polymer membrane, a frit, or a fiber might be used instead. The tip material must be inert and insoluble in the working solution. The issues of contamination raised in Attribute 4 can be very significant, because leakage can be expected in both directions at the tip. See Section 2.5.1 for a practical discussion of leakage. Attribute 5, relating to the acceptance of current by the reference electrode, is usually not a concern in contemporary practice with three-electrode cells, because the only current through the reference cell is that required for continuous measurement of the potential (Sections 1.5.3 and 1.5.4). Present-day instrumentation handles this function with very high input impedance (Section 16.4); hence, that current is tiny, and the reference electrode remains negligibly polarized. However, the picture can be different for experiments in two-electrode cells, where the reference electrode is also the counter electrode. In this case, the reference electrode must accept all of the current flowing at the working electrode. Depending on the size of the working electrode, the demand on the reference electrode can range from negligible to very significant. It can be helpful to use a reference electrode of larger area and with greater amounts of each redox component. In the historic literature (into the 1960s), reference electrodes were often the

73

74

2 Potentials and Thermodynamics of Cells

counter electrodes in two-electrode cells. They sometimes had to handle quite a bit of current, so they were large by current standards, with sizable electrodes and substantial masses of any solids in equilibrium. (c) Aqueous Reference Electrodes Based on Solubility Equilibria

Many practical reference electrodes are based on solubility equilibria involving aqueous electrolytes. The most widely used, by far, is Ag/AgCl with saturated KCl as the electrolyte: Ag∕AgCl∕KCl (sat’d) E = 0.197 V vs. NHE at 25∘ C (2.1.53) ref

The potential given here is that recommended in references on the basis of direct measurements (16, 18, 19).10 This electrode is available from many commercial sources, but is also easily fabricated in the laboratory by anodizing a silver wire in a concentrated KCl solution, then packaging the electrode and electrolyte generally as shown in Figure 1.5.4a. The solubility equilibria for KCl and AgCl assure that the activities of Ag+ and Cl− remain fixed at constant temperature, giving the reference electrode a stable potential. There is a modest temperature coefficient for the reference potential, but it is not an issue for most work. The saturated calomel electrode (SCE), Hg∕Hg Cl ∕KCl (sat’d) E = 0.244 V vs. NHE at 25∘ C (2.1.54) 2

2

ref

was, for decades, the reigning practical reference (15–18).10,11 A great fraction of the literature published before the 1980s involves potential scales referenced to this electrode. In recent years, it has lost out to Ag/AgCl, because (a) its construction is more complicated than that of Ag/AgCl, (b) it is bulkier and more awkward to use, and (c) it is based on Hg, whose toxicity has generated greater concern. Even so, the SCE remains commercially available and fairly widely used, especially with ion-selective electrodes (Section 2.4), because it is very well behaved. Like the Ag/AgCl electrode, the SCE rests on solubility equilibria to fix the activities of the and Cl− ). The potential of the SCE varies ionic species involved in the electrode reaction (Hg2+ 2 with temperature, somewhat more strongly than for Ag/AgCl. One also finds use in the literature of the normal calomel electrode (NCE) (15, 18, 19):10 Hg∕Hg Cl ∕KCl (1 M) E = 0.280 V vs. NHE at 25∘ C (2.1.55) 2

2

ref

When even minor leakage of chloride is not acceptable, a mercurous sulfate electrode may be used (16, 19):10 Hg∕Hg SO ∕K SO (sat’d) E = 0.651 V vs. NHE at 25∘ C (2.1.56) 2

4

2

4

ref

For strongly basic working solutions, a mercuric oxide electrode might be preferable (20, 21): Hg∕HgO∕NaOH (0.1 m) E = 0.164 V vs. NHE at 25∘ C (2.1.57) ref

(d) Reversible Hydrogen Electrode

Classic hydrogen electrodes were discussed above with the concluding comment that they are generally too cumbersome for routine work. Even so, they are used in situations where 10 The presented value is justified in the cited references on the basis of direct measurements. It includes a junction potential at the reference electrode tip, which is uncertain by 1 mV or more (Section 2.3.5). Because of this uncertainty, one will sometimes see slightly different values of Eref used in the literature. 11 One occasionally sees the potential of the SCE given as 0.2412 V vs. the “old” NHE (15, 19). This is a calculated value based on the standard potential for the couple and an experimentally determined 𝛾 ± for saturated KCl. The discrepancy between calculated and measured values has been ascribed (15) to a difference between 𝛾Cl− and 𝛾 ± , as well as to the junction potential. The measured value given in (2.1.54) has been recommended as more relevant to a practical SCE in use.

2.1 Basic Electrochemical Thermodynamics

investigators desire to avoid introducing ionic or solvent contamination from a reference electrode tip. The arrangement is just as described previously, so that the reference electrode can be written Pt∕H2 ∕aqueous test solution of known pH∕

(2.1.58)

The platinized Pt electrode is inserted directly into a sample of the solution used at the working electrode [called the “test solution” in (2.1.58)]. The pH of the test solution must be well defined, either by buffering or by adequate concentration of H+ . Molecular hydrogen is a necessary solute at the reference electrode, but must usually be kept from the working electrode; therefore, a porous separator, such as a frit, would be interposed. If the ionic composition on both sides of the separator is the same, there would be no liquid junction potential (Section 2.3). This reference is called a reversible hydrogen electrode (RHE). It is not the same as an NHE (SHE), which is a concept, not a realizable device. The potential of the RHE is given by ERHE =

0 EH + ∕H 2

a2H+ 2.303RT + log 2F aH

(2.1.59)

2

If the gas is taken as ideal, then we obtain PH 2.303RT 2.303RT pH − ln 02 (V vs. NHE) (2.1.60) F 2F P where PH is the partial pressure of H2 and P0 is the standard-state pressure (1 bar or 105 Pa). 2 Commercial versions of the RHE are available, the designers of which have worked to eliminate the need for the hydrogen gas bottle supporting the classic hydrogen electrode. These devices have internal hydrogen supplies (generally electrolytically generated), which deliver H2 through a platinized gas diffusion electrode. They are as compact as many of the more common reference electrodes.12 ERHE = −

(e) Reference Electrodes for Nonaqueous Systems

With nonaqueous solvents like acetonitrile, methylene chloride, or DMF, one might be concerned especially with the leakage of water from an aqueous reference electrode, because water is a reactant toward electrogenerated species that one commonly desires to investigate in an aprotic environment. In that case, a reference electrode such as Ag∕Ag+ (0.01 M in CH3 CN)

(2.1.61)

might be preferred. Such an electrode can be packaged much like the Ag/AgCl electrode in Figure 1.5.4a; however, the internal electrolyte may need periodic renewal because of the volatility of CH3 CN. Normally, Ag+ would be introduced as the perchlorate or tetrafluoroborate salt. Electrodes like these are available commercially in kit form, allowing the user to fill the electrode with an Ag+ solution in the solvent actually chosen for the cell. Much work in nonaqueous media is carried out with quasireference electrodes (QRE), covered in Section 2.5.2. (f) Interconversion of Scales

Figure 2.1.2 shows the relationship between the NHE, Ag/AgCl, and SCE scales. Also shown are potentials on the “absolute scale” and the corresponding Fermi energies, which are concepts introduced in Section 2.2.5. 12 Sometimes, these devices are called dynamic hydrogen electrodes.

75

76

2 Potentials and Thermodynamics of Cells

–0.76

E0(Zn2+/Zn)

0

NHE Ag/AgCl SCE

E0(Fe3+/(Fe2+)

–0.96

–1.00

3.6

–3.6

–0.197

–0.244

4.4

–4.4

–0.047

4.6

–4.6

0.197

0

0.244

0.047

0

4.6

–4.6

0.77

0.57

0.53

5.2

–5.2

E/V E/V E/V vs. NHE vs. Ag/AgCl vs. SCE

E a/V absolute

EF /eV Fermi energy

Figure 2.1.2 Potentials on the NHE, Ag/AgCl, SCE, and “absolute” scales, together with corresponding Fermi energies.

The standard potentials of some couples can be measured vs. NHE with a precision better than 1 mV, so one will sometimes see values legitimately reported to four decimal places (as in Table C.1). However, it is generally inappropriate to convert or to report any electrode potential to four decimal places vs. Ag/AgCl or SCE, because the potential of either of these reference electrodes, in practical form and use, is uncertain by a millivolt or more [Sections 2.1.8(c) and 2.3.5]. 2.1.9

Potential–pH Diagrams and Thermodynamic Predictions

We learned in Section 2.1.4 that half-cell potentials are useful for characterizing the behavior of electrochemical cells and for evaluating the spontaneity of reactions. Potential–pH diagrams (also called Pourbaix diagrams) are graphical representations of half-cell potentials useful for predicting the behavior of systems (22). In these diagrams, the potential vs. NHE is represented vertically, and the pH, horizontally. A half-reaction that involves transfer of electrons and protons appears as a sloping line on this diagram. The potential is given by the corresponding Nernst equation, where the activities of all species other than H+ and OH− are assumed to have unit value. Thus, the generic reaction involving reduction of an oxidized form, O, to the reduced form, R, with involvement of protons, O + qH+ + ne ⇌ R

(2.1.62)

is represented by E = E0 + 2.303q(RT∕nF) log aH+ = E0 − 0.059(q∕n)pH

(V vs. NHE)

(2.1.63)

On a graph of E vs. pH, this equation describes a line with a slope of −59(q/n) mV per pH unit (at 25 ∘ C) and an intercept at pH = 0 of E0 . An electron-transfer half-reaction that does not

2.1 Basic Electrochemical Thermodynamics

involve protons is represented by a horizontal line at E = E0 . Purely acid–base reactions are represented by vertical lines, because such processes have no potential dependence. (a) E–pH Diagram for Water

One can illustrate this approach with the E − pH diagram for water (Figure 2.1.3), assuming that reduction produces hydrogen, H+ + e ⇌ 1/2H2

E0 = 0.0 V vs. NHE

(2.1.64)

E0 = 1.229 V vs. NHE

(2.1.65)

and oxidation produces oxygen, O2 + 4H+ + 4e ⇌ 2H2 O

From the generic expression, (2.1.63), these reactions are presented on the diagram as the two lines E = 0.0 − 0.059pH

(line A)

(2.1.66)

E = 1.229 − 0.059pH

(line B)

(2.1.67)

and

Equations 2.1.66 and 2.1.67 can be used to compute the reversible potentials for reduction and oxidation of water as a function of pH. Potential–pH diagrams are sometimes called predominance area diagrams, because they indicate which species predominates at a selected pH and potential. For example, water is thermodynamically stable between the two lines in Figure 2.1.3, but oxygen predominates in the area above the water oxidation line, and hydrogen predominates in the area below the water 1.5 O2

B 1.0 OER

ORR

E/ V vs. NHE

0.5 H 2O 0.0

A

HER

–0.5 H2 –1.0

–1.5

0

2

4

6

8

10

12

14

pH

Figure 2.1.3 Potential–pH diagram for the water system. The reactions corresponding to the arrows are important chemical processes: the hydrogen evolution reaction (HER, H2 O or H+ to H2 ), the oxygen evolution reaction (OER, H2 O to O2 ), and the oxygen reduction reaction (ORR, O2 to H2 O or OH− ). Arrows just show the direction of change; the reactions can take place up and down the pH scale.

77

2 Potentials and Thermodynamics of Cells

reduction line. These two products of water electrolysis receive the focus here because they are the most stable thermodynamically and generally are found in actual experiments. However, one might well ask about other possible species generated by the oxidation or reduction of water or oxygen, such as superoxide ion, O2 −∙ or H2 O2 . We will discuss this matter later, when we cover the limitations of thermodynamic predictions. (b) E–pH Diagram for Iron

Diagrams are available for most elements and some alloys in compilations (22), encyclopedias (7), and journal publications. The one for Fe (Figure 2.1.4) is particularly useful, because it bears on the corrosion of this important material. In very acidic solutions, ferrous and ferric ions do not form insoluble hydroxides or oxides and the diagram is governed by the half-reactions Fe2+ + 2e ⇌ Fe

E0 = −0.44 V vs. NHE

(2.1.68)

Fe3+ + e ⇌ Fe2+

E0 = 0.77 V vs. NHE

(2.1.69)

Since these processes do not involve proton transfer, they are represented in Figure 2.1.4 as horizontal lines (A and B). The Fe3+ zone (above line A) extends rightward to a pH where the system will precipitate Fe(OH)3 or the hydrous oxide (Fe2 O3 ), according to the solubility product, Ksp = 4 × 10−38 = aFe3+ a3OH− . A unit-activity solution of Fe3+ (approximately 1 M) will start to precipitate at 1.5

C Fe3+

1.0

A O2

0.5 E/V vs. NHE

78

Fe2+

H2O

0.0

H2O

Fe(OH)3

D

H2 G

E B

–0.5

F

Fe(OH)2

Fe –1.0

–1.5

0

2

4

6

8

10

12

14

pH

Figure 2.1.4 Potential–pH diagram for the iron system in water (solid lines). Line labels are discussed in the text. Dashed lines are for the water system. Predominance areas related to the latter are identified at the far right (upper line) or far left (lower line).

2.1 Basic Electrochemical Thermodynamics

pH = 1.5. Since the solubility product expression does not involve electrons, the boundary between Fe3+ and Fe(OH)3 is a vertical line at this pH (line C). It does not extend below line A, where Fe2+ predominates. The reaction separating the zone of predominance for Fe2+ from that for Fe(OH)3 is Fe(OH)3 + 3H+ + e ⇌ Fe2+ + 3H2 O

(2.1.70)

which is represented by the line E = E0 − (3)0.059pH

(2.1.71)

It must pass through the bottom right vertex of the zone for Fe3+ , where Fe(OH)3 and Fe2+ can coexist. Given this point and the slope of −0.177 V per pH unit, one can draw line D. For half reaction 2.1.70, E0 can be determined by extrapolating the line to pH = 0. By extrapolating to pH 14, one can obtain E0 for the reaction written in terms of OH− : Fe(OH)3 + e ⇌ Fe2+ + 3OH−

(2.1.72)

Ferrous iron also precipitates as the pH rises, in this case according to Fe(OH)2 ⇌ Fe2+ + 2OH−

(2.1.73)

with Ksp = 1.6 × 10−15 = aFe2+ a2OH− . Using the procedure employed above for precipitation of Fe3+ , the result is a new vertical line (E) at pH = 6.6, representing the boundary between zones dominated by Fe2+ and Fe(OH)2 . Its lower limit is the intersection with line B, because metallic iron predominates below line B. The upper limit is the intersection with line D, because Fe(OH)3 predominates above that point. We still have two boundaries to place on the right side of the diagram, viz. that separating the zones for Fe and Fe(OH)2 and that dividing the zones for Fe(OH)2 and Fe(OH)3 . In the former case, the relevant process is Fe(OH)2 + 2H+ + 2e ⇌ Fe + 2H2 O

(2.1.74)

which gives rise to a boundary line having a slope of −0.059 V per pH unit. It must pass through the intersection point of lines B and E, where Fe and Fe(OH)2 coexist. Given this point and the slope, we can draw boundary line F. The last boundary corresponds to the process, Fe(OH)3 + H+ + e ⇌ Fe(OH)2 + H2 O

(2.1.75)

It also has a slope of −0.059 V per pH unit and must pass through the intersection of lines D and E. This is line G in Figure 2.1.4. The number of zones and the precise locations of the boundaries depend upon the species used in the formulation of an E − pH diagram. For the iron system, for example, there are many types of iron oxides. Generally, the mineral form hematite, Fe2 O3 , is used for the iron(III) oxide; however, other phases could also be included [e.g., magnetite, Fe3 O4 , a mixed Fe(II)–Fe(III) oxide]. Also, higher oxidation states of Fe are stable in very alkaline solutions and might be shown in a very complete diagram. The E − pH lines for water (Figure 2.1.3) are also included in Figure 2.1.4 and allow one to make predictions about reactions. For example, Fe(III) species lie in the potential region where oxygen is predominant, which indicates that metallic Fe would be oxidized (i.e., would corrode) in this region when exposed to oxygen. The diagram also indicates that elemental iron will dissolve as Fe2+ in acidic solutions, with production of hydrogen.

79

80

2 Potentials and Thermodynamics of Cells

(c) Limitations of Predictions from E–pH Diagrams

Since the data in Pourbaix diagrams are purely thermodynamic, predictions are fully valid only for thermodynamically reversible systems. For example, in the water diagram (Figure 2.1.3), if one interprets the lines as applying to all actual electrodes, one would assume that the water-splitting reaction to hydrogen and oxygen would always require 1.23 V under conditions of negligible cell resistance. However, this is never observed, because, as will be seen in the next chapter, kinetic limitations (lack of reversibility) require overpotentials to drive the reactions. In fact, the actual electrochemical behavior depends critically on electrode material. The line between water and H2 accurately describes behavior at a Pt electrode, but is very far off with Hg. There is no electrode material that allows reversible water oxidation. These matters are discussed in detail in Chapter 15.

2.2 A More Detailed View of Interfacial Potential Differences For the thermodynamic considerations of the previous section, we were not required to advance a mechanism for the observable differences in potentials across phase boundaries. However, a mechanistic concept aids chemical thinking, so let us now consider how these differences might arise. 2.2.1

The Physics of Phase Potentials

One can readily speak of the electric potential13 at any point, (x, y, z), within a phase. That quantity, 𝜙(x, y, z), is defined in physics as the work required to bring a unit positive charge, without material interactions, from an infinite distance to (x, y, z).14 From electrostatics, we have assurance that 𝜙(x, y, z) is independent of the path of the test charge (23). The work is done against a coulombic field; hence, we can express the electric potential generally as 𝜙(x, y, z) =

x,y,z

∫∞

E ∙ dl −E

(2.2.1)

where E is the electric field strength vector (i.e., the force exerted on a unit charge at any point), and dl is an infinitesimal tangent to the path in the direction of movement. The integral is carried out over any path to (x, y, z). The difference in potential between points (x′ , y′ , z′ ) and (x, y, z) is then 𝜙(x′ , y′ , z′ ) − 𝜙(x, y, z) =

x′ ,y′ ,z′

∫x,y,z

E ∙ dl −E

(2.2.2)

In general, the electric field strength is not zero everywhere between two points and the integral does not vanish; hence, some potential difference usually exists. A conducting phase, such as a metal, a semiconductor, or an electrolyte solution, has mobile charge carriers. When no current passes through the phase (i.e., when it is at electrical equilibrium), there is no net movement of charge carriers, so the electric field at all interior points must be zero. If it were not, the carriers would move to eliminate the field. From (2.2.2), one can see that the difference in potential between any two points in the interior of the phase must also be zero under these conditions; thus, the entire phase is an equipotential volume. The electric 13 Also called the electrostatic potential. 14 Although 𝜙(x, y, z) is rigorously defined, it is not rigorously measurable. Only differences of electrical potential can be precisely measured, and then only under particular conditions (Section 2.2.3).

2.2 A More Detailed View of Interfacial Potential Differences

potential applying at every interior location in a conducting phase, 𝛼, at equilibrium is known as the inner potential (or Galvani potential), 𝜙𝛼 , of the phase. For a phase not at electrical equilibrium, an inner potential is undefined.15 A principal factor determining the inner potential is any excess charge on the phase itself, because a test charge coming from an infinite distance would have to work against the coulombic field arising from that charge. The test charge might also have to work against other fields arising from charged bodies outside the sample. If so, they would also be contributors to the inner potential of the phase we are discussing. If the charge distribution throughout the system holds constant, the phase potential will remain constant, but alterations in charge distributions inside or outside the phase will change the phase potential. Thus, we have our first indication that differences in potential arising from chemical interactions between phases have some sort of charge separation as their basis. An interesting question concerns the location of any excess charge on a conducting phase. The Gauss law from elementary electrostatics (24) is extremely helpful here. It says that if we enclose a volume with an imaginary surface (a Gaussian surface), we will find that the net charge, q, inside the surface is given by an integral of the electric field over the surface: q = 𝜀0

∮

E ∙ dS

(2.2.3)

where 𝜀0 is a proportionality constant,16 and dS is an infinitesimal vector normal outward from the surface. Now consider a Gaussian surface located within a conductor that is uniform in its interior (i.e., without voids or interior phases). If no current flows, E is zero at all points on the Gaussian surface; hence, the net charge within the boundary is zero. The situation is depicted in Figure 2.2.1. This conclusion applies to any Gaussian surface, even one situated just inside the phase boundary; thus, we must infer that any excess charge resides on the surface of the conducting phase.17 – –

–

– –

–

Interior Gaussian surface

Charged conducting phase

–

– –

– –

–

–

Zero included charge

Figure 2.2.1 Cross-section of a three-dimensional conducting phase containing a Gaussian enclosure. Illustration that the excess charge resides on the surface of the phase. 15 When current is passing through a phase, the potential within the phase is not constant. A difference in 𝜙 is needed between any two interior points to drive whatever current flows through the electrical resistance between those points. Such situations are encountered and discussed in Sections 1.5 and 4.2, but we will not address them in this chapter, which is focused on equilibrium. 16 The parameter 𝜀0 is called the vacuum electric permittivity or the permittivity of free space and has the value 8.85419 × 10−12 C2 N−1 m−2 . See the footnote in Section 14.3.1 for a fuller explanation of electrostatic conventions followed in this book. 17 There can be a finite thickness to this surface layer. The critical aspect is the size of the excess charge with respect to the bulk carrier concentration in the phase. If the charge is established by drawing carriers from a significant

81

82

2 Potentials and Thermodynamics of Cells

A view of the way in which phase potentials are established is now beginning to emerge: 1) The potential of a conducting phase can be changed by altering charge distributions on or around the phase. 2) If the phase undergoes a change in its excess charge, its charge carriers adjust such that the excess becomes wholly distributed over the entire boundary of the phase. 3) The surface distribution is such that the electric field strength within the phase is zero under null-current conditions. 4) The interior of the phase features a constant potential, 𝜙. The excess charge needed to change the inner potential of a conductor to an electrochemically significant degree is sometimes not very large. Consider, for example, a spherical mercury drop of 0.5 mm radius. Changing its potential requires only about 5 × 10−14 C/V (about 300,000 electrons/V), if it is suspended in air or in a vacuum (23). 2.2.2

Interactions Between Conducting Phases

When two conductors, e.g., a metal and an electrolyte, are placed in contact, the situation becomes complicated by the coulombic interaction between the phases. Charging one phase to change its inner potential tends to alter the inner potential of the neighboring phase as well. This point is illustrated in Figure 2.2.2, which portrays a situation where there is a charged metal sphere of macroscopic size, perhaps a mercury droplet 0.5 mm in radius, surrounded by a layer of uncharged electrolyte a few millimeters in thickness. This assembly is suspended in a vacuum. We know that the charge on the metal, qM , resides on its surface. This unbalanced charge (negative in the diagram) attracts an excess cation concentration near the electrode in the solution. What can we say about the magnitudes and distributions of the obvious charge imbalances in solution? Consider the integral in (2.2.3) over the Gaussian surface shown in Figure 2.2.2. Since this surface is in a conducting phase where current is not flowing, E at every point is zero and the net enclosed charge is also zero. We could place the Gaussian surface just outside the boundary between the metal and solution, and we would reach the same conclusion. Thus, we know now that the excess positive charge in the solution, qS , resides at the metal/solution interface and Electrolyte layer with no net charge

Surrounding vacuum – –

– + + – – – – + + – – – + – + +

Figure 2.2.2 Cross-sectional view of the interaction between a metal sphere and a surrounding electrolyte layer. The Gaussian enclosure is a sphere containing the metal phase and part of the electrolyte.

+

–

Metal with charge qM

–

–

– –

Gaussian surface

volume, thermal processes will impede the compact accumulation of the excess strictly on the surface. Then, the charged zone is called a space charge region, because it has three-dimensional character. Its thickness in electrolytes and semiconductors can range from a few tenths of a nm to thousands of nm. In metals, it is negligibly thick. See Chapters 14 and 21 for more detailed discussions.

2.2 A More Detailed View of Interfacial Potential Differences

exactly compensates the excess metal charge. That is, qS = −qM

(2.2.4)

This fact is very useful in the treatment of interfacial charge arrays, which we have already seen as electrical double layers (Section 1.6 and Chapter 14).18 Alternatively, we might move the Gaussian surface to a location just inside the outer boundary of the electrolyte. The enclosed charge must still be zero; yet we know that the net charge on the whole system is qM . A negative charge equal to qM must, therefore, reside at the outer surface of the electrolyte. Figure 2.2.3 is a display of electric potential vs. distance from the center of this assembly, i.e., the work done to bring a unit positive test charge from infinitely far away to a given distance from the center. As the test charge is brought from the right side of the diagram, it is attracted by the charge on the outer surface of the electrolyte; thus, negative work is required to traverse any distance toward the electrolyte surface in the surrounding vacuum, and the electric potential steadily drops in that direction. Within the electrolyte, E is zero everywhere, so there is no work in moving the test charge, and the potential is constant at 𝜙S . At the metal/solution interface, there is a strong field because of the double layer there, and it is oriented such that negative work is done in taking the positive test charge through the interface. Thus, there is a sharp change in potential from 𝜙S to 𝜙M over the distance scale of the double layer.19 Since the metal is a field-free volume, the electric potential is constant in its interior. If we were to increase the negative charge on the metal, we would naturally lower 𝜙M , but we would also lower 𝜙S , because the excess negative charge on the outer boundary of the solution would increase, and the test charge would be attracted more strongly to the electrolyte layer at every point on the path through the vacuum. The difference 𝜙M − 𝜙S , called the interfacial potential difference, depends on the charge imbalance at the interface and the physical size of the interface. That is, it depends on the charge density (C/cm2 ) at the interface. Making a change in this interfacial potential difference 0

0

Distance

ϕS ϕ ϕM Metal

Electrolyte

Vacuum

Figure 2.2.3 Electric potential profile through the system shown in Figure 2.2.2. Distance is measured radially from the center of the metallic sphere. 18 Here we are considering the problem on a macroscopic distance scale, and it is accurate to think of qS as residing at the metal–solution interface. On a scale of 1 μm or finer, the picture is more detailed. One finds that qS is still near the metal–solution interface, but is distributed in one or more zones that can be thicker than 100 nm (Section 14.3). 19 The diagram is drawn on a macroscopic scale, so the transition from 𝜙S to 𝜙M appears vertical. The theory of the double layer (Section 14.3) indicates that most of the change occurs over a distance equivalent to one to several solvent monolayers, with a smaller portion being manifested over the diffuse layer in solution.

83

84

2 Potentials and Thermodynamics of Cells

requires sizable alterations in charge density. For the spherical mercury drop considered above (A = 0.03 cm2 ), now surrounded by 0.1 M strong electrolyte, one would need about 10−6 C (or 6 × 1012 electrons) for a 1-V change. These numbers are more than 107 times larger than for the case where the electrolyte is absent. The difference appears because the coulombic field of any surface charge is counterbalanced to a very large degree by polarization of the adjacent electrolyte. In practical electrochemistry, metallic electrodes are partially exposed to an electrolyte and partially insulated. For example, one might use a 0.1-cm2 platinum disk electrode attached to a platinum lead that is almost fully sealed in glass. It is interesting to consider the location of excess charge used in altering the inner potential of such a phase. Of course, the charge must be distributed over the entire surface, including both the insulated and the electrochemically active areas. However, we have seen that the coulombic interaction with the electrolyte is so strong that essentially all of the charge at any value of potential will lie adjacent to the solution, unless the percentage of the metal area in contact with electrolyte is really minuscule.20 Practical means exist for placing an excess charge on a phase. An important one is simply to pump electrons into or out of a metal or a semiconductor with a power supply. Indeed, we will make great use of this approach to control the kinetics of electrode processes. There are also chemical mechanisms for charging a phase. For example, we know from experience that a platinum wire dipped into a solution containing ferricyanide and ferrocyanide will have its potential shift toward a predictable equilibrium value given by the Nernst equation. This process occurs because the electron affinities of the two phases initially differ; hence, there is a transfer of electrons from the metal to the solution or vice versa [Section 2.2.5(d)]. Ferricyanide is reduced, or ferrocyanide is oxidized. The transfer of charge continues until the electrode potential reaches the equilibrium point, where the electron affinities of the solution and the metal are equal. Compared to the total charge that could be transferred to or from ferri- and ferrocyanide in a typical system, only a tiny charge is needed to establish the equilibrium at Pt; consequently, the net chemical effects on the solution are unnoticeable. By this mechanism, the metal adapts to the solution and reflects its composition. Electrochemistry is full of situations like this one, in which charged species (electrons or ions) cross interfacial boundaries. These processes create a net transfer of charge that sets up the potential differences that we observe. Considering them in more detail must, however, await the development of additional concepts (Section 2.3 and Chapter 3). Actually, interfacial potential differences can develop without an excess charge on either phase. Consider an aqueous electrolyte in contact with an electrode. Since the electrolyte interacts with the metal surface (e.g., wetting it), the water dipoles in contact with the metal generally have some preferential orientation. From a coulombic standpoint, this situation is equivalent to charge separation across the interface, because the dipoles are not randomized with time. Moving a test charge through the interface requires work; therefore, the interfacial potential difference is not zero (25–28). 2.2.3

Measurement of Potential Differences

The interfacial potential difference, Δ𝜙, between two phases in contact is of primary importance to electrochemical processes occurring at their interface. Its influence is partly based on the interfacial electric field, which embodies the high rate of change in 𝜙 at the boundary. This field can reach a strength as high as 107 V/cm. It can be large enough to distort electroreactants and alter their reactivity, and it can affect the dynamics of charge transport across the boundary. 20 As it can be with an ultramicroelectrode (Section 6.7).

2.2 A More Detailed View of Interfacial Potential Differences

Figure 2.2.4 Two devices for measuring the potential of a cell containing the Zn/Zn2+ interface.

V

Zn

V

Zn

Cu/Zn/Zn2+, Cl–/Cu

Zn/Zn2+, Cl–/Zn

(a)

(b)

However, a more important aspect for electrochemistry is the direct influence of the potential difference on the relative energies of electrons on either side of the interface. It controls the relative electron affinities of the two phases; hence, it controls the rate, and even the direction, of any electrochemical reaction at the interface. Unfortunately, Δ𝜙 is not measurable for a single interface, because one cannot sample the electrical properties of the system without introducing at least one more interface. This is true because devices for measuring potential differences (e.g., high-impedance voltmeters or electrometers) can be calibrated only to register potential differences between two phases of the same composition. Consider Δ𝜙 at the interface Zn/Zn2+ , Cl− . Shown in Figure 2.2.4a is the simplest approach that one could make to obtain Δ𝜙 using a potentiometric instrument with copper contacts. The measurable potential difference between the copper phases clearly includes—in addition to Δ𝜙—interfacial potential differences at the Zn/Cu interface and the Cu/electrolyte interface. We might simplify matters by constructing a voltmeter wholly from zinc, but, as shown in Figure 2.2.4b, the measurable voltage would still contain contributions from two separate interfacial potential differences. By now we realize that a measured cell potential is a sum of several interfacial differences, none of which we can evaluate independently. For example, one can sketch the “stairstep” potential profile through the cell Cu∕Zn∕Zn2+ , Cl− ∕AgCl∕Ag∕Cu′

(2.2.5)

according to Vetter’s representation (26) in the manner of Figure 1.1.2, where we see the mea′ surable equilibrium potential difference, E, defined as 𝜙Cu − 𝜙Cu . Even with these complications, it is still possible to focus on a single interfacial potential difference, such as that between zinc and the electrolyte in (2.2.5). If we can maintain constant interfacial potentials at all other junctions in the cell, then any change in the measurable quantity, E, must be wholly attributed to a change in Δ𝜙 at the zinc/electrolyte boundary. Keeping the other junctions at a constant potential difference is not so difficult, for the metal/metal junctions always remain constant (at constant temperature), and the silver/electrolyte junction is fixed if the activities of the participants in its half-reaction remain fixed. When this idea is understood, the rationale behind half-cell potentials and the choice of reference electrodes becomes much clearer. 2.2.4

Electrochemical Potentials

Let us continue to consider the interface Zn/Zn2+ , Cl− (aqueous), while focusing on zinc ions in metallic zinc and in solution. In the metal, Zn2+ is fixed in a crystalline array, with mobile electrons permeating the structure. In solution, zinc ion is hydrated and may interact with Cl− . The energy state of Zn2+ clearly depends on the local chemical environment, which manifests itself through short-range forces that are mostly electrical. In addition, there is the energy required simply to bring the +2 charge, disregarding the chemical effects, to the location in question. This second energy is proportional to the local electric potential, 𝜙; hence, it depends on the electrical properties of an environment very much larger than the ion itself. Although one cannot experimentally separate these two components for a single species, the differences in the

85

86

2 Potentials and Thermodynamics of Cells

distance scales of the two environments make it plausible to separate the energy components mathematically (25–28). Butler (29) and Guggenheim (30) developed the conceptual separation and introduced the electrochemical potential, 𝜇 j , for species j with charge zj at a given physical location, (x, y, z): 𝜇j (x, y, z) = 𝜇j (x, y, z) + zj F𝜙(x, y, z) In (2.2.6), 𝜇j is the familiar chemical potential, ( ) 𝜕G 𝜇j = 𝜕nj

(2.2.6)

(2.2.7)

T,P,nk≠j

where nj is the number of moles of j in the local volume element. Analogously, the electrochemical potential would be ( ) 𝜕G 𝜇j = (2.2.8) 𝜕nj T,P,nk≠j

in which the electrochemical free energy, G, differs from the chemical free energy, G, by the inclusion of effects from the large-scale electrical environment. If species j is in a homogeneous conducting phase, 𝛼, at equilibrium, then 𝜇j , 𝜇j , and 𝜙 do not vary over all interior points, and 𝜙 is the inner potential, 𝜙𝛼 . In that situation, (2.2.6) becomes 𝜇𝛼j = 𝜇j𝛼 + zj F𝜙𝛼

(2.2.9)

which applies to the whole phase, not just to a particular location. (a) Properties of the Electrochemical Potential

Important relationships involving electrochemical potentials include the following: 1) For an uncharged species at any location: 𝜇j = 𝜇j . 2) For any substance at any location: 𝜇j = 𝜇j0 + RT ln aj , where 𝜇j0 is the standard chemical potential, and aj is the local activity of species j. 3) For a pure phase, 𝛼, at equilibrium and at unit activity (e.g., solid Zn, AgCl, Ag, or H2 ): 𝜇𝛼j = 𝜇j0𝛼 . 4) For electrons (z = − 1) in a metal phase, 𝛼: 𝜇𝛼e = 𝜇e0𝛼 − F𝜙𝛼 . There is no separate activity term because the electron concentration never changes appreciably. 𝛽 5) For equilibrium of species j between phases 𝛼 and 𝛽: 𝜇 𝛼j = 𝜇j . (b) Reactions in a Single Phase

Within a single conducting phase, 𝛼, the inner potential exerts no effect on a chemical equilibrium. The 𝜙𝛼 -terms drop out of relations involving electrochemical potentials, and only chemical potentials remain. To see this effect, consider the acid–base equilibrium: HOAc ⇌ H+ + OAc−

(2.2.10)

which requires that 𝜇HOAc = 𝜇H+ + 𝜇OAc−

(2.2.11)

𝜇HOAc = 𝜇H+ + F𝜙𝛼 + 𝜇OAc− − F𝜙𝛼

(2.2.12)

𝜇HOAc = 𝜇H+ + 𝜇OAc−

(2.2.13)

2.2 A More Detailed View of Interfacial Potential Differences

(c) Reactions Involving Two Phases Without Charge Transfer

Let us now examine the solubility equilibrium AgCl (crystal) ⇌ Ag+ + Cl−

(solution, S)

(2.2.14)

for which we can write 0AgCl

𝜇AgCl = 𝜇 SAg+ + 𝜇SCl−

(2.2.15)

Expanding, we obtain 0AgCl

𝜇AgCl = 𝜇0S + + RT ln aS

Ag+

Ag

0S S S + F𝜙S + 𝜇Cl − + RT ln a − − F𝜙 Cl

(2.2.16)

and rearrangement gives 0AgCl

0S S 𝜇AgCl − 𝜇0S + − 𝜇Cl − = RT ln(a

Ag+

Ag

aSCl− ) = RT ln Ksp

(2.2.17)

where K sp is the solubility product. The 𝜙S -terms canceled in (2.2.16). Since the final result depends only on chemical potentials, the equilibrium is unaffected by the potential difference across the interface. This is a general feature of interphase reactions without transfer of charge (either ionic or electronic). (d) Equilibrium Between Two Metals

Consider the connection of a Cu wire to a Zn electrode. At equilibrium, Zn 𝜇 Cu e = 𝜇e

(2.2.18)

𝜇e0Cu − F𝜙Cu = 𝜇e0Zn − F𝜙Zn

(2.2.19)

Rearranging, we obtain 𝜙Zn − 𝜙Cu = (𝜇e0Zn − 𝜇e0Cu )∕F

(2.2.20)

which indicates that an electric potential difference, 𝜙Zn − 𝜙Cu , is established between the Zn and Cu at equilibrium. This potential difference, called a contact potential, is proportional to the difference in standard chemical potentials of electrons in the two metals.21 The equilibrium is established by the transfer of electrons immediately upon contact between the metals. As established in Section 2.2.3, one cannot measure a potential difference at a single interface with thermodynamic rigor; however, means exist for estimating the values of contact potentials. Figure 1.1.2, showing the potential distribution across an entire electrochemical cell, includes contact potentials at Cu/Zn and Ag/Cu interfaces. (e) Formulation of a Cell Potential

When net charge transfer occurs across an interface, the 𝜙-terms do not cancel and the interfacial potential difference strongly affects the chemical process. One can use that potential difference either to probe or to alter the equilibrium position. As an example, consider cell (2.2.5), for which the cell reaction can be written Zn + 2AgCl + 2e(Cu′ ) ⇌ Zn2+ + 2Ag + 2Cl− + 2e(Cu)

(2.2.21)

At equilibrium, ′

AgCl

Ag

Cu 𝜇 Zn = 𝜇S Zn + 2𝜇 AgCl + 2𝜇 e ′

Cu S 2(𝜇Cu e − 𝜇e ) = 𝜇

Zn2+

Zn2+

+ 2𝜇Ag + 2𝜇SCl− + 2𝜇Cu e

Ag

AgCl

+ 2𝜇Ag + 2𝜇SCl− − 𝜇Zn Zn − 2𝜇 AgCl

21 Or the difference in work functions [(Section 2.2.5(d)].

(2.2.22) (2.2.23)

87

88

2 Potentials and Thermodynamics of Cells

But, ′

′

Cu Cu − 𝜙Cu ) = −2FE 2(𝜇Cu e − 𝜇 e ) = −2F(𝜙

(2.2.24)

Expanding (2.2.23), we have 0Ag

−2FE = 𝜇0S 2+ + RT ln aS

Zn2+

Zn

0S + 2F𝜙S + 2𝜇Ag + 2𝜇Cl − 0AgCl

0Zn + 2RT ln aSCl− − 2F𝜙S − 𝜇Zn − 2𝜇AgCl

−2FE = ΔG0 + RT ln aS

Zn2+

(2.2.25)

(aSCl− )2

(2.2.26)

where 0Ag

0AgCl

0S 0Zn ΔG0 = 𝜇0S 2+ + 2𝜇Cl = −2FE0 − + 2𝜇Ag − 𝜇Zn − 2𝜇 AgCl

(2.2.27)

Zn

Thus, we arrive at RT E = E0 − ln(aS 2+ )(aSCl− )2 (2.2.28) Zn 2F which is the Nernst equation for the cell. This corroboration of an earlier result displays the general utility of electrochemical potentials for treating interfacial reactions with charge transfer. They are powerful tools. They are easily used, for instance, to consider whether the two cells Cu∕Pt∕Fe2+ , Fe3+ , Cl− ∕AgCl∕Ag∕Cu′

(2.2.29)

Cu∕Au∕Fe2+ , Fe3+ , Cl− ∕AgCl∕Ag∕Cu′

(2.2.30)

would have the same cell potential. This point is left to the reader as Problem 2.8. 2.2.5

Fermi Energy and Absolute Potential

(a) The Absolute Scale

For most purposes in electrochemistry, it is sufficient to refer electrode potentials and half-cell emfs arbitrarily to the NHE, but it is sometimes helpful to have an estimate of the absolute potential or single electrode potential. The idea of the absolute scale is to define a redox couple at equilibrium in terms of a half-reaction in which the electrons are in vacuo, rather than being on a metal electrode. The most important case is 2H+ (aq) + 2evac ⇌ H2 (g)

(2.2.31)

which relates to the NHE. If we can determine the change in standard free energy for (2.2.31), 0a 0a 0a ΔGH + ∕H , then we also have the corresponding standard potential, EH+ ∕H = −ΔGH+ ∕H ∕nF, 2

2

2

which is “absolute” in the sense that it is defined from first principles, independent of any reference electrode. The reason for the independence is that the electric potential for electrons in vacuo, (evac ), is a defined value, 𝜙vac = 0. In the normal H+ /H2 half-reaction, 2H+ (aq) + 2eM ⇌ H2 (g)

(2.2.32)

the electrons are on an electrode having an unknowable electric potential, 𝜙M , which can be handled only by reference to a second electrode. A good deal of effort has been devoted to the absolute potential of the NHE (10, 31–36), which 0a has been estimated as EH + ∕H = 4.4 ± 0.1 V. This result is built on both theory and experimental 2

0a results, but is not reached with thermodynamic rigor. By accepting this estimate for EH + ∕H , 2

0a = E 0a one can place any other couple, O/R, on the absolute scale. Its position is EO∕R + H+ ∕H 0 (vs.NHE). Figure 2.1.2 illustrates the principle. EO∕R

2

2.2 A More Detailed View of Interfacial Potential Differences

(b) Equivalence of Fermi Level and Electrochemical Potential of Electrons

If we subtract (2.2.32) from (2.2.31), we have 2evac ⇌ 2eM

(2.2.33)

vac Per mole of electrons, the free energy change for this process is 𝜇M e − 𝜇 e ; however, by definiM tion, 𝜇vac e = 0. Per electron, the free energy change becomes 𝜇 e ∕NA , which must be the same

as the average energy level occupied in the metal upon addition and equilibration of an electron from vacuum (or the average energy level vacated by loss of an electron into vacuum). This energy of “transferable” electrons is known as the Fermi energy, EM .22 F The process that we used to reach this conclusion for the NHE is broadly applicable. For any phase 𝛼 at equilibrium, E𝛼F = 𝜇𝛼e ∕NA

(2.2.34)

The Fermi energy is an equivalent representation of the electrochemical potential of electrons in a phase. Consequently, −EM is the energy needed to remove a single electron from the metal F into vacuum, producing an equilibrated vacancy. Removal of an electron from Pt/H2 /H+ (a = 1) to vacuum requires about 4.4 eV, or 425 kJ/mol. (c) Relationship between Absolute Potential and Fermi Level

One can re-express the free energy change for (2.2.33) in terms of the absolute potential by 0a 0a 0 recognizing ΔGH + ∕H = −2FE H+ ∕H for (2.2.31) and ΔG = 0, by definition, for (2.2.32). 2

2

The difference between these two, −2FE0a , is the free energy change for (2.2.33). Division H+ ∕H 2

by 2N A gives the free energy change per electron as −eE0a , which is also the Fermi energy H+ ∕H 2

for the NHE. If both values are expressed in eV, EM for an NHE at equilibrium is numerically F 0a the same as −EH+ ∕H . In fact, this relationship applies to any half-cell at equilibrium, as one 2

can see in Figure 2.1.2.23 (d) Equilibrium in a Half-Cell

For an inert metal in contact with a solution, the condition for electrical (or electronic) equilibrium is that the Fermi energies of the two phases be equal: ESF = EM F

(2.2.35)

This condition is equivalent to saying that the electrochemical potentials of electrons in both phases are equal, or that the average energies of transferable electrons are the same in both phases (31). Unlike a metal, the solution phase has no “free electrons.” Electrons transferable to the electrode are localized on the reduced half of a redox couple (e.g., Fe2+ ). However, the fact that these electrons are bound to a specific chemical species does not preclude a rigorously defined Fermi energy for the solution (31). When a metal is brought into contact with a solution containing Fe3+ and Fe2+ , for example, the Fermi energies of metal and solution will not usually be equal. Equality is attained by the transfer of electrons between the phases, with electrons flowing from the phase with the higher 22 This energy is also frequently called the Fermi level. More exactly, this is the energy where the occupation probability is 0.5 in the Fermi–Dirac distribution of electrons among the various energy levels. Sections 3.5.5 and 20.1.4 cover EF in more detail. If one adds an electron to the phase, it will equilibrate to the Fermi energy. If one removes an electron, it will “come from” (i.e., its vacancy will equilibrate to) the Fermi energy. 23 The potential and the Fermi energy of an electrode have different signs because the potential is based on energy changes involving a positive test charge, while the Fermi energy refers to a negative electron.

89

90

2 Potentials and Thermodynamics of Cells

Fermi energy (higher 𝜇e or more energetic electrons) to the phase with the lower Fermi energy. This electron flow adjusts the interfacial charge and causes the potential difference between the phases (and the electrode potential) to shift. As equilibrium is established, the electrode potential stabilizes. The total amount of charge transferred in this process is normally quite small (Section 2.2.2) and does not significantly change the composition of the cell. Equilibrium in this example requires 𝜇S

Fe3+

S + 𝜇M e =𝜇

(2.2.36)

Fe2+

With rearrangement of (2.3.36) and expansion of the electrochemical potentials, we obtain aFe3+ 0S 0S 𝜇M − F𝜙S (2.2.37) e = 𝜇Fe2+ − 𝜇Fe3+ − RT ln a 2+ Fe showing that 𝜇M e (i.e., the Fermi energy of the metal) does indeed reflect the standard chemical potentials and activities of Fe3+ and Fe2+ . We can rewrite (2.2.37) in terms of the measurable electrode potential (E vs. a reference electrode) and the standard potential of the redox couple, E0 3+ 2+ . To do so, we employ three Fe

∕Fe

definitions previously established in this chapter: 0M M 𝜇M e = 𝜇e − F𝜙

(2.2.38)

E = 𝜙M − 𝜙ref

(2.2.39)

E0

Fe3+ ∕Fe2+

= −(𝜇0S2+ − 𝜇0S3+ )∕F Fe

Fe

(2.2.40)

Substitution of all three into (2.2.37), with subsequent rearrangement, yields, E = E0

Fe3+ ∕Fe2+

+

0M RT aFe3+ 𝜇e ln + − (𝜙ref − 𝜙S ) F aFe2+ F

(2.2.41)

where 𝜇e0M is the standard chemical potential of the electron in the metal, and 𝜙ref − 𝜙S is the electric potential difference between the reference electrode and the solution. Equation 2.2.41 3+ and Fe2+ . confirms that E, like 𝜇M e , depends on the activities of Fe In (2.2.38), each term has units of energy per mole. Division by the Avogadro constant, N A , allows us to express each in terms of energy per electron. In the process, we recognize (a) that M F/N A = e (where e is the electronic charge), (b) that 𝜙M = E + 𝜙ref , (c) that 𝜇 M e ∕NA = EF , 0M and (d) that 𝜇e ∕NA = −ΦM (where ΦM is the work function of the electrode material)(37).24 Thus, (2.2.38) is equivalent to EM = −ΦM − e(E + 𝜙ref ) F

(2.2.42)

Three important points can be drawn from this relationship: • The value of EM reflects both the electronic structure of the metal, as quantified by the work F function, and the state of charge on phases in the cell, including the metal, as given by the last term in (2.2.42). • The relationship between EM and any measurable E cannot be rigorously evaluated because F ref 𝜙 , the potential of a single electrode, is unknowable. 24 The work function is the energy required to remove one electron from an uncharged surface of a solid into vacuum. For typical metals, the work function is 2–6 eV. The work function includes a surface energy in the definition of ΦM , which we have neglected for this discussion, and which depends on the state of the surface, including crystallographic orientation, surface relaxation, adsorption, and solvent interactions (28, 34).

2.3 Liquid Junction Potentials

• As E is shifted toward more positive values, the electron energy in the metal, EM , decreases F by a corresponding magnitude (and vice versa). This difference in directionality of EM and E F is shown in Figure 2.1.2.

2.3 Liquid Junction Potentials So far in this chapter, we have examined only systems at equilibrium, and we have learned that the potential differences in equilibrated electrochemical systems can be treated exactly by thermodynamics. However, many real cells are never at equilibrium, because they feature different electrolytes around the working and reference electrodes. There is an interface between the two solutions, where mass transport processes work to mix the solutes. Unless the solutions are the same initially, this liquid junction will not be at equilibrium, because net flows of mass occur continuously across it. 2.3.1

Potential Differences at an Electrolyte–Electrolyte Boundary

Consider the cell 2+ − ′ Cu∕Zn∕Zn2+ (0.1 M), NO− 3 (0.2 M)∕Cu (0.1 M), NO3 (0.2 M)∕Cu 𝛼 𝛽

(2.3.1)

for which we can depict the equilibrium processes as in Figure 2.3.1. The overall cell potential at null current is E = (𝜙Cu − 𝜙𝛽 ) − (𝜙Cu − 𝜙𝛼 ) + (𝜙𝛽 − 𝜙𝛼 ) ′

(2.3.2)

The first two components of E are the expected interfacial potential differences at the copper and zinc electrodes. The third term shows that the measured cell potential depends also on the potential difference between the electrolytes, that is, on the liquid junction potential. This discovery threatens our system of electrode potentials, which is based on the idea that all contributions to E can be assigned to one electrode or to the other. The junction potential cannot be so assigned. We must evaluate the importance of the phenomenon. 2.3.2

Types of Liquid Junctions

The reality of junction potentials is easily understood by considering the boundary shown in Figure 2.3.2a. At the junction, there is a steep concentration gradient in H+ and Cl− ; hence, both ions tend to diffuse from right to left. However, H+ has a much larger mobility than chloride, so it initially penetrates the dilute phase at a higher rate. This process gives a positive charge to the dilute phase and a negative charge to the concentrated one, with the result that a boundary potential difference develops. The corresponding electric field then retards the movement of H+ and speeds up the passage of Cl− until the two cross the boundary at equal rates. Because of the Cu

α

Zn e

e

Zn2+

Zn2+

β Cu2+

Cu′ Cu2+

Figure 2.3.1 Phase boundaries in (2.3.1). Equilibrium is established for certain charge carriers as shown, but equilibrium is not reached at the liquid junction between 𝛼 and 𝛽.

91

92

2 Potentials and Thermodynamics of Cells

Type 1 0.01 M HCl

Type 2

0.1 M HCl

0.1 M HCl H+

Type 3

0.1 M KCl

H+

0.1 M HCl

0.05 M KNO–3

H+

Cl–

K+

Cl– K+ NO–3

–

+

–

(a)

+ (b)

–

+ ( c)

Figure 2.3.2 Types of liquid junctions: (a) Type 1, (b) Type 2, (c) Type 3. Vectors show the direction of net transfer for each ion, with lengths indicating relative mobilities. The polarity of the junction potential is shown in each case by the circled signs. [Adapted from Lingane (3), John Wiley & Sons.]

charge separation, a detectable potential difference develops, which is not due to an equilibrium process (3, 26, 38, 39). From its origin, this interfacial potential is sometimes called a diffusion potential. Lingane (3) classified liquid junctions into three types: • Two solutions of the same electrolyte at different concentrations, as in Figure 2.3.2a. • Two solutions at the same concentration with different electrolytes having an ion in common, as in Figure 2.3.2b. • Two solutions not satisfying conditions 1 or 2, as in Figure 2.3.2c. This classification will prove useful in the treatments of junction potentials given below. Even though the boundary region cannot be at equilibrium, it has a composition that is effectively constant over long periods of time, and the reversible transfer of charge through the region can be considered. 2.3.3

Conductance, Transference Numbers, and Mobility

When an electric current flows in an electrochemical cell, it is carried in solution by the movement of ions. For example, take the following cell: ⊖Pt∕H2 (1 bar)∕H+ , Cl− ∕H+ , Cl− ∕H2 (1 bar)∕Pt′ ⊕ 𝛼(a1 ) 𝛽(a2 )

(2.3.3)

where a2 > a1 When the cell operates galvanically (i.e., spontaneously), an oxidation occurs at the left electrode, .25

H2 → 2H+ (𝛼) + 2e(Pt)

(2.3.4)

and a reduction happens on the right, 2H+ (𝛽) + 2e(Pt′ ) → H2

(2.3.5)

There is a tendency to build up positive charge in the 𝛼 phase and negative charge in 𝛽. This tendency is overcome by the movement of ions: H+ to the right and Cl− to the left. For each mole of electrons passed, 1 mole of H+ is produced in 𝛼, and 1 mole of H+ is consumed in 𝛽. The total amount of H+ and Cl− migrating across the boundary between 𝛼 and 𝛽 must equal 1 mole. 25 A cell like (2.3.3), having electrodes of the same type on both sides, but with differing activities of one or both redox forms, is called a concentration cell.

2.3 Liquid Junction Potentials

The fractions of the current carried by H+ and Cl− are called their transference numbers.26 If we let t + be the transference number for H+ and t − be that for Cl− , then, t+ + t − = 1

(2.3.6)

In general, for an electrolyte containing many ions, j, each ion carries some of the current, so ∑ tj = 1 (2.3.7) j

Schematically, the process in the cell can be represented as shown in Figure 2.3.3. Initially, there is a higher activity of hydrochloric acid on the right (Figure 2.3.3a); hence, discharge spontaneously produces H+ on the left and consumes it on the right. Assume that five units of H+ are produced and consumed as shown in Figure 2.3.3b. For hydrochloric acid, t + ≈ 0.8 and t − ≈ 0.2; therefore, four units of H+ migrate to the right and one unit of Cl− to the left, to maintain electroneutrality. This process is depicted in Figure 2.3.3c, and the final state of the solution is represented in Figure 2.3.3d. A charge imbalance like that suggested in Figure 2.3.3b could not actually occur, because a very large electric field would be established, which would work to erase the imbalance. On a macroscopic scale, electroneutrality is steadily maintained throughout the solution. The migration represented in Figure 2.3.3c occurs simultaneously with the electron-transfer reactions. Transference numbers are determined by the details of ionic conduction, which have become understood mainly through measurements of the resistance to current flow in solution, or of its reciprocal, the conductance, G (39, 40). The value of G for a segment of solution immersed in an electric field is directly proportional to the cross-sectional area perpendicular to the field vector, A, and is inversely proportional to the length, l, of the segment along the field. The proportionality constant is the conductivity, 𝜅, which is an intrinsic property of the solution: G = 𝜅A∕l

(2.3.8)

is given in units of siemens (S = Ω−1 ), and 𝜅

is in S cm−1

or Ω−1

cm−1 .

where G Since the passage of current through the solution is accomplished by the independent movement of different species, 𝜅 is the sum of contributions from all ionic species, j. It is intuitive that the jth component of 𝜅 is proportional to the concentration of the ion, the magnitude of its charge |zj |, and some index of its migration velocity. (a)

Pt H2

+ + + + + + – – – – – – H2 Pt

+ + – –

5e (b)

– Pt H2

5e + + + + + + + + – – – – – – – –

(c)

Pt H2 + + + + + + + – –

(d)

Pt H2

+ + + – – –

+ – – – – – –

H2 Pt

+

H2 Pt

+ + + + + – – – – – H2 Pt

Figure 2.3.3 Redistribution of charge during electrolysis of a system featuring a high concentration of HCl on the right and a low concentration on the left (+ as H+ , − as Cl− ). (a) Initial state; (b) faradaic changes caused by passage of 5e at the electrodes; (c) required ionic movements; (d) final state. 26 Or transport numbers.

93

94

2 Potentials and Thermodynamics of Cells

Direction of movement

Drag force

Figure 2.3.4 Forces on a charged particle moving in solution under the influence of an electric field. The forces balance at the terminal velocity.

Electric force

That index is the mobility, uj , which is the limiting velocity of the ion in an electric field of unit strength. Mobility usually carries dimensions of cm2 V−1 s−1 (i.e., cm/s per V/cm). When a field of strength E is applied to an ion, it will accelerate under the force imposed by the field until the frictional drag exactly counterbalances the electric force. Then, the ion continues its motion at that terminal velocity. This balance is represented in Figure 2.3.4. The magnitude of the force exerted by the field is |zj |eE, where e is the electronic charge. The frictional drag can be approximated from the Stokes law as 6𝜋𝜂rv, where 𝜂 is the viscosity of the medium, r is the radius of the ion, and v is the velocity. When the terminal velocity is reached, we have by equation and rearrangement, |zj |e v = (2.3.9) E 6𝜋𝜂r The proportionality factor relating an individual ionic conductivity to charge, mobility, and concentration turns out to be the Faraday constant; thus, ∑ 𝜅=F |zj |uj Cj (2.3.10) uj =

j

The transference number for species j is the contribution to conductivity made by that species divided by the total conductivity: |zj |uj Cj tj = ∑ |zk |uk Ck

(2.3.11)

k

For solutions of simple, pure electrolytes (i.e., one positive and one negative ionic species), such as KCl, CaCl2 , and HNO3 , conductance is often quantified in terms of the equivalent conductivity, Λ, which is defined by Λ=

𝜅 Ceq

(2.3.12)

where C eq is the concentration of positive (or negative) charges (often called the equivalent concentration). Thus, Λ expresses the conductivity per unit concentration of charge. Since |z|C = C eq for either ionic species in these systems, one finds from (2.3.10) and (2.3.12) that Λ = F(u+ + u− )

(2.3.13)

where u+ refers to the cation and u− to the anion. This relation suggests that Λ could be regarded as the sum of individual equivalent ionic conductivities, Λ = 𝜆+ + 𝜆−

(2.3.14)

hence, we find 𝜆j = Fuj

(2.3.15)

2.3 Liquid Junction Potentials

In these simple solutions, then, the transference number t j is given by tj =

𝜆j

(2.3.16)

Λ or, alternatively, uj tj = u+ + u−

(2.3.17)

Transference numbers can be measured by several approaches (39, 40), and numerous data for pure solutions appear in the literature. Frequently, transference numbers are measured by noting concentration changes caused by electrolysis (Problem 2.11), as in the experiment shown in Figure 2.3.3. Table 2.3.1 displays a few values for aqueous solutions at 25 ∘ C. From results of this sort, one can evaluate the individual ionic conductivities, 𝜆j . Both 𝜆j and t j vary with the concentration of the pure electrolyte, because interactions between ions tend to alter the mobilities (39, 40, 42). Lists of 𝜆j values, like Table 2.3.2, usually give 𝜆0j , which are limits obtained by extrapolation to infinite dilution. In the absence of measured transference numbers, it is convenient to use values of 𝜆0j to estimate t j for pure solutions by (2.3.16), or for mixed electrolytes by the following equivalent to (2.3.11), |zj |Cj 𝜆j tj = ∑ |zk |Ck 𝜆k

(2.3.18)

k

Solid electrolytes also exist and are sometimes used in electrochemical cells. Examples include sodium β-alumina, the silver halides, and polymers like polyethylene oxide/LiClO4 (44, 45). In these materials, ions move under the influence of an electric field, even in the absence of solvent. For example, the conductivity of a single crystal of sodium β-alumina at room temperature is 0.035 S/cm, a value similar to that of aqueous solutions. Solid electrolytes are technologically important in the fabrication of batteries and other electrochemical devices. In some of these materials (e.g., 𝛼-Ag2 S and AgBr)—and unlike essentially all liquid electrolytes—there is electronic conductivity as well as ionic conductivity. The relative Table 2.3.1 Cation Transference Numbers for Aqueous Solutions at 25 ∘ C(a). Concentration (C eq )(b) Electrolyte

0.01

0.05

0.1

0.2

HCl

0.8251

0.8292

0.8314

0.8337

NaCl

0.3918

0.3876

0.3854

0.3821

KCl

0.4902

0.4899

0.4898

0.4894

NH4 Cl

0.4907

0.4905

0.4907

0.4911

KNO3

0.5084

0.5093

0.5103

0.5120

Na2 SO4

0.3848

0.3829

0.3828

0.3828

K2 SO4

0.4829

0.4870

0.4890

0.4910

(a) From MacInnes (41) and references cited therein. (b) Moles of positive (or negative) charge per liter.

95

96

2 Potentials and Thermodynamics of Cells

Table 2.3.2 Ionic Properties at Infinite Dilution in Aqueous Solutions at 25 ∘ C. Ion

𝝀0 (cm2 𝛀−1 equiv−1 )(a)

u (cm2 sec−1 V−1 )(b)

H+

349.82

3.625 × 10−3

K+

73.52

7.619 × 10−4

Na+

50.11

5.193 × 10−4

Li+

38.69

4.010 × 10−4

NH+ 4

73.4

7.61 × 10−4

(1/2)Ca2+

59.50

6.166 × 10−4

OH−

198

2.05 × 10−3

Cl−

76.34

7.912 × 10−4

Br−

78.4

8.13 × 10−4

I−

76.85

7.96 × 10−4

NO− 3 OAc−

71.44

7.404 × 10−4

40.9

4.24 × 10−4

ClO− 4

68.0

7.05 × 10−4

79.8

8.27 × 10−4

44.48

4.610 × 10−4

101.0

1.047 × 10−3

110.5

1.145 × 10−3

(1∕2)SO2− 4 HCO− 3

(1∕3)Fe(CN)3− 6 (1∕4)Fe(CN)4− 6

(a) From MacInnes (43). (b) Calculated from 𝜆0 .

contribution of electronic conduction through the solid electrolyte can be found by applying a potential that is too small to drive electrochemical reactions and noting the magnitude of the (nonfaradaic) current. Alternatively, an electrolysis can be carried out and the faradaic contribution determined separately (Problem 2.12). 2.3.4

Calculation of Liquid Junction Potentials

Imagine concentration cell (2.3.3) connected to a power supply as shown in Figure 2.3.5. The voltage from the supply opposes that from the cell, and one finds experimentally that it is possible to oppose the cell voltage exactly, so that no current flows through the galvanometer. If the magnitude of the opposing voltage is reduced very slightly, the cell

–

Figure 2.3.5 System for demonstrating reversible flow of charge through a cell with a liquid junction. Device G is a galvanometer, a high-sensitivity ammeter able to show the existence and direction of a current. One finds the electrical balance point by adjusting the potentiometer (Figure 1.2.4) so that G shows null current.

+

G

– Pt/H2/HCl(𝛼)/HCl(𝛽)/H2/Pt′

+

2.3 Liquid Junction Potentials

α

Figure 2.3.6 Reversible charge transfer through the liquid junction in Figure 2.3.5.

β

t+H+ (a1)

(a2) t–Cl–

operates spontaneously as described above, and electrons flow from Pt to Pt′ in the external circuit. The process occurring at the liquid junction is the passage of an equivalent negative charge from right to left. If the opposing voltage is increased from the null point, the entire process reverses, including charge transfer through the interface between the electrolytes. The fact that an infinitesimal change in the driving force can reverse the direction of charge passage implies that the electrochemical free energy change for the whole process is zero. These events can be divided into those involving the chemical transformations at the metal/solution interfaces, 1/2H 2

⇌ H+ (𝛼) + e(Pt)

(2.3.19)

H+ (𝛽) + e(Pt′ ) ⇌ 1/2H2

(2.3.20)

and that effecting charge transport at the liquid junction depicted in Figure 2.3.6, t+ H+ (𝛼) + t− Cl− (𝛽) ⇌ t+ H+ (𝛽) + t− Cl− (𝛼)

(2.3.21)

Since (2.3.19) and (2.3.20) are at strict equilibrium under the null-current condition, the electrochemical free energy change for each of them individually is zero. Of course, this is also true for their sum, H+ (𝛽) + e(Pt′ ) ⇌ H+ (𝛼) + e(Pt)

(2.3.22)

which describes the chemical change in the system. The sum of this equation and the charge transport relation, (2.3.21), describes the overall cell operation. Since we have just learned that the electrochemical free energy changes for both the overall process and (2.3.22) are zero, we must conclude that the electrochemical free energy change for (2.3.21) is also zero. In other words, charge transport across the junction occurs in such a way that the electrochemical free energy change vanishes, even though it is not a process at equilibrium. This important conclusion permits an approach to the calculation of junction potentials. Let us focus first on the net chemical reaction, (2.3.22). Since the electrochemical free energy change is zero, 𝛽

𝛼 Pt 𝜇 H+ + 𝜇Pt e = 𝜇 H+ + 𝜇 e ′

𝛽

FE = F(𝜙Pt − 𝜙Pt ) = 𝜇 H+ − 𝜇𝛼H+ ′

E=

RT a2 ln + (𝜙𝛽 − 𝜙𝛼 ) F a1

(2.3.23) (2.3.24) (2.3.25)

The first component of E in (2.3.25) is the Nernst relation for the reversible chemical change, and 𝜙𝛽 − 𝜙𝛼 is the liquid junction potential, Ej . In general, for a chemically reversible system under null current conditions, Ecell = ENernst + Ej The junction potential is an additive perturbation onto the nernstian response.

(2.3.26)

97

98

2 Potentials and Thermodynamics of Cells

To evaluate Ej , we consider (2.3.21), for which 𝛽

𝛽

t+ 𝜇𝛼H+ + t− 𝜇Cl− = t+ 𝜇H+ + t− 𝜇 𝛼Cl−

(2.3.27)

Thus, 𝛽

𝛽

t+ (𝜇𝛼H+ − 𝜇H+ ) + t− (𝜇Cl− − 𝜇𝛼Cl− ) = 0

(2.3.28)

𝛽 ⎡ ⎤ ⎡ ⎤ a − a𝛼 + 𝛽 − 𝜙𝛼 )⎥ = 0 t+ ⎢RT ln H + F(𝜙𝛼 − 𝜙𝛽 )⎥ + t− ⎢RT ln Cl − F(𝜙 𝛽 ⎢ ⎥ ⎢ ⎥ a𝛼Cl− aH + ⎣ ⎦ ⎣ ⎦

(2.3.29)

Activity coefficients for single ions cannot be measured with thermodynamic rigor (2, 38, 46, 47); hence, they are usually equated to a measurable mean ionic activity coefficient (Section 𝛽 𝛽 2.1.5). Under this procedure, a𝛼H+ = a𝛼Cl− = a1 and aH+ = aCl− = a2 . Since t + + t − = 1, we have Ej = (𝜙𝛽 − 𝜙𝛼 ) = (t+ − t− )

RT a1 ln F a2

(2.3.30)

for a Type 1 junction involving 1:1 electrolytes. Consider, for example, HCl solutions with a1 = 0.01 and a2 = 0.1. We can see from Table 2.3.1 that t + = 0.83 and t − = 0.17; hence, at 25 ∘ C ( ) 0.01 = −39 mV (2.3.31) Ej = (0.83 − 0.17)(59.1) log 0.1 For the total cell, a E = 59.1 log 2 + Ej = 59 − 39 = 20 mV (2.3.32) a1 thus, the junction potential is a substantial component of the measured cell potential. In the derivation above, we implicitly assumed that the transference numbers were constant throughout the system. This is a good approximation for junctions of Type 1; hence, (2.3.30) is not seriously compromised. For Type 2 and Type 3 systems, it clearly cannot be true. To consider these cases, one must imagine the junction region to be sectioned into an infinite number of volume elements having compositions that range smoothly from that of pure 𝛼 to that of pure 𝛽. Transporting charge across one of these elements involves every ionic species in the element, and t j /|zj | moles of species j must move for each mole of charge passed. Thus, the passage of positive charge from 𝛼 toward 𝛽 might be depicted as in Figure 2.3.7. One can see that the change in electrochemical free energy upon moving any species is (tj ∕zj )d𝜇 j (recall that zj is signed ); therefore, the overall differential electrochemical free energy is ∑ tj dG = d𝜇 (2.3.33) zj j j Integrating from the 𝛼 phase to the 𝛽 phase, we have 𝛽 ∑ 𝛽 tj dG = 0 = d𝜇 ∫𝛼 ∫𝛼 zj j j

(2.3.34)

If 𝜇j0 for the 𝛼 phase is the same as that for the 𝛽 phase (e.g., if both are aqueous solutions), ∑ j

𝛽

∫𝛼

⎛∑ ⎞ 𝛽 RT d ln aj + ⎜ tj ⎟ F d𝜙 = 0 ⎜ ⎟ ∫𝛼 zj ⎝ j ⎠ tj

(2.3.35)

2.3 Liquid Junction Potentials

tj /zj mole of each cation

–tj /zj mole of each anion

x _ μj

Location Electrochemical potential

x + dx _ _ μj + dμj

Figure 2.3.7 Transfer of net positive charge from left to right through an infinitesimal segment of a junction region. Each species must contribute tj moles of charge per mole of overall charge transported; hence, tj /|zj | moles of that species must migrate.

Since

∑ t j = 1, Ej = 𝜙𝛽 − 𝜙𝛼 = −

𝛽 t j RT ∑ d ln aj ∫ F j 𝛼 zj

(2.3.36)

which is the general expression for the junction potential. It is easy to see now that (2.3.30) is a special case for Type 1 junctions between 1:1 electrolytes having constant t j . Note that Ej for a Type 1 junction is a strong function of t + − t − and vanishes if t + = t − . Thus, the cell Ag∕AgCl∕KCl (0.1 M)∕KCl (0.01 M)∕AgCl∕Ag

(2.3.37) ∘ which has t + = 0.49 and t − = 0.51, shows Ej = − 1.2 mV at 25 C, a figure 30 times smaller than for the HCl-based cell discussed above. While Type 1 junctions can be treated with some rigor and are independent of the method of forming the junction, Type 2 and Type 3 junctions have potentials that depend on the technique of junction formation (e.g., static or flowing) and can be treated only in an approximate manner. Different approaches to junction formation apparently lead to different profiles of t j through the junction, which in turn lead to different integrals for (2.3.36). Approximate values for Ej can be obtained by assuming that • Molar concentrations of ions everywhere in the junction are numerically equivalent to activities.9 • The concentration of each ion follows a linear transition between the two phases. One can then integrate (2.3.36) to give the Henderson equation (26, 38): ∑ |zj |uj ∑ [Cj (𝛽) − Cj (𝛼)] |zj |uj Cj (𝛼) zj j j RT Ej = ∑ ln ∑ |zj |uj [Cj (𝛽) − Cj (𝛼)] F |zj |uj Cj (𝛽) j

j

where uj is the mobility of species j, and C j is its molar concentration (without units).

(2.3.38)

99

100

2 Potentials and Thermodynamics of Cells

For Type 2 junctions between 1:1 electrolytes, this equation collapses to the Lewis–Sargent relation: Ej = ±

RT Λ𝛽 ln F Λ𝛼

(2.3.39)

where the positive sign corresponds to a junction with a common cation in the two phases, and the negative sign applies to the case with a common anion. As an example, consider the cell Ag∕AgCl∕HCl (0.l M)∕KCl (0.1 M)∕AgCl∕Ag

(2.3.40) ∘ for which Ecell is essentially Ej . The measured value at 25 C is 28 ± 1 mV, depending on the technique of junction formation (38), while the estimated value from (2.3.39) and the data of Table 2.3.2 is 26.8 mV. 2.3.5

Minimizing Liquid Junction Potentials

In most electrochemical experiments, the junction potential is a bothersome factor, so attempts are often made to minimize it. Alternatively, one hopes that it is small or that it at least remains constant. A method for minimizing Ej is to replace the junction, for example, HCl (C1 )∕NaCl (C2 )

(2.3.41)

with a system featuring a concentrated solution in an intermediate salt bridge, where the solution in the bridge has ions of nearly equal mobility. Such a system is HCl (C1 )∕KCl (C)∕NaCl (C2 )

(2.3.42)

Table 2.3.3 lists some measured junction potentials for the cell, Hg∕Hg2 Cl2 ∕HCl (0.1 M)∕KCl (C)∕KCl (0.1 M)∕Hg2 Cl2 ∕Hg

(2.3.43)

As C increases, Ej falls markedly, because ionic transport at the two junctions is dominated more extensively by the massive amounts of KCl. The series junctions become more similar in magnitude and have opposite polarities; hence, they tend to cancel. Solutions used in aqueous salt bridges usually contain KCl (t + = 0.49, t − = 0.51) or, where Cl− is deleterious, KNO3 (t + = 0.51, t − = 0.49). Other concentrated solutions with equitransferent ions that have been suggested (52) for salt bridges include CsCl (t + = 0.5025), RbBr (t + = 0.4958), and NH4 I (t + = 0.4906). For many measurements, such as the determination of pH, it is sufficient if the junction potential remains constant between calibration (e.g., with a standard buffer or solution) and measurement. However, variations in Ej of 1–2 mV can be expected and should be considered in any interpretations made from potentiometric data. Table 2.3.3 Effect of a Salt Bridge on Measured Junction Potentials(a). KCl Concentration (C/M)

0.1

0.2

0.5

1.0

2.5

3.5

4.2 (sat’d)

Ej /mV

27

20

13

8.4

3.4

1.1

>> Na+ > K+ , Rb+ , Cs+ >> Ca2+ , (b) sodium-sensitive electrodes with the order Ag+ > H+ > Na+ >> K+ , Li+ >> Ca2+ , and (c) a more general cation-sensitive electrode with a narrower selectivity range in the order H+ > K+ > Na+ >NH+ , Li+ >> Ca2+ . 4 There is a large literature on the design, performance, and theory of glass electrodes (16, 26, 47, 59–68). The interested reader is referred to it for more advanced discussions. 2.4.3

Other Ion-Selective Electrodes

The principles that we have just reviewed also apply to other types of selective membranes (59–65, 68, 69). The most common types fall into two categories. (a) Solid-State Membranes

Like the glass membrane, which is a member of this group, the remaining common solidstate membranes are electrolytes having tendencies toward the preferential adsorption of certain ions.

2.4 Ion-Selective Electrodes

Consider, for example, the single-crystal LaF3 membrane, which is doped with EuF2 to create fluoride vacancies that allow ionic conduction by fluoride. Its surface selectively accommodates F− to the virtual exclusion of other species, except OH− . Other membranes are made from precipitates of insoluble salts, such as AgCl, AgBr, AgI, Ag2 S, CuS, CdS, and PbS, usually pressed into pellets. The silver salts all conduct by mobile Ag+ ions, but the heavy metal sulfides are usually mixed with Ag2 S, since they are not very conductive. The surfaces of these membranes are generally sensitive to the ions comprising the salts, as well as to other species that tend to form very insoluble precipitates with a constituent ion. For example, the Ag2 S membrane responds to Ag+ , S2− , and Hg2+ . Likewise, the AgCl membrane is sensitive to Ag+ , Cl− , Br− , I− , CN− , and OH− . (b) Plastic Membranes

An alternative approach involves a hydrophobic polymer membrane (usually called a plastic membrane in the commercial literature) as the sensing element. As shown in Figure 2.4.4, the membrane separates an aqueous internal filling solution from an aqueous test solution. A chelating agent with selectivity toward an ion of interest is dissolved in the polymer, and it provides the mechanism for selective charge transport across the boundaries of the membrane. A device based on these principles is a calcium-selective electrode. The polymer is typically poly(vinylchloride) (PVC), and the chelating agent might be the sodium salt of an alkyl phosNa+ , where R is an aliphatic chain having 8–18 carbons. The membrane phate ester, (RO)2 PO− 2 2+ is sensitive to Ca , as well as to interfering ions (Table 2.4.1). “Water hardness” electrodes are based on similar agents, but are designed to show virtually equal responses to Ca2+ and Mg2+ . ISEs featuring polymer membranes are also available for anions, including NO− , ClO− 4, 3 − and Cl . Nitrate and perchlorate can be sensed by membranes based on chelating agents like alkylated 1,10-phenanthroline complexes of Ni2+ or Fe2+ . All three ions are active at other membranes based on quaternary ammonium salts. Most polymer-membrane ISEs feature charged chelating agents, and their operation is based on ion-exchange equilibria. A different type of device involves an uncharged chelating agent, called a neutral carrier, which enables the transport of charge by selectively complexing certain ions. For example, potassium-selective electrodes can be constructed with the natural macrocycle valinomycin as a neutral carrier. This membrane has a much higher sensitivity to K + than Figure 2.4.4 A typical plastic-membrane ISE. [Courtesy of ThermoFisher Scientific.]

Electrical contact

Module housing Aqueous reference solution Reference element (Ag/AgCl)

Ion selective membrane

107

108

2 Potentials and Thermodynamics of Cells

Table 2.4.1 Typical Commercially Available Ion-Selective Electrodes. Species

Type(a)

Conc. Range (M)

pH Range

Interferences(b)

Ammonia (NH3 )

GS

1 to 5 × 10−7

>11

Volatile amines

Ammonium (NH+ ) 4 Bromide (Br− )

P

1 to 5 × 10−7

0–8

K+ , Na+ , Mg2+

S

1 to 2 × 10−6

0–14

Ag+ , S2− , CN− , I−

Cadmium (Cd2+ )

S

10−1 to 10−7

3–7

Ag+ , S2− , Cu2+ , Hg2+ , Pb2+ , Fe3+

Calcium (Ca2+ )

P

1 to 5 × 10−7

2.5–11

Al3+ , Pb2+ , Hg2+ , H+ , Sr2+ , Fe2+ , Mg2+

Carbon dioxide (CO2 )

GS

10−2 to 10−4

4.8–5.2

NO2 , NO− , SO2 , HSO− , 2 3 HOAc, HCOOH

Chloride (Cl− )

S

1 to 3 × 10−5

1–12

Ag+ , S2− , CN− , I− , Br−

to 10−8

2–12

Ag+ , S2− , Hg2+ , Br− , Cl−

to 10−6

10–14

Ag+ , S2− , I−

Cupric (Cu2+ )

S

10−1

Cyanide (CN− )

S

10−2

Fluoride (F− )

S

Sat’d to 10−6

4–8

OH−

G

1 to 10−14

0–14

Na+

Hydrogen (H+ ) (pH electrode)

(sodium-free) 1 to 10−12 (sodium base)

0–12

Iodide (I− )

S

1 to 5 × 10−8

0–12

Ag+ , S2− , CN−

Lead (Pb2+ )

S

10−1 to 10−6

3–7

Ag+ , S2− , Cu2+ , Fe2+ , Fe3+ Hg2+ , Cd2+

Nitrate (NO− ) 3

P

1 to 5 × 10−6

2–11

Nitrite (NO− ) 2

I− , Br− , Cl− , HCO− , NO− , 3 2 2− − − OAc , F , SO4

P

10−2 to 4 × 10−6

4–8

Perchlorate (ClO− ) 4

P

0.1 to 2 × 10−6

0–11

Potassium (K+ )

P

1 to 10−6

1–11

Rb+ , Cs+ , NH+ , H+ , Na+ , 4 2+ 2+ Ca , Mg

Silver (Ag+ )

CN− , OAc− , F− ,Cl− , NO− , 3 SO2− 4 SCN− , I− , NO− , Cl− , 3 3− − PO4 , OAc

S

1 to 10−7

2–12

S2− , Hg2+

Sodium (Na+ )

G

Sat’d to 10−6

6–12

Ag+ , Li+ , K+ , Tl+ , NH+ 4

Sodium (Na+ )

P

0.1 to 2 × 10−6

3–10

K+ , NH+ , Ca2+ , Mg2+ 4

Sulfide (S2− )

S

1 to 10−7

12–14

Ag+ , Hg2+

Tetrafluoroborate (BF− ) 4

P

1 to 7 × 10−6

2.5–11

ClO− , I− ,ClO− , CN− , Br− , 4 3 − , HCO − NO− , NO 2 3 3

Thiocyanate (SCN− )

S

0.1 to 2 × 10−5

2–12

Ag+ , S2− , Cl− , I− , Br−

Water hardness (Ca2+ , Mg2+ )

P

1 to 10−5

5–10

Cu2+ , Zn2+ , Ni2+ , Fe2+

(a) G = glass; GS = gas-sensing; P = plastic membrane; S = solid-state. Typical temperature ranges are 0–40 ∘ C for plastic-membrane and 0–80 ∘ C for solid-state electrodes. (b) Lists are illustrative, with interference generally weakening from left to right. Identified interferences vary from manufacturer to manufacturer.

2.4 Ion-Selective Electrodes

to Na+ , Li+ , Mg2+ , Ca2+ , or H+ ; but Rb+ and Cs+ are sensed to much the same degree as K+ . The selectivity rests on the recognition of the target ion in the complexing site of the carrier. (c) Commercial Devices

Table 2.4.1 is a listing of typical commercial ion-selective electrodes, together with the pH and concentration ranges over which they operate and typical interferences. Selectivity coefficients for many of these electrodes are available from reviews (68, 70–72) or the manufacturers’ literature. (d) Detection Limits

As shown in Table 2.4.1, the lower limit for detection of an ion with an ISE is commonly 10−6 to 10−7 M. This limit is often governed by the leaching of ions from the internal electrolyte into the zone of sample solution just outside the sensing surface (64, 73). The leakage can be prevented by using a lower concentration of the ion of interest in the internal electrolyte, so that the concentration gradient established in the membrane causes an ion flux from the sample to the inner electrolyte. This low concentration can be maintained with an ion buffer, that is, a mixture of the metal ion with an excess of a strong complexing agent. In addition, a high concentration of a second potential-determining ion is added to the internal solution. Under these conditions, the lower detection limit can be improved by orders of magnitude (64, 74). Commercial ISEs are generally constructed to operate with sample volumes of 10–250 mL, although special products are marketed for operation in smaller volumes and in flow streams. The principles of operation remain the same, so we do not cover details of special products here. Tiny, specially fabricated ISEs can be employed to determine local ion concentrations with high spatial resolution. This is an important practice in SECM and is addressed in Section 18.6. (e) Solid-Contact Ion-Selective Electrodes

All conventional ISEs feature ionic electrical contact on both sides of the selective membrane (with the test solution on the outer face, and with the electrolyte of the internal reference on the inner face). The two reference electrodes (internal and external) transduce the ionic membrane potential into an electronic potential difference measurable in the external circuit. There is high interest in an alternative concept featuring “solid,” rather than “liquid” contact (usually meaning electronic, rather than ionic, contact) between the inner electrode and the ion-selective membrane (64, 75, 76), as illustrated in Figure 2.4.5. Solid contact is an idea with some important advantages. The internal reference electrode and filling solution would not be needed, so the overall structure of the ISE would be simplified, and possibilities would be enhanced for miniaturization, as well as for use in mobile environments. Moreover, ionic leakage from the internal filling solution into the test solution would be avoided, opening the possibility of an extended working range vs. conventional ISEs. However, the conditions for success are formidable. To achieve practical stability, one must aspire to a system capable of responding essentially at equilibrium (as a conventional ISE does). Consequently, it must 1) Be able to equilibrate electrons between the inner electrode and the membrane. 2) Be able to equilibrate the target ion selectively between the test solution and at least an outer zone of the membrane. 3) Have enough electronic conductivity throughout the membrane to support electrical communication between the ion-exchange zone and the inner electrode. These requirements suggest a need for a biconductive membrane (i.e., both electronically and ionically conducting). Hundreds of papers have been written about experimental devices based on varying ideas about the structure and composition of the membrane.

109

2 Potentials and Thermodynamics of Cells

Solution level

Inner electrode (e.g., Pt, Au)

Electron exchange

Insulating sheath

External connector (e.g., Cu, alloy)

Insulating sheath

110

Solution level

Biconductive membrane

Selective ion exchange (e.g., K+) Test solution (e.g., a clinical sample containing K+)

Figure 2.4.5 Concept of a solid-contact ISE. An inert inner electrode interfaces directly with the ion-selective membrane. This type of ISE would be paired with an external reference electrode to make a complete cell, just as in the case of a conventional ISE. However, there is no internal reference electrode within the ISE assembly.

Successes have been achieved with membranes based on electronically conducting polymers [e.g., poly(3,4-ethylenedioxythiophene), polypyrrole, poly(3-octylthiophene), and polyaniline].31 While these materials fulfill conditions 1 and 3 in the list above, they generally do not, by themselves, provide a basis for selective ion exchange with the test solution (condition 2). The ionic selectivity must be added through careful design. One might, for example, choose a charge-compensating ion for the conducting polymer that is also a neutral carrier for the ion of interest. Alternatively, one might modify the electronically conducting membrane to incorporate a lipophilic ion exchanger in a thin zone at the outer surface. Or one might even establish a distinct ion-exchanging polymer overlayer on the conducting polymer, much like a plastic membrane found in devices that we discussed above. In this last strategy, stability requires good ionic communication with the conducting polymer. That can sometimes be achieved using the ions employed for charge compensation in the conducting polymer. The original solid-contact design based on a conducting polymer was built on the last strategy (77). 31 Such materials are very different from the polymers that one normally encounters, which are either electrically insulating [e.g., poly(ethylene), poly(styrene) or TeflonTM ] or purely ionically conducting [e.g., poly(styrenesulfonate) or NafionTM ]. Electronic conductivity can be engendered by partially oxidizing or partially reducing a polymer having electronically communicating (usually aromatic) monomer units (Section 20.2.3). Partial oxidation or partial reduction means that only a fraction of the monomer units are oxidized or reduced. Charge-compensating ions must enter the material during the oxidation or reduction process, which normally occurs electrochemically (often coincidentally with polymerization).

2.4 Ion-Selective Electrodes

Functional solid-contact membranes have also been created from carbon structures, including nanotubes and graphene, combined with ion-exchanging surfactants. Successful systems have provided good stability, as well as working ranges comparable with those of conventional electrodes; however, no solid-contact ISE has yet become commercially available. 2.4.4

Gas-Sensing ISEs

Figure 2.4.6 depicts the structure of a typical potentiometric gas-sensing ISE (78). In general, such a device involves a glass pH electrode that is protected from the test solution by a polymer diaphragm. Between the glass membrane and the diaphragm is a small volume of electrolyte. Small molecules, such as SO2 , NH3 , and CO2 , can penetrate the membrane and interact with the trapped electrolyte by reactions that produce changes in pH. The glass electrode responds to the alterations in acidity. Electrochemical cells that use a solid electrolyte composed of zirconium dioxide containing Y2 O3 (yttria-stabilized zirconia) are available to measure the oxygen content of gases at high temperature. Sensors of this type are widely used to monitor the exhaust gas from motor vehicles, so that the air-to-fuel mixture can be controlled to minimize the emission of pollutants such as CO and NOx . This solid electrolyte shows good conductivity only at high temperatures (500–1,000 ∘ C), where the conduction process is based on mobile oxide ions. A typical sensor is composed of a tube of zirconia with Pt electrodes deposited on the inside and outside of the tube. The outside electrode contacts air with a known partial pressure of oxygen, Pa , and serves as the reference electrode. The inside of the tube is exposed to the hot exhaust gas with a lower oxygen partial pressure, Peg . The cell configuration can, therefore, be written Pt∕O2 (air, Pa )∕ZrO2 + Y2 O3 ∕O2 (exhaust gas, Peg )∕Pt

(2.4.21)

The potential of this oxygen concentration cell can be used to measure Peg (Problem 2.19). The widely employed Clark oxygen electrode differs fundamentally from these devices (17, 79). The Clark device is similar in construction to the apparatus of Figure 2.4.6, in that a polymer membrane traps an electrolyte against a sensing surface. However, the sensor is a platinum Figure 2.4.6 Structure of a gas-sensing electrode. [Courtesy of ThermoFisher Scientific.]

Outer body

Reference element

Inner body

Internal filling solution

O-ring

Spacer

Bottom cap Sensing element

Membrane

111

112

2 Potentials and Thermodynamics of Cells

electrode, and the analytical signal is the steady-state current due to the faradaic reduction of molecular oxygen. Amperometric sensors are discussed in Section 17.8.3.

2.5 Lab Note: Practical Use of Reference Electrodes Section 2.1.8 introduces the important attributes of reference electrodes and the most practical systems. This note is devoted to two matters relevant to daily work. 2.5.1

Leakage at the Reference Tip

Ionic conduction is required at the reference tip for a functioning electrode; therefore, one must expect leakage of both ions and solvent through the tip in both directions. The important thing is that any leakage remain inconsequential—not affecting the electrode process at the working electrode, the integrity of the working solution, or the properties of the reference electrode. Often, this is the reality. However, one must be wary of complications. For example, Ag/AgCl reference electrodes leak small amounts of water into nonaqueous electrolytes, and that contamination can degrade the 2− test solution unacceptably. A more insidious problem comes from traces of AgCl− 2 and AgCl3 leaked into aqueous working solutions. These species can make their way to the working electrode, where they can be electrodeposited, contaminating the surface. In experiments sensitive to the surface structure and surface composition of the working electrode, this effect can be serious. A double junction can be helpful toward preventing leakage. For example, in a cell with a Pt working electrode and an Ag/AgCl reference electrode, this arrangement would take the form, |←double junction→| Ag∕AgCl∕KCl(sat d)∕test solution∕test solution∕Pt ′

(2.5.1)

where the “test solution” is the solution of interest at the working electrode. The double junction consists of the reference tip itself, a captured sample of test solution, and a second ionically conducting barrier, such as a frit. It can be achieved simply by slipping a piece of tubing, plugged at one end by a porous frit and filled with test solution, over the reference electrode tip. With a double junction, leakage from the reference electrode occurs initially into the trapped test solution. Eventually, the contamination might make its way through the frit into the compartment with the working electrode, but this takes time—ideally, enough time for the experiments of interest to be accomplished without serious interference. Leakage in the other direction, from the working solution into the reference electrode, can cause precipitation within the tip. This process can gradually reduce the quality of ionic contact, sometimes to the point where the tip can no longer fulfill its electrical role. The reference electrode may begin to drift and become unreliable, or the high impedance at a clogged tip may introduce unacceptable noise. Such behavior often marks the end of usable life for the reference electrode. 2.5.2

Quasireference Electrodes

For nonaqueous media, many workers use QREs,32 because aqueous reference electrodes tend to degrade a nonaqueous working solution, and because nonaqueous reference electrodes 32 Quasi implies that it is “almost” or “seemingly” a reference electrode.

2.6 References

[Section 2.1.8(e)] are inconvenient, especially with volatile solvents. A QRE is often just a metal wire, Ag or Pt, used in the expectation that, in experiments where there is essentially no change in the bulk composition of the test solution, the potential of this wire, although unknown, will not change appreciably during a series of measurements. One might ask how a QRE can accept the current involved in a measurement of potential [Attribute 5 in Section 2.1.8(b)], since there is no electrochemical equilibrium capable of charge delivery. The answer is that the QRE is stabilized by its own double-layer capacitance, which can deliver or receive the tiny amounts of charge required by contemporary instrumentation for measurements of potential in three-electrode cells. But there are limits to the stability. One can expect a QRE to drift over the course of a long series of measurements as the charge on its double layer changes. Attempts have been made to develop poised QREs. An alternative (80) is to employ a reference electrode in which Fc and Fc+ are immobilized at a known concentration ratio in a polymer layer on a conducting surface. Quite favorable performance is reported (81) for a QRE in which polypyrrole is electrodeposited on Pt or stainless steel and then left in a partially oxidized state. Silver wires have been oxidized in aqueous base, covering them with AgO, in the hope that they maintain better stability through a poising effect. The actual potential of a QRE vs. a true reference electrode must be calibrated before reporting potentials; uncalibrated data are meaningless. Typically, calibration is achieved vs. the Ag/AgCl or SCE simply by measuring vs. the QRE (e.g., by SSV or CV) the standard or formal potential of a couple whose potential is already known vs. Ag/AgCl or SCE under the same conditions. There are two approaches: 1) One might add an “internal reference redox system” to the solution to be investigated (82). Ferrocene/ferrocenium (Fc/Fc+ ) is commonly used, since it has reversible behavior in many solvents (83). One just adds Fc to the test solution and measures the position of its reversible voltammetric response vs. the QRE. Of course, it is important that the internal reference not interfere with the electrochemistry of primary interest. 2) Many investigations are based on systems that can furnish the internal reference without addition of any other substance. If the CV remains well behaved and involves a couple whose standard potential vs. Ag/AgCl or SCE is already well known, one can calibrate the QRE directly from the CV. A good example is found in Figure 7.5.2, where there are CV waves for two couples, both with well-known formal potentials vs. good reference electrodes. A QRE is not suitable in experiments where changes in the composition of the bulk solution can cause concomitant variations in the potential of the QRE.

2.6 References 1 The arguments presented here follow those given earlier by (a) D. A. MacInnes, “The Prin-

ciples of Electrochemistry,” Dover, New York, 1961, pp. 110–113 and by (b) J. J. Lingane, “Electroanalytical Chemistry” 2nd ed., Wiley–Interscience, New York, 1958, pp. 40–45; (c) Experiments like those described here were actually carried out by H. Jahn, Z. Physik. Chem., 18, 399 (1895). 2 I. M. Klotz and R. M. Rosenberg, “Chemical Thermodynamics,” 7th ed., Wiley, Hoboken, NJ, 2008. 3 J. J. Lingane, “Electroanalytical Chemistry,” 2nd ed., Wiley–Interscience, New York, 1958, Chap. 3. 4 F. C. Anson, J. Chem. Educ., 36, 394 (1959).

113

114

2 Potentials and Thermodynamics of Cells

5 A. J. Bard, R. Parsons, and J. Jordan, Eds., “Standard Potentials in Aqueous Solutions,”

Marcel Dekker, New York, 1985. 6 National Institute of Standards and Technology, U. S. Department of Commerce, “NIST

Chemistry WebBook, SRD 69,” http://webbook.nist.gov/ (accessed 30 August 2021). 7 A. J. Bard and H. Lund, Eds., “Encyclopedia of Electrochemistry of the Elements” (16 vols),

Marcel Dekker, New York, 1973–1986. 8 M. W. Chase, Jr., “NIST–JANAF Thermochemical Tables,” 4th ed., American Chemical Soci-

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

ety, Washington, and American Institute of Physics, New York, for the National Institute of Standards and Technology, 1998. L. R. Faulkner, J. Chem. Educ., 60, 262 (1983). R. Parsons in A. J. Bard, R. Parsons, and J. Jordan, Eds., op.cit., Chap. 1. R. A. Alberty and F. Daniels, “Physical Chemistry,” 5th ed., Wiley, New York, 1979, pp 175–179. I. M. Klotz, “Chemical Thermodynamics,” Benjamin, New York, 1964, p. 417. W. J. Moore, “Physical Chemistry,” 3rd ed., Prentice–Hall, Englewood Cliffs, NJ, 1964, p. 351. R. Parsons, op. cit., p. 5. D. J. G. Ives and G. J. Janz, Eds., “Reference Electrodes,” Academic, New York, 1961. H. Galster, “pH Measurement: Fundamentals, Methods, Applications, Instrumentation,” VCH, Wenheim, 1991 (English translation of “pH Messung,” VCH, Wenheim, 1990). D. T. Sawyer, A. Sobkowiak, and J. L. Roberts, Jr., “Electrochemistry for Chemists,” 2nd ed., Wiley, New York, 1995. H. Kahlert in “Electroanalytical Chemistry,” F. Scholz, Ed., 2nd ed., Springer-Verlag, Berlin, Heidelberg, 2010, Chapter III.2. T. J. Smith and K. J. Stevenson in “Handbook of Electrochemistry,” C. G. Zoski, Ed., Elsevier, Amsterdam, 2007, Chap. 4. P. Longhi, T. Mussini, R. Orsenigo, and S. Rondinini, J. Appl. Electrochem. 17, 505 (1987). R. A. Nickell, W. H. Zhu, R. U. Payne, D. R. Cahela, and B. J. Tatarchuk, J. Power Sources, 161, 1217 (2006). M. Pourbaix, “Atlas of Electrochemical Equilibria in Aqueous Solutions,” 2nd English ed., J. A. Franklin, transl., National Association of Corrosion Engineers, Houston, 1974. J. Walker, D. Halliday and R. Resnick, “Fundamentals of Physics,” 10th ed., Wiley, Hoboken, NJ, 2014, Chap. 24. Ibid., Chap. 23. J. O’M. Bockris and A. K. N. Reddy, “Modern Electrochemistry,” Vol. 2, Plenum, New York, 1970, Chap. 7. K. J. Vetter, “Electrochemical Kinetics,” Academic, New York, 1967. B. E. Conway, “Theory and Principles of Electrode Processes,” Ronald, New York, 1965, Chap. 13. R. Parsons, Mod. Asp. Electrochem., 1, 103 (1954). J. A. V. Butler, Proc. Roy. Soc., London, 112A, 129 (1926). E. A. Guggenheim, J. Phys. Chem., 33, 842 (1929); 34, 1540 (1930). H. Reiss, J. Phys. Chem, 89, 3783 (1985). H. Reiss and A. Heller, J. Phys. Chem., 89, 4207 (1985). S. Trasatti, Pure Appl. Chem., 58, 955 (1986). A. A. Isse and A. Gennaro, J. Phys. Chem. B, 114, 7894 (2010). W. A. Donald and E. R. Williams, Electroanal. Chem., 25, 1, 2013. J. Ho, M. L. Coote, C. J. Cramer, and D. G. Truhlar in “Organic Electrochemistry,” 5th ed., O. Hammerich and B. Speiser, Eds., CRC Press, Boca Raton, FL, 2016, Chap. 4.

2.6 References

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

J. Janata and M. Josowicz, Anal. Chem., 69, 293A (1997). D. A. MacInnes, “The Principles of Electrochemistry,” Dover, New York, 1961, Chap. 13. J. O’M. Bockris and A. K. N. Reddy, op. cit., Vol. 1, Chap. 4. D. A. MacInnes, op. cit., Chap. 4. Ibid., p. 85. Ibid., Chap. 18. Ibid., p. 342. D. O. Raleigh, Electroanal. Chem., 6, 87 (1973). G. Holzäpfel, “Solid State Electrochemistry” in “Encyclopedia of Physical Science and Technology,” R. A. Meyers, Ed., Academic, New York, 1992, Vol. 15, p. 471. J. O’M. Bockris and A. K. N. Reddy, op. cit., Vol. 1, Chap. 3. R. G. Bates, “Determination of pH,” 2nd ed., Wiley–Interscience, New York, 1973. J. J. Lingane, “Electroanalytical Chemistry,” Wiley–Interscience, New York, 1958, p. 65. H. A. Fales and W. C. Vosburgh, J. Am. Chem. Soc., 40, 1291 (1918). E. A. Guggenheim, J. Am. Chem. Soc., 52, 1315 (1930). A. L. Ferguson, K. Van Lente, and R. Hitchens, J. Am. Chem. Soc., 54, 1285 (1932). P. R. Mussini, S. Rondinini, A. Cipolli, R. Manenti and M. Mauretti, Ber. Bunsenges. Phys. Chem., 97, 1034 (1993). P. Vanýsek, “Electrochemistry on Liquid/Liquid Interfaces,” Springer, Berlin, 1985. H. H. J. Girault and D. J. Schiffrin, Electroanal. Chem., 15, 1 (1989). H. H. J. Girault, Mod. Asp. Electrochem., 25, 1 (1993). A. G. Volkov, D. W. Deamer, D. L. Tanelian, and V. S Markin, “Liquid Interfaces in Chemistry and Biology,” Wiley–Interscience, New York, 1998. R. A. Iglesias and S. A. Dassie, “Ion Transfer at Liquid/Liquid Interfaces,” Nova Science Publishers, New York, 2010. E. Grunwald, G. Baughman, and G. Kohnstam, J. Am. Chem. Soc., 82, 5801 (1960). H. Freiser, “Ion-Selective Electrodes in Analytical Chemistry,” Plenum, New York, Vol. 1, 1979; Vol. 2, 1980. J. Koryta and K. Štulík, “Ion-Selective Electrodes,” 2nd ed., Cambridge University Press, Cambridge, 1983. A. Evans, “Potentiometry and Ion Selective Electrodes,” Wiley, New York, 1987. D. Ammann, “Ion-Selective Microelectrodes: Principles, Design, and Application,” Springer, Berlin, 1986. E. Lindner, K. Toth, and E. Pungor, “Dynamic Characteristics of Ion-Sensitive Electrodes,” CRC, Boca Raton, FL, 1988. E. Bakker and E. Pretsch, Electroanal. Chem., 24, 1 (2011). K. N. Mikhelson, “Ion-Selective Electrodes,” Springer-Verlag, Berlin, Heidelberg, 2013 M. Dole, “The Glass Electrode,” Wiley, New York, 1941. G. Eisenman, Ed., “Glass Electrodes for Hydrogen and Other Cations,” Marcel Dekker, New York, 1967. Y. Umezawa, Ed., “CRC Handbook of Ion-Selective Electrodes,” CRC, Boca Raton, FL 1990. R. P. Buck and E. Lindner, Accts. Chem. Res., 31, 257 (1998). Y. Umezawa, P. Buhlmann, K. Umezawa, K. Tohda, S. Amemiya, Pure Appl. Chem., 72, 1851 (2000). Y. Umezawa, K. Umezawa, P. Buhlmann, N. Hamada, H. Aoki, J. Nakanishi, M. Sato, K. P. Xiao, Y. Nishimura, Pure Appl. Chem., 74, 923 (2002). Y. Umezawa, P. Buhlmann, K. Umezawa, N. Hamada, Pure Appl. Chem., 74, 995 (2002). S. Mathison and E. Bakker, Anal. Chem., 70, 303 (1998). T. Sokalski, A. Ceresa, T. Zwicki, and E. Pretsch, J. Am. Chem. Soc., 119, 11347 (1997).

115

116

2 Potentials and Thermodynamics of Cells

75 J. Bobacka, A. Ivaska, and A. Lewenstam, Chem. Rev., 108, 329 (2008). 76 E. Zdrachek and E. Bakker, Anal. Chem., 91, 2 (2019). 77 A. Cadogan, Z. Gao, A. Lewenstam, A. Ivaska, and D. Diamond, Anal. Chem., 64, 2496

(1992). 78 J. W. Ross, J. H. Riseman, and J. A. Krueger, Pure Appl. Chem., 36, 473 (1973). 79 L. C. Clark, Jr., Trans. Am. Soc. Artif. Intern. Organs, 2, 41 (1956). 80 (a) P. Peerce and A. J. Bard, J. Electroanal. Chem., 108, 121 (1980); (b) R. M. Kannuck, J. M.

Bellama, E. A Blubaugh, and R. A. Durst, Anal. Chem., 59, 1473 (1987). 81 J. Ghilane, P. Hapiot, and A. J. Bard, Anal. Chem., 78, 6868 (2006). 82 A. A. J. Torriero, S. W. Feldberg, J. Zhang, A. N. Simonov, and A. M. Bond, J. Solid State

Electrochem., 17, 3021 (2013). 83 G. Gritzner and J. Kuta, Pure Appl. Chem., 56, 461 (1984).

2.7 Problems 2.1

Devise electrochemical cells in which the following reactions could be made to occur. If liquid junctions are necessary, note them in the cell schematic appropriately, but neglect their effects. (a) H2 O ⇌ H+ + OH− (b) 2H2 + O2 ⇌ 2H2 O (c) 2PbSO4 + 2H2 O ⇌ PbO2 + Pb + 4H+ + 2SO2− 4 + (d) An−∙ + TMPD ∙ ⇄ An + TMPD (in acetonitrile), where An and An−∙ are anthracene + and its anion radical, and TMPD and TMPD ∙ are N,N,N ′ ,N ′ -tetramethyl-p-phenylenediamine and its cation radical (structures in Figure 1). Use anthracene potentials for DMF solutions given in Appendix C.3. (e) 2Ce3+ + 2H+ + BQ ⇌ 2Ce4+ + HQ (aqueous), where BQ is p-benzoquinone and HQ is p-hydroquinone (Figure 1). (f ) Ag+ + I− ⇌ AgI (aqueous) (g) Fe3+ + Fe(CN)4− ⇌ Fe2+ + Fe(CN)3− (aqueous) 6 6 (h) Cu2+ + Pb ⇌ Pb2+ + Cu (aqueous) (i) An−∙ + BQ ⇌ BQ−∙ + An (in DMF), where BQ, An, and An−∙ are defined above and BQ−∙ is the anion radical of p-benzoquinone. Use BQ potentials in acetonitrile given in Appendix C.3. Which half-reactions take place at the electrodes in each cell? What is the standard cell potential in each case? Which electrode is negative? Would the cell operate electrolytically or galvanically in carrying out a net reaction from left to right? Be sure your decisions accord with chemical intuition.

2.2

Several hydrocarbons and carbon monoxide have been studied as possible fuels for use in fuel cells. From thermodynamic data in references [5–8, 15], derive E0 for each of the following reactions at 25 ∘ C: (a) CO(g) + H2 O(l) → CO2 (g) + 2H+ + 2e (b) CH4 (g) + 2H2 O(l) → CO2 (g) + 8H+ + 8e (c) C2 H6 (g) + 4H2 O(l) → 2CO2 (g) + 14H+ + 14e (d) C2 H2 (g) + 4H2 O(l) → 2CO2 (g) + 10H+ + 10e Even though a reversible emf could not be established (Why not?), which half-cell would ideally yield the highest cell voltage when coupled with the standard oxygen half-cell in

2.7 Problems

acid solution? Which of the fuels above would yield the highest net work per mole of fuel oxidized? Which would give the most net work per gram? Which would release the least CO2 per unit of net work? 2.3

Devise a cell in which the following is the overall cell reaction (T = 298 K): 2Na+ + 2Cl− → 2Na(Hg) + Cl2 (aqueous) where Na(Hg) symbolizes the amalgam. Is the reaction spontaneous or not? What is the standard free energy change? Take the standard free energy of formation of Na(Hg) as −85 kJ/mol. From a thermodynamic standpoint, another reaction should occur more readily at the cathode of your cell. What is it? It is observed that the reaction written above takes place with good current efficiency. Why? Could your cell have a commercial value?

2.4

What are the cell reactions and their emfs in the following systems? Are the reactions spontaneous? Assume that all systems are aqueous. (a) Ag/AgCl/K+ , Cl− (1 M)/Hg2 Cl2 /Hg (b) Pt/Fe3+ (0.01 M), Fe2+ (0.1 M), HCl (1 M)//Cu2+ (0.1 M), HCl (1 M)/Cu (c) Pt/H2 (1 bar)/H+ , Cl− (0.1 M)//H+ , Cl− (0.1 M)/O2 (0.2 bar)/Pt (d) Pt/H2 (1 bar)/Na+ , OH− (0.1 M)//Na+ , OH− (0.1 M)/O2 (0.2 bar)/Pt (e) Ag/AgCl/K+ , Cl− (1 M)//K+ , Cl− (0.1 M)/AgCl/Ag (f ) Pt∕Ce3+ (0.01 M), Ce4+ (0.1 M), H2 SO4 (1 M)∕∕ Fe2+ (0.01 M), Fe3+ (0.1 M), HCl (1 M)∕Pt

2.5

Consider the cell in part (f ) of Problem 2.4. What would the composition of the system be at the end of a galvanic discharge to an equilibrium condition? What would the cell potential be? What would the potential of each electrode be vs. NHE? vs. SCE? Assume equal volumes on both sides.

2.6

Devise a cell for evaluating the solubility product of PbSO4 . Calculate the solubility product from the appropriate E0 values (T = 298 K).

2.7

Obtain the dissociation constant of water from the parameters of the cell constructed for reaction (a) in Problem 2.1 (T = 298 K).

2.8

Consider the cell: Cu∕M∕Fe3+ , Fe2+ , H+ ∕∕Cl− ∕AgCl∕Ag∕Cu′ Would the cell potential be independent of the identity of M (e.g., graphite, gold, platinum) as long as M is chemically inert? Use electrochemical potentials to prove your point.

2.9

Given the half-cell of the standard hydrogen electrode, Pt∕H2 (a = 1)∕H+ (a = 1) (soln) H2 ⇌ 2H+ (soln) + 2e(Pt) Show that although the emf of the cell half-reaction is taken as zero, the potential difference between the platinum and the solution, that is, 𝜙Pt − 𝜙S , is not zero.

117

118

2 Potentials and Thermodynamics of Cells

2.10 Devise a thermodynamically sound basis for obtaining the standard potentials for new half-reactions by taking linear combinations of other half-reactions. As two examples, calculate E0 values for CuI + e ⇌ Cu + I− (a) . O2 + 2H+ + 2e ⇌ H2 O2

(b) .

given the following half-reactions and values for E0 vs. NHE at T = 298 K: Cu2+ + 2e ⇌ Cu

0.340 V

Cu2+

0.86 V

+ I−

+ e ⇌ CuI

O2 + 4H+ + 4e ⇌ 2H2 O

1.229 V

+

1.763 V

H2 O2 + 2H + 2e ⇌ 2H2 O

2.11 Transference numbers are sometimes measured by the Hittorf method, illustrated in this problem. Consider the three-compartment cell: L

C

R

⊖Ag∕AgNO3 (0.100 M)∕∕AgNO3 (0.100 M)∕∕AgNO3 (0.100 M)∕Ag⊕ where the double slashes (//) signify sintered glass disks that divide the compartments and prevent mixing, but not ionic movement. The volume of AgNO3 solution in each compartment (L, C, R) is 25.00 mL. An external power supply is connected to the cell with the polarity shown, and current is applied until 96.5 C have passed, causing Ag to deposit on the left Ag electrode and Ag to dissolve from the right Ag electrode. (a) How many grams of Ag have deposited on the left electrode? How many mmol of Ag have deposited? (b) If the transference number for Ag+ were 1.00 (i.e., tAg+ = 1.00, tNO− = 0.00), what 3

would the concentrations of Ag+ be in the three compartments after electrolysis? (c) Suppose the transference number for Ag+ were 0.00 (i.e., tAg+ = 0.00, tNO− = 1.00), 3

what would the concentrations of Ag+ be in the three compartments after electrolysis? (d) In an actual experiment like this, it is found experimentally that the concentration of Ag+ in the anode compartment R has increased to 0.121 M. Calculate tAg+ and tNO− . 3

2.12 Suppose one wants to determine the contribution of electronic (as opposed to ionic) conduction through doped AgBr, a solid electrolyte. A cell is prepared with a film of AgBr between two Ag electrodes, each of mass 1.00 g, that is, ⊖Ag/AgBr/Ag⊕. After passage of 200 mA through the cell for 10.0 min, the cell was disassembled, and the cathode was found to have a mass of 1.12 g. If Ag deposition is the only faradaic process that occurs at the cathode, what fraction of the current through the cell represents electronic conduction in AgBr? 2.13 Calculate the individual junction potentials at T = 298 K on either side of the salt bridge in (2.3.43) for the first two concentrations in Table 2.3.3. What is the sum of the two potentials in each case? How does it compare with the corresponding entry in the table? 2.14 Estimate the junction potentials for the following situations (T = 298 K): (a) HCl (0.1 M)/NaCl (0.1 M) (b) HCl (0.1 M)/NaCl (0.01 M)

2.7 Problems

(c) KNO3 (0.01 M)/NaOH (0.1 M) (d) NaNO3 (0.1 M)/NaOH (0.1 M) 2.15

One often finds pH meters with direct readout to 0.001 pH unit. Comment on the accuracy of these readings in making comparisons of pH from test solution to test solution. Comment on their meaning in measurements of small changes in pH in a single solution (e.g., during a titration).

2.16

pot are typical for interferents j at a sodium-selective glass Na+ ,j + + −5 electrode: K , 0.001; NH4 , 10 ; Ag+ , 300; H+ , 100. Calculate the concentrations of each interferent that would cause a 10% error when the concentration of Na+ is estimated to be 10−3 M from a potentiometric measurement. Assume that activity coefficients are

The following values of k

unity. 2.17

Would Na2 H2 EDTA be a good ion-exchanger for a polymer membrane electrode? How about Na2 H2 EDTA - R, where R designates a C20 alkyl substituent? Why or why not?

2.18

Comment on the feasibility of developing selective electrodes for the direct potentiometric determination of uncharged substances.

2.19

Consider the exhaust gas analyzer based on the oxygen concentration cell, (2.4.21). The electrode reaction that occurs at high temperature at both Pt/ZrO2 + Y2 O3 interfaces is O2 + 4e ⇌ 2O2− Write the equation that governs the potential of this cell as a function of the pressures, Peg and Pa . What would the cell voltage be when the partial pressure of oxygen in the exhaust gas is 0.01 bar (1000 Pa)?

2.20

For the half reaction, Fe(III) + e ⇌ Fe(II): (a) Predict the formal potential in 0.1 M HClO4 [a noncomplexing medium for Fe(III)/Fe(II)] using the standard potential from Table C.1 and activity coefficients from Table 2.1.1. (b) Using the formal potential given in Table C.2, estimate the ratio 𝛾Fe3+ ∕𝛾Fe2+ in 1 M HClO4 . How does it compare with the ratio calculated in a) for 0.1 M HClO4 ? Account in chemical terms for these ratios. (c) Why are the formal potentials of Fe(III)/Fe(II) in 1 M HCl and 10 M HCl shifted further from the standard potential than in HClO4 ?

119

121

3 Basic Kinetics of Electrode Reactions In Chapter 1, we established a proportionality between the current and the net rate of an electrode reaction. We also recognized that for a given electrode process, current does not flow in some potential regions; yet, it flows to a varying degree in others. The reaction rate is, therefore, a strong function of potential. In this chapter, our goal is to devise a theory that can quantitatively rationalize the observed behavior of electrode kinetics with respect to potential and concentration. Once constructed, the theory will support our understanding of kinetic effects in new situations. We begin with a brief review of certain aspects of homogeneous kinetics, which offer both a familiar starting ground and a basis for the construction, through analogy, of an electrochemical kinetic theory.

3.1 Review of Homogeneous Kinetics 3.1.1

Dynamic Equilibrium

Consider two substances, A and B, that are linked by simple unimolecular elementary reactions.1 kf

−−− ⇀ A↽ −B kb

(3.1.1)

Both elementary reactions are always active, and the rates of the forward and backward processes, vf and vb (mol L−1 s−1 ), are vf = kf CA vb = kb CB

(3.1.2a) (3.1.2b)

The rate constants, k f and k b , have units of s−1 , and one can easily show that they are the reciprocals of the mean lifetimes of A and B, respectively (Problem 3.6). The net conversion rate of A to B is vnet = kf CA − kb CB At equilibrium, the net conversion rate is zero; hence, kf C =K = B kb CA

(3.1.3)

(3.1.4)

The kinetic theory, therefore, predicts a fixed concentration ratio at equilibrium, just as thermodynamic theory does. 1 An elementary reaction describes an actual, discrete chemical event. Many chemical reactions, as written, are not elementary, because the transformation of products to reactants involves several steps. The steps are the elementary reactions that comprise the mechanism for the overall process. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

122

3 Basic Kinetics of Electrode Reactions

Such agreement is required of any kinetic theory. In the limit of equilibrium, the kinetic equations must collapse to relations of the thermodynamic form. Kinetics describe the evolution of the system, including both the approach to equilibrium and the dynamic maintenance of that state. Thermodynamics describe only equilibrium. Understanding of a system is not even at a crude level unless the kinetic and thermodynamic views agree on the properties of the equilibrium state. On the other hand, thermodynamics provide no information about the mechanism required to maintain equilibrium, whereas kinetics describe the balance quantitatively. In the example above, equilibrium features nonzero rates of conversion of A to B (and vice versa), but those rates are equal. Sometimes they are called the exchange velocity of the reaction, v0 : v0 = kf (CA )eq = kb (CB )eq

(3.1.5)

The idea of exchange velocity is important in treatments of electrode kinetics. 3.1.2

The Arrhenius Equation and Potential Energy Surfaces

It is an experimental fact that most rate constants vary with temperature in a common fashion: ln k is nearly always linear with 1/T. Arrhenius was first to recognize the generality of this behavior, and he proposed that rate constants be expressed in the form: k = Ae−EA ∕RT

(3.1.6)

where EA has units of energy. Since the exponential factor is reminiscent of the probability of using thermal energy to surmount an energy barrier of height EA , that parameter has been known as the activation energy. If the exponential expresses the probability of surmounting the barrier, then A must be related to the frequency of attempts on it; thus, A is known commonly as the frequency factor. As usual, these ideas turn out to be oversimplifications, but they carry an essence of truth and are useful for casting a mental image of the ways in which reactions proceed. The idea of activation energy has led to pictures of reaction paths in terms of potential energy along a reaction coordinate (1–5). An example is shown in Figure 3.1.1. In a simple unimolecular process, e.g., the cis–trans isomerization of stilbene, the reaction coordinate might be an easily recognized molecular parameter, such as the twist angle about the central double bond in stilbene. More generally, the reaction coordinate expresses progress along a favored path on a multidimensional surface describing potential energy as a function of all relevant atomic coordinates for a single set of reactants and products (e.g., one molecule of stilbene). One zone of this surface corresponds to the configuration we call “reactant,” and another corresponds to the “product.” Both must be found at minima on the energy surface, because they are the only configurations possessing a significant lifetime. Even though other configurations are possible, they must lie at higher energies and lack the energy minimum required for even momentary stability. As the reaction takes place, the coordinates change from those of the reactant to those of the product. Since the path along the reaction coordinate connects two minima, it must rise, pass over a maximum, then fall into the product zone. Very often, the height of the maximum above a valley or “hollow” is identified with an activation energy, either EA,f or EA,b , for the forward or backward reaction, respectively.2 2 The energy surface ideally comprehends the positions of all atoms affecting the energetics of the processes of interest, not just those defining A and B. Also relevant are atoms in immediate and more remote solvation spheres and atoms that might be involved in ion-pairing, adsorption sites, or other forms of close association with either or both primary participants, A and B. Thus, the “reactant zone” on the surface is a “hollow” describing A in its normal range of low-energy configurations, and as found in its normal range of “habitats,” for example in a solution or adsorbed on a surface. Likewise, the “product zone” is a different hollow corresponding to B in its usual habitats.

Potential energy

3.1 Review of Homogeneous Kinetics

Reactants

Products

Reaction coordinate

Figure 3.1.1 Simple representation of potential energy changes during a reaction.

In another notation, we can understand EA as the change in standard internal energy in going from one of the minima to the maximum, which is often called the transition state or activated complex. We might then identify that EA as the standard internal energy of activation, ΔE‡ . The standard enthalpy of activation, ΔH ‡ , would then be ΔE‡ + Δ(PV )‡ , but Δ(PV ) is usually negligible in a condensed-phase reaction, so that ΔH ‡ ≈ ΔE‡ . Thus, the Arrhenius equation could be recast as k = Ae−ΔH

‡ ∕RT

(3.1.7)

We are free also to factor the coefficient A into the product A′ exp(ΔS‡ /R), because the exponential involving the standard entropy of activation, ΔS‡ , is a dimensionless constant. Then k = A′ e−(ΔH

‡ −TΔS‡ )∕RT

(3.1.8)

or k = A′ e−ΔG

‡ ∕RT

(3.1.9)

where ΔG‡ is the standard free energy of activation.3 This relation, like (3.1.7), is an equivalent statement of the Arrhenius equation, (3.1.6), which itself is an empirical generalization of reality. Equations 3.1.7 and 3.1.9 are derived from (3.1.6), but only by the interpretation we apply to the phenomenological constant EA . Nothing we have written so far depends on a specific theory of kinetics. 3.1.3

Transition State Theory

Many theories of kinetics have been constructed to illuminate the factors controlling reaction rates. A prime goal is to predict the values of A and EA for specific chemical systems in terms of quantitative molecular properties. An important general theory that has been adapted for electrode kinetics is the transition state theory (1–5), also known as the activated complex theory. 3 We are using standard thermodynamic quantities here (Section 2.1.5), because the free energy and the entropy of a species are concentration dependent. The rate constant is not concentration dependent in dilute systems; thus, the argument that leads to (3.1.9) needs to be developed in the context of a standard state of concentration. The choice of standard state is not critical to the discussion. To simplify notation, we omit the superscript “0” from ΔE‡ , ΔH ‡ , ΔS‡ , and ΔG‡ , but normally understand them to be referred to the standard state of concentration.

123

3 Basic Kinetics of Electrode Reactions

Activated complex Standard free energy

124

Reactant

ΔG‡f ΔG‡b ΔG0rxn Product

Reaction coordinate

Figure 3.1.2 Free-energy changes during a reaction. The activated complex (or transition state) is the configuration of maximum free energy on the favored reaction path.

Central to this approach is the idea that reactions proceed through a well-defined transition state or activated complex, as shown in Figure 3.1.2. The standard free-energy change in going from the reactants to the complex is ΔGf‡ , while the complex is elevated above the products by ΔGb‡ . Let us consider the system of (3.1.1), in which two substances, A and B, are linked by unimolecular reactions. First, we focus on the special condition in which the entire system—A, B, and all other configurations—is at thermal equilibrium. For this situation, the concentration of activated complexes can be calculated according to either of two equilibrium constants: [Complex] 𝛾A ∕C 0 = K = [A] 𝛾‡ ∕C 0 f [Complex] 𝛾B ∕C 0 = K = [B] 𝛾‡ ∕C 0 b

𝛾A 𝛾‡ 𝛾B 𝛾‡

exp(−ΔGf‡ ∕RT)

(3.1.10)

exp(−ΔGb‡ ∕RT)

(3.1.11)

where C 0 is the concentration of the standard state, and 𝛾 A , 𝛾 B , and 𝛾 ‡ are dimensionless activity coefficients (Section 2.1.5). Normally, we assume that the system is ideal, so that the activity coefficients approach unity and fall out of (3.1.10) and (3.1.11). The activated complexes decay into either A or B according to a combined rate constant, ′ k , and can be divided into four fractions: (a) those created from A and reverting back to A, f AA , (b) those arising from A and decaying to B, f AB , (c) those created from B and decaying to A, f BA , and (d) those arising from B and reverting back to B, f BB . Thus, f AA + f AB + f BA + f BB = 1, and the rates of transforming A into B and vice versa are kf [A] = fAB k ′ [Complex] ′

kb [B] = fBA k [Complex]

(3.1.12a) (3.1.12b)

Since we require k f [A] = k b [B] at equilibrium, f AB and f BA must be the same. In the simplest version of the theory, both are taken as 1/2. This assumption implies that f AA = f BB = 0; thus, complexes are not considered as reverting to the source state. Instead, any system reaching the activated configuration continues to proceed in the direction opposite the source. In a more flexible version, the fractions f AB and f BA are equated to 𝜅/2, where 𝜅, the transmission coefficient, can take a value from zero to unity.

3.2 Essentials of Electrode Reactions

Substitution for the concentration of the complex from (3.1.10) and (3.1.11) into (3.1.12a) and (3.1.12b), respectively, leads to the rate constants: 𝜅k ′ −ΔGf‡ ∕RT e 2 𝜅k ′ −ΔGb‡ ∕RT kb = e 2

(3.1.13a)

kf =

(3.1.13b) ′

Statistical mechanics can be used to predict k . In general, it depends on the shape of the energy ′ surface in the region of the complex, but for simple cases k can be shown to be 2 kT∕h, where, k and h are the Boltzmann and Planck constants. Thus, both rate constants in (3.1.13) might be expressed in the form: k=𝜅

kT −ΔG‡ ∕RT e h

(3.1.14)

which is the expression most frequently seen for a rate constant in the transition state theory. To reach (3.1.14), we considered only a system at equilibrium. However, the rate constant for an elementary process is fixed for a given temperature and pressure and does not depend on the reactant and product concentrations. Equation 3.1.14 is, therefore, a general expression. If it holds at equilibrium, it will hold away from equilibrium. The assumption of equilibrium, though useful in the derivation, does not constrain the equation’s range of validity (2).4

3.2 Essentials of Electrode Reactions Any accurate kinetic picture of a dynamic process must yield an equation of the thermodynamic form in the limit of equilibrium. For an electrode reaction, equilibrium is characterized by the Nernst equation, which links the electrode potential to the bulk concentrations of the participants. For the simple case kf

−−− ⇀ O + ne ↽ − R

(3.2.1)

kb

this equation is ′

Eeq = E0 +

∗ RT CO ln ∗ nF CR

(3.2.2) ′

∗ and where Eeq is the equilibrium potential, E0 is the formal potential (Section 2.1.7), and CO CR∗ are the bulk concentrations. Any valid theory of electrode kinetics must predict this result for corresponding conditions. 4 Since kT/h has units of s−1 and the exponential is dimensionless, (3.1.14) is dimensionally correct for a first-order rate constant. For a second-order reaction, the equilibrium expression leading to (3.1.10) would place the concentrations of two reactants in the denominator on the left side and the activity coefficient for each of those species divided by the standard-state concentration, C 0 , in the numerator on the right. Thus, C 0 no longer divides out altogether and is carried to the first power into the denominator of the final expression. Since it normally has a unit value (usually 1 M), its presence has no effect numerically, but it does dimensionally. The overall result is to create a prefactor having a numeric value equal to kT/h, but with units of M−1 s−1 , as required. This point is often omitted in applications of transition state theory to processes more complicated than unimolecular decay. See Section 2.1.5 and reference (2).

125

126

3 Basic Kinetics of Electrode Reactions

The theory must also explain the observed dependence of the current on potential under various circumstances. Especially relevant is the case of low current and efficient stirring, where the current is entirely controlled by interfacial electron-transfer kinetics. Early studies of such systems showed that the current is often related exponentially to the overpotential, 𝜂 = E − Eeq . That is, i = a′ e𝜂∕b

′

(3.2.3)

or, as given by Tafel in 1905 (6, 7), 𝜂 = a + b log i

(3.2.4)

A successful model of electrode kinetics must explain the frequent validity of (3.2.4), known as the Tafel equation. Reaction 3.2.1 has forward and backward paths as shown. The forward component proceeds at a rate, vf , which must be proportional to the concentration of O at the electrode surface ∗ ). Since we express the concentration at (often different from the bulk concentration, CO distance x from the surface and at time t as C O (x, t), the surface concentration is C O (0, t). The rate constant, k f , links the forward reaction rate to C O (0, t); thus, vf = kf CO (0, t) =

ic

(3.2.5) nFA This equation also relates vf to a cathodic current, ic , that crosses the interface as a consequence of the forward reaction. Likewise, we have for the backward reaction i vb = kb CR (0, t) = a (3.2.6) nFA where ia is the corresponding anodic current. The net reaction rate is vnet = vf − vb = kf CO (0, t) − kb CR (0, t) =

ic − ia nFA

(3.2.7)

and we have, overall: i = ic − ia = nFA[kf CO (0, t) − kb CR (0, t)]

(3.2.8)

The net current, i, is measurable in an external circuit. The component currents, ic and ia , are not independently measurable, but have conceptual value. Reaction velocities in heterogeneous systems refer to unit interfacial area; hence, they have units of mol s−1 cm−2 . Consequently, heterogeneous rate constants carry units of cm/s when the concentrations are expressed in mol/cm3 .

3.3 Butler–Volmer Model of Electrode Kinetics Experience demonstrates that the potential of an electrode strongly affects the kinetics of reactions occurring on its surface. Hydrogen evolves rapidly at some potentials, but not at others. Copper dissolves from a metallic sample in a clearly defined potential range; yet, the metal is stable at more negative potentials. And so it is for all faradaic processes. Because the interfacial potential difference can be used to control reactivity, we want to understand the precise way in which k f and k b depend on potential. In this section, we will develop a predictive model based purely on classical concepts (8–19). Even though it has significant limitations, it is very widely used in the electrochemical literature and must be understood by any student of the field. Section 3.5 will cover more fundamental models based on microscopic ideas about electron transfer.

3.3 Butler–Volmer Model of Electrode Kinetics

3.3.1

Effects of Potential on Energy Barriers

We saw in Section 3.1 that reactions can be visualized in terms of progress along a reaction coordinate connecting a reactant configuration to a product configuration on an energy surface. This idea applies to electrode reactions too, but the shape of the energy surface becomes a function of electrode potential. One can see the effect easily by considering the following reaction at a mercury electrode, Hg

−−− ⇀ Na+ + e ↽ − Na(Hg)

(3.3.1)

where Na(Hg) represents sodium amalgam, and Na+ is dissolved in acetonitrile or DMF. If we think of the reaction coordinate as the distance of the sodium nucleus from the interface; then the free-energy profile along the reaction coordinate would resemble Figure 3.3.1a. To the right is Na+ + e. This configuration has an energy that depends little on the nuclear position in solution unless the electrode is approached so closely that the ion must become partially or wholly desolvated. To the left, the configuration corresponds to a sodium atom dissolved in mercury. Within the mercury phase, the energy does not depend on position, but if the atom leaves the interior, its energy rises as the favorable mercury–sodium interaction is lost. The curves corresponding to these reactant and product configurations intersect at the transition state, and the heights of the barriers to oxidation and reduction determine the relative rates of those processes. When the rates are equal, as in Figure 3.3.1a, the system is at equilibrium, and the potential is Eeq . Now suppose the electrode potential is changed to a more positive value. The main effect is to lower the energy of the “reactant” electron, which resides on the electrode. The curve corresponding to Na+ + e drops with respect to that corresponding to Na(Hg), and the situation resembles that of Figure 3.3.1b. The barrier for reduction is raised and that for oxidation is lowered, so there is a net conversion of Na(Hg) to Na+ + e. Setting the potential to a value more negative than Eeq raises the energy of the electron and shifts the curve for Na+ + e to higher energies, as shown in Figure 3.3.1c. Since the reduction barrier drops and the oxidation barrier rises, relative to the condition at Eeq , a net cathodic current flows. These arguments show qualitatively how the potential affects the net rates and directions of electrode reactions. By considering the model more closely, we can establish a quantitative theory. 3.3.2

One-Step, One-Electron Process

Let us now consider the simplest possible electrode process, wherein two solutes, O and R, engage in a one-electron transfer at the interface without being involved in any other chemical step, kf

−−− → O+e ← − R

(3.3.2)

kb

Suppose also that the standard free-energy profiles along the reaction coordinate have the parabolic shapes shown in Figure 3.3.2. The upper frame of that figure depicts the full path from reactants to products, while the lower frame is an enlargement of the region near the transition state. It is not important for this discussion that we know the shapes of these profiles in detail. In developing a theory of electrode kinetics, it is convenient to express the potential, E, vs. a point of significance to the chemistry of the system, rather than vs. an arbitrary

127

3 Basic Kinetics of Electrode Reactions

Reduction

Oxidation

(a) Na+ + e

Na(Hg)

Oxidation Standard free energy

128

(b) Na(Hg) Na+ + e

Reduction

(c) Na+ + e Na(Hg)

Amalgam

Solution Reaction coordinate

Figure 3.3.1 Simple representation of standard free-energy changes during a faradaic process. (a) At a potential corresponding to equilibrium. (b) At a more positive potential than the equilibrium value. (c) At a more negative potential than the equilibrium value.

external reference, such as an Ag/AgCl electrode. There are two natural reference points: the equilibrium potential of the system and the standard (or formal) potential of the couple under consideration. Indeed, we used the equilibrium potential as a reference point in the discussion of the Section 3.3.1, and we will use it again below. However, it is possible to do so only when both members of the couple are present, so that an equilibrium is defined. The more general ′ reference point is E0 . Suppose the upper curve on the O + e side of Figure 3.3.2 applies when ′ the electrode potential is equal to E0 . The corresponding cathodic and anodic activation ‡ ‡ energies are ΔG0c and ΔG0a , respectively. If the potential is changed to a new value, E, the energy per mole of the electrons resident on ′ the electrode changes by −FΔE = − F(E − E0 ); hence, the O + e curve moves up or down by

Standard free energy

3.3 Butler–Volmer Model of Electrode Kinetics

At E0′ ΔG‡0a

ΔG‡0c

ΔG‡a

At E

ΔG‡c

F(E – E0′)

O+e

R

Reaction coordinate

Standard free energy

(a)

(1 – 𝛼)F(E – E0′) At E0′ F(E – E0′) 𝛼F(E – E0′) At E O+e

R

Reaction coordinate (b)

Figure 3.3.2 Effects of a potential change on the standard free energies of activation for oxidation and reduction. The shaded, boxed area in (a) is magnified in (b).

that amount.5 The lower curve on the left side of Figure 3.3.2 shows this effect for a positive ΔE. ‡ The barrier for oxidation, ΔGa‡ , has clearly become less than ΔG0a by a fraction of the total energy change. Let us call that fraction 1 − 𝛼, where 𝛼, the transfer coefficient, can range from zero to unity. Thus, ′

‡ ΔGa‡ = ΔG0a − (1 − 𝛼)F(E − E0 )

(3.3.3)

Study of Figure 3.3.2 also reveals that the cathodic barrier at potential E, ΔGc‡ , is higher than ′ ‡ ΔG0c by the amount 𝛼F(E − E0 ); therefore, ′

‡ ΔGc‡ = ΔG0c + 𝛼F(E − E0 )

(3.3.4)

Now let us assume that the rate constants k f and k b have an Arrhenius form: ‡

kf = Af e−ΔGc ∕RT

(3.3.5a)

kb = Ab e

(3.3.5b)

−ΔGa‡ ∕RT

5 The energy required to take a charge q through a difference in potential ΔE is qΔE. The charge on a mole of electrons is −F, where F is the Faraday constant; thus, the energy change for the electrons is −FΔE (J/mol).

129

130

3 Basic Kinetics of Electrode Reactions

Inserting the activation energies, (3.3.3) and (3.3.4), gives ‡

kf = Af e−ΔG0c ∕RT e−𝛼f (E−E kb = Ab

0′ )

‡ 0′ e−ΔG0a ∕RT e(1−𝛼)f (E−E )

(3.3.6a) (3.3.6b)

where f = F/RT. In each of these expressions, the first two factors form a product that is both ′ independent of potential and equal to the corresponding rate constant at E = E0 .6 Now let us think about the special case in which the interface is at equilibrium with a solu∗ = C ∗ . In this situation, E = E 0′ , C (0, t) = C ∗ , and C (0, t) = C ∗ . Because tion in which CO O R R O R ∗ = k C ∗ , and k must equal k . We now see that equilibrium applies, i = 0. Consequently, kf CO b R f b ′ E0 is the potential where the forward and reverse rate constants have the same value, called the standard rate constant, k 0 .7 Although we used a special case to discover this point, the rate ′ constants do not depend on concentration; hence, the equality of k f and k b at E0 holds for all concentrations. In fact, we made this discovery without using any detail concerning k f or k b .8 ′ Only the principles of mass action were involved. The fact that k f = k b = k 0 at E0 depends only on the form of the electrochemical reaction given in (3.3.2). The rate constants at other potentials can now be expressed simply in terms of k 0 : kf = k 0 e−𝛼f (E−E

0′ )

kb = k 0 e(1−𝛼)f (E−E

(3.3.7a) 0′ )

Insertion of these relations into (3.2.8) yields the current–potential characteristic: [ ] 0′ 0′ i = FAk 0 CO (0, t)e−𝛼f (E−E ) − CR (0, t) e(1−𝛼)f (E−E )

(3.3.7b)

(3.3.8)

This relation is very important. It, or a variation derived from it, is used in the treatment of a great many situations requiring an account of heterogeneous kinetics. Section 3.4 will cover some of its ramifications. The results in (3.3.7) and (3.3.8) and the inferences derived from them are known broadly as the Butler–Volmer (BV) formulation of electrode kinetics, in honor of the pioneers in this area (20, 21). One can derive the BV kinetic expressions by an alternate method based on electrochemical potentials (13, 14, 16, 22–24). Such an approach can be more convenient for more complicated cases, such as those requiring the inclusion of double-layer effects. The first edition developed it in detail.9 3.3.3

The Standard Rate Constant

The physical interpretation of k 0 is straightforward: It is a measure of the heterogeneous kinetic facility of a redox couple. A system with a large k 0 can achieve equilibrium quickly, but a system with small k 0 is sluggish. 6 In other electrochemical literature, k f and k b are designated as k c and k a or as k ox and k red . Sometimes, kinetic equations are written in terms of a complementary transfer coefficient, 𝛽 = 1 − 𝛼. If one sees 𝛽 in kinetic expressions, it is important to discern whether it is meant to be what we call 𝛼 or what we call 1 − 𝛼. See also footnote 10. 7 The standard rate constant is also designated by k s,h or k s in the electrochemical literature. Sometimes, it is also called the intrinsic rate constant. ′ 8 Nor did we invoke the Nernst equation. We only made use of E0 as an identifier for the equilibrium potential of the special case. It remains for us to demonstrate that this theory will predict a nernstian balance at equilibrium (Section 3.4.1). 9 First edition, Section 3.4.

3.3 Butler–Volmer Model of Electrode Kinetics

The largest standard rate constants measured to date are in the range of 10–40 cm/s (25–31) and relate to Ru(NH3 )3+ ∕Ru(NH3 )2+ and various ferrocenium/ferrocene couples. Also very 6 6 3+ 2+ facile are Ru(bpy)3 ∕Ru(bpy)3 and many couples linking large aromatic hydrocarbons to their anion and cation radicals. All of these processes involve only electron transfer and resolvation. There are no significant alterations in the molecular frames. Similarly, some electrode reactions involving the formation of amalgams [e.g., the couples Na+ /Na(Hg), Cd2+ /Cd(Hg), and Hg2+ ∕Hg] are kinetically facile (32, 33). In contrast, heterogeneous reactions involving 2 significant molecular rearrangement upon electron transfer, such as the reduction of molecular oxygen to hydrogen peroxide or water, or the reduction of protons to molecular hydrogen, can be very sluggish (7, 8, 32–34). Systems like these typically involve multistep mechanisms (Section 3.7; Chapters 13 and 15). Values of k 0 significantly lower than 10−9 cm/s have been reported (35, 36); hence, electrochemistry deals with a range of more than 10 orders of magnitude in kinetic reactivity. Note from (3.3.7) that k f and k b can be made quite large, even if k 0 is small, by using a suf′ ficiently extreme potential relative to E0 . In effect, one drives the reaction by lowering the activation energy electrically. This idea is explored more fully in Section 3.4. 3.3.4

The Transfer Coefficient

In the Butler–Volmer kinetic model, the transfer coefficient, 𝛼, is assumed to be independent of potential. Like k 0 , it is a fixed parameter that depends on the chemical identity of the system. The transfer coefficient is a measure of the symmetry of the energy barrier for a single electron-transfer step.10 Let us develop this idea in terms of the geometry of the intersection region, as shown in Figure 3.3.3. If the free-energy curves are locally linear, then the angles 𝜃 and 𝜙 are defined by tan 𝜃 = 𝛼FE∕x

(3.3.9a)

tan 𝜙 = (1 − 𝛼)FE∕x

(3.3.9b)

Standard free energy

E=0

(1 – 𝛼)FE ϕ

𝜃 O+e

𝛼FE

E=E

𝜃 Length x R

O+e Reaction coordinate

Figure 3.3.3 Relationship of the transfer coefficient to the angles of intersection of the free-energy curves. 10 Sometimes 𝛼 for a one-step, one-electron reaction is called the symmetry factor, and sometimes it is symbolized by 𝛽 (7). However, the use of the symbol 𝛼 and the name transfer coefficient for the concept identified in this section are much more commonly found in the literature.

131

3 Basic Kinetics of Electrode Reactions

Standard free energy

132

α =1 – 2

α 1 – 2

O+e

O+e

O+e R

R

R

Reaction coordinate

Figure 3.3.4 The transfer coefficient as an indicator of the symmetry of the barrier to reaction. The dashed lines show the shift in the curve for O + e as the potential is made more positive.

Hence, 𝛼=

tan 𝜃 tan 𝜙 + tan 𝜃

(3.3.10)

If the intersection is symmetrical, 𝜙 = 𝜃, and 𝛼 = 1/2. Otherwise 0 ≤ 𝛼 < 1/2 or 1/2 < 𝛼 ≤ 1, as shown in Figure 3.3.4. In most experimental cases, 𝛼 turns out to lie between 0.3 and 0.7, and is often taken to be 0.5 in the absence of actual measurements. The assumption in the BV model that 𝛼 is constant with potential is equivalent to assuming linear free-energy curves, so that the geometry of intersection never changes with potential. Since real free-energy profiles are likely to be nonlinear over large ranges of the reaction coordinate, the angles 𝜃 and 𝜙 can be expected to change as the intersection between the reactant and product curves shifts with potential. Consequently, one must generally expect 𝛼 to be a potential-dependent factor. Nevertheless, 𝛼 actually appears constant in most experiments, if only because the potential range over which kinetic data can be collected is often fairly narrow. The intersection point corresponding to accessible measurements then varies only over a small domain, such as the boxed region in Figure 3.3.2, where the curvature in the (parabolic) profiles can hardly be seen. The usable potential range is limited because the rate constant for electron transfer rises exponentially with potential. Not far beyond the potential where a process first produces a detectable current, mass transfer becomes rate limiting, and the electron-transfer kinetics no longer control the experiment. The competition between kinetics and mass transfer will be encountered throughout the remainder of this book. Other models for heterogeneous electron transfer (Section 3.5) do predict a potentialdependent transfer coefficient. In systems where electron-transfer kinetics can be measured over very wide ranges of potential, sizable variations of 𝛼 with potential can be seen [Section 3.5.4(c)].

3.4 Implications of the Butler–Volmer Model for the One-Step, One-Electron Process In this section, we develop operational relationships that will prove valuable in the interpretation of electrochemical experiments. Each is derived under the assumption that the electrode reaction is the simple one-step, one-electron process for which primary relations were derived above. Multistep processes, involving additional electron transfers or chemical steps, are more complicated, and the findings of this section cannot be assumed to apply to them. They are treated in Section 3.7 and Chapters 13 and 15.

3.4 Implications of the Butler–Volmer Model for the One-Step, One-Electron Process

3.4.1

Equilibrium Conditions and the Exchange Current

At equilibrium, the net current is zero, and the electrode is known to adopt a potential based on the bulk concentrations of O and R as dictated by the Nernst equation (Section 2.1.6). Let us see now if the kinetic model yields that thermodynamic relation as a special case. From (3.3.8) we have, at zero current, FAk 0 CO (0, t)e

′

−𝛼f (Eeq −E0 )

= FAk 0 CR (0, t)e

′

(1−𝛼)f (Eeq −E0 )

(3.4.1)

Since equilibrium applies, surface concentrations do not differ from those of the bulk; hence, upon substitution and rearrangement, we obtain ′

f (Eeq −E0 )

e

=

∗ CO

(3.4.2)

CR∗

which is an exponential form of the Nernst relation: ′

Eeq = E0 +

∗ RT CO ln ∗ F CR

(3.4.3)

The theory has passed its first test. Even though the net current is zero at equilibrium, we still envision balanced faradaic activity that can be expressed in terms of the exchange current, i0 , which is equal in magnitude to either component current, ic or ia . Using ic , we have ∗e i0 = FAk 0 CO

′

−𝛼f (Eeq −E0 )

(3.4.4)

If both sides of (3.4.2) are raised to the −𝛼 power, we obtain ( ∗ )−𝛼 ′ CO −𝛼f (Eeq −E0 ) e = CR∗

(3.4.5)

Substitution of (3.4.5) into (3.4.4) gives11 (1−𝛼) ∗𝛼 CR

∗ i0 = FAk 0 CO

(3.4.6)

The exchange current is proportional to k 0 and can often be substituted for k 0 in kinetic ∗ = C∗ = C∗, equations. For the case where CO R i0 = FAk 0 C ∗

(3.4.7)

Often the exchange current is normalized to unit area to provide the exchange current density, j0 = i0 /A. 3.4.2

The Current–Overpotential Equation

An advantage of working with i0 rather than k 0 is that the current can be described in terms ′ of the deviation from the equilibrium potential, rather than the formal potential, E0 . Dividing (3.3.8) by (3.4.6), we obtain C (0, t)e−𝛼f (E−E i = O i0 ∗(1−𝛼) C ∗𝛼 CO R

0′ )

−

CR (0, t)e(1−𝛼)f (E−E

11 The same equation can be derived from ia .

(1−𝛼) ∗𝛼 CR

∗ CO

0′ )

(3.4.8)

133

134

3 Basic Kinetics of Electrode Reactions

i/il 1.0

il

0.8 0.6 ic

400

300

200

Total current

0.4 0.2

100 Eeq

–100 –0.2

i0

–0.4

–ia

–0.6

–200

–300

–400 𝜂(mV)

–0.8 –il

–1.0

Figure 3.4.1 Current–overpotential curves for the system O + e ⇌ R with 𝛼 = 0.5, T = 298 K, il, c = − il, a = il and i0 /il = 0.2. The dashed lines show the component currents, ic and ia .

or

( ∗ )𝛼 ( ∗ )−(1−𝛼) CO (0, t) −𝛼f (E−E0′ ) CO CR (0, t) (1−𝛼)f (E−E0′ ) CO i = e − e ∗ i0 CO CR∗ CR∗ CR∗

(3.4.9)

∗ ∕C ∗ )𝛼 and (C ∗ ∕C ∗ )−(1−𝛼) are easily evaluated from (3.4.2) and (3.4.5), and by The ratios (CO R O R substitution we obtain [ ] CO (0, t) −𝛼f 𝜂 CR (0, t) (1−𝛼)f 𝜂 i = i0 e − e (3.4.10) ∗ CO CR∗

where 𝜂 = E − Eeq . This equation, known as the current–overpotential equation, will be used frequently in later discussions. The first term describes the cathodic component current at any potential, and the second gives the anodic contribution.12 The behavior predicted by (3.4.10) is displayed in Figure 3.4.1, where the solid curve shows the actual total current, which is the sum of the components ic and ia , shown as dashed traces. For large negative 𝜂, the anodic component is negligible; hence, the total current curve merges with that for ic . At large positive 𝜂, the cathodic component is negligible, and the total current is essentially the same as ia . Going in either direction from Eeq , the magnitude of the current rises rapidly, because the exponential factors dominate behavior, but at extreme 𝜂, the current levels off. In these level regions, the current is limited by mass transfer rather than heterogeneous kinetics. The exponentials in (3.4.10) become ∗ and C (0, t)∕C ∗ , which manifest the reactant supply moderated by the factors CO (0, t)∕CO R R (Section 3.4.6).

12 Since double-layer effects have not been included in this treatment, k 0 and i0 are, in Delahay’s nomenclature (16), apparent constants of the system. Both depend on double-layer structure to some extent and are functions of the potential at the outer Helmholtz plane, 𝜙2 , relative to the solution bulk. This point will be discussed in more detail in Section 14.7.

3.4 Implications of the Butler–Volmer Model for the One-Step, One-Electron Process

3.4.3

Approximate Forms of the i–𝜼 Equation

(a) No Mass-Transfer Effects

If the solution is well stirred, or currents are kept so low that the surface concentrations do not differ appreciably from the bulk values, then (3.4.10) becomes i = i0 [e−𝛼f 𝜂 − e(1−𝛼)f 𝜂 ]

(3.4.11)

which is historically known as the Butler–Volmer equation. It is a good approximation of (3.4.10) when i is less than about 10% of the smaller limiting current, il, c or |il, a |. ∗ and C (0, t)∕C ∗ will then be between 0.9 Equations 1.4.10 and 1.4.19 show that CO (0, t)∕CO R R and 1.1. Figure 3.4.2 shows the behavior of (3.4.11) for systems in which the exchange current densities differ, but 𝛼 = 0.5 consistently. The notable feature is the degree to which the inflection at Eeq depends on the exchange current density. Figure 3.4.3 illustrates the effect of 𝛼 in a similar manner for cases in which the exchange current density remains the same. j(μA/cm2) 8

(b)

6 (a) 4 400

300

200

100

(c)

2

–2

–100

–200

–300

–400 𝜂(mV)

–4 –6 –8

Figure 3.4.2 Effect of exchange current density on the activation overpotential required to deliver net current densities. (a) j0 = 10−3 A/cm2 (curve is indistinguishable from the current axis), (b) j0 = 10−6 A/cm2 , (c) j0 = 10−9 A/cm2 . For all cases the reaction is O + e ⇌ R with 𝛼 = 0.5, T = 298 K. j(μA/cm2) 8

𝛼 = 0.5

𝛼 = 0.75

6 𝛼 = 0.25

4 200

150

100

50

2

–2

–50

–100

–150

–200 𝜂(mV)

–4 –6 –8

Figure 3.4.3 Effect of the transfer coefficient on the symmetry of the current–overpotential curves for O + e ⇌ R with T = 298 K and j0 = 10−6 A/cm2 .

135

136

3 Basic Kinetics of Electrode Reactions

Since mass-transfer effects are not included here, the overpotential associated with any given current serves solely to alter the activation energy as required to drive the heterogeneous process at the rate reflected by the current. The smaller the exchange current, the more sluggish the kinetics; hence, the larger this activation overpotential must be for any given net current. If the exchange current is large, as for Figure 3.4.2a, then the system can supply large currents, even the mass-transfer-limited current, with insignificant activation overpotential. In that case, any observed overpotential is associated with changing surface concentrations of species O and R. It is called a mass-transfer overpotential and reflects the activation energy required to drive mass transfer at the rate needed to support the current. If the concentrations of O and R are ′ comparable, then Eeq will be near E0 , and the limiting currents for both the anodic and cathodic ′

segments will be reached within a few tens of millivolts of E0 . On the other hand, one might have a system with an exceedingly small exchange current, because k 0 is very low, as for Figure 3.4.2c. In that circumstance, no significant current flows unless a large activation overpotential is applied. At a sufficiently extreme potential, the heterogeneous process can be driven fast enough that mass transfer controls the current, and a limiting plateau is reached. When mass-transfer effects start to manifest themselves, a concentration overpotential will also contribute, but the bulk of the overpotential will be for activation of charge transfer. In this kind of system, the reduction wave occurs at much more negative ′ potentials than E0 , and the oxidation wave lies at much more positive values. The exchange current can be viewed as a kind of “idle rate” for charge exchange across the interface. If one wants to draw a net current that is only a small fraction of this bidirectional idle current, only a tiny overpotential will be required to extract it. Even at equilibrium, the system is delivering charge across the interface at rates much greater than we require on a net basis. The role of the slight overpotential is to slightly unbalance the rates in the two directions so that one of them predominates. However, if we ask for a net current that exceeds the exchange current, the job is much harder. We must drive the system to deliver charge at the required rate, and we can do that only by applying a significant overpotential. From this perspective, we see that the exchange current is a measure of any system’s ability to deliver a net current without a significant energy loss due to activation. Exchange current densities in real systems reflect the wide range in k 0 , amplified by the wide range in possible concentrations. Values may exceed 10 A/cm2 or be less than 1 pA/cm2 . (b) Linear Characteristic at Small 𝜼

For small values of x, an exponential ex can be approximated as 1 + x; hence, for sufficiently small 𝜂, (3.4.11) can be re-expressed as i = −i0 f 𝜂

(3.4.12)

Thus, the net current is linearly related to overpotential in a narrow potential range near Eeq . The ratio −𝜂/i has units of resistance and is often called the charge-transfer resistance, Rct : Rct =

RT Fi0

(3.4.13)

This parameter is the negative reciprocal slope of the i − 𝜂 curve at the origin (𝜂 = 0, i = 0). It can be evaluated directly in some experiments, and it serves as a convenient index of kinetic facility. For very large k 0 , it approaches zero (Figure 3.4.2a). (c) Tafel Behavior at Large 𝜼

For large values of 𝜂 (either negative or positive), one of the bracketed terms in (3.4.11) becomes negligible. For example, at large negative overpotentials, exp(−𝛼f𝜂) ≫ exp[(1 − 𝛼)f𝜂] and (3.4.11)

3.4 Implications of the Butler–Volmer Model for the One-Step, One-Electron Process

becomes i = i0 e−𝛼f 𝜂

(3.4.14)

or 𝜂=

RT RT ln i0 − ln i 𝛼F 𝛼F

(3.4.15)

Thus, we find that the kinetic treatment outlined above does yield a relation of the Tafel form, as required by common observation for the appropriate conditions. The empirical Tafel constants (see equation 3.2.4) can now be identified from theory as13 2.303RT log i0 (3.4.16a) 𝛼F −2.303RT b= (3.4.16b) 𝛼F The Tafel form can be expected to hold whenever the back reaction (i.e., the anodic process, when a net reduction is considered) contributes less than 1% of the current, or a=

e(1−𝛼)f 𝜂 = ef 𝜂 ≤ 0.01 (𝜂 < 0) (3.4.17) e−𝛼f 𝜂 which implies that |𝜂| > 118 mV at 25 ∘ C.14 When electrode kinetics are sluggish and significant activation overpotentials are required, good Tafel relationships can be seen. This point underscores the fact that Tafel behavior is an indicator of totally irreversible kinetics. Such systems allow no significant current flow except at high overpotentials, where the faradaic process is effectively unidirectional. If the electrode kinetics are facile, the system will approach the mass-transfer-limited current by the time such an overpotential is established. Tafel relationships cannot be observed for such cases, because (3.4.14) rests upon the absence of mass-transfer effects on the current. (d) Tafel Plots

A plot of 𝜂 vs. log i or (more commonly) log i vs. 𝜂—either known as a Tafel plot—is a classical tool for evaluating kinetic parameters (7). In the form of log i vs. 𝜂, there is an anodic branch with slope (1 − 𝛼)F/2.303RT and a cathodic branch with slope −𝛼F/2.303RT. As shown in Figure 3.4.4, both linear segments extrapolate to an intercept of log i0 . The plots deviate sharply from linear behavior as 𝜂 approaches zero, because the back reaction begins to contribute appreciably to the net current. The transfer coefficient and the exchange current are readily accessible from this kind of presentation when it indeed applies to an elementary reaction. For complex electrode reactions, Tafel plots remain useful for mechanistic diagnosis, but quantitative interpretation becomes more complicated by the need to include intermediates in the treatment (Section 15.2.2). Some real Tafel plots are shown in Figure 3.4.5 for the Mn(IV)/Mn(III) system in concentrated acid (37). The negative deviations from linearity at very large overpotentials arise from limitations imposed by mass transfer. The region of very low overpotentials shows sharp falloffs for the reason outlined just above. A modified approach allows the use of data obtained at low overpotentials (38). Equation 3.4.11 can be rewritten as i = i0 e−𝛼f 𝜂 (1 − ef 𝜂 )

(3.4.18)

13 For 𝛼 = 0.5, −b = 0.118 V per decade of current, a value that is sometimes quoted as a “typical” Tafel slope. 14 The same criterion is reached by considering the anodic branch.

137

3 Basic Kinetics of Electrode Reactions

log |i| Slope =

–3.5

(1 – 𝛼)F 2.303RT

Slope =

– 𝛼F 2.303RT

–4.5 log i0

–5.5

–6.5 200

150

100

–50

50

–100

–150

–200

𝜂(mV)

Figure 3.4.4 Tafel plots for anodic and cathodic branches of the current–overpotential curve for O + e ⇌ R with 𝛼 = 0.5, T = 298 K, and i0 = 10−6 A. –2

–3

log j(A/cm2)

138

CMn(III) = 10–2 M CMn(IV)(M)

–4

10–2 3 × 10–3 10–3

–5

–6

1.6

1.4

1.2

1.0

0.8

0.6

E/V vs. NHE

Figure 3.4.5 Tafel plots for the reduction of Mn(IV) to Mn(III) at Pt in 7.5 M H2 SO4 at 298 K. The dashed line corresponds to 𝛼 = 0.24. [From Vetter and Manecke (37).]

or log

𝛼F𝜂 i = log i0 − 2.303RT 1 − e f𝜂

(3.4.19)

so that a plot of log[i/1 − e f𝜂 ] vs. 𝜂 yields an intercept of log i0 and a slope of −𝛼F/2.303RT. This method is applicable to electrode reactions that are not totally irreversible, i.e., those in which both the anodic and cathodic processes contribute significantly to the currents measured in the overpotential range where mass-transfer effects are not important. Such systems are often termed quasireversible, because the opposing charge-transfer reactions must both be considered; yet, a noticeable activation overpotential is required to drive a net current through the interface.

3.4 Implications of the Butler–Volmer Model for the One-Step, One-Electron Process

3.4.4

Exchange Current Plots

From equation 3.4.4, we recognize that the exchange current can be restated as ′ 𝛼F 𝛼F ∗ + log i0 = log FAk 0 + log CO E0 − E (3.4.20) 2.303RT 2.303RT eq ∗ should be linear with a slope of −𝛼F/2.303RT. Therefore, a plot of log i0 vs. Eeq at constant CO The equilibrium potential, Eeq , can be varied experimentally by changing the bulk concentration of species R, while that of species O is held constant. This kind of plot is useful for obtaining 𝛼 from experiments in which i0 is measured essentially directly (Sections 9.6 and 11.4). Another means for determining 𝛼 is suggested by rewriting (3.4.6) as ∗ + 𝛼 log C ∗ log i0 = log FAk 0 + (1 − 𝛼) log CO R

Thus,

( (

)

𝜕 log i0 ∗ 𝜕 log CO

𝜕 log i0

(3.4.21)

)

𝜕 log CR∗

=1−𝛼

(3.4.22a)

=𝛼

(3.4.22b)

C∗ R

C∗

O

∗ or C ∗ constant, is An alternative equation, which does not require holding either CO R ∗) d log(i0 ∕CO ∗) d log(CR∗ ∕CO

=𝛼

(3.4.23)

which is easily derived from (3.4.6). 3.4.5

Very Facile Kinetics and Reversible Behavior

For any kinetic model, an important limiting case is where the electrode kinetics require a negligible driving force. As we noted above, this situation corresponds to a very large exchange current, which in turn reflects a big standard rate constant, k 0 . Let us rewrite the current–overpotential equation, (3.4.10), as C (0, t) C (0, t) i = O ∗ e−𝛼f 𝜂 − R ∗ e(1−𝛼)f 𝜂 (3.4.24) i0 CO CR and consider its behavior when i0 becomes very large compared to any current of interest. Because i/i0 → 0, we can rearrange (3.4.24) to CO (0, t) CR (0, t)

=

∗ CO

CR∗

e

f (E−Eeq )

and, by substitution from the Nernst equation in form (3.4.2), we obtain ′ CO (0, t) f (E −E0 ) f (E−Eeq ) = e eq e CR (0, t) or CO (0, t) 0′ = ef (E−E ) CR (0, t)

(3.4.25)

(3.4.26)

(3.4.27)

This equation can be rearranged to a very important result: ′

E = E0 +

RT CO (0, t) ln F CR (0, t)

(3.4.28)

139

140

3 Basic Kinetics of Electrode Reactions

Thus, we see that the electrode potential and the surface concentrations of O and R are linked by an equation of the Nernst form, regardless of current flow. No kinetic parameters are present in (3.4.28) because the kinetics are so facile that no experimental manifestations can be seen. In effect, the potential and the surface concentrations are always kept in equilibrium by the fast charge-transfer processes, and the thermodynamic equation, (3.4.28), characteristic of equilibrium, always holds. Net current flows only because the surface concentrations are not at equilibrium with the bulk. Mass transfer continuously moves the electroreactant to the surface, where it must be reconciled to the potential by electrochemical change. We have already seen that a system that is always at equilibrium is termed a reversible system (Section 2.1.1); thus, it is logical that an electrochemical system in which the charge-transfer interface is always at equilibrium be called reversible (or, alternatively, nernstian). These terms are simply labels for cases in which the interfacial redox kinetics are so fast that activation effects cannot be seen. Many such systems exist in electrochemistry, so we will consider this case frequently under different experimental circumstances. We will also see that any given system may appear reversible, quasireversible, or totally irreversible, depending on the demands we make on the charge-transfer kinetics. 3.4.6

Effects of Mass Transfer

∗ A more complete i − 𝜂 relation can be obtained from (3.4.10) by substituting for CO (0, t)∕CO ∗ and CR (0, t)∕CR according to (1.3.11) and (1.3.20): ( ) ( ) i i i = 1− e−𝛼f 𝜂 − 1 − e(1−𝛼)f 𝜂 (3.4.29) i0 il,c il,a

In Problem 3.2, the reader has the opportunity to show that (3.4.29) can be rearranged to give i explicitly vs. 𝜂 over the whole range of 𝜂: i=

e−𝛼f 𝜂 − e(1−𝛼)f 𝜂 1 e−𝛼f 𝜂 e(1−𝛼)f 𝜂 + − i0 il,c il,a

(3.4.30)

In Figure 3.4.6, one can see i − 𝜂 curves drawn from (3.4.30) for several ratios of i0 /il , where il = il,c = − il,a . As we have now learned to expect, the anodic and cathodic branches separate on the potential axis when the demanded net current exceeds the exchange current. For Figure 3.4.6, the scale of net current is defined by il (the largest possible net current, given the mass-transfer conditions). When i0 /il is large, the system is reversible, and when i0 /il is very small, total irreversibility applies. For i0 /il on the order of unity, the system is quasireversible. For small overpotentials, the behavior can be linearized. The complete Taylor expansion (Section A.2) of (3.4.24) gives, for 𝛼f𝜂 ≪ 1, C (0, t) CR (0, t) F𝜂 i = O ∗ − − i0 RT CO CR∗ which can be substituted from (1.3.11) and (1.3.20) and rearranged to give ) ( RT 1 1 1 𝜂 = −i + − F i0 il,c il,a

(3.4.31)

(3.4.32)

In terms of the charge- and mass-transfer pseudoresistances defined in (1.3.29) and (3.4.13), this equation is 𝜂 = −i(Rct + Rmt,c + Rmt,a )

(3.4.33)

3.4 Implications of the Butler–Volmer Model for the One-Step, One-Electron Process

i/il 1.0

∞

0.8

1 0.1

0.6

0.01

0.4 400

300

200

100

0.2 –100 –0.2

–200

–300

– 400 𝜂(mV)

–0.4 –0.6 –0.8 –1.0

Figure 3.4.6 Relationship between the activation overpotential and net current demand relative to the exchange current. The reaction is O + e ⇌ R with 𝛼 = 0.5, T = 298 K, and il, c = − il, a = il . Numbers by curves show i0 /il .

Now we see very clearly that when i0 is much greater than the limiting currents, Rct ≪ Rmt,c + Rmt,a , so that the overpotential, even near Eeq , is a concentration overpotential. On the other hand, if i0 is much less than the limiting currents, then Rmt,c + Rmt,a ≪ Rct , and the overpotential near Eeq is due to activation of charge transfer. In the Tafel regions, other useful forms of (3.4.29) can be obtained. For the cathodic branch at large 𝜂, the anodic contribution is insignificant, and (3.4.29) becomes ( ) i i = 1− e−𝛼f 𝜂 (3.4.34) i0 il,c or 𝜂=

i RT RT (il,c − i) ln 0 + ln 𝛼F il,c 𝛼F i

(3.4.35)

This equation can be useful for obtaining kinetic parameters for systems in which the normal Tafel plots are complicated by mass-transfer effects. Not given here, but easily developed, are the analogous results for the anodic branch. 3.4.7

Limits of Basic Butler–Volmer Equations

If only O and R participate in a 1e elementary process, the results we have developed here are generally applicable, even if O or R (or both) is other than a solute, perhaps being bound to the electrode surface or incorporated into a three-dimensional lattice. However, one must take care with the expression of concentrations. In the i − E characteristic, (3.3.8), the units chosen for k 0 determine the representation of the two surface concentrations, and both must be in the same units. If, as in our treatment above, k 0 is in cm/s, then both C O (0, t) and C R (0, t), are three-dimensional concentrations in mol/cm3 . Alternatively, k 0 can be in s−1 when two-dimensional concentrations [e.g., ΓO (t) and ΓR (t) in mol/cm2 ] are more appropriate, as in a case where O and R are bound to the surface (Section 17.6.2). The requirement that the same units apply to both surface concentrations

141

142

3 Basic Kinetics of Electrode Reactions

complicates the kinetic relationships for some elementary electrode reactions, such as the electrodeposition of a metal or a reaction where O is a solute, but R is adsorbed. A formulation of BV kinetics in terms of activities can be more functional in such situations. Another notable point is that elementary electrode reactions often involve participants other than O and R, such as ligands, protons, leaving groups, and adsorption sites. The BV results derived to this point do not accommodate elementary reactions involving such species. Only O and R participate in the underlying model, so it is not appropriate to apply the results to cases that involve participants other than O and R. One can adapt the BV model to address them, but we defer examples to Chapter 15.

3.5 Microscopic Theories of Charge Transfer In Sections 3.3 and 3.4, we explored the most common description of heterogeneous electrontransfer kinetics, the Butler–Volmer model, in which the rate of a reaction is expressed in terms of two phenomenological parameters, k 0 and 𝛼. The BV approach has proven useful for organizing experimental results and for providing diagnostic information about reaction mechanisms; however, it cannot support detailed chemical thinking about the reacting species, the solvent, the electrode material, or adsorbed layers on the electrode. To go further, one needs a microscopic theory describing the effects of molecular structure and environment on the electron-transfer step. The goal is to make predictions that can be tested by experiments, so that one can grasp the fundamental factors causing reactions to be kinetically facile or sluggish. With such understanding, there would be a firmer basis for designing superior systems for many scientific and technological purposes. Major contributions in that direction have been made by Marcus (39–43), Hush (44–46), Levich (47), Dogonadze (48), and many others. Comprehensive reviews are available (49–57). The presentation below is largely based on the Marcus model, which has been widely applied in electrochemical studies and has demonstrated the ability to support useful predictions with minimal computation. Marcus was recognized with the 1992 Nobel Prize in Chemistry for his contributions. 3.5.1

Inner-Sphere and Outer-Sphere Electrode Reactions

Taube (58) introduced the terms inner-sphere and outer-sphere to describe broad classes of homogeneous electron-transfer reactions of coordination compounds (Figure 3.5.1). In his usage, “outer-sphere” denotes a reaction in which the original coordination spheres of the reactants remain intact in the activated complex, so that the reaction can be understood as “electron transfer from one primary bond system to another” (58). In distinction, a homogenous “inner-sphere” reaction occurs in an activated complex where the ions share a ligand [“electron transfer within a primary bond system” (58)]. By analogy, we transfer these ideas into the domain of electrode reactions: • In an elementary outer-sphere electrode reaction, the reactant and product do not interact chemically with the electrode surface, but remain at least a solvent layer away (Figure 3.5.1a). A typical example is the heterogeneous reduction of Ru(NH3 )3+ , where the reactant at the 6 electrode surface is considered to be essentially the same as in the bulk. • In an elementary inner-sphere electrode reaction, the electrode surface becomes a chemical component of the reacting system, and the path from reactants to products involves specific adsorption (Sections 1.6.3 and 14.3.4). One type of inner-sphere reaction features a specifically adsorbed anion that serves as a ligand bridge (59) to a metal ion (Figure 3.5.1b). The mechanisms for oxygen reduction and hydrogen evolution at Pt involve inner-sphere elementary steps (Figure 3.5.1c and Chapter 15).

3.5 Microscopic Theories of Charge Transfer

Homogeneous electron transfer Outer-sphere: Co(NH3)63+ + Cr(bpy)32+ → Co(NH3)62+ + Cr(bpy)33+ Inner-sphere: Co(NH3)5Cl2+ + Cr(H2O)62+ → (NH3)5Co Cl Cr(H2O)54+ Heterogeneous electron transfer (a) Outer-sphere

M

(b) Inner-sphere

(c) Inner-sphere

M

O

O

Electrode Solvent

Figure 3.5.1 Outer-sphere and inner-sphere reactions. The inner-sphere homogeneous reaction produces, with loss of H2 O, a ligand-bridged complex (shown above), which decomposes to CrCl(H2 O)2+ and 5 Co(NH3 )5 (H2 O)2+ . Heterogeneous reactions: (a) A metal ion (M) surrounded by ligands, spaced away from the electrode by a solvent layer. (b) A metal complex having a bridging ligand that also adsorbs on the electrode (dark shaded), e.g., the oxidation of Cr(H2 O)2+ at a mercury electrode in the presence of halide. (c) End-on 5 adsorption of O2 on a metal electrode during the oxygen reduction reaction.

The kinetics of inner-sphere reactions are typically strongly dependent on the electrode material, while those of inner-sphere reactions are much less so.15 Outer-sphere processes can be treated generally, and the remaining discussion in this section focuses on them. Inner-sphere electrode reactions are deferred to Chapter 15. These processes resist generalization, because they implicate specific, varied chemical interactions. Even so, a theoretical approach to inner-sphere electrode reactions remains an important goal—not least because they are critical in practical applications, such as fuel cells and batteries. A useful treatment must comprehend specific adsorption, as well as other factors commonly encountered in heterogeneous catalysis (60). 3.5.2

Extended Charge Transfer and Adiabaticity

It is worthwhile to consider whether a reactant can undergo electron transfer at different distances from an electrode and, if so, how the rate might depend on distance and on the nature of the intervening medium. (a) Electron Tunneling

The electron transfer itself can be understood as the tunneling of an electron between a state in the electrode and a state on the reactant. The probability of tunneling declines as exp(−𝛽x) with increasing distance, x, where the factor 𝛽 depends upon the height of the energy barrier and the nature of the medium between the states. For example, for tunneling between two pieces of metal through vacuum (61) 𝛽 ≈ 4𝜋(2mΦ)1∕2∕h ≈ 10.2 nm−1 eV−1∕2 × Φ1∕2

(3.5.1)

(9.1 × 10−28

where m is the mass of the electron g), and Φ is the work function of the metal, typically given in eV. For Pt, where Φ ≈ 5.7 eV, 𝛽 is about 24 nm−1 . 15 Even if there is not a strong interaction with the electrode, the kinetics of an outer-sphere reaction can depend on the electrode material, because of (a) double-layer effects (Section 14.7), (b) the effect of the metal on the structure of the Helmholtz layer, or (c) the effect of the energy and distribution of electronic states in the electrode (Section 3.5.5).

143

144

3 Basic Kinetics of Electrode Reactions

Electroactive centers

Alkylthiol chains

Gold electrode

Figure 3.5.2 Schematic diagram of an adsorbed monolayer of alkane thiol containing similar molecules with attached electroactive groups held by the film at a fixed distance from the electrode surface.

It is possible to study electron transfer to an electroreactant held at a fixed distance (1–3 nm) from an electrode surface by a spacer, such as an adsorbed monolayer (Section 17.6.2). One approach is based on the use of a blocking layer, such as a self-assembled monolayer of an alkane thiol or an insulating oxide film, to define the distance of closest approach of a dissolved reactant to the electrode (62–64). This strategy requires knowledge of the precise thickness of the blocking layer and assurance that the layer is free of pinholes and defects, through which a dissolved electroreactant might penetrate (Section 17.6.1). Alternatively, the adsorbed monolayer may itself contain electroactive groups (65–73). A typical layer of this kind involves an alkane thiol (RSH) with a terminal ferrocene group (−Fc), i.e., HS(CH2 )n OOCFc (often written as HSCn OOCFc; typically n = 8 − 18) (Figure 3.5.2). Such molecules are often diluted in the monolayer film with similar electroinactive molecules (e.g., HSCn CH3 ). The rate constant is measured as a function of the length of the alkyl chain, and the slope of the plot of ln k vs. n or x allows determination of 𝛽. For saturated chains, 𝛽 is typically in the range of 10–12 nm−1 . Since exp(−𝛽x) is greater at any distance for a smaller value of 𝛽, the difference between this through-bond value and that for vacuum (through-space), ∼20 nm−1 , suggests the facilitation of tunneling by the intervening molecular bonds. Even smaller 𝛽 values (3–6 nm−1 ) have been seen with 𝜋-conjugated spacers [e.g., those with phenyleneethynyl (−Ph − C ≡ C−) units] (69, 70). In electron-transfer theory, the tunneling probability is included in the prefactor of the rate constant as the electronic transmission coefficient, 𝜅 el (57) [Sections 3.1.3 and 3.5.3(b)]. Often, its dependence on distance is expressed as: 𝜅el (x) = exp[−𝛽(x − xa )]

(x ≥ xa )

(3.5.2)

𝜅el (x) = 1

(x < xa )

(3.5.3)

where xa is the distance from the electrode where electronic coupling between the electroreactant and the electrode becomes sufficient for the reaction to be adiabatic, a concept that we will develop next. Thus, 𝜅 el = 1 when x becomes as small as xa (or even smaller). For larger separations than xa , the tunneling probability falls off exponentially. This approximation fits real cases (65, 66, 68–70), but more complex functions of distance may be used instead (63).

3.5 Microscopic Theories of Charge Transfer

(b) Adiabatic and Nonadiabatic Reactions

In the theory of reaction kinetics, the extent of electronic interaction between two reactants in mutual proximity (or, in our case, between a reactant and the electrode) has important effects on the energy surfaces describing the system. Because of the electronic coupling, the surfaces do not make a simple intersection, such as we drew in Figures 3.3.1 and 3.3.2. Instead, they avoid crossing and separate at the point of intersection (Figure 3.5.3). The gap between the surfaces at the reaction coordinate where intersection would otherwise have occurred is called the surface splitting or curve splitting. Its magnitude is 2H ab , where H ab is the electronic coupling matrix element (74). The coupling lowers the energy of the activated complex along the main reaction path by H ab and raises the energy of the minimum formed by the upper parts of the intersecting surfaces by the same amount. There are two limiting situations: • If the electronic coupling is strong,16 the gap, 2H ab , exceeds kT. Figure 3.5.3a depicts the effect for the heterogeneous case. There is a lower surface (or curve) proceeding continuously from O to R and an upper surface (or curve) describing an excited state. Since the two surfaces are energetically well separated, a reacting system will nearly always remain on the lower surface, with momentum along the q-axis assuring that the system proceeds from O to R (or vice versa) with 𝜅 el → 1. The reaction is said to be adiabatic, meaning that it remains on a single surface. • If the electronic coupling is small (e.g., when the electroreactant is spaced away from the electrode surface), the splitting of the energy surfaces at the point of intersection is less than kT, as in Figure 3.5.3b. Momentum along the q-axis and the lack of an appreciable energy barrier lead the system to remain on the original O + e surface—or, actually, to cross from the ground-state surface to the excited-state surface. Accordingly, there is a reduced likelihood

Large splitting

Small splitting

G 0(q)

G 0(q) O+e

R

O+e

R

Reaction coordinate, q

Reaction coordinate, q

(a )

(b)

Figure 3.5.3 Splitting of energy curves (energy surfaces) for an outer-sphere electrode reaction in the intersection region. The splittings shown here are equal to 2Hab . (a) A strong interaction between O and the electrode leads to a well-defined, continuous curve (surface) connecting O + e with R. If the reacting system reaches the transition state, the probability is high that it will proceed into the valley corresponding to R, as indicated by the curved arrow. (b) A weak interaction leads to a splitting less than kT. When the reacting system approaches the transition state from the left, it tends to remain on the O + e curve, as indicated by the straight arrow. The probability of crossover to the R curve can be small. These curves are drawn for an electrode reaction, but the principle is the same for a homogeneous reaction, where the reactants and products might be O + R′ and R + O′ , respectively. 16 But not so strong as to produce chemical binding. The discussion here is about outer-sphere electrode reactions, so the electroreactant is outside the solvent layer at the electrode surface.

145

146

3 Basic Kinetics of Electrode Reactions

that the system will proceed from O to R. The process is said to be nonadiabatic, meaning that it involves a crossing from one surface to another. The probability of successful reaction (O becoming R, or vice versa) per passage through the intersection region depends on H ab and is expressed by 𝜅 el < 1 (57). If, for example, an electroreactant were spaced away by 1 nm from the plane of adiabatic reaction at an electrode (i.e., if x − xa = 1 nm), and if 𝛽 were 10 nm−1 , then according to (3.5.2), the tunneling probability would be 𝜅 el = e−10 = 4.5 × 10−5 . On average, the reactants would have to pass through the intersection region (reach the transition state) about 22,000 times for every successful electron transfer. For electrode reactions involving dissolved electroreactants, one can view the reaction as occurring over a range of distances, with the rate constant falling exponentially with distance. Electron transfer occurs in a three-dimensional zone near the electrode, rather than only in a single plane of closest approach; however, the effect for dissolved electroreactants is predicted to be distinguishable experimentally only under restricted circumstances (e.g., DO < 10−10 cm2 /s) (75). In systems with normal diffusivities, the integral of events taking place at extended distances (mostly with 𝜅 el ≪ 1) does not appreciably add to those taking place at closest approach (where 𝜅 el = 1), because all molecules in the extended electron-transfer zone (within several times 1/𝛽) can make a close approach on the relevant timescale, which is the reaction lifetime of a molecule at the surface (Problem 3.6). If the diffusivity is very small, this is no longer true, and extended electron transfer can contribute appreciably. 3.5.3

The Marcus Microscopic Model

An outer-sphere, 1e transfer from an electrode to species O, forming the product R, is closely related to the homogeneous reduction of O in a bimolecular reaction involving a reductant, R′ : O + R′ → R + O′

(3.5.4)

We will find it convenient to consider the two situations in the same theoretical context. The act of electron transfer, whether heterogeneous or homogeneous, is a radiationless electronic rearrangement of reacting species. Accordingly, there are many common elements between theories of electron transfer and those of deactivation in excited molecules (76). Since the transfer is radiationless, the electron must move from an initial state (on the electrode or in the reductant, R′ ) to a receiving state of the same energy (in species O). This demand for isoenergetic electron transfer is a fundamental aspect with extensive consequences. Another important aspect of most microscopic theories of electron transfer is the assumption that the reacting system does not alter its configuration during the actual transfer (77). This idea rests on the Franck–Condon principle, which says that nuclear momenta and positions do not change on the time scale of electronic transitions. Thus, the reactant and the product—O and R—are assumed to share a common nuclear configuration at the instant of transfer. (a) The Precursor State

In the early development of the Marcus theory,17 the reactants were assumed to encounter each other by hard-sphere Brownian motion, and the pre-exponential factor for the rate constant was written in terms of a collision number (39–41). Later (57, 78, 79), the model became restated in terms of a precursor state—a reactive pair (either O plus the electrode surface or O plus R′ 17 The presentation below is based on the theory founded by Marcus in 1956, then elaborated by Marcus and others over later decades, to describe thermally activated electron-transfer reactions (citations above). An important contributor was Hush, who gave early attention to electrode reactions. The overall body of work is often also called the Marcus–Hush theory.

3.5 Microscopic Theories of Charge Transfer

in solution) situated at the right distance for reaction, but not necessarily in the conformation of the activated complex. The precursor state is assumed to be at equilibrium with the separated reactants and to have a concentration described by an equilibrium constant, K P,O . While formed, the precursor state acts unimolecularly—as a single entity, sampling different conformations on its potential energy surface and possibly achieving the conformation corresponding to the activated complex, where electron transfer can occur, producing R from O. The equilibrium constant, K P,O , differs for the heterogeneous and homogeneous cases: • For a heterogeneous reduction, the precursor state, Opre , is species O situated near the electrode, perhaps separated by the solvated radius of O, aO . Then, KP,O = ΓO ∕CO (0, t), where pre

ΓO

pre

is the two-dimensional concentration (in mol/cm2 ) of Opre .18 Consequently, K P,O has

units of cm when C O (0, t) is in mol/cm3 . • For a homogeneous reduction of O, the precursor state, also denoted Opre , is the assembly OR′ , where the separation might be the sum of the solvated radii, aO + aR′ . Then KP,O = CO ∕CO CR′ , which has units of M−1 , if the concentrations are expressed conventionally. pre

(b) Form of the Rate Constant

Whether the reaction is heterogeneous or homogeneous, the rate is given by the unimolecular decay of Opre to produce R, which is described by a first-order rate constant, k f,pre , having units of s−1 . This rate constant operates on the concentration of Opre , which is two dimensional for the heterogeneous case, but three dimensional for the homogeneous case; thus, the reaction rate is vf = kf,pre ΓO

pre

vf = kf,pre CO

pre

(heterogeneous, mol cm−2 s−1 ) −1 −1

(homogeneous, mol L s )

(3.5.5) (3.5.6)

In either case, the concentration of Opre can be re-expressed in terms of the equilibrium constant, K P, O , vf = kf,pre KP,O CO (0, t)

(heterogeneous)

(3.5.7)

vf = kf,pre KP,O CO CR′

(homogeneous)

(3.5.8)

vf = kf CO (0, t)

(heterogeneous)

(3.5.9)

vf = kf CO CR′

(homogeneous)

(3.5.10)

In conventional form, we write

Thus, we see that the overall forward electron-transfer rate constant, k f , has a consistent expression for both cases: (3.5.11)

kf = kf,pre KP,O If k f, pre adheres to the Arrhenius form, (3.1.9), then kf = KP,O A′ exp(−ΔGf‡ ∕RT)

(3.5.12)

where ΔGf‡ is the free energy of activation for the “forward” process (the reduction of O), and A′ is a frequency factor with units of s−1 . 18 In the precursor state, species O is positioned, on a molecular scale, in the proper relationship to the electrode for reaction (at the right distance, perhaps in a favored orientation). The symbol ΓO quantifies the two-dimensional pre

concentration of species O in this condition, which would be a subset of O molecules near the electrode and included in the three-dimensional concentration, C O (0,t).

147

148

3 Basic Kinetics of Electrode Reactions ′

The prefactor, A , has been factored and expressed in many ways in the literature. Here, we ′ follow a common notation (57) and set A = 𝜈 n 𝜅 el , where 𝜈 n is the nuclear frequency factor (s−1 ), representing the frequency of attempts on the energy barrier (generally associated with changes in bond lengths, rotations about bonds, and solvent motion); and 𝜅 el is the electronic transmission coefficient [related to the probability of electron tunneling; Section 3.5.2(a)]. With this elaboration, the rate constant for reduction can be expressed as kf = KP,O 𝜈n 𝜅el exp(−ΔGf‡ ∕RT)

(3.5.13)

As noted, this relationship applies to either a heterogeneous reduction at an electrode or a homogeneous electron transfer in which O is reduced by another reactant in solution. For the heterogeneous case, K P,O has units of cm, so k f is in cm/s, as required. For a homogeneous process, K P, O has units of M−1 ; therefore, k f is in M−1 s−1 , again as required. The pre-exponential factors have been reviewed systematically (57, 80).19 Methods for estimating them are available, but there is considerable uncertainty in their values. Normally, 𝜅 el is taken as unity for an adiabatic process—a reaction in which the reactant is close to the electrode and significant electronic coupling exists between the reactant and the electrode.20 Nonadiabatic electron-transfer reactions, generally involving tunneling across a spatial separation of the reactant and electrode (or of two homogeneous reactants), are treated by using (3.5.2) or some other expression to define 𝜅 el . In such cases, 𝜅 el is usually orders of magnitude less than unity. (c) Conformational Change and Activation Energy

Now let us focus on events in the precursor state, which evolves on a multidimensional surface defining the standard free energy of the system in terms of the relative positions of the atoms. Before the electron transfer occurs, we envision the precursor state essentially as species O at the reactive distance from either the electrode or from R′ . Let us call this the “O configuration.” After electron transfer, the same assembly of atoms is, in essence, species R at the reactive distance from either the electrode or O′ . This we call the “R configuration.” A transition from the O configuration to the R configuration, or vice versa, corresponds to electron transfer, which can happen only when the precursor state distorts itself into a form that is common to the O and R configurations.21 The mean separation of the two main components of the precursor state (e.g., the O/R frame plus the electrode) is assumed to remain constant during the life of the precursor state. Thus, changes in nuclear coordinates come about from vibrational and rotational motion, rather than relative translation of the two main parts. Marcus examined the manner in which the free-energy surfaces of the O and R configurations depend upon all vibrational and rotational modes (41). He was able to show that, along the most 0 and G0 , behave likely reaction path, the standard free energies of the two configurations, GO R simply, depending quadratically on a single effective coordinate, q (41, 54): 0 GO (q) = (k∕2)(q − qO )2 GR0 (q) = (k∕2)(q − qR )2 + ΔG0

(J∕mol) (J∕mol)

(3.5.14) (3.5.15)

19 The pre-exponential term sometimes also includes a nuclear tunneling factor, Γn , which is disregarded here. It arises from a quantum mechanical treatment that accounts for electron transfer for nuclear configurations with energies below that of the transition state (50, 56, 80). 20 But not chemical bonding. See Section 3.5.2(b). 21 In the literature, the O configuration is sometimes called the precursor complex, while the R configuration is termed the successor complex. This language is appropriate if one consistently thinks of the O configuration as “the

3.5 Microscopic Theories of Charge Transfer

Here, qO and qR are the values of the coordinate for the equilibrium O and R configurations, and k has the character of a force constant. It is natural to refer to the Marcus coordinate, q, as the ′ “reaction coordinate,” and we employ that term below. In (3.5.15), ΔG0 is either F(E − E0 ) for an electrode reaction22 or the overall free-energy change for a homogeneous electron transfer, depending on the case under consideration.23 On the basis of (3.5.14) and (3.5.15), one can represent a reacting system in terms of two intersecting parabolas in a plot of standard free energy vs. the Marcus coordinate, as illustrated in Figure 3.5.4a for the heterogeneous case. The activated complex is the position where (b) O3+

R2+ +e

(c)

(a) Standard free energy

e transfer

O+e qO q‡

R

qR

O+e 𝜆 G‡ GO0(qO) GR0(qR)

R

ΔGf‡ ΔGb‡

F(E − E0′)

qO

q‡

qR

Marcus coordinate, q

Figure 3.5.4 (a) Standard free energy vs. Marcus coordinate for an electron-transfer reaction in a precursor complex, e.g., for Ru(NH3 )3+ + e → Ru(NH3 )2+ . Labeled for a heterogeneous reaction, but also applicable to 6 6 the homogeneous case (see text). The curve for the O configuration includes the energy of an electron on the electrode at the Fermi energy corresponding to potential E. Curve splitting near the activated complex is depicted. It creates a lower-energy path connecting O and R (solid curve), plus an excited state (dashed line). (b) Schematic depiction of structural alterations accompanying electron transfer, e.g., changes in bond lengths within the electroactive species or the restructuring of the solvent shell. (c) As the reaction path is traversed from qO to qR (or vice versa), the chemical identity of the complex changes as indicated in the shaded bar. Electron transfer occurs only when the precursor state attains a conformation that is common to the energy surfaces for both O + e and R. reactant” and the R configuration as “the product.” We are generally interested in reactions in both directions, so we use one name, precursor state, for the reactive assembly, while distinguishing the O and R configurations. 22 In the literature (and in the second edition of this book), one can find ΔG0 for an electrode reaction given as ′ F(E – E0 ). Since the rate expressions are based on concentrations, it is more appropriate to use F(E – E0 ). Marcus himself employed the formal potential in his original work on heterogeneous kinetics (41). 23 The development of Marcus theory given here essentially implies that free energies are expressed in J/mol, ′ as indicated in (3.5.14) and (3.5.15). This point is implicit in (3.5.13) and in the identification of ΔG0 as F(E – E0 ) for ′ a heterogeneous reaction, since F and E are always expressed in C/mol and V, respectively, the product F(E – E0 ) is in J/mol.

149

150

3 Basic Kinetics of Electrode Reactions

distorted O and R configurations have the same structure and energy, viz. the intersection point at q‡ . The reaction path for the forward process proceeds rightward along the O + e curve from qO , through the activated complex at q‡ , where the electron transfer happens, then along the R curve until the minimum at qR is reached. Figure 3.5.4a also applies to the homogeneous case, where the curve for the O configuration describes O and its reactant partner, R′ , while the curve for the R configuration describes R and O′ together. Then, the energy difference between the minima is ΔG0 for the reaction. Figure 3.5.4b is a stylized representation of the stable configurations of the participants (perhaps Ru(NH3 )3+ and Ru(NH3 )2+ as species O and R at an electrode). This diagram also sym6 6 bolically expresses the change in nuclear configuration upon reduction. The free energies at the transition state are given by24 0 ‡ GO (q ) = (k∕2)(q‡ − qO )2

(3.5.16)

GR0 (q‡ ) = (k∕2)(q‡ − qR )2 + ΔG0 0 (q‡ ) = G0 (q‡ ), (3.5.16) and (3.5.17) can be solved for q‡ Since GO R (q + ΔG0 R qO ) ‡ q =

2

+

(3.5.17) with the following result:

k(qR − qO )

(3.5.18)

The free energy of activation for reduction is given by 0 ‡ ΔGf‡ = GO (q )

(3.5.19)

0 (q ) = 0, as defined in (3.5.14). Substitution for q‡ since all free energies are referenced to GO O from (3.5.18) into (3.5.16) followed by rearrangement yields [ ]2 k(qR − qO )2 2ΔG0 ‡ ΔGf = 1+ (3.5.20) 8 k(qR − qO )2

Defining 𝜆 = (k/2)(qR − qO )2 , we have [ ]2 ′ F(E − E0 ) 𝜆 ‡ 1+ ΔGf = 4 𝜆 ΔGf‡ =

( )2 𝜆 ΔG0 1+ 4 𝜆

(electrode reaction)

(3.5.21)

(homogeneous reaction)

(3.5.22)

If work is required to bring the reactants or the products together (e.g., to bring a positively charged reactant to a reactive position at a positively charged electrode), that work is part of the overall standard free energy of reaction, but is not included in Figure 3.5.4a, which describes only the precursor state. Expressed more completely, the energy difference between the minima in Figure 3.5.4 is ΔG0 + wR − wO , where wO and wR are the work terms expressing, respectively, the free energy required to form the O configuration of the precursor complex from separated O-side components (O and either the electrode or R′ ) and the free energy required to form the R configuration from separated R-side components (R and either the electrode or O′ ). For simplicity, the work terms were omitted in the derivation above, but they can be included by replacing ΔG0 everywhere in (3.5.15)–(3.5.22) with ΔG0 + Δw, where Δw = wR − wO . For an 24 This treatment does not include the effect of electronic coupling [Section 3.5.2(b)], which lowers the energy of the transition state by the amount H ab . This term can be easily added and would eventually manifest itself as a constant factor in the pre-exponential of the rate constant. We omit it here for simplicity. Nothing in the later discussion is appreciably affected by the omission.

3.5 Microscopic Theories of Charge Transfer ′

electrode reaction, the replacement is F(E − E0 ) + Δw. The complete equations, including the work terms, are, therefore,25 ( )2 ′ F(E − E0 ) + Δw 𝜆 ‡ ΔGf = 1+ (electrode reaction) (3.5.23) 4 𝜆 ΔGf‡

( )2 𝜆 ΔG0 + Δw = 1+ 4 𝜆

(homogeneous reaction)

(3.5.24)

(d) Reorganization Energy

A key parameter in the Marcus theory is 𝜆, the reorganization energy, which quantifies the energy requirement for restructuring the precursor complex. The broad, gray arrow on the right side of Figure 3.5.4a illustrates the scale of 𝜆, which is not the same as the activation energy in either direction, but is a parameter determining the activation energies. We saw just above that 𝜆 is defined as (k/2)(qR − qO )2 ; thus, it is physically the free-energy change required to distort the O configuration, without transferring an electron, from its energy minimum at qO to the coordinate, qR , corresponding to the energy minimum of the R configuration. This relationship is depicted in Figure 3.5.4a. The reorganization energy is usually separated into “inner” and “outer” components, 𝜆i and 𝜆o , respectively: 𝜆 = 𝜆i + 𝜆o

(3.5.25)

where 𝜆i relates to the central elements of the precursor state (reactants, products, or electrode), and 𝜆o , to peripheral elements, mainly the solvent.26 To the extent that the normal modes of the precursor complex remain harmonic over the range of distortion needed, one can, in principle, calculate 𝜆i by summing over the normal vibrational modes of the complex, that is, ∑ 2 1/2k (q 𝜆i = (3.5.26) j O,j − qR,j ) j

where the k’s are force constants, and the q’s are displacements in the normal mode coordinates. Typically, 𝜆o is computed by assuming that the solvent is a dielectric continuum and the reactant is a sphere (41). For an electrode reaction in which species O is reduced, ) ( )( e2 1 1 1 1 − − (3.5.27) 𝜆o = 8𝜋𝜀0 aO R 𝜀op 𝜀s where aO is the solvated radius of species O, and 𝜀op and 𝜀s are the optical and static dielectric constants, respectively. The distance R is twice the distance from the center of the molecule to the electrode, which is the distance between the center of the reactant and the center of its image charge in the electrode.27 In the usual case, where the electrode reaction is assumed to happen 25 The work terms are the standard free energy changes for the precursor equilibria; therefore, wO = −RTln K P,O and wR = −RTln K P,R . 26 One must not confuse the inner and outer components of 𝜆 with the concepts of inner- and outer-sphere reaction. In the treatment under consideration, we are dealing with an outer-sphere reaction, and 𝜆i and 𝜆0 simply apportion the reorganization energy between the main participants and the solvation sphere. 27 In some treatments of electron transfer, the assumption is made that the reactant charge is largely shielded by counter ions in solution, so that an image charge does not form in the electrode. In that case, R is the distance between the center of the reactant molecule and the electrode (26, 44).

151

152

3 Basic Kinetics of Electrode Reactions

at the plane of closest approach, R = 2aO . For a homogeneous electron-transfer reaction: ) ( )( e2 1 1 1 1 1 𝜆o = + − − (3.5.28) 4𝜋𝜀0 2a1 2a2 d 𝜀op 𝜀s where a1 and a2 are the solvated radii of the reactants (O and R′ in equation 3.5.4) and d is the distance between the centers of the reactants in the precursor complex. Normally, d = a1 + a2 . Typical values of 𝜆 are in the range of 30–150 kJ/mol (0.3–1.5 eV/molecule, normally expressed as “0.3–1.5 eV”).28 The component 𝜆o is commonly 0.3–0.5 eV. For an electroreactant near an electrode, evidence has been presented (81) that the value of 𝜆o is spatially dependent and can become as small as 0.1 eV when the separation from the electrode is reduced to 0.4 nm. In classic Marcus theory, the reorganization energy is the same on both sides of the reaction path. A recent adaptation (82) allows the O configuration and the R configuration to have different reorganization energies, in recognition that significantly different sets of force constants might apply on the two sides of the precursor complex. 3.5.4

Implications of the Marcus Theory

In principle, it is possible to estimate the rate constant for an electrode reaction by computation of the pre-exponential terms and the 𝜆 values; however, this is rarely done in practice. The greater value of the Marcus theory is in the chemical and physical insight that it affords, which is based on the predicted behavior of activation energy with changes in 𝜆 or electrode potential. (a) Comparison to Butler–Volmer Kinetics

For electrochemistry, a principal point of interest is how the Marcus theory’s description of heterogeneous kinetics differs from that of the Butler–Volmer model. To address that issue, we begin with the same electrode reaction used for construction of the BV model, kf

−−− ⇀ O+e ↽ − R

(3.5.29)

kb

From Marcus theory, we can re-express ΔGf‡ from (3.5.23) as ′

′

𝜆 F(E − E0 ) + Δw [F(E − E0 ) + Δw]2 + + (3.5.30) 4 2 4𝜆 where Δw = wR − wO . Equation 3.5.30 can be expanded, then factored to isolate the potentialdependent terms, with the following result: [ ] ′ ′ 𝜆 Δw Δw2 1 F(E − E0 ) Δw ‡ ΔGf = + + + + + F(E − E0 ) (3.5.31) 4 2 4𝜆 2 4𝜆 2𝜆 ΔGf‡ =

If we identify 𝛼 as the factor in brackets, ′

𝛼=

1 F(E − E0 ) Δw + + 2 4𝜆 2𝜆

(3.5.32)

28 The development of the Marcus theory often involves energies expressed in joules (footnote 23), but much analysis is based on energies expressed on a molecular basis. The most convenient energy unit for that purpose is the electron-volt (eV). One can convert an energy from J/mol to eV/molecule by dividing by both the Avogadro constant, N A (molecule/mol), and the charge on an electron, e (C/electron). Since the product of the divisors, N A e, is the Faraday constant, the conversion factor is simply F. A single molecule or single electron often remains unexpressed in units, so energies meant as eV/molecule are usually just written and spoken as eV. For the same reason, e and F are normally given in C and C/mol, respectively, rather than C/electron and C molecule mol−1 electron−1 .

3.5 Microscopic Theories of Charge Transfer

then ′ 𝜆 Δw Δw2 + + + 𝛼F(E − E0 ) (3.5.33) 4 2 4𝜆 For the backward reaction, we find from (3.5.16) and (3.5.17)—or just from inspection of Figure 3.5.4—that

ΔGf‡ =

′

ΔGb‡ = ΔGf‡ − ΔG0 = ΔGf‡ − F(E − E0 ) − Δw

(3.5.34)

Therefore, ΔGb‡ =

′ 𝜆 Δw Δw2 − + − (1 − 𝛼)F(E − E0 ) 4 2 4𝜆

(3.5.35)

The expressions for ΔGf‡ and ΔGb‡ derived from Marcus theory, (3.5.33) and (3.5.35), have the same form as in the BV model [(3.3.3) and (3.3.4)], except that 𝛼 is not a constant in the Marcus theory, but depends on potential according to (3.5.32). Using (3.5.13) with (3.5.33) and (3.5.35), we can express the two rate constants as 2 ∕4𝜆RT

kf = KP,O 𝜈n 𝜅el e−𝜆∕4RT e−Δw∕2RT e−Δw kb = KP,R 𝜈n 𝜅el e−𝜆∕4RT eΔw∕2RT e

−Δw2 ∕4𝜆RT

e

e−𝛼f (E−E

0′ )

(3.5.36)

′ (1−𝛼)f (E−E0 )

(3.5.37)

0′

When E = E , these rate constants comprise only the factors preceding the last exponential. Since KP,O = e−wO ∕RT and KP,R = e−wR ∕RT (footnote 25), we find the same result in either case: ′

2 ∕4𝜆RT

′

kf (E0 ) = kb (E0 ) = k 0 = 𝜈n 𝜅el e−𝜆∕4RT e−(wO +wR )∕2RT e−Δw

(3.5.38)

0′

Thus, we have verified the equality of k f and k b at E , which must be true for any valid kinetic model, as shown in Section 3.3.2. Now, we can write (3.5.36) and (3.5.37) in the familiar forms: kf = k 0 e−𝛼f (E−E

0′ )

kb = k 0 e(1−𝛼)f (E−E

(3.5.39a) 0′ )

(3.5.39b)

From (3.5.39a,b) flow the implications of the BV model presented in Sections 3.3 and 3.4. Many, but not all, important BV equations also apply to Marcus kinetics. Here is a summary of particulars: 1) The BV i − E characteristic, (3.3.8), is valid for Marcus kinetics, but with a potentialdependent 𝛼 defined by (3.5.32). 2) The Nernst equation, (3.4.3), falls out of the i − E characteristic for Marcus kinetics at i = 0, just as for BV kinetics. 3) The general BV equation for the exchange current, (3.4.6), also applies for Marcus kinetics, but 𝛼 must be taken as 𝛼 eq , the value at E = Eeq . The BV equation for the special case of ∗ = C ∗ , (3.4.7), is valid for Marcus kinetics. CO R 4) The BV current–overpotential equation, (3.4.10), is not generally applicable to Marcus kinetics, even with recognition of the potential-dependent 𝛼. One can show (Problem 3.13) that the i − 𝜂 equation for Marcus kinetics is [ ] ′ CR (0, t) (1−𝛼)f 𝜂 −F 2 𝜂(Eeq −E0 )∕4𝜆RT CO (0, t) −𝛼f 𝜂 i = i0 e e − e (3.5.40) ∗ CO CR∗

153

154

3 Basic Kinetics of Electrode Reactions ′

For the important special case of Eeq = E0 , the i − 𝜂 equation for Marcus kinetics adopts the same form as for BV kinetics. To use (3.5.40), one needs 𝛼 as a function of 𝜂. Equation 3.5.32 is easily re-expressed as ′

0 1 F𝜂 Δw F(Eeq − E ) 𝛼= + + + 2 4𝜆 2𝜆 4𝜆

(3.5.41)

5) In general, BV equations written in terms of 𝜂 or i0 [including the Tafel equation, (3.4.15), the exchange–current relationships in Section 3.4.4, and the mass-transfer-limited relation′ ships in Section 3.4.6] are not directly applicable to Marcus kinetics, except when Eeq = E0 . Complications arise in the use of BV equations for Marcus kinetics whenever potentials are ′ referenced in the equations to Eeq ≠ E0 . 6) The linear BV relationships for small 𝜂, (3.4.12) and (3.4.13), are valid for Marcus kinetics. 7) In the reversible case, the i − 𝜂 equation for Marcus kinetics, (3.5.40), collapses to the nernstian relationship linking the electrode potential and surface concentrations, (3.4.28), just as for BV kinetics. (b) Effect of the Reorganization Energy on the Standard Rate Constant

Let us assume two reacting systems having the same values of 𝜈 n , 𝜅 el , wO , and wR , but with different values of reorganization energy, 𝜆1 and 𝜆2 . The ratio of the corresponding standard rate constants, k10 and k20 , can be written from (3.5.38) as 2 ∕4𝜆 RT Δw2 ∕4𝜆 RT 2 1 e

k20 ∕k10 = e−(𝜆2 −𝜆1 )∕4RT e−Δw

(3.5.42)

Since Δw2

is normally much less than 4𝜆RT, the last two exponential factors are approximately unity. Their product is even closer to unity, because the exponents are offsetting; thus, (3.5.42) becomes k20 ∕k10 ≈ e−(𝜆2 −𝜆1 )∕4RT

(3.5.43)

or log

k20 k10

≈−

𝜆2 − 𝜆1 2.303 × 4RT

(𝜆1 and 𝜆2 in J∕mol)

(3.5.44)

It is more usual for 𝜆1 and 𝜆2 to be considered in eV per molecule. Applying the conversion factor (division by the Faraday constant; footnote 28) to both numerator and denominator, we obtain k0 F(𝜆2 − 𝜆1 ) log 20 = − (𝜆1 and 𝜆2 in eV) (3.5.45) 2.303 × 4RT k1 At 25 ∘ C, this relationship is log

k20 k10

=−

(𝜆2 − 𝜆1 ) 0.236

(𝜆1 and 𝜆2 in eV)

(3.5.46)

For each 236 meV difference between 𝜆1 and 𝜆2 , there is an order of magnitude difference in the standard rate constants at 25 ∘ C, with the larger value of k 0 corresponding to the smaller 𝜆. In general, k 0 will be larger for reactions in which O and R have similar structures. Electron transfers involving large structural alterations (such as sizable changes in bond lengths or bond angles) tend to be slower. Solvation also has an impact through its contribution to 𝜆. Large molecules (large aO ) tend to show lower solvation energies, and smaller changes in solvation upon reaction, by comparison with smaller species. On this basis, one can understand why

3.5 Microscopic Theories of Charge Transfer

electron transfers to small molecules, such as the reduction of O2 to O2 −∙ in aprotic media, tend to be slower than the reduction of Ar to Ar−∙, where Ar is a large aromatic molecule like anthracene. The role of the solvent in an electron transfer goes beyond its contribution to 𝜆o . There is evidence that the dynamics of solvent reorganization, often represented in terms of a solvent longitudinal relaxation time, 𝜏 L , contributes inversely to the pre-exponential factor in (3.5.13) (50, 53, 83–86). Since 𝜏 L is roughly proportional to the viscosity, the implication is that k 0 for an outer-sphere electrode reaction would decrease as the solution viscosity increases (i.e., as the diffusion coefficient of the reactant decreases). Such behavior has been reported in several cases (87–89). (c) Behavior of the Transfer Coefficient

′

Equation 3.5.32 predicts that 𝛼 for any outer-sphere electrode reaction is 0.5 + Δw/2𝜆 at E0 . ′ Since Δw/2𝜆 is normally small, (3.5.32) implies that 𝛼 is typically near 0.5 at E0 , but changes linearly with potential, becoming smaller toward more negative potentials. Figure 3.5.5 illustrates this behavior for three different values of reorganization energy in the expected range for outer-sphere electrode reactions [lines (b)–(d)]. The variation in 𝛼 predicted for Marcus kinetics can cover a sizable span and is stronger for smaller 𝜆. Figure 3.5.5a represents 𝛼 for BV kinetics, for which the transfer coefficient is invariant with potential and is not constrained in value. It is a fully adjustable parameter for fitting data to the BV model. ′ For Marcus kinetics, the transfer coefficient has a fixed value 0.5 + Δw/2𝜆 at E = E0 . The fully adjustable parameters are 𝜆 and Δw. In some cases, the latter can be justifiably neglected. Experimental results consistent with these expectations of Marcus kinetics have been reported (82, 90–93). (d) Rate Constants and Current–Potential Curves

The potential dependence of 𝛼 in Marcus kinetics has very significant consequences for the heterogeneous rate constants, k f and k b . Curves (b) and (c) in Figure 3.5.6 illustrate the effect on 1.0

0.8

(b) (c)

0.6 (a)

α

(d)

0.4

0.2

0.0 500

250

0

−250

−500

E − E0′(mV)

Figure 3.5.5 Transfer coefficient vs. potential. (a) BV kinetics, 𝛼 = 0.5; (b) Marcus kinetics, 𝜆 = 0.4 eV, Δw = 0 eV; (c) Marcus kinetics, 𝜆 = 0.8 eV, Δw = 0 eV; (d) Marcus kinetics, 𝜆 = 1.2 eV, Δw = 0 eV.

155

156

3 Basic Kinetics of Electrode Reactions

4.0 (a)

2.0

(c) (b)

0.0

log k0

−2.0 log kf

−4.0 −6.0 −8.0

−10.0 500

250

0

−250

−500

E − E0′(mV)

Figure 3.5.6 Dependence of kf on potential for k0 = 10−2 cm/s. (a) BV kinetics, 𝛼 = 0.5; (b) Marcus kinetics, 𝜆 = 0.4 eV, Δw = 0 eV; (c) Marcus kinetics, 𝜆 = 0.8 eV, Δw = 0 eV. The finely dotted section in (b) shows the prediction for the inverted region [Section 3.5.4(e)] based on activation energy alone. Nonadiabaticity would further suppress the rate constant in the inverted region, but no attempt is made to illustrate it here. Inversion is inapplicable at a metal electrode [Section 3.5.4(f )].

k f for two cases involving Marcus kinetics. In contrast with the result for the BV model [curve (a)], k f for the Marcus cases rolls off on both the positive and negative wings. The roll-off is greater for the smaller reorganization energy, reflecting the stronger potential-dependence of 𝛼 for smaller 𝜆. For k b , the picture is essentially the same, but in mirror image, with large values of k b at positive potentials and small values on the negative side. For BV kinetics, i − E curves are derived from the rate constants as described in Sections 3.3 and 3.4. On the basis of the behavior discovered here for Marcus kinetics, we can expect i − E or ′ i − 𝜂 curves that largely coincide with those given by BV kinetics within perhaps 100 mV of E0 , but with significantly smaller currents on the wings. Voltammetric responses under Marcus kinetics should, therefore, be more drawn out. These expectations are confirmed in theoretical steady-state voltammograms comprehending both kinetic and mass-transfer effects (94). In Figure 3.5.7, one can see examples computed using the method of Section 3.4.6, but with rate constants as given by Marcus kinetics. As expected (Section 3.4.6), the curves show an exaggerated inflection at 𝜂 = 0 when the exchange current, i0 , is much smaller than the mass-transfer-limited current. This behavior appears because k f or k b must be activated by a significant overpotential to deliver currents much larger in magnitude than i0 . The two frames of Figure 3.5.7 show predictions for two different ratios of i0 /il . In each frame, one can compare the expectations from BV kinetics and Marcus kinetics. An interesting prediction (94) for sufficiently small values of i0 /il and 𝜆 is that Marcus kinetics never become sufficiently activated to deliver the mass-transfer-limited current, il . At high overpotentials, the current instead reaches a plateau at a level below il . One can see this behavior in Figure 3.5.7. Experimental work (63, 64) has confirmed the prediction. This behavior is in sharp contrast with the BV model, where there is no limit to rate activation at high overpotentials.

3.5 Microscopic Theories of Charge Transfer

1.0

1.0

i0 /il = 0.1

i/il 0.0

i/il 0.0

–1.0

–1.0

500

250

0

–250

–500

i0 /il = 0.01

500

250

0

𝜂(mV)

𝜂(mV)

(a)

(b)

–250

–500

Figure 3.5.7 Steady-state current–overpotential curves analogous to those in Figure 3.4.6. The reaction is ′ O + e ⇌ R with T = 298 K, il, c = − il, a = il , and E eq = E 0 . Values of i0 /il are indicated in each frame. Solid curves: BV kinetics with 𝛼 = 0.5 (same as in Figure 3.4.6). Dashed curves: Marcus kinetics with 𝜆 = 0.4 eV and Δw = 0.0 eV [finely dotted sections for inverted regions, inapplicable at a metal electrode; Section 3.5.4(f )]. Dotted curves: Marcus kinetics with 𝜆 = 0.8 eV and Δw = 0.0 eV. The 𝛼-functions for the Marcus cases are presented in Figure 3.5.5.

Because 𝛼 changes continuously in Marcus kinetics, Tafel plots are curved, not linear, as shown in Figure 3.5.8 (7).29 (e) Inverted Regions

The Marcus theory makes a notable prediction about reactions having large driving forces. To illuminate it, let us express the cathodic activation energy for an electrode reaction in a dimensionless format: ( )2 ′ ΔGf‡ 1 E − E0 + Δw = 1+ (energies in eV) (3.5.47) 𝜆 4 𝜆 This equation is obtained from (3.5.23) by dividing both sides by 𝜆 and by understanding 𝜆, Δw, and ΔGf‡ to be in eV. In a similar manner, we can also express the anodic activation energy dimensionlessly: ( )2 ′ ′ ΔGb‡ 1 E − E0 + Δw E − E0 + Δw = 1+ − (energies in eV) (3.5.48) 𝜆 4 𝜆 𝜆 ′

Figure 3.5.9, showing ΔGf‡ ∕𝜆 and ΔGb‡ ∕𝜆 vs. (E − E0 + Δw)/𝜆, is a universal presentation, applicable to all outer-sphere O/R electrode reactions, regardless of their particular values of 𝜆, Δw, ′ or E0 . Let us focus now on the behavior of ΔGf‡ (solid curve). At the vertical axis, the potential is ′

E0 − Δw. As E moves more positively from there (toward the left), ΔGf‡ rises monotonically, and the corresponding rate constant, k f , would steadily fall, generally as one sees in Figure 3.5.6 for a specific case, and qualitatively the same as for BV kinetics. 29 This statement applies to a one-step, one-electron process. Tafel plots for complex mechanisms (Section 3.7 and Chapter 15) are often curved for other reasons (7).

157

158

3 Basic Kinetics of Electrode Reactions

1.0 (a) 0.0 –1.0

(c)

–2.0

(b)

log i

–3.0 log i0

–4.0 –5.0

–100

0

–200

–300

–400

–500

𝜂(mV) ′

Figure 3.5.8 Tafel plots for O + e → R with i0 = 10−4 A/cm2 , T = 298 K, and E eq = E 0 . (a) BV kinetics with 𝛼 = 0.5. Dotted section shows extrapolation for 𝜂 < 118 mV. (b) Marcus kinetics with 𝜆 = 0.4 eV and Δw = 0.0 eV [dotted section for inverted region, inapplicable at a metal electrode; Section 3.5.4(f )]. (c) Marcus kinetics with 𝜆 = 0.8 eV and Δw = 0.0 eV. The 𝛼-functions for the Marcus cases are presented in Figure 3.5.5.

3.0

ΔGf‡/𝜆 (cathodic)

ΔGb‡/𝜆 (anodic)

2.0 ΔG‡/𝜆 1.0

0.0 Cathodic

Normal

Inverted

Inverted 2.5

2

1.5

Anodic

Normal

–1.0 1

0.5

0

–0.5

–1

–1.5

–2

–2.5

(E − E 0′ + Δw)/𝜆

Figure 3.5.9 Dimensionless presentation of activation energies for an electrode reaction vs. potential, also indicating the normal and inverted regions for cathodic and anodic directions.

3.5 Microscopic Theories of Charge Transfer

As the potential is moved negatively from the vertical axis, the behavior is more complicated. If the potential remains only modestly negative, ΔGf‡ falls, and k f becomes steadily larger, also as

seen in Figure 3.5.6, and qualitatively as expected from the BV model. However, ΔGf‡ reaches ′

′

zero when (E − E0 + Δw)/𝜆 = − 1 (i.e., when E − E0 ≈ − 𝜆), then increases at more extreme potentials. A consequence is that k f would not rise monotonically in the negative region (as for ′ BV kinetics), but would flatten out near E − E0 ≈ − 𝜆, and then fall beyond that point.30 This behavior is seen in Figure 3.5.6b, which corresponds to 𝜆 = 0.4 eV. It is not seen in Figure 3.5.6c, ′ corresponding to 𝜆 = 0.8 eV, because the depicted range of E − E0 does not extend to −0.8 V. This feature of the Marcus theory can be understood in terms of the free-energy surfaces for O + e and R, discussed earlier in connection with Figure 3.5.4. In Figure 3.5.10, there is an elaboration of Figure 3.5.4 showing how the relationship between the free-energy surfaces (depicted as curves) shifts as the electrode potential is changed. Since the curve for O + e includes the energy of an electron on the electrode, this curve has vertical position that depends on E. Several different curves for O + e are presented in Figure 3.5.10, each corresponding to a particular potential. These curves would not change shape or location along the reaction coordinate unless the potential were to influence the structure of the precursor state. For an outer-sphere electrode reaction, any such effect is assumed to be negligible. In contrast, the curve for R is independent of potential, because R is the only participant on its side of the process, and it is located entirely in solution. There is no sensitivity to the difference in potential across the interface. In any of the cases presented in Figure 3.5.10, the cathodic activation energy, ΔGf‡ , is the height difference between the minimum of O + e curve and its intersection with the R curve. Likewise, the anodic activation energy, ΔGb‡ , is the height difference between the minimum of the R curve and that same intersection. ′ For Case 0 (solid line for O + e), the potential, E, is at E0 for the couple. Since work terms are neglected in this discussion, the minima for O + e and R are at the same energy, and ΔGf‡ = ΔGb‡ . From this starting point, we proceed through the other cases: ′

• For Cases N1, N2, and N3 (dashed curves), E is progressively more negative than E0 , and ′ ΔGf‡ declines from Case 0 through Case N2. In the latter instance, E = E0 − 𝜆, and the intersection is exactly at the minimum of the O + e curve, so that ΔGf‡ = 0. For any more negative potential, as in Case N3, the intersection point is on the “back side” of the O + e curve (q‡ < qO ), and ΔGf‡ becomes steadily greater as the potential grows more extreme.

Thus, we have encountered a reversal of normal trends for both ΔGf‡ and k f as the driving ′

force for reaction increases. The potential region where this occurs, E < E0 − 𝜆, is called the inverted region for the cathodic process. For the anodic process, there is no inversion: ΔGb‡ is steadily larger, and k b is steadily smaller through Cases N1−N3. ′ • For Cases P1−P3 (dotted curves), we consider progressively more positive potentials vs. E0 . The behavior with respect to ΔGb‡ in this sequence is essentially the same as for ΔGf‡ in Cases N1−N3. From Case 0 through Case P2, ΔGb‡ declines. In the latter instance, the potential is ′

E0 + 𝜆, and the intersection of the O + e and R curves lies exactly at the minimum of the R curve, where ΔGb‡ = 0. At any more positive potential, including Case P3, the system operates 30 A rollover in the rate constant at high overpotential is an important prediction of the Marcus theory, amplified carefully in the next paragraphs. However, the reader is cautioned that the effect is not expected to appear in all experimental situations. This point is discussed in Section 3.5.4(f ), after the basic presentation of the effect.

159

3 Basic Kinetics of Electrode Reactions

E < E0′ − 𝜆 qO min. O + e

qR min. R

E = E0′ − 𝜆 E = E0′

Free energy

160

E = E0′ + 𝜆 N3 E > E0′ − 𝜆 N2 N1

0

P1

P2

P3

Curve for R All others for O + e

Reaction coordinate, q

Figure 3.5.10 Curves representing the intersection of free-energy surfaces for O + e and R in the outer-sphere electrode reaction O + e ⇌ R. The broad gray curve, having a minimum at qR , represents R at all potentials. The other curves, all having minima at qO , represent O + e at different potentials (indicated at right). These curves, and their intersections with the R curve, are labeled to facilitate discussion in the text. Work terms are ignored in this figure and the accompanying discussion. To preserve clarity, the curve splitting expected at each intersection is not depicted.

in the inverted region for the anodic process, where the intersection point is on the “back side” of the R curve (q‡ > qR ). In this region, ΔGb‡ becomes progressively larger at more extreme ′

potentials. Thus, we see the same kind of trend reversal for the anodic process at E > E0 + 𝜆 ′ that we saw for the cathodic process at E < E0 − 𝜆. For the cathodic process, there is no inver‡ sion in Cases P1−P3: ΔGf is steadily larger with increasing potential, maintaining normal behavior. The essence of inversion is that activation energy increases (and the corresponding rate constant decreases) with increasing driving force for a reaction. Normal behavior is precisely opposite: Activation energy decreases with increasing driving force, and the rate constant grows. There are two physical reasons for the effect of inversion: • A large driving force implies that the products are required to accept the liberated energy very quickly in vibrational modes, and the probability for doing so declines as −ΔG0 exceeds 𝜆 (Section 20.5.1). • The inverted region always features an intersection on the “back side” of the curve for the reactant configuration, where the curve splitting resulting from electronic coupling causes the reactant minimum and the product minimum to lie on different surfaces (Figure 3.5.11). Consequently, an electron-transfer reaction always becomes nonadiabatic in an inverted region. The momentum of a reacting system along the q-axis must actually be reversed for reaction to occur in the inverted region. Since 𝜅 el ≪ 1, the inhibition of reaction rate should be much greater than one would expect from the reversed trend in activation energy alone.

3.5 Microscopic Theories of Charge Transfer

O+e

R

O+e

O+e minimum

Standard free energy

R R minimum q‡

qO

qR

Marcus coordinate, q ′

Figure 3.5.11 Marcus free-energy curves in the cathodic inverted region (E < E 0 − 𝜆). The minimum energy configurations for O + e and for R lie on different surfaces, so the reduction is nonadiabatic. This situation applies to any reaction in an inverted region.

(f) Effect of the Electrode Material

For heterogeneous electron transfer, the Marcus theory entails an implicit assumption that electrons react from a narrow range of states on the electrode corresponding to the Fermi energy (caption to Figure 3.5.4). All implications of the theory presented above rest on that idea. However, the electrode material can introduce some additional effects especially relevant to kinetic behavior in the inverted regions. For example, inversion is expected to be unobservable for electron transfer at a metal electrode because of a mechanism that we have not yet recognized (43). Although the reaction rate at the Fermi energy is indeed predicted to show inversion at very negative overpotentials, there remains a continuum of occupied states in the metal below the Fermi energy, and they can transfer an electron to O without inversion. Any low-level vacancy created in the metal by such a transfer would be filled ultimately with an electron from the Fermi energy, with dissipation of the difference in energy as heat; thus, the overall energy change would be as expected from thermodynamics. A parallel argument holds for oxidations at metals, where unoccupied states are always available above the Fermi energy to receive the transferred electron. Here we have our first indication that the electronic structure of the electrode can be important in the kinetics of an outer-sphere electrode reaction. In Sections 3.5.5 and 21.2, we will develop that idea much more fully. Even though rate inversion is not expected for electron transfer at metals, the flattening of heterogeneous rate constants at high overpotential is still predicted by Marcus theory for such systems and has been observed experimentally (65, 66, 93). In principle, rate inversion can occur for heterogeneous electron transfer at nonmetallic electrodes, such as semiconductors (Chapter 20) or conducting polymers (Section 20.2.3), if the electronic structure of the electrode does not support the mechanism outlined above

161

162

3 Basic Kinetics of Electrode Reactions

for metals. Examples of rate inversion at nonmetallic electrodes have indeed been reported (95–100); however, experimental manifestations of inversion remain rare in heterogeneous electrochemistry. (g) Predictions for Homogeneous Reactions

The Marcus theory also describes the relationship between the homogeneous and heterogeneous kinetics for a given reactant. Consider the rate constant for the self-exchange reaction, kex

O + R −−−→ R + O

(3.5.49)

in comparison with k 0 for the related electrode reaction, O + e → R. One can determine k ex by labeling O isotopically and measuring the rate at which the isotope appears in R, or sometimes by other methods, like ESR or NMR. A comparison of (3.5.27) and (3.5.28), where aO = a1 = a2 = a and R = d = 2a, yields 𝜆el = 𝜆ex ∕2

(3.5.50)

where 𝜆el and 𝜆ex are the values of 𝜆o for the electrode reaction and the self-exchange reaction, respectively. For self-exchange, ΔG0 = 0, so (3.5.21) gives ΔGf‡ = 𝜆ex ∕4, as long as 𝜆o dominates 𝜆i in the reorganization energy. For the electrode reaction, k 0 corresponds to E = E0 , so (3.5.22) gives ΔGf‡ = 𝜆el ∕4, again with the condition that 𝜆i is negligible. From (3.5.50), one can express

ΔGf‡ for the homogeneous and heterogeneous reactions in common terms, and one finds that k ex is related to k 0 by the expression (kex ∕Aex )1∕2 = k 0 ∕Ael

(3.5.51)

where Aex and Ael are the pre-exponential factors for the self-exchange and electrode reactions, respectively.31 Roughly, Ael is expected to be 104 –105 cm/s, while Aex is anticipated as 1011 –1012 M−1 s−1 (101). The Marcus theory also predicts the existence of an inverted region for homogeneous electron transfer, where ΔGf‡ increases with the thermodynamic driving force for the electron transfer, ΔG0 . Curves like those in Figure 3.5.10 can be drawn for homogeneous reactions and can be analyzed essentially as presented above for that figure. When ΔG0 = − 𝜆, ΔGf‡ is zero, and the forward rate constant is predicted to be at a maximum. For any ΔG0 less exothermic than this value, the forward reaction is in the normal region, where ΔGf‡ decreases with increasing driving force, and the forward rate constant increases. However, if ΔG0 is more exothermic than −𝜆 (i.e., for very strongly driven reactions), the activation energy becomes steadily larger with increased driving force, and the rate constant becomes smaller. This is the inverted region for the forward reaction. Likewise, there is an inverted region for the backward reaction when ΔG0 is more endothermic than +𝜆. The existence of the inverted region accounts for the phenomenon of electrogenerated chemiluminescence (Section 20.5) and has also been clearly seen by other means for electron-transfer reactions in solution (102). An inverted region should be observable for interfacial electron transfer at the boundary between immiscible electrolyte solutions, with an oxidant, O, in one phase, and a reductant, R′ , in the other (103, 104). Experimental studies bearing on this issue have been reported (105). 3.5.5

A Model Based on Distributions of Energy States

An alternative theoretical approach to heterogeneous kinetics is based on the overlap between electronic states of the electrode and those of the reactants in solution (47–49, 53, 106, 107). 31 With rate constants factored as in (3.5.13), Aex = (K P,O 𝜈 n 𝜅 el )hom and Ael = (K P,O 𝜈 n 𝜅 el )het , where the subscripts “hom” and “het” indicate that the factors apply to the homogeneous and heterogeneous cases, respectively.

3.5 Microscopic Theories of Charge Transfer

Electron energy (eV)

–2

Unoccupied states

–3 f(E) EF –4

–5

DO(𝜆, E)

4𝓀T

𝜆 𝜆

Occupied States

0

1 Electrode states

E0′

DR(𝜆, E)

0 Reactant states

Figure 3.5.12 Relationships among electronic states at an interface between a metal electrode and a solution containing species O and R at equal concentrations. The vertical axis is electron energy, E, on the absolute scale. Indicated on the electrode side is a zone, 4kT wide, centered on the Fermi energy, EF , where the probability of occupancy, f (E), makes the transition from a value of nearly 1 below the zone to a value of virtually zero above (graph in shaded area at left). On the solution side, the state density distributions are shown for O and R. These are gaussians having the same shapes as the probability density functions, W O (𝜆, E) ′ and W R (𝜆, E). The electron energy corresponding to the formal potential, E0 , is −3.8 eV. 𝜆 = 0.3 eV. The Fermi ′ energy corresponds to an electrode potential of −250 mV vs. E 0 . Filled states are denoted on both sides of the interface by dark shading. Filled electrode states overlap with (empty) O states, so reduction can proceed. Since the (filled) R states overlap only with filled electrode states, oxidation is blocked.

The concept is presented graphically in Figure 3.5.12, which will be discussed extensively in this section. This model is rooted in contributions from Gerischer (106, 107) and is particularly useful for treating electron transfer at semiconductor electrodes (Section 20.2), where the electronic structure of the electrode is important. (a) Gerischer Model

The main idea is that an electron transfer can take place from any occupied energy state that is matched in energy, E, with an unoccupied receiving state. If the process is a reduction, the occupied state is on the electrode and the receiving state is on an electroreactant, O. Conversely for an oxidation, the occupied state is on species R in solution, and the receiving state is on the electrode. In general, the eligible states extend over a range of energies, and the total rate is an integral of the rates at each energy. On the electrode, the number of electronic states in the energy range between E and E + dE is given by A𝜌(E)dE, where A is the area exposed to the solution, and 𝜌(E) is the density of states [having units of (area-energy)−1 , such as cm−2 eV−1 ]. The total number of states in a broad energy range is, of course, the integral of A𝜌(E) over the range. If the electrode is a metal, the density of states is large and continuous, but if it is a semiconductor, there is a sizable energy range, called the band gap, where the density of states is very small. (See Section 20.1 for a discussion of the electronic properties of materials.) Electrons fill states in the electrode from lower energies to higher ones until all electrons are accommodated. Any material has more states than are required for the electrons, so there are always empty states above the filled ones. If the temperature were to approach 0 K, the highest filled state would correspond to the Fermi energy (or the Fermi level), EF , and all states above the Fermi energy would be empty. At any higher temperature, thermal energy elevates some of the

163

164

3 Basic Kinetics of Electrode Reactions

electrons into states above EF and creates vacancies below. The filling of the states at thermal equilibrium is described by the Fermi function, f (E), f (E) = {1 + exp[(E − EF )∕kT]}−1

(3.5.52)

which is the probability that a state of energy E is occupied by an electron. For energies much lower than the Fermi energy, the occupancy is virtually unity, and for energies much higher than the Fermi energy, the occupancy is practically zero (Figure 3.5.12). States within a few kT of EF have intermediate occupancy, graded from unity to zero as the energy rises through EF , where the occupancy is 0.5. This intermediate zone is shown in Figure 3.5.12 as a band 4kT wide (about 100 meV at 25 ∘ C). The number of electrons in the energy range between E and E + dE is the number of occupied states, AN occ (E)dE, where N occ (E) is the density function Nocc (E) = f (E)𝜌(E)

(3.5.53)

Like 𝜌(E), N occ (E) has units of (area-energy)−1 , typically cm−2 eV−1 , while f (E) is dimensionless. In a similar manner, we can define the density of unoccupied states as Nunocc (E) = [1 − f (E)]𝜌(E)

(3.5.54)

As the potential is changed, the energy levels on the electrode move linearly, with the change being toward higher energies at more negative potentials, and vice versa. The Fermi energy tracks the potential in the same way (Section 2.2.5). On a metal electrode, these changes occur not by the filling or emptying of many additional states, but mostly by charging the metal, so that all states are shifted by the effect of potential (Section 2.2). While charging does involve a change in the total electron population on the metal, the change is a tiny fraction of the total (Section 2.2.2). Consequently, the same set of states exists near the Fermi energy at all potentials. For this reason, it is more appropriate to think of 𝜌(E) as a consistent function of E − EF , nearly independent of the value of EF . Since f (E) behaves in the same way, so do N occ (E) and N unocc (E). The picture is more complicated at a semiconductor, as discussed in Sections 20.1 and 20.2. States in solution are described by similar concepts, except that filled and empty states correspond to different chemical species, namely the two components of a redox couple, R and O, respectively. These states differ from those of the metal by being localized. The R and O species cannot communicate with the electrode without approaching it closely. Since R and O can exist in the solution inhomogeneously and our concern is with the mix of states near the electrode surface, it is better to express the density of states in terms of concentration, rather than total number. At any moment, the removable electrons on R species in solution in the vicinity of the electrode are distributed over an energy range according to a concentration density function, DR (𝜆, E), having units of (volume-energy)−1 , such as cm−3 eV−1 . Thus, the number concentration of R species near the electrode in the range between E and E + dE is DR (𝜆, E)dE. Because this small element of the R population should be proportional to the overall surface concentration of R, C R (0, t), we can factor DR (𝜆, E) in the following way: DR (𝜆, E) = NA CR (0, t)WR (𝜆, E)

(3.5.55)

where N A is the Avogadro constant, and W R (𝜆, E) is a probability density function with units of (energy)−1 . Since the integral of DR (𝜆, E) over all energies must yield the total number concentration of all states, which is N A C R (0, t), we see that W R (𝜆, E) is a normalized function ∞

∫−∞

WR (𝜆, E) dE = 1

(3.5.56)

3.5 Microscopic Theories of Charge Transfer

Similarly, the distribution of vacant states, represented by O species, is given by DO (𝜆, E) = NA CO (0, t)WO (𝜆, E)

(3.5.57)

where W O (𝜆, E) is normalized, as indicated for its counterpart in (3.5.56). In Figure 3.5.12, the state distributions for O and R are depicted as gaussians for reasons that we will discover below. Now let us consider the rate at which O is reduced from occupied states on the electrode in the energy range between E and E + dE. This is only a part of the total rate of reduction, so we call it a local rate for energy E. In a time interval Δt, electrons from occupied states on the electrode can make the transition to states on species O in the same energy range, and the rate of reduction is the number that succeed divided by Δt. This rate is the instantaneous rate if Δt is short enough that • The reduction does not appreciably alter the number of unoccupied states on the solution side. • Individual O molecules do not appreciably change the energy of their unoccupied levels by internal vibrational and rotational motion. Thus, Δt is at or below the time scale of vibration. The local rate of reduction can be written as P (E)ANocc (E)dE (3.5.58) Local rate(E) = red Δt where AN occ (E)dE is the number of electrons available for the transition and Pred (E) is the probability of transition to an unoccupied state on O. It is intuitive that Pred (E) is proportional to the density of receiving states, DO (𝜆, E). Defining 𝜀red (E) as the proportionality function, we have 𝜀 (E)DO (𝜆, E)ANocc (E)dE Local rate(E) = red (3.5.59) Δt where 𝜀red (E) has units of volume-energy (e.g., cm3 eV). The total rate of reduction is the sum of the local rates in all infinitesimal energy ranges; thus, it is given by the integral Rate = 𝜈

∞

∫−∞

𝜀red (E)DO (𝜆, E)ANocc (E) dE

(3.5.60)

where we have expressed Δt in terms of a frequency, 𝜈 = 1/Δt. The limits on the integral cover all energies, but the integrand has a significant value only where there is overlap between occupied states on the electrode and states of O in the solution. In Figure 3.5.12, the relevant range is roughly −4.0 to −3.5 eV. Substitution from (3.5.53) and (3.5.57) gives Rate = 𝜈ANA CO (0, t)

∞

∫−∞

𝜀red (E)WO (𝜆, E)f (E)𝜌(E) dE

(3.5.61)

This rate is expressed in molecules or electrons per second. Division by AN A gives the rate more conventionally in mol cm−2 s−1 , and further division by C O (0, t) provides the rate constant, kf = 𝜈

∞

∫−∞

𝜀red (E)WO (𝜆, E)f (E)𝜌(E) dE

(3.5.62)

In an analogous way, one can easily derive the rate constant for the oxidation of R. On the electrode side, the empty states are candidates to receive an electron; hence, N unocc (E) is the distribution of interest. The density of filled states on the solution side is DR (𝜆, E), and the probability

165

166

3 Basic Kinetics of Electrode Reactions

for electron transfer in the time interval Δt is Pox (E) = 𝜀ox (E)DR (𝜆, E). Proceeding exactly as in the derivation of (3.5.62), we arrive at ∞

kb = 𝜈

∫−∞

𝜀ox (E)WR (𝜆, E)[1 − f (E)]𝜌(E) dE

(3.5.63)

In Figure 3.5.12, the distribution of states for species R does not overlap the zone of unoccupied states on the electrode, so the integrand in (3.5.63) is practically zero everywhere, and k b is negligible compared to k f . The electrode is in a reducing condition with respect to the O/R couple. By changing the electrode potential to a more positive value, we shift the position of the Fermi energy downward, and we can reach a position where the R states begin to overlap unoccupied electrode states, so that the integral in (3.5.63) becomes significant, and k b is enhanced. The literature contains many versions of equations 3.5.62 and 3.5.63 manifesting different notation and involving wide variations in the interpretation applied to the integral prefactors and the proportionality functions, 𝜀red (E) and 𝜀ox (E). For example, it is common to see a tunneling probability, 𝜅 el , or a precursor equilibrium constant, K P, O or K P, R , extracted from the 𝜀-functions and placed in the integral prefactor. Often, the frequency 𝜈 is identified with 𝜈 n in (3.5.13). Sometimes, the prefactor encompasses things other than the frequency parameter, but is still expressed as a single symbol. The treatment offered here is general and can be accommodated to most views about how the fundamental properties of the system determine 𝜈, 𝜀red (E), and 𝜀ox (E). With (3.5.62) and (3.5.63), it is apparently possible to account for kinetic effects of the electronic structure of the electrode by using an appropriate density of states, 𝜌(E), for the electrode material. Efforts in that direction have been reported. However, one must be on guard for the possibility that 𝜀red (E) and 𝜀ox (E) also depend on 𝜌(E).32 (b) Marcus–Gerischer Kinetics

To define the probability densities W O (𝜆, E) and W R (𝜆, E), the Marcus theory can be added into the Gerischer model. The key is to recognize that the derivation leading to (3.5.21) is based implicitly on the idea that electron transfer occurs entirely from the Fermi energy. In the context that we are now considering, the local rate at the Fermi energy must, therefore, be proportional to the activation energy given by Marcus theory. We can rewrite (3.5.21) in terms of electron energy as33 ( ′ )2 𝜆 E − E0 ‡ ΔGf = 1− (3.5.64) 4 𝜆 32 Consider, for example, a simple model based on the idea that, in the time interval Δt, all electrons in the energy range between E and E + dE redistribute themselves among all available states with equal probability. A refinement allows for the possibility that the states on species O participate with different weight from those on the electrode. If the states on the electrode are given unit weight and those in solution are given weight 𝜅 red (E), then Pred (E) =

𝜅red (E)DO (𝜆, E)𝛿 𝜌(E) + 𝜅red (E)DO (𝜆, E)𝛿

= 𝜀red (E)DO (𝜆, E).

where 𝛿 is the average distance across which electron transfer occurs, and 𝜅 red (E) is dimensionless and can be identified with the tunneling probability, 𝜅 el , used in other representations of k f . If the electrode is a metal, 𝜌(E) is orders of magnitude greater than 𝜅 red (E)DO (𝜆,E)𝛿; hence, the rate constant becomes kf = 𝜈

∞

∫−∞

𝜅red (E)𝛿WO (𝜆, E)f (E)dE

which has no dependence on the electronic structure of the electrode. 33 A difference in potential, ΔE, corresponds to a difference in electron energy −ΔE (Section 2.2.5); therefore, ′ ′ ′ ′ E – E0 corresponds to –(E – E0 ), and E = E0 – 𝜆 corresponds to E = E0 + 𝜆.

3.5 Microscopic Theories of Charge Transfer ′

where E0 is the energy corresponding to the formal potential of the O/R couple. From the discussion in Section 3.5.4(e), one can understand that ΔGf‡ reaches a minimum of zero at ′

E = E0 + 𝜆. Thus, the maximum local rate of reduction at the Fermi energy is found where ′ EF = E0 + 𝜆. When the Fermi energy is at any other energy, E, the local rate of reduction at the Fermi energy can be expressed, according to (3.5.13), (3.5.59), and (3.5.64), in terms of the following ratios [ ] ( ′ )2 𝜆 E − E0 𝜈n 𝜅el exp − 1− 𝜆 4kT Local rate (EF = E) = 𝜈n 𝜅el Local rate (EF = E0 + 𝜆) =

𝜀red (E)DO (𝜆, E)f (EF )𝜌(EF ) 𝜀red (E0 + 𝜆)DO (𝜆, E0 + 𝜆)f (EF )𝜌(EF )

Assuming that 𝜀red does not depend on the position of EF , we can simplify this to [ ] ′ DO (𝜆, E) (E − E0 − 𝜆)2 = exp − 4𝜆kT DO (𝜆, E0 + 𝜆)

(3.5.65)

(3.5.66)

′

This is a gaussian distribution (Section A.3) having a mean at E = E0 + 𝜆 and a standard devi′ ′ ation of (2𝜆kT)1∕2 . From (3.5.57), DO (𝜆, E)/DO (𝜆, E0 + 𝜆) = W O (𝜆, E)/W O (𝜆, E0 + 𝜆). Also, ′ since W O (𝜆, E) is normalized, the exponential prefactor, W O (𝜆, E0 + 𝜆), is quickly identified (Section A.3) as (2𝜋)−1/2 times the reciprocal of the standard deviation; therefore, [ ] ′ (E − E0 − 𝜆)2 −1∕2 WO (𝜆, E) = (4𝜋𝜆kT) exp − (3.5.67) 4𝜆kT In a similar manner, one can show that [ WR (𝜆, E) = (4𝜋𝜆kT)

−1∕2

′

(E − E0 + 𝜆)2 exp − 4𝜆kT

] (3.5.68) ′

thus, the distribution for R has the same shape as that for O, but is centered on E0 − 𝜆, as depicted in Figure 3.5.12. Any model of electrode kinetics is constrained by the requirement that kb 0′ 0′ = ef (E−E ) = e−(E−E )∕kT (3.5.69) kf which is easily derived from the need for convergence to the Nernst equation at equilibrium (Section 3.3.2). The development of the Gerischer model up through equations 3.5.62 and 3.5.63 is general, and one can imagine that the various component functions in those two equations might come together in different ways to fulfill this requirement. By later including results from the Marcus theory without work terms, we were able to define the distribution functions, W O (𝜆, E) and W R (𝜆, E). In this Marcus–Gerischer model, (3.5.69) requires that 𝜀red (E) and 𝜀ox (E) be identical functions. However, this will not necessarily be true for related models including work terms. (c) Effect of Reorganization Energy

In the Marcus–Gerischer model, the reorganization energy, 𝜆, has great influence on the predicted current–potential response. Figure 3.5.13a illustrates behavior for 𝜆 = 0.3 eV, a value near the lower limit found experimentally. For this reorganization energy, an overpotential of

167

3 Basic Kinetics of Electrode Reactions

–2

Electron energy (eV)

168

(a) 𝜆 = 0.3 eV

–3

–2

(b) 𝜆 = 1.5 eV

–3

O 3

3 –4 EF,eq –5

1

O

2

R

–4 EF,eq

E0′ –5

–6

–6

Electrode states

E0′

2 4

4

–7

1

Solution states

–7

R

Electrode states

Solution states

Figure 3.5.13 Effect of 𝜆 on kinetics in the Marcus–Gerischer representation. (a) 𝜆 = 0.3 eV. (b) 𝜆 = 1.5 eV. Species O and R are at equal concentrations, so the Fermi energy corresponding to the equilibrium potential, ′ EF, eq , is the electron energy at the formal potential, E0 . Fermi energy shifts with electrode potential: (1) 𝜂 = − 300 mV, (2) 𝜂 = + 300 mV, (3) 𝜂 = − 1000 mV, and (4) 𝜂 = + 1000 mV. In each case, the Fermi energy is the central line in the group, and the lighter lines mark EF ± kT. On the solution side: DO (𝜆, E) is a dashed curve, DR (𝜆, E) is solid.

−300 mV (case 1) places the Fermi energy opposite the peak of the state distribution for O; hence, rapid reduction would be seen. Likewise, an overpotential of +300 mV (case 2) brings the Fermi energy down to match the peak in the state distribution for R and enables rapid oxidation. An overpotential of −1000 mV (case 3) creates a situation in which DO (𝜆, E) overlaps entirely with filled states on the electrode, and for 𝜂 = +1000 mV (case 4), DR (𝜆, E) overlaps only empty states on the electrode. These latter two cases correspond to very strongly enabled reduction and oxidation, respectively.34 Figure 3.5.13b shows the very different situation for the large reorganization energy of 1.5 eV. In this case, an overpotential of −300 mV is not enough to elevate the Fermi energy into a condition where filled states on the electrode overlap DO (𝜆, E), nor is an overpotential of +300 mV enough to lower the Fermi energy into a condition where empty states on the electrode overlap DR (𝜆, E). It takes 𝜂 ≈ − 1000 mV to enable reduction significantly, and 𝜂 ≈ + 1000 mV to do the same for oxidation. For this reorganization energy, the anodic and cathodic branches of the i − E curve would be widely separated, as exemplified in Figure 3.4.2c or Figure 3.5.7b. Since this formulation of heterogeneous kinetics in terms of overlapping state distributions is linked directly to the basic Marcus theory, it is not surprising that many of its predictions are compatible with those of the Sections 3.5.3 and 3.5.4. The principal difference is that this formulation allows explicitly for contributions from states far from the Fermi energy, which can be important in reactions at semiconductor electrodes (Section 20.2) or involving bound monolayers on metals (Section 17.6.2). Work confirming the existence of inversion in heterogeneous reactions at semiconductor electrodes relied on the Marcus–Gerischer formulation for interpretation (95–99).

3.6 Open-Circuit Potential and Multiple Half-Reactions at an Electrode The open-circuit potential (OCP) of an electrode is defined as the potential where the net current is zero. Normally, it is measured between a working and reference electrode using a high-input-impedance device without any other connections to the working electrode. 34 Cases 3 and 4 in Figure 3.5.13 illustrate the point made in Section 3.5.4(f ) that an inverted region is not expected at a metal electrode. Electron transfer is not limited to the Fermi energy at a metal but, instead, can involve the continuum of states above and below the Fermi energy.

3.6 Open-Circuit Potential and Multiple Half-Reactions at an Electrode

The OCP is the potential where the anodic and cathodic half-reaction currents offset each other precisely. When both halves of a single redox couple are present and other electroactive species do not contribute significantly to current flow, the system behaves as shown in Figure 3.4.1, and the OCP is the equilibrium potential defined by the Nernst equation for that couple. The OCP is said to be well poised under these conditions, (i.e., it does not drift with time or with slight changes in solution conditions). However, not all experimental systems are so simple. There is a need for a more general description of the OCP that can explain the potential measured when an inert working electrode is immersed in electrolyte in the absence of obvious faradaic processes, or when the solution concentrations for the redox species making up a couple are so small that the OCP is no longer close to the Nernst potential (108). 3.6.1

Open-Circuit Potential in Multicomponent Systems

In general, electrochemical systems involve several (often many) electroactive species. Beyond deliberately added solutes, which usually are of primary interest, electroactivity may also arise from the solvent, the supporting electrolyte, the material of the working electrode, or contaminants, such as oxygen or impurities introduced with the solvent or the supporting electrolyte. At any given potential and at any given time, the working electrode passes a total current that is the sum of all individual currents arising from the anodic and cathodic half-reactions taking place there. Thus, ∑ ∑ ic,j (E, t) − ia,j (E, t) (3.6.1) i(E, t) = j

j

where the sums run over all relevant electrode reactions, and ic, j and ia, j are the cathodic and anodic component currents for each such process. At the OCP, i = 0, therefore, ∑ ∑ ic,j (E, t) = ia,j (E, t) (3.6.2) j

j

If the individual component currents vary with time, the zero-current point can drift. When the cathodic and anodic component currents are not dominated by a single couple, but arise from electrode reactions based on different couples, the OCP is not an equilibrium potential, but is instead called a mixed potential. The net cathodic current at the OCP (equal in magnitude to the net anodic current) is sometimes called the mixed current, imix . In principle, the partial currents might be calculated from theory, such as Butler–Volmer kinetics; however, the contributing species for the relevant reactions may not be known. Consider, for example, a Pt electrode immersed in deaerated 0.1 M KCl. Possible cathodic half-reactions include the reduction of protons, water, or trace oxygen, while anodic reactions might involve the oxidation of water or impurities. Curves are shown schematically in Figure 3.6.1a for a situation in which the predominant half-reactions are well separated on the potential scale—meaning that there is a gap between the regions where the anodic current for the oxidizable species and the cathodic current for the reducible species reach toward their mass-transfer limits. This case applies when the standard 0 , is on the positive side of the standard potential for the potential for the oxidizable species, EOx 0 . While the net process leading to the mixed current is not thermodyreducible species, ERed namically spontaneous, it can happen modestly because the feet of the curves describing ia for Ox and ic for Red do overlap. The mixed current at the OCP is small—orders of magnitude less than the mass-transfer limit for either ia or ic . 0 < E 0 , as in Figure 3.6.1b, the net faradaic reaction In the opposite situation, where EOx Red describing the mixed process becomes spontaneous, and the mixed current can be much larger. In effect, the mixed process manifests a galvanic cell.

169

3 Basic Kinetics of Electrode Reactions

OCP

ic i

imix OCP

E 0Ox

i

0

E Red

0

0

E Red

imix

ic

0

E Ox

0

ia

ia 0.5

–0.5

0.0

0.5

0.0

E(V)

E(V)

(a)

(b)

–0.5

Figure 3.6.1 Cathodic and anodic half-reactions contributing to the OCP. In both (a) and (b), the anodic 0 , while the cathodic component current, ia (dashed curve), arises from a couple with standard potential EOx 0 . (a) A system in component current, ic (solid curve) arises from a different couple with standard potential ERed 0 0 which EOx > ERed (e.g., reduction of protons or water and oxidation of a trace impurity at a Pt surface). At the 0 < E 0 (e.g., oxygen reduction and Fe OCP, ic = |ia | = imix , which must be small. (b) A system in which EOx Red oxidation at an iron surface). Here, ia for Ox and ic for Red overlap in the zone where both are strongly rising. At the OCP, ic = |ia | = imix , which can even be at the mass-transfer limit for Ox or Red. In this case, imix is about 65% of the limiting value for ic .

3.6.2

Establishment or Loss of Nernstian Behavior at an Electrode 3−∕4−

Now suppose both forms of a redox couple, e.g., Fe(CN)6 , are added to the Pt/0.1 M KCl system depicted schematically by Figure 3.6.1a. The half-reaction currents for the ferri- and ferrocyanide would add to all other currents in (3.6.1) and (3.6.2), but if their contributions remained negligible, the OCP would not change much. However, as the added concentration of 3−∕4− the Fe(CN)6 increases over orders of magnitude, the half-reaction currents from this couple eventually overwhelm the currents from background processes, and the OCP shifts toward the nernstian value for the redox couple. Figure 3.6.2 shows experimental results demonstrating this behavior. When the total concen3−∕4− tration of Fe(CN)6 is in the μM to nM range, the OCP remains near that of deaerated 0.1 M 450

E0′

350 OCP/mV vs. NHE

170

Poised by Fe(CN)63–/4–

Weak effect of Fe(CN)63–/4–

250 150 50 OCP in Blank

–50 0

1

2

3 –log

4

5

([Fe(CN)63–]

6

7

8

9

+ [Fe(CN)6

4–])

Figure 3.6.2 OCP vs. total redox concentration for 0.1 M KCl with added equimolar concentrations of ferri- and ferrocyanide. [Reprinted with permission from Percival and Bard (108). © 2017, American Chemical Society.]

3.7 Multistep Mechanisms 3−∕4−

KCl; however, it attains the Nernst potential of Fe(CN)6 just below the mM range. In this ′ example, equimolar amounts of ferri- and ferrocyanide were always added, so Eeq = E0 in the poised state. Of course, a poised electrode can become unpoised by reversal of the process just described. 3−∕4− If, for example, an Fe(CN)6 solution in the mM range is progressively diluted by orders 3−∕4−

of magnitude, the contributions of Fe(CN)6 to the mix of currents at the electrode will become progressively less significant, and the potential will fall down the curve in Figure 3.6.2. 3−∕4− It is the competition from background currents that eventually prevents Fe(CN)6 from maintaining its nernstian balance at the electrode. 3.6.3

Multiple Half-Reaction Currents in i–E Curves

When faradaic currents flow, contributions are normally made to the total current from more than one process. Almost always, there is some background current from the sources described above for the “pure” supporting electrolyte solution. This can usually be accounted for by observing the behavior in the absence of the redox species under study. More extreme examples of mixed currents are found for processes that occur close to, or even beyond, the solvent/supporting electrolyte background limits, caused, for example, by water oxidation and proton reduction. A classic example is the Kolbe electrolysis, which occurs with considerable oxygen evolution at the electrode. This process involves the oxidation of a carboxylate to form hydrocarbons, e.g., CH3 COO− − e → CO2 + CH3 ∙

(3.6.3)

2CH3 ∙ → C2 H6

(3.6.4)

Another multicurrent situation is found in the hydrogen evolution reaction, when a metal electrode, e.g., Mn, can simultaneously oxidize during the reductive production of H2 . Determining the different contributions of processes to the net current in such a case generally requires the use of a second electrode, as in SECM (Chapter 18), to monitor the production of one of the products.

3.7 Multistep Mechanisms In Sections 3.3–3.5, we achieved a qualitative and quantitative understanding of the major features of electrode kinetics, and we developed a set of relations that we can expect to fit many real chemical systems, such as Fe(CN)3− + e ⇌ Fe(CN)4− 6 6

(3.7.1)

Hg

−−−−−⇀ Tl+ + e − ↽ −− Tl(Hg)

(3.7.2)

Anthracene + e ⇌

(3.7.3)

Anthracene−∙

However, we must now recognize that most electrode processes are mechanisms of steps. For example, the important reaction 2H+ + 2e ⇌ H2

(3.7.4)

clearly must involve several elementary reactions. The hydrogen nuclei are separated in the oxidized form, but are combined by reduction. Somehow during reduction, there must be a pair of charge transfers and some chemical process linking the two nuclei.

171

172

3 Basic Kinetics of Electrode Reactions

Consider also the reduction Sn(IV) + 2e ⇌ Sn(II)

(3.7.5)

Is it realistic to regard two electrons as tunneling simultaneously through the interface? Or must we consider reduction and oxidation as sequential 1e processes proceeding through the ephemeral intermediate Sn(III)?35 Much effort has been spent on the mechanisms of complex electrode reactions, and there is a large literature on the subject (7–13, 110), because many such reactions are of great fundamental and technological importance. With this edition, the authors seek to provide a more detailed treatment of complex electrode processes. Toward that end, it is convenient to distinguish three categories:36 1) Outer-sphere heterogeneous electron transfer coupled to homogeneous reactions. The product of a heterogeneous electron transfer often engages in subsequent solution phase reactions (e.g., protonation, decomposition, change in ligation, or radical coupling). There are also cases in which the electroreactant is involved in a homogeneous equilibrium or is itself the product of a preceding homogeneous reaction. For electrode reactions in this category, one can view the electrode as an initiator or interrogator of the homogeneous chemistry. Mechanisms are varied and can be simple or complex. Chapter 13 is devoted largely to processes in this category. 2) Multiple outer-sphere heterogeneous electron transfers. Many species, including Sn(IV) and Sn(II), mentioned above, undergo multielectron reduction or oxidation at an electrode. Even if all reactants, products, and intermediates are chemically stable, these processes can become complex. Chapter 13 addresses them. 3) Inner-sphere electrode reactions. In this category, there is chemical binding of at least one participant to the electrode surface, and the overall reaction kinetics are sensitive to the chemical identity of the electrode. Adsorbed intermediates are frequently involved, as in the important example of H+ /H2 , identified above. The electroplating of metals falls in this category, too. Chapter 15, new in this edition, is dedicated to inner-sphere electrode reactions. The chemical variety of electrode processes is too great to capture under just three headings, but these categories encompass a large range of behavior. Discussion of specific cases is reserved for Chapters 13 and 15. In the remainder of this section, we address only a few broadly applicable points concerning multistep processes. 3.7.1

The Primacy of One-Electron Transfers

A widely held concept in electrochemistry is that truly elementary electron-transfer reactions always involve the exchange of one electron, so that an overall process involving a change of n electrons must involve n distinct electron-transfer steps.37 Equations similar to those derived 35 For this reaction in acidic bromide solution, the question was recently resolved (109) by electrochemical detection of Sn(III). 36 In this book, inner-sphere and outer-sphere electrode reactions (Section 3.5.1) are frequently distinguished. In an outer-sphere process, the reactants, products, and intermediates do not interact chemically with the electrode surface. In an inner-sphere process, one or more participants do. 37 Gileadi and Kirowa-Eisner (7) have suggested that this principle might not apply to the incorporation of an ion from solution into a metallic lattice (e.g., as with the two-electron reduction of Cu2+ on a copper electrode). The electrons required, though written stoichiometrically, might be viewed as just part of the conduction-band population, which adjusts to accommodate the deposited ion. In this respect, the electrodeposition of copper could be a quite different process than the reduction of, say, anthracene to the radical anion.

3.7 Multistep Mechanisms

above for the one-step, one-electron process can be used to describe each of these elementary steps, although the concentrations must often be understood as applying to intermediates, rather than to starting species or final products. Of course, the overall process may also involve other elementary reactions, such as adsorption, desorption, or various chemical reactions away from the interface. 3.7.2

Rate-Determining, Outer-Sphere Electron Transfer

In the study of chemical kinetics, one often simplifies the prediction and analysis of behavior by recognizing that a single step, much more sluggish than all others, controls the rate of the overall reaction. If the mechanism is an electrode process, this rate-determining step (RDS) can be a heterogeneous electron-transfer reaction. Consider an overall process in which O and R are coupled in an overall multielectron process O + ne ⇌ R

(3.7.6)

by a mechanism having the following general character: ′

0 (net result of steps preceding RDS, Epre )

O + n′ e ⇌ O′ kf

′

−−−−−−− ⇀ O′ + e ↽ − R′

0 (outer − sphere RDS, Erds )

kb

′

(3.7.8) ′

0 (net result of steps following RDS, Epost )

R′ + n′′ e ⇌ R

(3.7.7)

(3.7.9)

′′

where n + n + 1 = n.38 The i − E characteristic can be written as [ ] 0′ ) 0′ ) −𝛼rds f (E−Erds (1−𝛼rds ) f (E−Erds 0 i = nFAkrds CO′ (0, t)e − CR′ (0, t) e

(3.7.10)

′

0 , 𝛼 , and E 0 apply to the RDS. This relation is (3.3.8) written for the RDS and where krds rds rds multiplied by n, because each net conversion of O′ to R′ results in the flow of n electrons, not just one electron, across the interface. The concentrations C O′ (0, t) and C R′ (0, t) are controlled not only by the interplay between mass transfer and the kinetics of heterogeneous electron transfer, as we found in Section 3.4, but also by the properties of the preceding and following reactions. The situation can become quite complicated, so we will make no attempt to discuss the general problem. However, a few important simple cases exist, and we will develop them briefly now. 39

3.7.3

Multistep Processes at Equilibrium

Even though a process may involve a complex mechanism, it still must obey the Nernst equation whenever true equilibrium can be established [i = 0 and Cj (0, t) = Cj∗ for all j]. The electrode 38 The discussions that follow hold if either or both of n′ or n′′ are zero. 39 Regarding this point, let us offer a warning: In the first edition and in much of the older literature, one finds treatments of multielectron processes in which na is assumed to be the n-value of the rate-determining step. Consequently, na appears in many kinetic expressions. Since na is probably always 1 (if there is a rate-determining electron transfer), it is a redundant symbol and has been dropped here. The current–potential characteristic for a multistep process has often been expressed as [ ] 0′ 0′ i = nFAk 0 CO (0, t)e−𝛼na f (E−E ) − CR (0, t)e(1−𝛼)na f (E−E ) This is rarely, if ever, a valid form of the i–E characteristic for a multistep mechanism.

173

174

3 Basic Kinetics of Electrode Reactions

potential is then determined by the concentrations of the initial reactant and final product. For solutes O and R linked by n electrons, Eeq

∗ RT CO =E + ln ∗ nF CR 0′

(3.7.11)

′

where E0 is the formal potential of the overall reaction. This is a consequence of the thermodynamics of the system and microscopic reversibility.40 Mechanistic details are irrelevant at equilibrium. 3.7.4

Nernstian Multistep Processes

If all steps in a complex electrode process are facile, so that the exchange velocities of all steps are large compared to the net reaction rate, the concentrations of all species participating in them are always essentially at local equilibrium, even though a net current flows. Since it is only at the electrode surface that the forward and backward processes can happen for a heterogeneous electrode process, it is only at the surface that the thermodynamic balance can be continuously maintained. For chemically reversible systems with facile kinetics, the surface concentrations of reactants and products (even those of initial reactants and final products in a multistep process) are always tied to the electrode potential by an equation of the Nernst form. For solutes O and R linked by n electrons:41 ′

E = E0 +

RT CO (0, t) ln nF CR (0, t)

(3.7.12)

A great many real systems satisfy these conditions, and electrochemical examination of them can yield a rich variety of chemical information (Section 5.3.2). An example is the reduction of the ethylenediamine (en) complex of Cd(II) at a mercury electrode: Hg

−−−−−⇀ Cd(en)2+ + 2e − ↽ −− Cd(Hg) + 3en 3 3.7.5

(3.7.13)

Quasireversible and Irreversible Multistep Processes

If a multistep process is neither nernstian nor at equilibrium, the details of the kinetics influence its behavior in electrochemical experiments, and one can use the results to diagnose the mechanism and to quantify kinetic parameters. As in the study of homogeneous kinetics, one proceeds by devising a hypothetical mechanism, predicting experimental behavior according to the hypothesis, and comparing the predictions against results. In the electrochemical sphere, an important part of predicting behavior is developing the i − E characteristic in terms of controllable parameters, such as the concentrations of participating species. If the RDS is a heterogeneous electron-transfer step, then the current–potential characteristic has the form of (3.7.10). For most mechanisms, this equation is of limited direct utility, because O′ and R′ are intermediates, whose concentration cannot be controlled directly. Still, (3.7.10) can serve as the basis for a more practical current–potential relationship, because one can sometimes use the presumed mechanism to re-express CO′ (0, t) and CR′ (0, t) in terms of the concentrations of more controllable species, such as O and R (111). 40 See the second edition, Section 3.5.2, where (3.7.11) is derived. 41 See the second edition, Section 3.5.3, where (3.7.12) is derived.

3.7 Multistep Mechanisms

The general case can easily become too complex for practical application. Even for the simple mechanism in (3.7.7)–(3.7.9), where the pre- and post-reactions are assumed to be kinetically facile enough to remain in local equilibrium, one must have ways to find out the individual 0′ , E 0′ , and E 0′ before one can evaluate the kinetics of the RDS in a fully values of n′ , n′′ , Epre post rds quantitative way.42 This is normally a difficult requirement. More readily usable results arise from some simpler situations. (a) Totally Irreversible Initial Step

Suppose the RDS is the first step in the mechanism and is also a totally irreversible outer-sphere heterogeneous electron transfer: kf

′

O + e → R′

0 (outer-sphere RDS, Erds )

(3.7.14)

R′ + n′′ e → R

(net result of steps following RDS)

(3.7.15)

The chemistry following (3.7.14) has no effect on the electrochemical response, except to add ′′ ′′ n electrons per molecule of O that reacts. Thus, the current is n = 1 + n times bigger than the current arising from step (3.7.14) alone. The overall result is given by the first term of (3.7.10) with C O′ (0, t) = C O (0, t), i = nFAk0rds CO (0, t)e

′

0 ) −𝛼rds f (E−Erds

(3.7.16)

Examples exist, such as the polarographic reduction of chromate in 0.1 M NaOH: − − CrO2− 4 + 4H2 O + 3e → Cr(OH)4 + 4OH

(3.7.17)

Despite the obvious mechanistic complexity of this system, it behaves as though it has an irreversible electron transfer as the first step. (b) Chemically Reversible Processes Near Equilibrium

Some experimental methods, such as impedance spectroscopy (Chapter 11), are based on the application of small perturbations to a system otherwise at equilibrium. These methods often provide the exchange current in a relatively direct manner. It is worthwhile for us to consider the exchange properties of a multistep process at equilibrium. The example that we will take is the overall process O + ne ⇌ R, effected by the general mechanism in (3.7.7)–(3.7.9) and having ′ a formal potential E0 . At equilibrium, all steps in the mechanism are individually at equilibrium, and each has an exchange velocity. The electron-transfer reactions have exchange velocities that can be expressed as exchange currents. There is also an exchange velocity for the overall process that can be expressed as an exchange current. In a serial mechanism with a single RDS, such as we are now considering, the overall exchange velocity is limited by the exchange velocity through the RDS. One can readily show that43 ∗ i0 = nFAk0app CO

[1−(n′ +𝛼rds )∕n]

CR∗

[(n′ +𝛼rds )∕n]

(3.7.18)

0 is where the apparent standard rate constant, kapp 0 0 kapp = krds e

′

′

′

′

0 −E 0 ) 0 −E 0 ) n′ f (Epre 𝛼f (Erds

e

42 Details are derived in the second edition, Section 3.5.4. 43 Details are provided in the second edition, Section 3.5.4(d).

(3.7.19)

175

176

3 Basic Kinetics of Electrode Reactions

This relationship applies generally to mechanisms fitting the pattern of (3.7.7)–(3.7.9), but not to others, such as those involving purely homogeneous pre- or post-reactions, or those involving different RDSs in the forward and reverse directions. Even so, the principles used to obtain (3.7.18) can be employed to derive an analogous expression for any other pattern, provided that the steps are chemically reversible and equilibrium applies. It will be generally possible to express the overall exchange current in terms of an apparent standard rate constant and the bulk concentrations of the various participants. If the exchange current can be measured validly for a given process, the derived relationship can provide insight into details of the mechanism. For example, by an approach similar to that in Section 3.4.4, one obtains the following from (3.7.18): ( ) 𝜕 log i0 n′ + 𝛼rds = 1 − (3.7.20) ∗ n 𝜕 log CO CR∗ ) ( n′ + 𝛼rds 𝜕 log i0 = (3.7.21) n 𝜕 log CR∗ ∗ CO

Since n is often independently available from coulometry or from chemical knowledge of the ′ reactants and products, one can frequently calculate n + 𝛼 rds . From its magnitude, one may ′ be able to estimate separate values for n and 𝛼 rds , which can afford chemical insight into the participants in the RDS. 0 is usually not a simple kinetic parameter for a multistep process. As one can see above, kapp Interpreting it may require detailed understanding of the mechanism, including knowledge of standard potentials or equilibrium constants for various elementary steps. For a quasireversible mechanism having the pattern of (3.7.7)–(3.7.9), one can also show that the current–overpotential equation is44 C (0, t) C (0, t) ′′ ′ i = O ∗ e−(n +𝛼rds )f 𝜂 − R ∗ e(n +1−𝛼rds )f 𝜂 i0 CO CR

(3.7.22)

When the current is small or mass transfer is efficient, the surface concentrations do not differ from those of the bulk, and one has [ ] ′ ′′ i = i0 e−(n +𝛼rds )f 𝜂 − e(n +1−𝛼rds )f 𝜂 (3.7.23) which is analogous to (3.4.11). At small overpotentials, this relationship can be linearized to give i = −i0 nf 𝜂

(3.7.24)

which is the counterpart of (3.4.12). The charge-transfer resistance for this multistep system is then Rct =

RT nFi0

(3.7.25)

which is a generalization of (3.4.13). The arguments leading to (3.7.22)–(3.7.25) are particular to the assumed mechanistic pattern of (3.7.7)–(3.7.9), but similar results can be obtained by the same techniques for any quasireversible mechanism. In fact, (3.7.24) and (3.7.25) are general for quasireversible multistep processes, and they underlie the experimental determination of i0 via methods, such as impedance spectroscopy, based on small perturbations of systems at equilibrium. 44 Details are provided in the second edition, Section 3.5.4(d).

3.8 References

3.8 References 1 P. Atkins, J. de Paula, and J. Keeler, “Physical Chemistry,” 11th ed., Oxford University

Press, Oxford, 2018. 2 R. S. Berry, S. A. Rice, and J. Ross, “Physical Chemistry,” 2nd ed., Oxford University Press,

New York, 2000. 3 H. Eyring, S. H. Lin, and S. M. Lin, “Basic Chemical Kinetics,” Wiley, New York, 1980,

Chap. 4. 4 W. C. Gardiner, Jr., “Rates and Mechanisms of Chemical Reactions,” Benjamin, New York,

1969. 5 S. Glasstone, K. J. Laidler, and H. Eyring, “Theory of Rate Processes,” McGraw-Hill, 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

New York, 1941. J. Tafel, Z. Phys. Chem., 50, 641 (1905). E. Gileadi and E. Kirowa-Eisner, Corros. Sci., 47, 3068 (2005). E. Gileadi, “Electrode Kinetics,” Wiley-VCH, New York, 1993. R. G. Compton, Ed., “Electrode Kinetics: Reactions,” Elsevier, Amsterdam, 1987. C. H. Bamford and R. G. Compton, Eds., “Electrode Kinetics: Principles and Methodology,” Elsevier, Amsterdam, 1986. W. J. Albery, “Electrode Kinetics,” Clarendon, Oxford, 1975. H. R. Thirsk, “A Guide to the Study of Electrode Kinetics,” Academic, New York, 1972, Chap. 1. T. Erdey-Grúz, “Kinetics of Electrode Processes,” Wiley-Interscience, New York, 1972, Chap. 1. K. J. Vetter, “Electrochemical Kinetics,” Academic, New York, 1967, Chap. 2. B. E. Conway, “Theory and Principles of Electrode Processes,” Ronald, New York, 1965, Chap. 6. P. Delahay, “Double Layer and Electrode Kinetics,” Wiley-Interscience, New York, 1965, Chap. 7. C. N. Reilley in “Treatise on Analytical Chemistry,” Part I, Vol. 4, I. M. Kolthoff and P. J. Elving, Eds., Wiley-Interscience New York, 1963, Chap. 42. P. Delahay, “New Instrumental Methods in Electrochemistry,” Wiley-Interscience, New York, 1954, Chap. 2. J. E. B. Randles, Trans. Faraday Soc., 48, 828 (1952). J. A. V. Butler, Trans. Faraday Soc., 19, 729, 734 (1924). T. Erdey-Grúz and M. Volmer, Z. Phys. Chem., 150A, 203 (1930). R. Parsons, Trans. Faraday Soc., 47, 1332 (1951). J. O’M Bockris, Mod. Asp. Electrochem., 1, 180 (1954). D. M. Mohilner and P. Delahay, J. Phys. Chem., 67, 588 (1963). N. Koizumi and S. Aoyagui, J. Electroanal. Chem., 55, 452 (1974). H. Kojima and A. J. Bard, J. Am. Chem. Soc., 97, 6317 (1975). P. Sun and M. V. Mirkin, Anal. Chem., 78, 6526 (2006). M. Shen and A. J. Bard, J. Am. Chem. Soc., 133, 15737 (2011). N. Nioradze, R. Chen, N. Kurapati, A. Khvataeva-Domanov, S. Mabic, and S. Amemiya, Anal. Chem., 87, 4836 (2015). J. Kim and A. J. Bard, J. Am. Chem. Soc., 138, 975 (2016). Y. Yu, T. Sun, and M. V. Mirkin, Anal. Chem., 88, 11758 (2016). K. J. Vetter, op. cit., Chap. 4. T. Erdey-Grúz, op. cit., Chap. 4. P. Delahay, “Double Layer and Electrode Kinetics,” op. cit., Chap. 10. R. Parsons, “Handbook of Electrochemical Data,” Butterworths, London, 1959.

177

178

3 Basic Kinetics of Electrode Reactions

36 A. J. Bard and H. Lund, “Encyclopedia of the Electrochemistry of the Elements,” Marcel 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

Dekker, New York, 1973–1986. K. J. Vetter and G. Manecke, Z. Physik. Chem. (Leipzig), 195, 337 (1950). P. A. Allen and A. Hickling, Trans. Faraday Soc., 53, 1626 (1957). R. A. Marcus, J. Chem. Phys., 24, 966 (1956). R. A. Marcus, Annu. Rev. Phys. Chem., 15, 155 (1964). R. A. Marcus, J. Chem. Phys., 43, 679 (1965). R. A. Marcus, Electrochim. Acta, 13, 955 (1968). R. A. Marcus (Nobel Lecture), Angew. Chem. Intl. Ed., 32, 1111 (1993). N. S. Hush, J. Chem. Phys., 28, 962 (1958). N. S. Hush, Trans. Faraday Soc., 57, 557 (1961). N. S. Hush, Electrochim. Acta, 13, 1005 (1968). V. G. Levich, Adv. Electrochem. Electrochem. Eng., 4, 249 (1966) and references cited therein. R. R. Dogonadze in “Reactions of Molecules at Electrodes,” N. S. Hush, Ed., Wiley-Interscience, New York, 1971, Chap. 3 and references cited therein. W. Schmickler and E. Santos, “Interfacial Electrochemistry,” 2nd ed., Springer, Heidelberg, 2010. M. D. Newton in “Electron Transfer in Chemistry,” V. Balzani, Ed., Wiley-VCH, Weinheim, 2001, Vol. I, Part 1, Chap. 1. N. S. Hush, J. Electroanal. Chem., 470, 170 (1999). P. F. Barbara, T. J. Meyer, and M. A. Ratner, J. Phys. Chem., 100, 13148 (1996). C. J. Miller in “Physical Electrochemistry. Principles, Methods, and Applications,” I. Rubinstein, Ed., Marcel Dekker, New York, 1995, Chap. 2. R. A. Marcus and P. Siddarth, “Photoprocesses in Transition Metal Complexes, Biosystems and Other Molecules,” E. Kochanski, Ed., Kluwer, Amsterdam, 1992. A. M. Kuznetsov, Mod. Asp. Electrochem., 20, 95 (1989). M. J. Weaver in “Comprehensive Chemical Kinetics,” R. G. Compton, Ed., Elsevier, Amsterdam, Vol. 27, 1987, Chap. 1. N. Sutin, Acc. Chem. Res., 15, 275 (1982). H. Taube, “Electron Transfer Reactions of Complex Ions in Solution,” Academic, New York, 1970, p. 27. J. J. Ulrich and F. C. Anson, Inorg. Chem., 8, 195 (1969). G. A. Somorjai, “Introduction to Surface Chemistry and Catalysis,” Wiley, New York, 1994. C. J. Chen, “Introduction to Scanning Tunneling Microscopy,” Oxford University Press, New York, 1993, p. 5. H. O. Finklea, Electroanal. Chem., 19, 109 (1996). C. M Hill, J. Kim, N. Bodappa, and A. J. Bard, J. Am. Chem. Soc., 139, 6114 (2017). M. Velický, S. Hu, C. R. Woods, P. S. Tóth, V. Zólyomi, A. K. Geim, H. D. Abruña, K. S. Novoselov, and R. A. W. Dryfe, ACS Nano, 14, 993 (2020). C. E. D. Chidsey, Science, 251, 919 (1991). R. J. Forster and L. R. Faulkner, J. Am. Chem. Soc., 116, 5444 (1994). J. F. Smalley, S. W. Feldberg, C. E. D. Chidsey, M. R. Linford, M. D. Newton, and Y.-P. Liu, J. Phys. Chem., 99, 13141 (1995). J. N. Richardson, S. R. Peck, L. S. Curtin, L. M. Tender, R. H. Terrill, M. T. Carter, R. W. Murray, G. K. Rowe, and S. E. Creager, J. Phys. Chem., 99, 766 (1995). S. B. Sachs, S. P. Dudek, R. P. Hsung, L. R. Sita, J. F. Smalley, M. D. Newton, S. W. Feldberg, and C. E. D. Chidsey, J. Am. Chem. Soc., 119, 10563 (1997).

3.8 References

70 S. Creager, S. J. Yu, D. Bamdad, S. O’Conner, T. MacLean, E. Lam, Y. Chong, G. T. Olsen,

J Luo, M. Gozin, and J. F. Kayyem, J. Am. Chem. Soc., 121, 1059 (1999). 71 K. Slowinski, K. U. Slowinska, and M. Majda, J. Phys. Chem. B, 103, 8544 (1999). 72 J. F. Smalley, H. O. Finklea, C. E. D. Chidsey, M. R. Linford, S. E. Creager, J. P. Ferraris,

73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

K. Chalfant, T. Zawodzinsk, S. W. Feldberg, and M. D. Newton, J. Am. Chem. Soc., 125, 2004 (2003). P. R. Bueno, G. Mizzon, and J. J. Davis, J. Phys. Chem. B, 116, 8822 (2012). R. J. Cave and M. D. Newton, J. Chem. Phys., 106, 9213 (1997). S. W. Feldberg, J. Electroanal. Chem., 198, 1 (1986). S. F. Fischer and R. P. Van Duyne, Chem. Phys., 26, 9 (1977). W. F. Libby, J. Phys. Chem., 56, 863 (1952). N. Sutin, Acc. Chem. Res., 1, 225 (1968). J. T. Hupp and M. J. Weaver, J. Electroanal. Chem., 152, 1 (1983). B. S. Brunschwig, J. Logan, M. D. Newton, and N. Sutin, J. Am. Chem. Soc., 102, 5798 (1980). R. E. Bangle, J. Schneider, E. J. Piechota, L. Troian-Gautier, and G. J. Meyer, J. Am. Chem. Soc., 142, 674 (2020). E. Laborda, M. C. Henstridge, and R. G. Compton, J. Electroanal. Chem., 667, 48 (2012). D. F. Calef and P. G. Wolynes. J. Phys. Chem., 87, 3387 (1983). J. T. Hynes in “Theory of Chemical Reaction Dynamics,” M. Baer, Ed., CRC, Boca Raton, FL, 1985, Chap. 4. H. Sumi and R. A. Marcus, J. Chem. Phys., 84, 4894 (1986). M. J. Weaver, Chem. Rev., 92, 463 (1992). X. Zhang, J. Leddy, and A. J. Bard, J. Am. Chem. Soc., 107, 3719 (1985). X. Zhang, H. Yang, and A. J. Bard, J. Am. Chem. Soc., 109, 1916 (1987). M. E. Williams, J. C. Crooker, R. Pyati, L. J. Lyons, and R. W. Murray, J. Am. Chem. Soc., 119, 10249 (1997). J. M. Savéant and D. Tessier, J. Electroanal. Chem., 65, 57 (1975). J. M. Savéant and D. Tessier, Faraday Discuss. Chem. Soc., 74, 57 (1982). A. M. Bond, P. J. Mahon, E. A. Maxwell, K. B. Oldham, and C. G. Zoski, J. Electroanal. Chem., 370, 1 (1994). M.C. Henstridge, E. Laborda, and R. G. Compton, J. Electroanal. Chem., 674, 90 (2012). S. W. Feldberg, Anal. Chem., 82, 5176 (2010). H. Lu, J. N. Preiskorn, and J. T. Hupp, J. Am. Chem. Soc., 115, 4927 (1993). X. Dang and J. T. Hupp, J. Am. Chem. Soc., 121, 8399 (1999). D. A. Gaal and J. T. Hupp, J. Am. Chem. Soc., 122, 10956 (2000). T. W. Hamann, F. Gstrein, B. S. Brunschwig, and N. S. Lewis, J. Am. Chem. Soc., 127, 7815 (2005). T. W. Hamann, F. Gstrein, B. S. Brunschwig, and N. S. Lewis, J. Am. Chem. Soc., 127, 13949 (2005). M. Rudolph and E. L. Ratcliff, Nat. Comm., 8, 1 (2017). R. A. Marcus, J. Phys. Chem., 67, 853 (1963). G. L. Closs and J. R. Miller, Science, 240, 440 (1988). R. A. Marcus, J. Phys. Chem., 94, 1050 (1990). R. A. Marcus, J. Phys. Chem., 95, 2010 (1991). M. Tsionsky, A. J. Bard, and M. V. Mirkin, J. Am. Chem. Soc., 119, 10785 (1997). H. Gerischer, Adv. Electrochem. Electrochem. Eng., 1, 139 (1961). H. Gerischer in “Physical Chemistry: An Advanced Treatise,” Vol. 9A, H. Eyring, D. Henderson, and W. Jost, Eds., Academic, New York, 1970.

179

180

3 Basic Kinetics of Electrode Reactions

108 109 110 111 112

S. J. Percival and A. J. Bard, Anal. Chem., 89, 9843 (2017) J. Chang and A. J. Bard, J. Am. Chem. Soc., 136, 311 (2014). P. Delahay, “Double Layer and Electrode Kinetics,” op. cit., Chaps. 8–10. K. B. Oldham, J. Am. Chem. Soc., 77, 4697 (1955). G. Scherer and F. Willig, J. Electroanal. Chem., 85, 77 (1977).

3.9 Problems 3.1

∗ = C∗ = Consider the electrode reaction O + ne ⇌ R. Under the conditions that CO R 1 mM, k 0 = 10−7 cm/s, 𝛼 = 0.3, and n = 1: (a) Calculate the exchange current density, j0 = i0 /A, in μA/cm2 . (b) Draw a current density–overpotential curve for this reaction for current densities up to 600 μA/cm2 anodic and cathodic. Neglect mass-transfer effects. (c) Draw log|j| vs. 𝜂 curves (Tafel plots) for the current ranges in (b).

3.2 (a) Derive (3.4.30) from (3.4.29). (b) Use (3.4.30) and a spreadsheet to repeat Problem 3.1(b,c), including the effects of mass transfer. Assume mO = mR = 10−3 cm/s. 3.3

Use a spreadsheet to calculate and plot current vs. potential and ln(current) vs. potential for the general i − 𝜂 expression in (3.4.30). (a) Show a table of results [potential, current, ln(current), overpotential] and graphs of i ∗ = 1 × 10−3 mol/cm3 ; C ∗ = 1 × 10−5 mol/cm3 ; vs. 𝜂 and ln|i| vs. 𝜂 for A = 1 cm2 ; CO R ′ n = 1; 𝛼 = 0.5; k 0 = 1.0 × 10−4 cm/s; mO = 0.01 cm/s; mR = 0.01 cm/s; E0 = − 0.5 V vs. NHE. (b) Show the i vs. E curves for a range of k 0 values with the other parameters as in (a). At what values of k 0 are the curves indistinguishable from the nernstian case? (c) Show the i vs. E curves for a range of 𝛼 values with the other parameters as in (a).

3.4

Consider 1e electrode reactions for which 𝛼 = 0.50 and 𝛼 = 0.10. Calculate the relative error in current resulting from the use in each case of: (a) The linear i − 𝜂 characteristic for overpotentials of 10, 20, and 50 mV. (b) The Tafel (totally irreversible) relationship for overpotentials of 50, 100, and 200 mV.

3.5

The exchange current density, j0 , for Pt∕Fe(CN)3− (2.0 mM), Fe(CN)4− (2.0 mM), 6 6 ∘ 2 NaCl (1.0 M) at 25 C is reported to be 2.0 mA/cm (112). The transfer coefficient, 𝛼, for this system is about 0.50. Calculate: (a) The value of k 0 . (b) The expected j0 for a solution 0.1 M each in the two complexes. (c) The charge-transfer resistance of a 0.1-cm2 electrode in a solution 10−4 M each in ferricyanide and ferrocyanide.

3.6

(a) Show that for a first-order homogeneous reaction A −−−→ B, the average lifetime of A is 1/k f . (b) Derive an expression for the average lifetime of a collection of electroreactant molecules, O, when they are “at the electrode surface” and able to undergo the

kf

3.9 Problems kf

heterogeneous reaction O + e −−−→ R. Assume that O can react when it is within distance d of the surface. Consider a hypothetical system in which the solution phase extends only d (perhaps 1.0 nm) from the surface. (c) What value of k f would be needed for a lifetime of 1 ms? Are lifetimes as short as 1 ns possible? 3.7

Discuss the mechanism by which the potential of a 0.1-cm2 platinum electrode becomes poised by immersion into a solution of Fe(II) and Fe(III) in 1 M HCl. Approximately how much charge is required to shift the electrode potential by 100 mV? Assume C d = 20 μF/cm2 . Why does the potential become uncertain at low concentrations of Fe(II) and Fe(III), even if the ratio of their concentrations is held near unity? Does this experimental fact reflect thermodynamic considerations? Do your answers to these issues apply to the establishment of potential at an ion-selective electrode?

3.8 The following data were obtained for the reduction of species R to R− in a stirred solution at a 0.1-cm2 electrode. The solution contained 0.01 M R and 0.01 M R− . 𝜂 (mV)

−100

−120

−150

−500

−600

i (μA)

45.9

62.6

100

965

965

Calculate: i0 , k 0 , 𝛼, Rct , il,c , mO , Rmt 3.9

From results in Figure 3.4.5 for 10−2 M Mn(III) and 10−2 M Mn(IV), estimate j0 and k 0 . What is the predicted j0 for a solution 1 M in both Mn(III) and Mn(IV)?

3.10

The magnitude of the solvent term (1/𝜀op − 1/𝜀s ) is about 0.5 for most solvents. Calculate the value of 𝜆o and the free energy of activation (in eV) due only to solvation for a molecule of radius 0.7 nm spaced 0.7 nm from an electrode surface.

3.11

Derive (3.5.63).

3.12

Show from the equations for DO (𝜆, E) and DR (𝜆, E) that the equilibrium energy of a ∗ and C ∗ , and E0′ by an expressystem, Eeq , is related to the bulk concentrations, CO R sion resembling the Nernst equation. How does this expression differ from the Nernst ′ equation written in terms of potentials, Eeq and E0 ? How do you account for the difference?

3.13

Derive (3.5.40) using (3.3.8) and (3.4.6) as expressed for Marcus kinetics, plus (3.5.32).

181

183

4 Mass Transfer by Migration and Diffusion In Chapters 1 and 3, we came to understand that electrode reactions proceed at rates determined not only by reaction kinetics, but also by mass-transfer processes controlling the delivery of reactants to an electrode surface. In this chapter, we focus on the fundamentals of mass transfer, including important partial differential equations, which we will employ frequently in later chapters to develop theory for a large range of electrochemical techniques.

4.1 General Mass-Transfer Equations In Section 1.3, we learned that mass transfer in solution occurs by diffusion, migration, and convection. Both diffusion and migration result from a gradient in electrochemical potential, 𝜇 (Section 2.2.4), but convection is caused by a local imbalance of mechanical forces within the solution. Consider an infinitesimal portion of solution containing two points, r and s (Figure 4.1.1), where, for a certain species j, 𝜇j (r) ≠ 𝜇j (s). This difference of 𝜇j over a distance (a gradient of electrochemical potential) can arise because there is a difference in the concentration of species j (a concentration gradient), or because there is a difference of electric potential, 𝜙 (an electric field or electrical potential gradient). In general, movement of species j occurs in the direction of decreasing electrochemical potential, to alleviate any difference of 𝜇j . The movement is expressed as a flux, which is the rate of mass transfer per unit of area normal to the direction of transfer; thus, the flux Jj (mol s−1 cm−2 ) is a vector quantity proportional to the gradient of electrochemical potential, 𝛁𝜇j , Jj = − Kj 𝛁𝜇j

(4.1.1)

where K j is the constant of proportionality and 𝛁 is the familiar vector operator from mathematics. The minus sign appears in (4.1.1) because the flux vector of species j is in the direction opposite the electrochemical potential gradient, 𝛁𝜇j , as the flux acts to reduce the gradient. The form of 𝛁 depends on the coordinate system. For linear (one-dimensional) mass transfer, 𝛁 = i(𝜕/𝜕x), where i is the unit vector along the axis and x is distance. For mass transfer in a three-dimensional Cartesian space, 𝛁=i

𝜕 𝜕 𝜕 +j +k 𝜕x 𝜕y 𝜕z

(4.1.2)

Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

184

4 Mass Transfer by Migration and Diffusion

_ 𝜇j

Point s _ 𝜇j(s) = 𝜇0j + RT ln aj(s) + zjFϕ(s)

s

Point r _ 𝜇j(r) = 𝜇0j + RT ln aj(r) + zjFϕ(r)

r

Jj x

(a)

(b)

Figure 4.1.1 (a) Two points having different electrochemical potentials in a solution. (b) Gradient of electrochemical potential and corresponding flux vector in a one-dimensional system.

If, in addition to this 𝜇 gradient, the solution is moving, so that an element of solution with a concentration, C j , flows with a velocity vector, v, then an additional term is added to the flux equation: Jj = −Kj 𝛁 𝜇j + Cj v

(4.1.3)

where v can have a different direction than 𝛁𝜇j . For linear mass transfer, where all motion occurs in one dimension, (4.1.3) becomes [ ] 𝜕𝜇 j (x) + Cj v(x) Jj (x) = −Kj (4.1.4) 𝜕x From Section 2.2.4, we recognize that 𝜇j (x) = 𝜇j0 + RT ln aj (x) + zj F𝜙(x), where 𝜇j0 is the standard chemical potential for species j, zj is its charge, aj (x) is the activity of j, and 𝜙(x) is the electric potential. As indicated, 𝜇j , aj , and 𝜙 can be functions of position in the solution. From Section 2.1.5, we have aj = 𝛾j Cj (x)∕Cj0 , where Cj0 is the standard-state concentration of species j, and 𝛾 j is the activity coefficient. By substitution into (4.1.4), we obtain ⎧ ⎡ ⎤⎫ 𝛾j Cj (x) ⎪𝜕 ⎢ 0 ⎪ Jj (x) = −Kj ⎨ + zj F𝜙(x)⎥⎬ + Cj (x)v(x) 𝜇j + RT ln 0 ⎥ ⎢ 𝜕x Cj ⎪ ⎣ ⎦⎪ ⎩ ⎭

(4.1.5)

Since Cj0 is a constant and 𝛾 j usually can be regarded as independent of position, we can take derivatives and rearrange to obtain [ ] 𝜕𝜙(x) RT 𝜕Cj (x) Jj (x) = −Kj + zj F + Cj (x)v(x) (4.1.6) Cj (x) 𝜕x 𝜕x The two terms in brackets represent contributions to the flux from diffusion and migration, respectively. The diffusional contribution is given by Fick’s first law, which we will show in Section 4.4.2 to have the following form for the one-dimensional case: 𝜕Cj (x) Jj = −Dj (4.1.7) 𝜕x where Dj is the diffusion coefficient of species j. By comparing the first term of (4.1.6) with (4.1.7), we see that (4.1.8)

Kj = Dj Cj (x)∕RT Substitution into (4.1.6) yields the final result, Jj (x) = −Dj

𝜕Cj (x) 𝜕x

−

zj F RT

Dj Cj (x)

𝜕𝜙(x) + Cj (x)v(x) 𝜕x

(4.1.9)

4.1 General Mass-Transfer Equations

which is a modified Nernst–Planck equation.1 The general form of this transport equation is Jj = −Dj 𝛁Cj −

zj F RT

Dj Cj 𝛁𝜙 + Cj v

(4.1.10)

If species j is an ion, its diffusion coefficient, Dj , and its ionic mobility, uj , (Section 2.3.3) are linked by the Nernst–Einstein equation: Dj =

uj kT |zj |e

=

uj RT

(4.1.11)

|zj |F

Therefore, one can express the modified Nernst–Planck equation in terms of mobility: Jj = −Dj 𝛁Cj −

uj zj Cj |zj |

𝛁𝜙 + Cj v

(4.1.12)

In any of the three forms of the modified Nernst–Planck equation given above, the first, second, and third terms express the contributions to the flux due to diffusion, migration, and convection, respectively. To describe local concentrations and local fluxes in a system where mass transfer is occurring, one must solve this transport equation under specified mathematical conditions (initial and boundary conditions) that define the system at hand (Section 4.5). If all three modes of mass transfer are important, then all three terms must be retained in the modified Nernst–Planck equation. If one or two modes are known not to be important, the corresponding terms can be dropped, and a simplified transport equation can be used to describe the system. In this chapter, we focus on systems without convection, which is a topic deferred until Chapter 10. Under quiescent conditions (i.e., in an unstirred or stagnant solution), the solution velocity, v, is zero, so the Nernst–Planck equation applies, Jj = −Dj 𝛁Cj −

zj F

D C 𝛁𝜙 RT j j For linear mass transfer, this is 𝜕Cj (x)

zj F

(4.1.13)

𝜕𝜙(x) (4.1.14) 𝜕x RT 𝜕x If species j is charged, its flux is equivalent to a local current density. To show this, let us consider a linear system having a cross-sectional area, A, normal to the axis of mass flow, which occurs along dimension x. We define a positive current as the movement of positive charge toward lesser x (toward the left) or the movement of negative charge to the right. A positive flux of any species corresponds to movement toward greater x, i.e., toward the right. Then, J j (mol s−1 cm−2 ) is equal to −ij /zj FA [C/s per (C mol−1 cm2 )],2 where ij is the current component at any value of x arising from the flux of species j. Equation 4.1.14 can then be written as ij id,j im,j −Jj = = + (4.1.15) zj FA zj FA zj FA Jj (x) = −Dj

−

Dj Cj (x)

1 The Nernst–Planck equation encompasses only terms for diffusion and migration. In this book, we use the label modified Nernst–Planck equation for the more general transport equation, including a term for convection. 2 The minus sign is needed because of the direction chosen for positive current. A positive current resulting from movement of a species with zj = + 1 corresponds to a negative flux of that species (movement toward the left). If the moving species has zj = − 1, a positive current corresponds to a positive

185

186

4 Mass Transfer by Migration and Diffusion

with id,j

= Dj

zj FA im,j

=

zj FA

𝜕Cj

(4.1.16)

𝜕x

zj F RT

Dj Cj

𝜕𝜙 𝜕x

(4.1.17)

where id, j and im, j are diffusion and migration currents of species j, respectively. At any location in solution during electrolysis, the total current, i, is made up of contributions from all species; that is, ∑ i= ij (4.1.18) j

or, from (4.1.16)–(4.1.18), i = FA

∑

zj Dj

j

𝜕Cj 𝜕x

+

F 2 A 𝜕𝜙 ∑ 2 ⋅ z DC RT 𝜕x j j j j

(4.1.19)

where the current at that location for each species is made up of a diffusional component (from the corresponding term in the first summation) and a migrational component (from the corresponding term in the second summation).3 We now proceed to discuss migration and diffusion in electrochemical systems in more detail. The concepts and equations derived below date back to at least the work of Fick (1) and Planck (2). Further details concerning the general problem of mass transfer in electrochemical systems can be found in a number of reviews (3–8).

4.2 Migration in Bulk Solution In the bulk solution (away from an electrode), concentration gradients are generally small, and the total current is carried essentially entirely by migration. All charged species contribute. For species j in the bulk of a linear system having a cross-sectional area A, ij = im, j . Rearrangement of (4.1.17) yields zj2 F 2 ADj Cj

𝜕𝜙 ⋅ RT 𝜕x By substitution from (4.1.11), ij can be reexpressed as ij =

𝜕𝜙 𝜕x For a linear electric field, such as exists between two parallel-plate electrodes,

(4.2.1)

ij = |zj |FAuj Cj

(4.2.2)

𝜕𝜙 ΔE = 𝜕x l

(4.2.3)

3 The component currents assigned to diffusion and migration of a species are conceptual. They are useful for thinking about contributions to the overall current at a point in space, but they are not separately measurable or independently controllable.

4.3 Mixed Migration and Diffusion Near an Active Electrode

where ΔE/l is the gradient (V/cm) arising from the change in potential ΔE over distance l. Thus, ij =

|zj |FAuj Cj ΔE

l and the total current in bulk solution due to migration is given by ∑ FA ΔE ∑ i= ij = |zj |uj Cj l j j

(4.2.4)

(4.2.5)

which is (4.1.18) expressed for this situation. The conductance of the solution, G (Ω−1 ), which is the reciprocal of the resistance, R (Ω), is given by Ohm’s law, 1 i FA ∑ A G= = = |z |u C = 𝜅 (4.2.6) R ΔE l j j j j l where 𝜅, the conductivity (Ω−1 cm−1 ; Section 2.3.3), is ∑ 𝜅 = F |zj |uj Cj

(4.2.7)

j

Equally, one can write an equation for the solution resistance in terms of 𝜌, the resistivity (Ω cm), where 𝜌 = 1/𝜅: 𝜌l (4.2.8) A The fraction of the total current that a given ion, j, carries is t j , the transference number of j [see also (2.3.11) and (2.3.18)]: R=

tj =

|zj |uj Cj |zj |Cj 𝜆j =∑ =∑ i |zk |uk Ck |zk |Ck 𝜆k

ij

k

(4.2.9)

k

where the 𝜆-values are equivalent ionic conductivities (Section 2.3.3).

4.3

Mixed Migration and Diffusion Near an Active Electrode

The relative contributions of diffusion and migration to the flux of a species (and of the flux of that species to the total current) differ at a given time for different locations in solution (9–11). Near the electrode, the concentrations of electroreactants and electroproducts deviate from those of the bulk because of the local effects of the electrode reactions. There are concentration profiles smoothly connecting the surface concentrations with the bulk values. These profiles extend over distances that are usually small (1–100 μm) compared to the dimensions of the cell. Most of the solution is in the bulk, where mass transfer is by migration (Section 4.2). But near the electrodes, where concentrations vary with spatial location, an electroactive substance is, in general, transported by both migration and diffusion. The flux of an electroactive substance at the electrode surface is proportional to the rate of its reaction and, therefore, to the corresponding faradaic current flowing in the external circuit (Sections 1.3.1 and 4.4.3). That current can be conceptually separated into diffusion and migration currents reflecting the diffusive and migrational components to the flux of the electroactive species at the surface: i = id + im

(4.3.1)

187

188

4 Mass Transfer by Migration and Diffusion

Cu(CN)42– + 2e → Cu + 4CN–

Cu2++ 2e → Cu Cu2+

Cu(CN)42–

id

id

im

im

(a) i = id + ǀ imǀ

Cu(CN)2 + 2e → Cu + 2CN–

(b) i = id – ǀ imǀ

Cu(CN)2 id

(c) i = id

Figure 4.3.1 Examples of reduction processes with different contributions of the migration current: (a) positively charged reactant, (b) negatively charged reactant, (c) uncharged reactant. The effects illustrated here can be readily observed in steady-state voltammetry at a UME (Section 5.7).

The fluxes associated with id and im may be in the same or opposite directions, depending on the direction of the electric field and the charge on the electroactive species. Examples of three reductions—of a positively charged, a negatively charged, and an uncharged substance—are shown in Figure 4.3.1. The migrational component is in the same direction as id for reduction of a cationic species and for oxidation of an anionic species. It opposes id when anions are being reduced and when cations are being oxidized. 4.3.1

Balance Sheets for Mass Transfer During Electrolysis

To illustrate the mass-transfer effects in a whole cell, let us apply a “balance sheet” approach (9, 12) to several examples. (a) Electrolysis of HCl

Consider the cell in Figure 4.3.2a, and assume that a total current of 10e per unit time is passed through the cell, producing five H2 molecules at the cathode and five Cl2 molecules at the anode.4 From Table 2.3.2, we see that the equivalent ionic conductivity for H+ (𝜆+ ) is about four times larger than that for Cl− (𝜆− ), i.e., 𝜆+ ≈ 4𝜆− ; therefore, we find from (4.2.9) that t + = 0.8 and t − = 0.2. The total current is carried in the bulk solution by the movement, per unit time, of 8H+ toward the cathode and 2Cl− toward the anode (Figure 4.3.2b). To maintain a steady current, 10H+ must be supplied to the cathode per unit time, so an additional 2H+ must diffuse to the electrode, bringing along 2Cl− to maintain electroneutrality. Similarly, to supply 10Cl− per unit time at the anode, 8Cl− must arrive by diffusion, along with 8H+ . Thus, the different currents (in arbitrary e-units per unit time) are: for H+ , id = 2, im = 8; for Cl− , id = 8, im = 2. The total current, i, is 10. Equation 4.3.1 holds, with migration in this case being in the same direction as diffusion. If a single species is undergoing electrolysis at one of the electrodes, the total number of moles electrolyzed per second is |i/nF|. Of this amount, the number of moles being supplied or withheld per second by migration is |im /nF|, which we can define more fully by considering the behavior in the bulk. For mixtures of charged species, the fraction of current carried in the bulk by the jth species is t j , and the current carried by that species is t j i. The number of moles of the jth species migrating per second is then |t j i/zj F|. For the species being electrolyzed,

4 Actually, some O2 could also be formed at the anode; for simplicity we neglect this reaction.

4.3 Mixed Migration and Diffusion Near an Active Electrode

–

+

e

e Pt/H+, Cl–/Pt (a )

(Cathode)

–

10e

10e 10H

++

10Cl– – 10e → 5Cl2

10e → 5H2 10H

(Anode)

+

10Cl–

+

8H+ 2Cl– Diffusion

Diffusion

+

8Cl– 8H+

2H 2Cl– (b)

Figure 4.3.2 Balance sheet for electrolysis of HCl solution. (a) Cell schematic. (b) Various contributions to the current when 10e are passed in the external circuit per unit time.

this quantity is also |im /nF|, because we assume the migration current for any species to apply everywhere in the solution (in the bulk or near the electrode surface). Thus, | i | | tj i | | m| | | (4.3.2) | |=| | | nF | | zj F | | | | | If we take account of current direction and the sign of zj , this becomes5 n im = ± tj i zj From equation 4.3.1, ( ) nt j id = i 1 ∓ zj

(+ for reduction, − for oxidation)

(4.3.3)

(+ for reduction, − for oxidation)

(4.3.4)

The assumption that im is uniform everywhere in solution is equivalent to assuming that the transference numbers are the same everywhere. This will be true when the concentrations of ions in the solution are high, and only small fractional changes in local concentration are caused by the electrolytic generation or removal of ions. If the electrolysis significantly perturbs the ionic concentrations in the diffusion layer compared to those in the bulk solution, the t j values must vary near the electrode, as required by (4.2.9). In such cases, the treatment developed here provides only approximations of the component currents assignable to diffusion and migration.6 5 As discussed in connection with Figure 4.3.1, im is in the same direction as i when cations are reduced or anions are oxidized, but im opposes i when anions are reduced or cations are oxidized. 6 Section 5.7 covers experiments using ultramicroelectrodes (UMEs) in low ionic strength solutions, in which faradaic reactions significantly change the concentrations of both the redox species and inert electrolyte ions near the electrode surface, resulting in position-dependent transference numbers.

189

190

4 Mass Transfer by Migration and Diffusion

–

+

Hg/Cu(NH3)4Cl2(10–3 M), Cu(NH3)2 Cl(10–3 M), NH3 (0.1 M)/Hg (a) –

(Cathode)

+

6e 6Cu(II) + 6e → 6Cu(I)

(Anode) 6e

6Cu(II)

6Cu(II)

6Cu(I)

6Cu(I)

6Cu(I) – 6e → 6Cu(II)

3Cl– 1Cu(II) 1Cu(I) Diffusion

Diffusion

5Cu(II)

5Cu(II)

7Cu(I)

7Cu(I)

3Cl–

3Cl– (b)

Figure 4.3.3 Balance sheet for electrolysis of the Cu(II), Cu(I), NH3 system. (a) Cell schematic. (b) Various contributions to the current when 6e are passed in the external circuit per unit time; i = 6, n = 1. For Cu(II) at the cathode, |im | = (1/2)(1/3)(6) = 1 (equation 4.3.3), id = 6 − 1 = 5 (equation 4.3.4). For Cu(I) at the anode, |im | = (1/1)(1/6)(6) = 1, id = 6 + 1 = 7.

(b) Electrolysis of Copper Complexes Without Added Electrolyte

Now consider a cell containing 10−3 M Cu(NH3 )2+ , 10−3 M Cu(NH3 )+ , and 3 × 10−3 M Cl− in 4 2 0.1 M NH3 (Figure 4.3.3a). If the equivalent conductances of all ions are equal, i.e., 𝜆Cu(II) = 𝜆Cu(I) = 𝜆Cl− = 𝜆

(4.3.5)

the transference numbers in the bulk are obtained from (4.2.9) as t Cu(II) = 1/3, t Cu(I) = 1/6 and tCl− = 1∕2. With a current of 6e per unit time being passed through the cell, the current in bulk solution is carried by migration of one Cu(II) and one Cu(I) toward the cathode, and three Cl− toward the anode. The balance sheet for this system is shown in Figure 4.3.3b. At the cathode, one-sixth of the current for the reduction of Cu(II) is provided by migration and five-sixths by diffusion.7 The NH3 , being uncharged, does not contribute to the carrying of the current, but serves only to stabilize the copper species in the +1 and +2 states. The resistance of this cell would be relatively large, since the total concentration of ions in the solution is small. 7 A rigorous computation of the faradaic current in such a system would involve using the Nernst–Planck equation to obtain the fluxes of Cu(II) and other ionic species.

4.3 Mixed Migration and Diffusion Near an Active Electrode

–

+

e

e Hg/Cu(NH3)4Cl2(10–3 M), Cu(NH3)2 Cl(10–3 M)/Hg NH3 (0.1 M), NaClO4 (0.10 M) 2+ + Cu(NH3)4 (10–3 M), Cu(NH3)2 (10–3 M),

lons in cell:

–

Cl (3 × 10

–3

+

–

M), Na (0.1 M), ClO4 (0.1 M) (a)

–

(Cathode)

+

6e 6Cu(II) + 6e → 6Cu(I)

(Anode) 6e

6Cu(II)

6Cu(II)

6Cu(I)

6Cu(I)

6Cu(I) – 6e → 6Cu(II)

2.91 Na+ –

2.91 ClO4 0.0291 Cu(II) 0.0291 Cu(I) 0.0873 Cl– Diffusion

Diffusion

5.97 Cu(II)

5.97 Cu(II)

6.029 Cu(I)

6.029 Cu(I)

2.92 Na+

2.92 Na+

ClO–4

2.92 ClO–4

2.92

(b)

Figure 4.3.4 Balance sheet for the system in Figure 4.3.3, but with excess NaClO4 as a supporting electrolyte. (a) Cell schematic. (b) Various contributions to the current when 6e are passed in the external circuit per unit time (i = 6, n = 1). tCu(II) = [(2 × 10–3 )𝜆/(2 × 10–3 + 10–3 + 3 × 10–3 + 0.2)𝜆] = 0.0097. For Cu(II) at the cathode, |im | = (1/2)(0.0097)(6) = 0.03, id = 6 – 0.03 = 5.97.

(c) Electrolysis of Copper Complexes with Excess Electrolyte

Finally, let us consider the same cell as in the preceding example, but with the addition of 0.10 M NaClO4 (Figure 4.3.4a). Assuming that the equivalent ionic conductivities of all ionic species are equal to 𝜆, the transference numbers in the bulk become tNa+ = tClO− = 0.485, 4

t Cu(II) = 0.0097, t Cu(I) = 0.00485, tCl− = 0.0146. The Na+ and ClO− 4 do not participate in the electron-transfer reactions; but they carry 97% of the current in the bulk solution, because their concentrations are high. The balance sheet for this cell (Figure 4.3.4b) shows that most of the Cu(II) now reaches the cathode by diffusion, with only 0.5% of the total flux by migration.

191

192

4 Mass Transfer by Migration and Diffusion

4.3.2

Utility of a Supporting Electrolyte

The preceding example shows that an added excess of electroinactive ions (a supporting electrolyte) nearly eliminates the contribution of migration to the mass transfer of an electroactive species. This is a valuable, general result. The presence of a supporting electrolyte simplifies the mathematical treatment of an electrochemical system by rendering the migrational terms negligible in the mass transport equations for electroreactants and electroproducts near the electrode. Consequently, it has become standard electrochemical practice to add an excess supporting electrolyte, unless there are specific overriding considerations, such as mentioned below. A supporting electrolyte must be selected for compatibility with both the solvent and the electrode process of interest. Many acids, bases, and salts are available for aqueous solutions. For organic solvents with high dielectric constants, such as acetonitrile and N,N-dimethylformamide, normal practice is to employ tetra-n-alkylammonium salts, such as Bu4 NBF4 and Et4 NClO4 (Bu = n-butyl, Et = ethyl). Studies in low-dielectric solvents, such as benzene, inevitably involve solutions of high resistance, because most ionic salts do not appreciably dissolve in such solvents. Salts that do dissolve in apolar media, such as Hx4 NClO4 (where Hx = n-hexyl), undergo extensive ion pairing, which reduces conductivity. In addition to minimizing the contribution of migration, the supporting electrolyte serves other important functions: • It decreases the uncompensated ohmic potential drop between the working and reference electrodes (Section 1.5.4); consequently, the supporting electrolyte improves the accuracy with which the working electrode potential is controlled or measured [Sections 1.6.4(d) and 16.7]. • Improved conductivity in the bulk of the solution reduces the electrical power dissipated in the cell and can lead to simplifications in apparatus (Sections 12.1.3 and 16.7). • Frequently, the supporting electrolyte establishes important reaction conditions, such as pH, ionic strength, or ligand concentration. • In analytical applications, the supporting electrolyte (often also serving as a buffer) can reduce or eliminate sample matrix effects. • A supporting electrolyte ensures that the electrical double layer remains thin with respect to the diffusion layer (Section 14.3), and it establishes a uniform ionic strength throughout the solution, even when ions are produced or consumed at the electrodes. Supporting electrolytes also bring some disadvantages. Because they are used in large concentrations, their impurities can present serious interferences, for example, by giving rise to faradaic responses of their own, by reacting with the intended product of an electrode process, or by adsorbing on the electrode surface and altering kinetics. Also, a supporting electrolyte significantly alters the medium in the cell, so that its properties differ from those of the pure solvent. The difference can complicate the comparison of results obtained from electrochemical experiments (e.g., thermodynamic data) with data from other kinds of experiments, in which pure solvents are typically employed. Sometimes, one seeks, for good reasons, to examine systems with low concentrations of supporting electrolyte—even systems without any added electrolyte beyond the electroactive species to be studied. UMEs have made such work practical. In this arena, migration is always a major consideration and must be brought explicitly into the theory. Further discussion of such systems is deferred to Section 5.7.

4.4 Diffusion

4.4 Diffusion As we have just seen, one can restrict the mass transfer of an electroactive species near the electrode to the diffusive mode by using a supporting electrolyte and operating in a quiescent solution. Most electrochemical methods are built on the assumption that such conditions prevail; therefore, diffusion is a process of central importance. We now take a closer look at this phenomenon and the mathematical models describing it (13–17). 4.4.1

A Microscopic View

Diffusion works toward the homogenization of a mixture by a “random walk” process, which can be readily understood through a one-dimensional case. Consider a single solute molecule constrained to a linear path and buffeted by solvent molecules undergoing Brownian motion. The solute molecule moves randomly forward and backward in steps of length, l, with one step being made per unit time, 𝜏. We can ask, “Where will the molecule be after a time, t?” However, we can answer only by giving the probability that the molecule will be found at different locations. Equivalently, we can envision a large number of molecules concentrated in a line at t = 0 and ask what the distribution of molecules will be at time t. This is sometimes called the “drunken sailor problem,” in which a very drunk sailor emerges from a bar (Figure 4.4.1) and staggers randomly along the sidewalk, taking a step of size l every 𝜏 seconds. What is the probability that the sailor will get down the street a certain distance after time t? In a random walk, all paths that can be traversed in any elapsed period are equally likely; hence, the probability that a molecule (or sailor) has arrived at any given point is simply the number of paths leading to that point divided by the total of paths to all accessible points. This idea is developed in Figure 4.4.2. At time 𝜏, it is equally likely that the molecule is at +l and −l; and at time 2𝜏, the relative probabilities of being at +2l, 0, and −2l, are 1, 2, and 1, respectively. In general, the probability, P(m, r), that the molecule is at a particular location after m time units (m = t/𝜏) is given by the binomial coefficient, ( )m m! 1 P(m, r) = (4.4.1) r!(m − r)! 2 where the set of locations is defined by x = (–m + 2r)l, with r = 0, 1, … m. The mean square displacement of the molecule, Δ2 , can be calculated by summing the squares of the displacements achieved with each path, then dividing by the total number of paths (2m ). The squares of the displacements are used (just as when one obtains the standard deviation in statistics) because movement is possible in both the positive and negative directions, and the sum of the displacements is always zero. This procedure is shown in Table 4.4.1. Figure 4.4.1 The one-dimensional random walk or “drunken sailor problem.”

Harbor bar

–4l

–3l

–2l

–l

0

+l

+2l

+3l

+4l

193

194

4 Mass Transfer by Migration and Diffusion

–4l

–5l

–3l

–2l

–l

+l

0

+2l

+3l

+4l

+5l

t 0τ 1

1

1τ 1

2

1

2τ 1

3

3

1

3τ 1

4

6

4

1

+2l

+4l

4τ (a)

–4l

–2l

0 (b)

Figure 4.4.2 (a) Probability distribution for a one-dimensional random walk over four time units. The number printed over each allowed arrival point is the number of paths to that point. (b) Bar graph showing distribution at t = 4𝜏. At this time, the probability of being at x = 0 is 6/16, at x = ± 2l is 4/16, and at x = ± 4l is 1/16. Table 4.4.1 Distributions for a Random Walk Process(a) ∑

t

n(b)

𝚫(c)

𝚫2

𝚫2 =

0𝜏

1(= 20 )

0

0

0

1𝜏

2(= 21 )

± l(1)

2l2

l2

2𝜏

4(= 22 )

0(2), ± 2l(1)

8l2

2l2

3𝜏

8(= 23 )

± l(3), ± 3l(1)

24l2

3l2

4𝜏

16(= 24 )

0(6), ± 2l(4), ± 4l(1)

64l2

4l2

m𝜏

2m

mnl2 (= m2m l2 )

ml2

(∑

) 𝚫2 ∕n

(a) l = step size, 1/𝜏 = step frequency, t = m𝜏 = time interval. (b) n = total number of possibilities. (c) Δ = possible positions; relative probabilities are parenthesized.

In general, Δ2 is given by t Δ2 = ml2 = l2 = 2Dt 𝜏

(4.4.2)

where the diffusion coefficient, D, identified by Einstein (18, 19) as l2 /2𝜏, is a constant related to the step size and step frequency.8 It has units of length2 /time, usually cm2 /s. 8 The diffusion coefficient was introduced phenomenologically in 1855 by Fick (1), as he proposed the relationship now known as Fick’s first law (Section 4.4.2). In 1905, Einstein (18, 19) derived (4.4.2) and showed l2 2𝜏 to be the same as Fick’s D. Soon thereafter, Langevin (20) derived (4.4.2) by a completely different approach starting with Newton’s second law. Sometimes, D is given as f l2 /2, where f is the number of displacements per unit time (=1/𝜏).

4.4 Diffusion

The root-mean-square displacement at time t (often called the diffusion length) is, therefore, Δ = (2Dt)1∕2

(4.4.3)

This equation provides a handy rule of thumb for estimating the distance affected by a diffusion process (e.g., how far product molecules have moved, on the average, from an electrode in a certain time). A typical value of D for aqueous solutions is 5 × 10−6 cm2 /s; hence, a diffusion layer on the order of 10−4 cm is built up in 1 ms ( or 10−3 cm in 0.1 s, or 10−2 cm in 10 s). [Section 6.1.1(c).] As m becomes large, (4.4.1) takes on a continuous form. For N 0 molecules located at the origin at t = 0, a Gaussian curve describes the distribution at a later time, t. The number of molecules, N(x, t), in a segment Δx wide centered on position x is (21) ( 2) N(x, t) Δx −x = exp (4.4.4) N0 4Dt 2(𝜋Dt)1∕2 A similar treatment can be applied to two- and three-dimensional random walks, where the root-mean-square displacements are (4Dt)1/2 and (6Dt)1/2 , respectively (17, 22). The diffusion coefficient can be usefully related to the solution viscosity, 𝜂 by inserting the definition of ionic mobility, (2.3.9), into the Nernst–Einstein equation, (4.1.11). The result is the Stokes–Einstein equation: Dj = kT∕6𝜋𝜂r

(4.4.5)

where r is the radius of the diffusing species. Although the derivation just described is constructed on ionic mobility, the result does not depend on charge; thus, the equation is not limited to ions, but also applies to neutral species. A more molecular picture of diffusion emerges by considering the concepts of molecular and diffusional velocity (22). In a Maxwellian gas, a particle of mass, m, and average one-dimensional velocity, vx , has an average kinetic energy of 1/2mv2x . This energy can also be shown to be kT∕2 (23, 24); thus, the average instantaneous molecular velocity is vx = (kT∕m)1∕2 . For an O2 molecule (m = 5 × 10–23 g) at 300 K, one finds that vx = 3 × 104 cm/s; however, an O2 molecule can make progress in a given direction at this high velocity only over a short distance before it collides with another molecule and changes direction. The net movement by the random walk resulting from repeated collisions is much slower than vx and is governed by the process described above. In a liquid solution, a velocity distribution similar to that of a Maxwellian gas applies, but the distance traveled between collisions is much shorter than in the gas phase; thus, the net movement by a random walk in solution is much slower in the liquid. Values of D in liquids are typically 10,000 times smaller than in the gas phase. A “diffusional velocity,” vd , can be extracted from (4.4.3) as vd = Δ∕t = (2D∕t)1∕2

(4.4.6)

Care must be used in interpreting (4.4.6) as a true velocity, because there is a time dependence in vd arising from the random walk’s tendency to favor small displacements from a starting point vs. large ones. Values of vd based on accessible experimental measurements of Δ and t are generally much smaller than the instantaneous molecular velocity, vx . For a typical value of D (5 × 10−6 cm2 /s in water), vd is 3, 0.1, and 0.003 cm/s at t = 1 μs, 1 ms, and 1 s, respectively. The relative importance of migration and diffusion can be gauged by comparing vd with the steady-state migrational velocity, v, for an ion of mobility uj in an electric field (Section 2.3.3).

195

196

4 Mass Transfer by Migration and Diffusion

By definition, v = uj E, where E is the electric field strength felt by the ion. From the Nernst–Einstein equation, (4.1.11), v = |zj |FDj E∕RT

(4.4.7)

When v ≪ vd for a given species, diffusion dominates over migration at a given position and time. From (4.4.6) and (4.4.7), we find that this condition can be expressed as 2RT (4.4.8) (2Dj t)1∕2 E ≪ |zj |F The left side is the diffusion length times the field strength, which is the voltage drop in the solution over the length scale of diffusion. To ensure that migration is negligible compared to diffusion, this voltage drop must be smaller than about 2RT/|zj |F, which is 51.4/|zj | mV at 25 ∘ C. This is the same as saying that for migration to be neglected, the difference in electrical potential energy for the diffusing ion must be smaller than a few kT over the length scale of diffusion (typically 1–100 μm in electrochemical experiments). 4.4.2

Fick’s Laws of Diffusion

Fick’s laws (1) are differential equations describing the flux of a substance and its concentration as functions of time and position. In this section, we will derive them for linear (one-dimensional) diffusion and will extend the results to other geometries. The flux of a species, O, written as J O (x, t), is the number of moles of O that pass location x per second per cm2 of area normal to the axis of diffusion. The flux at x may vary with time; therefore, t is explicitly recognized in the function. Fick’s first law states that the flux is proportional to the concentration gradient, 𝜕C O /𝜕x: JO (x, t) = −DO

𝜕CO (x, t)

(4.4.9)

𝜕x

This equation was originally postulated by Fick (1), but it can be derived from the microscopic model by an argument originally due to Einstein (19, 25). Assume that at time t, N O (x) molecules are immediately to left of x and N O (x + Δx) molecules are immediately to the right (Figure 4.4.3). All of the molecules are understood to be within one step length, Δx, of location x. During the time increment, Δt, half of them move Δx in either direction by the random walk process; hence, the net flux through an area A at x is given by the difference between the number of molecules, per unit of area, moving from left to right vs. right to left: NO (x) NO (x + Δx) − 1 2 2 JO (x, t) = (4.4.10) A Δt Multiplying by Δx2 /Δx2 and noting that the concentration is C O = N O /AΔx, we derive −JO (x, t) =

Δx2 CO (x + Δx) − CO (x) 2Δt Δx NO(x + Δx)

NO(x)

NO(x + Δx) 2

NO(x) 2

x→

(4.4.11) Figure 4.4.3 Fluxes at plane x in solution.

4.4 Diffusion

dx

JO(x + dx, t)

JO(x, t)

x

x + dx

Figure 4.4.4 Fluxes into and out of an element at x.

Allowing Δx and Δt to approach zero, we obtain (4.4.9). As Einstein noted (19, 25), this derivation confirms the identity of Fick’s D as Δx2 /2Δt. Fick’s second law describes the change in the concentration of O with time:9 𝜕CO (x, t) 𝜕t

= DO

𝜕 2 CO (x, t) 𝜕x2

(4.4.12)

This equation is derived readily from the first law. The change in concentration at a location x is given by the difference in flux into and flux out of an element of width dx (Figure 4.4.4). 𝜕CO (x, t) 𝜕t

=

J(x, t) − J(x + dx, t) dx

(4.4.13)

As required, J/dx has units of (mol s−1 cm−2 )/cm, or concentration per unit time. The flux at x + dx can be given in terms of that at x by J(x + dx, t) = J(x, t) +

𝜕J(x, t) dx 𝜕x

(4.4.14)

From (4.4.9) we obtain 𝜕C (x, t) 𝜕J(x, t) 𝜕 = DO O 𝜕x 𝜕x 𝜕x Combination of (4.4.13)–(4.4.15) yields [ ] 𝜕CO (x, t) 𝜕C (x, t) 𝜕 = DO O 𝜕t 𝜕x 𝜕x −

(4.4.15)

(4.4.16)

When DO is not a function of x, (4.4.12) results. In most electrochemical systems, the changes in solution composition caused by electrolysis are sufficiently small that variations in the diffusion coefficient with x can be neglected. However, when the electroactive component is present at a high concentration, large changes in solution properties, such as the local viscosity, can occur during electrolysis (Section 5.8). For such systems, (4.4.12) is no longer appropriate, and more complicated treatments are necessary (26, 27). Under these conditions, migrational effects can also become important. The general formulation of Fick’s second law for any geometry is 𝜕CO 𝜕t

= DO ∇2 CO

9 Fick’s second law is often called a continuity or mass-conservation equation.

(4.4.17)

197

198

4 Mass Transfer by Migration and Diffusion

Table 4.4.2 Forms of the Laplacian Operator for Different Geometries(a). Type

Variables

𝛁2

Example

Linear

x

𝜕 2 /𝜕x2

Shielded disk electrode

Spherical

r

𝜕 2 /𝜕r2 + (2/r)(𝜕/𝜕r)

Hanging drop electrode

Cylindrical (axial)

r

𝜕 2 /𝜕r2 + (1/r)(𝜕/𝜕r)

Wire electrode

Disk

r, z

𝜕 2 /𝜕r2 + (1/r)(𝜕/𝜕r) + 𝜕 2 /𝜕z2

Inlaid disk ultramicroelectrode(b)

x, z

𝜕 2 /𝜕x2 + 𝜕 2 /𝜕z2

Inlaid band electrode(c)

Band

(a) See Crank (14) for more information. (b) r = radial distance measured from the center of the disk; z = distance normal to the disk surface. (c) x = distance in the plane of the band normal to the long axis; z = distance normal to the band surface.

95%

80% 20%

60% 40%

40% 60%

x

r

r

80%

95%

(a)

(b)

Figure 4.4.5 Diffusion in solution near different electrodes after an electroreactant has been consumed by reduction or oxidation for a given time. (a) Linear diffusion at a planar electrode. The shaded area is a planar electrode sealed in a cylindrical glass mantle (outer vertical lines). The solution is below the electrode inside the mantle. Each dashed line connects points of equal electroreactant concentration, where the indicated percentage is the fraction of bulk concentration. At the electrode surface, the electroreactant concentration is zero. (b) Radial diffusion at a spherical electrode. The dashed line closest to the electrode (shaded) corresponds to 20% of the bulk concentration.

where ∇2 is the Laplacian operator. Forms of ∇2 for different geometries are given in Table 4.4.2. Thus, for problems involving a planar electrode (Figure 4.4.5a), the linear diffusion equation, (4.4.12), is appropriate. For problems involving a spherical electrode (Figure 4.4.5b), such as the hanging mercury drop electrode (HMDE), the spherical form must be employed: ( ) 𝜕CO (r, t) 𝜕 2 CO (r, t) 2 𝜕CO (r, t) = DO + (4.4.18) 𝜕t r 𝜕r 𝜕r2 The difference between the linear and spherical equations arises because spherical diffusion takes place through an increasing area at larger r.

4.5 Formulation and Solution of Mass-Transfer Problems

4.4.3

Flux of an Electroreactant at an Electrode Surface

Now consider a linear system where species k is an electroreactant transported purely by diffusion to an electrode, where it undergoes reaction. The corresponding current is proportional to the flux of this species at the electrode surface, J k (0, t): [ ] 𝜕Ck (x, t) Jk (0, t) = −Dk (4.4.19) 𝜕x x=0 Because k is consumed at the surface, its concentration profile is positively sloped at x = 0, and J k (0, t) is always negative. The constant of proportionality between the flux magnitude and the current magnitude is nFA, as shown in (1.3.3); however, the current has a sign that depends on whether the reaction is cathodic or anodic. Accordingly, [ ] 𝜕Ck (x, t) ∓i = Jk (0, t) = −Dk (− for reduction, + for oxidation) (4.4.20) nFA 𝜕x x=0 or [ ] 𝜕Ck (x, t) i = ±Dk nFA 𝜕x x=0

(+ for reduction, − for oxidation)

(4.4.21)

This is an important relationship in electrochemistry, because it links the evolving concentration profile near the electrode to the current flowing in an electrochemical experiment. We will draw upon it many times in subsequent chapters. If several electroactive species are being simultaneously reduced or oxidized, the current is related to the sum of their fluxes at the electrode surface. Thus, for q reducible species, [ ] q q ∑ ∑ 𝜕Ck (x, t) i =− nk Jk (0, t) = nk Dk (4.4.22) FA 𝜕x x=0 k=1 k=1

4.5 Formulation and Solution of Mass-Transfer Problems Throughout the rest of this book, we will solve mass-transfer equations under diverse conditions. A typical goal is to predict the current at a working electrode in response to a programmed stimulus (e.g., a potential step). In the simplest cases, we evaluate the time-dependent current from the flux of the electroreactant, k, at the electrode surface using (4.4.21), as discussed just above. To do that, we need [𝜕C k (x, t)/𝜕x]x = 0 . This is the slope of the concentration profile for species k at the electrode surface, which is determined from the concentration function, C k (x, t), which is itself the solution of the relevant mass-transfer partial differential equation (PDE) (e.g., Fick’s second law). To get the theoretical results that we need, we must be able to solve that PDE. An example of this procedure is computing the time-dependent current at a disk electrode following application of a potential to oxidize or reduce a molecule at the diffusion-limited rate (Section 6.1). Unfortunately, one frequently does not have the luxury of solving a single mass-transfer PDE for a single participant, because electrode processes involve connections between species. Participants in electrode reactions can be interconverted (e.g., O into R, or vice versa, via the electrode process itself ), or can decay into inert products, or can even react with each other in solution. Reaching a mathematical solution for any participant typically requires a simultaneous solution for all participants. We must express a PDE for each participant,

199

200

4 Mass Transfer by Migration and Diffusion

defining its mass transfer and any reaction that it undergoes.10 Then, we must solve the full set of PDEs together. If, for example, one wished to consider a purely diffusive system in which the electrode reaction is O + ne ⇌ R and both O and R influence the behavior (e.g., through reversible kinetics), then two mass-transfer equations (one for each species) would be required to solve for the concentration profile or the current. If the experiment is arranged so that diffusion occurs only in one dimension,11 each PDE is the one-dimensional version of (4.4.17): 𝜕CO (x, t)

𝜕 2 CO (x, t)

𝜕CR (x, t)

𝜕 2 CR (x, t)

(4.5.1a,b) 𝜕t 𝜕t 𝜕x2 𝜕x2 A solution of (4.5.1a and 4.5.1b) requires more than just these two equations, because a mass-transfer PDE is a general relationship that must be placed in the specific context of the experiment under consideration (Sections A.1.1 and A.1.6). This is done by defining mathematical conditions that bind the solution at special points or boundaries in time and space. 4.5.1

= DO

= DR

Initial and Boundary Conditions in Electrochemical Problems

In diffusional PDEs based on Fick’s second law, derivatives of concentration are taken with respect to time, t, or a spatial coordinate, such as x or r. For each derivative on t, one must specify an initial condition defining the concentration profile for t = 0. Likewise, for each derivative on a spatial coordinate, one must specify a boundary condition giving the time dependence of the concentration (or its first derivative) at one of the boundaries of the spatial domain (commonly at the electrode surface or infinitely far away). To solve (4.5.1a and 4.5.1b), six conditions would be needed—two initial conditions and four boundary conditions.12 (a) Initial Conditions

For t = 0, conditions are of the form, C O (x, 0) = f (x). At the start of most experiments, O and ∗ and C ∗ , R are uniformly distributed throughout the solution at their bulk concentrations, CO R in which case the initial conditions would be ∗ CO (x, 0) = CO

CR (x, 0) = CR∗

(4.5.2a,b)

(b) Semi-infinite Boundary Conditions

An electrolysis cell is usually large compared to the length scale of diffusion; hence, the electrolyte at the walls of the cell (indeed, practically all of the electrolyte) remains unaltered by the process at the electrode (Section 6.1.1). One can normally assume that, at large distances from the electrode, the concentration reaches the bulk value: ∗ lim CO (x, t) = CO

x→∞

lim CR (x, t) = CR∗

x→∞

(4.5.3a,b)

These are known as semi-infinite conditions. For thin-layer electrochemical cells (Section 12.6), where the cell wall is at a distance, l, on the order of the diffusion length, one must use boundary conditions at x = l instead of those for x → ∞. 10 It is not necessary at this stage to understand how reaction terms are included in these equations. Chapter 13 will provide numerous examples. 11 It is often easy to do this. Section 6.1 fully covers the conditions, which are widely implemented. 12 There are two different derivatives on time in (4.5.1a and 4.5.1b), plus four different derivatives on concentration, including the second derivatives.

4.5 Formulation and Solution of Mass-Transfer Problems

(c) Boundary Conditions at the Electrode Surface

The conservation of matter in an electrode reaction must also be recognized. When O is converted to R at the electrode, and both O and R are soluble in the solution phase, then for each O that undergoes electron transfer at the electrode, an R must be produced. Thus, J O (0, t) = − J R (0, t), and [ ] [ ] 𝜕CO (x, t) 𝜕CR (x, t) DO + DR =0 (4.5.4) 𝜕x 𝜕x x=0 x=0 This condition is called the flux balance. Additional boundary conditions usually relate to concentrations or concentration gradients at the electrode surface. For example, if the potential is controlled in an experiment, one might have CO (0, t) = f (E)

(4.5.5)

or CO (0, t) CR (0, t)

= f (E)

(4.5.6)

where f (E) is some function of the electrode potential derived from the general current– potential characteristic or one of its special cases (e.g., the Nernst equation). Since E is often a function of time (as in the case of a potential sweep), f (E) might also be time-dependent. If the current is the controlled quantity, the corresponding boundary condition is expressed in terms of the flux at x = 0; for example, [ ] 𝜕CO (x, t) i = DO = f (t) (4.5.7) −JO (0, t) = nFA 𝜕x x=0 4.5.2

General Formulation of a Linear Diffusion Problem

Many mass-transfer problems encountered in this book have a common foundation: • The electrode process, O + ne ⇌ R, occurs at a planar electrode in a quiescent, initially homogeneous, solution where linear diffusion applies. • Both O and R are chemically stable and are soluble in the solution. ∗ and C ∗ . • A supporting electrolyte is present in great excess relative to CO R The mass-transfer problem requires Fick’s second law for both O and R, (4.5.1a and 4.5.1b). Of the six conditions required to solve these PDEs, five are common to nearly every problem: • The two initial conditions, (4.5.2a and 4.5.2b). • The two semi-infinite conditions, (4.5.3a and 4.5.3b) • The flux balance, (4.5.4). For easy later reference throughout this book, we give a specific name to this specific set of PDEs, initial conditions, and boundary conditions: the general formulation. For a special case in which either O or R is absent from the bulk, the general formulation ∗ = 0 in (4.5.2a and 4.5.2b) or (4.5.3a still applies. One need only recognize that CR∗ = 0 or CO and 4.5.3b). To achieve a solution for any problem built on the general formulation, we must add an eighth equation—a final boundary condition—conveying the type of experiment being treated and the nature of the electrode kinetics. It defines the specific problem at hand and might have the form of (4.5.5), (4.5.6), or (4.5.7).

201

202

4 Mass Transfer by Migration and Diffusion

4.5.3

Systems Involving Migration or Convection

In systems where added electrolyte is not in great excess vs. the concentrations of electroactive species, or where convection exists, diffusion is not the only important mode of mass transfer. Migration or convection, or both, must be included in the mass-transfer PDE written out for each participant at the beginning of the problem. A starting point is to define the modified Nernst–Planck equation applicable to the system, using the principles of Section 4.1. One must choose the form appropriate to the geometry of the system, then one must include the terms relevant to the system, with the diffusion term always being relevant. Suppose, for example, that one is working with a system of linear diffusion at a planar electrode in an unstirred solution with no added electrolyte beyond the electroactive species themselves (Section 5.7). The Nernst–Planck equation applies 𝜕Cj (x, t) zj F 𝜕𝜙(x, t) − D C (x, t) (4.5.8) Jj (x, t) = −Dj 𝜕x RT j j 𝜕x which is (4.1.9) with only the terms for diffusion and migration retained, and with time recognized explicitly as a variable. A Nernst–Planck equation defines the flux of a species and has a role like that of Fick’s first law for purely diffusional problems. To address mass-transfer problems in which the spatial profiles of concentration and electrical potential are needed, one requires an analogue to Fick’s second law. It can always be derived from the modified Nernst–Planck equation using the approach of Section 4.4.2. For a linear mass-transfer system, the general result from (4.4.13) to (4.4.14) is 𝜕Cj (x, t) 𝜕t

=−

𝜕Jj (x, t) 𝜕x

thus, for the example leading to (4.5.8), one has [ ] 𝜕Cj (x, t) 𝜕Cj (x, t) zj F 𝜕𝜙(x, t) 𝜕 Dj = + D C (x, t) 𝜕t 𝜕x 𝜕x RT j j 𝜕x

(4.5.9)

(4.5.10)

This is the mass-transfer PDE that would be written at the beginning of the problem for each participant (e.g., for O and R). Real problems involving migration or convection are treated in Section 5.7 and Chapter 10. 4.5.4

Practical Means for Reaching Solutions

After a problem is formulated by laying out the mass-transfer PDEs and a full set of initial and boundary conditions, it must be solved. There are three general ways to achieve results: (a) Analytical Solution

In simplified circumstances, one can work through the mathematics of a problem to reach a closed-form or series solution. A large fraction of the historical theory underlying electrochemistry has come from analytical treatments of simplified systems, many of which we will encounter in the chapters to come. The following simplifications are common: • The electrode geometry is often defined as planar or spherical, so that mass transfer remains one dimensional. • Convection and migration are suppressed in ways that we have already noted. • Timescales are restricted to avoid chemical or mass-transfer complications that might develop at longer times (e.g., chemical decay of the initial electroproduct or the onset of natural convection). • Kinetics are limited to simple cases, both for the electrode process and for any coupled process in solution.

4.5 Formulation and Solution of Mass-Transfer Problems

Appendix A introduces the Laplace transform method, which has been widely applied to mass-transfer problems. In Section A.1.6, this method is used to extend the general formulation, yielding broadly applicable results that we will often employ in later chapters as starting points for specific problems. The reader may wish to examine that section now. A special class of problem is the steady state, which is of practical interest for ultramicroelectrodes (Chapter 5) or convective systems (Chapter 10). Since 𝜕C j /𝜕t = 0 at steady state, Fick’s second law for species j becomes Laplace’s equation ∇2 Cj = 0

(4.5.11)

Initial conditions are no longer needed for a solution, because time is not a variable. Closed-form solutions can sometimes be achieved for systems in which transient behavior cannot be described simply. Solutions to mass-transfer problems can occasionally be found by searching the literature for analogous problems. For example, the conduction of heat involves PDEs of the same form as the diffusion equation (28, 29): 𝜕T∕𝜕t = 𝛼∇2 T

(4.5.12)

where T is the temperature, and 𝛼 is the thermal diffusivity (𝛼 = 𝜅/𝜌s, in which 𝜅 = thermal conductivity, 𝜌 = density, and s = specific heat). If one can find the solution of a problem of interest in terms of the temperature distribution or heat flux, one can easily transpose the results to give concentration profiles and currents. Electrical analogues also exist. For example, the steady-state diffusion equation, (4.5.11), is of the same form as that for the electric potential distribution in a region of space, ∇2 𝜙 = 0

(4.5.13)

If one can solve an electrical problem in terms of the current density, j, where j = −𝜅∇𝜙

(4.5.14)

(with 𝜅 as the conductivity), one can write the solution to an analogous diffusion problem (in terms of the function C O ) and find the flux from (4.4.9) or from the more general form, J = −DO ∇CO

(4.5.15)

This approach has been employed, for example, in determining the steady-state uncompensated resistance at a UME (30) and the solution resistance between an ion-selective electrode tip and a surface in a scanning electrochemical microscope (31, 32). It also is sometimes possible to model the mass transport and kinetics in an electrochemical system by a network of electrical components (33, 34). Since computer applications (e.g., SPICE) exist for the analysis of electrical circuits, this approach can be convenient for certain electrochemical problems. (b) Electrochemical Simulators

More complicated mass-transfer problems, often involving complex kinetics, lie beyond the reach of analytical solution. Many such cases have been addressed by digital simulation. In this approach, a numeric model of the system is constructed, and it evolves through computational iterations manifesting each relevant physical and chemical process. Feldberg (35) introduced simulation into electrochemistry using the finite-difference approach described in Appendix B. The coverage there is detailed enough to enable the reader to construct and experiment with finite-difference simulators (an activity recommended

203

204

4 Mass Transfer by Migration and Diffusion

for improving one’s grasp of electrochemical dynamics). Much primary literature has come from investigators who created their own software essentially as described in Appendix B. Digital simulation of this kind can be effective (a) for fairly complex kinetics (at the electrode or in solution), (b) for the treatment of convection or migration in systems of simple geometry, and (c) for multielectrode systems of simple geometry. Such modeling is now carried out largely with commercial electrochemical simulators. These packages all involve the creation of numeric models, but they employ more sophisticated computational approaches than the explicit simulation method discussed in Appendix B. All accommodate mechanistic versatility. The user defines the electrode reaction mechanism, as well as the geometry and experimental mode to be simulated. The mechanism is entered using chemical notation. All also permit comparisons of computational and experimental results. In earlier generations, these simulators were confined with respect to experimental mode (e.g., to cyclic voltammetry alone), geometry (only planar or spherical electrodes), and complexity of mechanism. In later generations, they have become capable of treating additional experimental modes, more complex geometries, multielectrode systems, and rather complex electrode processes, even including adsorbed species. (c) General Simulator/Solvers

The most versatile approach to the treatment of electrochemical systems is to employ a general modeling package, such as COMSOL Multiphysics . This kind of application uses finite-element simulations and other numerical methods to solve systems of simultaneous partial differential equations under defined initial and boundary conditions. It allows those equations to be drawn from many physical domains, including mechanics, mass transfer, hydrodynamics, and electrodynamics. Accordingly, the package can address a broad range of scientific and engineering problems, including many from electrochemistry. Complex kinetics, complex mass-transfer, irregular electrode shapes, and multielectrode arrays can all be accommodated effectively. Moreover, these applications can solve for functions such as the spatial variation of electric potential or the hydrodynamic velocity distribution that might be important to an electrochemical problem, but lie beyond the scope of an electrochemical simulator.

®

4.6 References 1 A. Fick, Poggendorff’s Annalen, 94, 59 (1855) [In Engl.: Phil. Mag., S.4, 10, 30 (1855)]. 2 M. Planck, Ann. Phys. Chem., 39, 161 (1890); 40, 561 (1890). 3 J. S. Newman and K. E. Thomas-Alyea, “Electrochemical Systems,” 3rd ed., Wiley, Hobo-

ken, NJ, 2004. J. Newman, Electroanal. Chem., 6, 187 (1973). J. Newman, Adv. Electrochem. Electrochem. Eng., 5, 87 (1967). N. Ibl, Chem. Ing. Tech., 35, 353 (1963). W. Vielstich, Z. Elektrochem., 57, 646 (1953). C. W. Tobias, M. Eisenberg, and C. R. Wilke, J. Electrochem. Soc., 99, 359C (1952). G. Charlot, J. Badoz-Lambling, and B. Tremillion, “Electrochemical Reactions,” Elsevier, Amsterdam, 1962, pp. 18–21, 27–28. 10 I. M. Kolthoff and J. J. Lingane, “Polarography,” 2nd ed., Interscience, New York, 1952, Vol. 1, Chap. 7. 11 J. Koryta, J. Dvoˇrák, and L. Kavan, “Principles of Electrochemistry,” 2nd ed., Wiley, Chichester, 1993, Chap. 2. 4 5 6 7 8 9

4.7 Problems

12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

J. Coursier, Thesis, Masson, Paris, 1954, as credited in G. Charlot, et al., op. cit. H. J. V. Tyrrell and K. R. Harris, “Diffusion in Liquids,” Butterworths, London, 1984. J. Crank, “The Mathematics of Diffusion”, 2nd ed. Clarendon, Oxford, 1975. W. Jost, Angew. Chem. Int. Ed., 3, 713 (1964). W. Jost, “Diffusion in Solids, Liquids, and Gases,” Academic, New York, 1960. S. Chandrasekhar, Rev. Mod. Phys., 15, 1 (1943). A. Einstein, Ann. Phys., 17, 549 (1905). A. Einstein, “Investigations on the Theory of the Brownian Movement,” R. Fürth, Ed., and A. D. Cowper, Trans., Dover, Mineola, NY, 1956 (an anthology of English translations of Einstein’s publications, with notes added by Fürth). P. Langevin, C.R. Acad. Sci., 146, 530 (1908). L. B. Anderson and C. N. Reilley, J. Chem. Educ., 44, 9 (1967). H. C. Berg, “Random Walks in Biology,” Princeton University Press, Princeton, NJ, 1983. N. Davidson, “Statistical Mechanics,” McGrawHill, New York, 1962, pp. 155–158. R. S. Berry, S. A. Rice, and J. Ross, “Physical Chemistry,” Wiley, New York, 1980, pp. 1056–1060. A. Einstein, Zeit. Elektrochem., 14, 235, 1908. R. B. Morris, K. F. Fischer, and H. S. White, J. Phys. Chem., 92, 5306 (1988). S. C. Paulson, N. D. Okerlund, and H. S. White, Anal. Chem., 68, 581 (1996). H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd ed., Clarendon, Oxford, 1959. M. N. Özi¸sik, “Heat Conduction,” Wiley, New York, 1980. K. B. Oldham in “Microelectrodes, Theory and Applications,” M. I. Montenegro, M. A. Queiros, and J. L. Daschbach, Eds., Kluwer, Amsterdam, 1991, p. 87. B. R. Horrocks, D. Schmidtke, A. Heller, and A. J. Bard, Anal. Chem., 65, 3605 (1993). C. Wei, A. J. Bard, G. Nagy, and K. Toth, Anal. Chem., 67, 1346 (1995). J. Horno, M. T. García-Hernández, and C. F. González-Fernández, J. Electroanal. Chem., 352, 83 (1993). A. A. Moya, J. Castilla, and J. Horno, J. Phys. Chem., 99, 1292 (1995). S. W. Feldberg, Electroanal. Chem., 3, 199 (1969).

4.7 Problems 4.1

Show that all three terms on the right-hand side of the one-dimensional modified Nernst–Planck equation, (4.1.9), have units of flux (mol cm−2 s−1 ).

4.2 Consider the electrolysis of a 0.10 M NaOH solution at platinum electrodes, where the reactions are: 2OH− → 1/2O2 + H2 O + 2e −

2H2 O + 2e → H2 + 2OH

(anode) (cathode)

Show the balance sheet for the system operating at steady state. Assume 20e are passed in the external circuit per unit time, and use the 𝜆0 values in Table 2.3.2 to estimate transference numbers.

205

206

4 Mass Transfer by Migration and Diffusion

4.3

Consider the electrolysis of a solution containing 0.1 M Fe(ClO4 )3 and 0.1 M Fe(ClO4 )2 at platinum electrodes: Fe2+ → Fe3+ + e Fe

3+

2+

+ e → Fe

(anode) (cathode)

Assume that both salts are completely dissociated, that the 𝜆 values for Fe3+ , Fe2+ , and ClO− 4 are equal, and that 10e are passed in the external circuit per unit time. Show the balance sheet for the steady-state operation of this system. 4.4

For a given electrochemical system to be described by equations involving linear diffusion and semi-infinite boundary conditions, the cell wall must be at least five “diffusion lengths” away from the electrode. For a substance with D = 10−5 cm2 /s, what distance between the working electrode and the cell wall is required for a 100-s experiment?

4.5

The mobility, uj , is related to the diffusion coefficient, Dj , by (4.1.11). a) From the mobility data in Table 2.3.2, estimate the diffusion coefficients of H+ , I− , and Li+ at 25 ∘ C. b) Write the equation for the estimation of Dj from the 𝜆j -value.

4.6

Using the procedure of Section 4.4.2, derive Fick’s second law for spherical diffusion, (4.4.18). [Hint: Because of the different areas through which diffusion occurs at r and at r + dr, it is more convenient to obtain the change of concentration in an element of width dr by considering the number of moles diffusing per second rather than the flux.]

207

5 Steady-State Voltammetry at Ultramicroelectrodes To this point, we have been laying a foundation for understanding electrochemical methods. Now, we begin an eight-chapter journey through the most widely used methods, covering their conceptual design, implementation, and interpretation. This chapter and the next two establish the methodological core: steady-state voltammetry in this chapter, basic potential-step methods in Chapter 6, and potential-sweep methods in Chapter 7. Most electrochemical methods involve the application of a stimulus to the working electrode (such as a step change in potential or current) and the observation of a response (such as the resulting change in current or potential). Usually, the response is time dependent; however, there are some circumstances in which the current at the working electrode quickly attains a steady value after a change in potential. There are great advantages in focusing on steady-state currents. Time dependences need not be addressed, either in theory or in experimental apparatus, so both implementation and interpretation are simplified. This simplicity underlies the authors’ decision to start the sequence of methodological chapters with this one, covering steady-state voltammetry. In Chapter 1, we developed an approximate treatment of steady-state voltammetry based on convection at a rotating disk electrode. Our focus here is on diffusion-based systems, for which practical steady-state experimentation became possible with the introduction of the ultramicroelectrode (UME). For now, let us just think of a UME as an exceptionally small working electrode, such as a disk or a sphere of radius smaller than ∼25 μm. The tiny size endows such an electrode with especially useful properties, including a quick convergence to a steady-state current after application of any potential (1, 2).1 In Section 5.1, we will see how steady-state voltammetry can be carried out at a spherical UME. In Section 5.2, we will discuss behavior at UMEs with other shapes.

5.1 Steady-State Voltammetry at a Spherical UME Consider an experiment in which a spherical working electrode of radius r0 is used in a three-electrode cell with a counter electrode and an Ag/AgCl reference electrode. The solution 1 In the methods covered in Chapters 5–11, the electrode area, A, is small enough, and the solution volume, V , is large enough that the passage of current does not alter the bulk concentrations of electroactive species. Such circumstances are known as small A/V conditions. Laitinen and Kolthoff (1, 2) invented the term microelectrode to describe the electrode’s role under small A/V conditions, which is to probe a system, rather than to effect compositional change. Their usage has been broadly employed for decades. To preserve the established term, the tiny working electrodes were given the name ultramicroelectrodes, which has also become general. In Chapter 12, we will explore large A/V conditions, where the electrode is intended to transform the bulk system. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

208

5 Steady-State Voltammetry at Ultramicroelectrodes

i(t) Ctr

Automated potentiostat

Waveform generator

Target E(t)

Potentiostat

Ref

Wk E controlled User interface

i(t) measured

Figure 5.1.1 Experimental arrangement for controlled-potential experiments. The potentiostat acts on the cell as described in the text. Many contemporary instruments are automated potentiostats (or electrochemical workstations) in which the waveform generator, the potentiostat, and a user interface are integrated into a single package as indicated by the pale box. These instruments operate in user-selected modes corresponding to the waveforms and data collection patterns used for common methods, such as cyclic voltammetry or chronocoulometry. ∗ . An electrode reaction, contains an electroactive species, O, at a bulk concentration, CO O + ne ⇌ R, allows species R to be generated at the working electrode, but R does not exist in the bulk (i.e., CR∗ = 0). The solution is unstirred and there is a supporting electrolyte, so convection and migration do not contribute. Mass transfer of O and R is solely by diffusion. Figure 5.1.1 represents the experimental system. A potentiostat (Section 1.9.1) has control of the voltage across the working and counter electrodes, and it automatically adjusts this voltage to keep the potential difference between the working and reference electrodes at a target value defined by a waveform generator.2 The target value can be fixed or time dependent. One can think of the potentiostat as an active element whose job is to force through the working electrode whatever current is required to achieve the targeted working-electrode potential at any time. In chemical terms, this current is the flow of electrons needed to support the active electrochemical processes at rates consistent with the potential. The response from the potentiostat (the current) is the experimental observable. Chapter 16 provides an introduction to the design of such apparatus.

5.1.1

Steady-State Diffusion

Suppose the potential of the working electrode, E, is held at a fixed value where O can be reduced to R. A current flows in proportion to the rate of reduction. Species O is consumed near the ∗ . Species R is generated electrode, and the surface concentration of O becomes less than CO at the electrode. A diffusive flux of O must move toward the electrode from the bulk, and a diffusive flux of R must move away from the electrode toward the bulk. The question we ask is whether the diffusion layer, where the O and R concentrations differ from those of the bulk, grows ever thicker over time, or eventually reaches a steady state in which the concentration profiles, the diffusive fluxes, and the current remain constant. The answer comes from Fick’s second law, which describes the evolution of the concentration profiles (Section 4.4.2). It is expressed for any geometry in dilute solutions of the electroactive 2 Or function generator. The target voltage was identified and discussed as Eappl in Sections 1.5 and 1.6.

5.1 Steady-State Voltammetry at a Spherical UME

species as 𝜕Cj

= Dj ∇2 Cj (5.1.1) 𝜕t where C j is the concentration of species j, which generally is a function of spatial coordinates and time, t. In spherical coordinates, Fick’s second law is (Table 4.4.2), ( 2 ) ( ) 𝜕Cj 𝜕 Cj 2 𝜕Cj 𝜕Cj 1 𝜕 = Dj 2 r2 (5.1.2) = Dj + 𝜕t r 𝜕r 𝜕r 𝜕r2 r 𝜕r where r is the radial distance from the center of the electrode. The angular spherical coordinates do not appear because the behavior is uniform over all angles. If the system can achieve a steady state, then time becomes irrelevant and 𝜕C j /𝜕t = 0. In that case, Fick’s second law becomes much simpler. From (5.1.2), we can write ( ) dC j 1 d 2 r =0 (5.1.3) dr r2 dr which is Laplace’s equation, ∇2 C j = 0, for this problem. If (5.1.3) has a solution, then a steady state exists. The form of Equation 5.1.3 requires that the parenthesized expression be a constant with respect to r. We identify the constant as Aj : r2

dC j

= Aj dr Let us now integrate indefinitely,

(5.1.4)

Aj

dr (5.1.5) ∫ r2 Aj Cj (r) = − + Bj (5.1.6) r in which Bj is the constant of integration, and the dependence of C j on r has now become explicit. For each species, O and R, we can evaluate Aj and Bj by applying boundary conditions. As r becomes very large, C j (r) must approach the bulk concentration [the semi-infinite condition; Section 4.5.1(b)]. Thus, ∫

dC j =

lim Cj (r) = Cj∗ = Bj

r→∞

(5.1.7)

∗ and B = 0. Consequently, BO = CO R At the electrode surface, the concentrations of O and R are C O (r = r0 ) and C R (r = r0 ), respectively. By writing (5.1.6) for the surface concentration and rearranging, we obtain

Aj = r0 [Cj∗ − Cj (r = r0 )]

(5.1.8)

Now we can substitute for Aj and Bj in (5.1.6) to obtain the concentration profiles for species O and R: r ∗ − 0 [C ∗ − C (r = r )] CO (r) = CO (5.1.9) O 0 r O r CR (r) = 0 CR (r = r0 ) (5.1.10) r

209

5 Steady-State Voltammetry at Ultramicroelectrodes

Figure 5.1.2 Steady-state concentration profiles ∗, at a spherical electrode for CO (r = r0 ) = 0.25CO ∗ , and D = D . For a 1e CR (r = r0 ) = 0.75CO O R reversible electrode reaction, they correspond to ′ E = E 0 − 28.2 mV at 25 ∘ C.

1.0 O

0.8

C(r)/CO*

210

0.6

0.4

0.2 Electrode surface

0.0 0

R

5

10 r/r0

15

20

Both C O (r) and C R (r) are independent of time, thereby proving that a steady state can arise at a spherical electrode. Figure 5.1.2 presents a pair of steady-state concentration profiles calculated from (5.1.9) and (5.1.10). Note that the diffusion layer extends over a distance many times greater than the radius of the electrode. At the right edge of the figure (19 times r0 from the surface), the concentration of species O is still only 96% of its bulk value. The concentration profiles for O and R are linked through the electrode reaction. We tie them together by recognizing the flux balance at the electrode surface [Section 4.5.1(c)]. Diffusion occurs along a downward slope of a concentration profile; thus, there is a flux of O into the electrode surface, J O (r0 ), and a flux of R out of the surface, J R (r0 ). [ ] dC O (r) JO (r0 ) = −DO (5.1.11a) dr r=r0 [ ] dC R (r) JR (r0 ) = −DR (5.1.11b) dr r=r 0

These fluxes represent the rate of reaction per unit area. The conservation of mass requires that they sum to zero: [ ] [ ] dC O (r) dC R (r) DO + DR =0 (5.1.12) dr dr r=r r=r 0

0

By evaluating the derivatives using (5.1.9) and (5.1.10), one finds that the surface concentrations of O and R are simply related through: ] D [ ∗ − CO (r = r0 ) (5.1.13) CR (r = r0 ) = O CO DR Therefore, we have from (5.1.10) ] r D [ ∗ CR (r) = 0 O CO − CO (r = r0 ) r DR

(5.1.14)

For the general case, where both O and R can be present in the bulk, the same treatment gives ] r [ ∗ − 0 C ∗ − C (r = r ) CO (r) = CO (5.1.15) O 0 O r

5.1 Steady-State Voltammetry at a Spherical UME

CR (r) = CR∗ +

r0 DO r DR

CR (r = r0 ) = CR∗ +

∗ − C (r = r )] [CO O 0

DO DR

(5.1.16)

∗ − C (r = r )] [CO O 0

(5.1.17)

These results show that C O (r = r0 ) and DO /DR fully determine the steady-state profiles for both ∗ + C ∗ at any r, as is true for Figure 5.1.2. O and R. If DO = DR , then CO (r) + CR (r) = CO R 5.1.2

Steady-State Current

If a system produces steady-state concentration profiles, then it also produces a steady flux of the electroreactant to the electrode surface and, therefore, a steady current, i.3 From principles in Section 4.4.2, [ ] dC O (r) i JO (r0 ) = − = −DO (5.1.18) nFA dr r=r 0

Evaluating the derivative using (5.1.15), we have [ ] ∗ nFADO CO CO (r = r0 ) i= 1− ∗ r0 CO

(5.1.19) ′

If the working electrode potential is sufficiently negative vs. E0 , then species O cannot coexist with the electrode and C O (r = r0 ) becomes effectively zero. The electroactive species is reduced as fast as it can possibly arrive. The corresponding current is the limit of (5.1.19) for very small C O (r = r0 ), which is4 id,c =

∗ nFADO CO

r0

(5.1.20)

This is the cathodic diffusion-limited steady-state current for reduction at a spherical electrode, which is the largest that the diffusive flux of O can support at steady state. Since, A = 4𝜋r02 , id,c at a sphere can also be written as ∗r id,c = 4𝜋nFDO CO 0

5.1.3

(5.1.21)

Convergence on the Steady State

Figure 5.1.2 shows that time must elapse in the establishment of a steady state, for the diffusion layer is depleted of the electroreactant over a distance many times the radius of the electrode. If ∗ everywhere and then one starts with a working electrode at open circuit in a solution having CO applies the potential relevant to Figure 5.1.2, a sizable transient current would flow, because O is readily available at the electrode surface. As electrolysis continues, the layer depleted of O becomes thicker, so the concentration profile of O near the electrode becomes steadily less steep. Thus, the flux of O at the electrode surface, and, therefore, the current, falls with time. 3 In this chapter, all occurrences of i signify steady-state currents. The symbol id is used for diffusion-limited steady-state currents. Usually, id is symbolized by id,c or id,a , distinguishing a cathodic or anodic diffusion-limited current. When there is no risk of confusion, the general symbol, id , is commonly used. 4 This is equation 1.3.10 with the mass transfer coefficient, mO , given by DO /r0 .

211

212

5 Steady-State Voltammetry at Ultramicroelectrodes

Eventually, the concentration profiles converge to those of Figure 5.1.2, and the current levels at the steady-state value. If one then changes the potential to a new value within the range of electroactivity, the result will generally be another transient current, decaying to a steady value as the diffusion layer evolves to a new steady state. We save the details of the transient response for Chapter 6. For now, it is only necessary to recognize that a current transient can be expected when a new potential is applied and that there is a delay in reaching the steady state. The time required to establish the steady state can be estimated from diffusion theory. In Section 4.4.1, we learned that the one-dimensional root-mean-square displacement of a collection of diffusing molecules is (2Dt)1/2 , where D is the diffusion coefficient and t is the elapsed time. This distance, sometimes called the diffusion length, is a useful measure of the distance scale that can be affected by a diffusive process over time t. From Figure 5.1.2, one can see that the thickness of the steady-state diffusion layer at a spherical electrode is in the range of 5r0 . Taking that value as the diffusion length, we find that the time needed to establish the steady-state diffusion layer (the steady-state renewal time, 𝜏 ss ) would be about 12.5r02 ∕D. A mercury droplet from a historical dropping electrode (Section 8.1) might have r0 = 250 μm. If DO = 10−5 cm2 /s (a typical figure for solutes in common solvents), it would take 800 s or longer to establish a steady state. Actually, the steady state probably could not be established at all, because adventitious convection from density gradients or vibration would disrupt a diffusion-based experiment lasting so long. To make practical use of steady-state currents, one must reach the steady state in much less time, and that requires a much smaller r0 . Table 5.1.1 shows that renewal times for electrodes with r0 ≤ 25 μm are all practical (and can become quite brief ). It now becomes clear why the focus in this chapter is entirely on UMEs.5 5.1.4

Steady-State Voltammetry

Let us now imagine an experiment, performed with the same system that we have been discussing, in which we apply a progression of potentials (E1 , E2 , E3 , …) and hold each one long enough to measure the steady-state current. Let us further suppose that the working electrode is a spherical UME with r0 = 5 μm, so that it takes only a second or two to reach steady state after a change of potential. In that transient period, the diffusion process has time to replace all species in the diffusion layer virtually entirely, so the effects of electrolysis at the preceding potential are fully erased, even if the electrode reaction is irreversible and yields an electroinactive product. Thus, the application of each potential represents a separate measurement based on its own steady-state concentration profile, without memory of earlier events in the sequence. Table 5.1.1 Steady-State Renewal Times for UMEs Dimension(a)

25 𝛍m

10 𝛍m

2.5 𝛍m

1 𝛍m

250 nm

100 nm

25 nm

10 nm

𝜏 ss (b)

8s

1.3 s

80 ms

13 ms

0.8 ms

130 μs

8 μs

1.3 μs

(a) Characteristic dimension: r0 for a sphere, hemisphere, disk, or cylinder; w for a band. (b) 12.5r02 ∕D or 12.5w2 /D with D = 1 × 10−5 cm2 /s. A sphere, hemisphere, or disk reaches a steady state; a cylinder or band does not. For the latter, renewal times relate to quasi-steady state (Sections 5.2.3 and 5.2.4).

5 Table 5.1.1 anticipates electrode shapes other than spheres. These are covered in Section 5.2. For now, just concentrate on the spherical case.

5.1 Steady-State Voltammetry at a Spherical UME

id,c

Steady-state current

Figure 5.1.3 Steady-state voltammogram for the O/R system at a spherical UME. Successive applied potentials are 50 mV apart, starting at −0.2 V and ending at −1.0 V. Every data point is a steady-state current for the given applied potential. It is not just the plateau current that is a steady-state value.

E1/2 0.5id,c

0 –0.2

–0.4

–0.6

–0.8

–1.0

E/V vs. reference

This experimental approach is called steady-state voltammetry (SSV), and the result—a plot of the recorded steady-state current vs. the potential—is a steady-state voltammogram (3). For the system under discussion, the outcome might resemble Figure 5.1.3, in which steadystate currents are plotted every 50 mV from −0.2 to −1.0 V. When E is more positive than the point where species O begins to be reduced, there is no electrode reaction, and the steady-state current is zero. Beyond −0.5 V, the reduction happens and draws down the surface concentration of O. As E is made more negative, that surface concentration falls further, so the concentration profile becomes steeper and the steady-state current becomes larger. Beyond −0.8 V, C O (r = r0 ) is driven essentially to zero for every potential, and the current becomes diffusion-limited at id,c , as given by (5.1.20) or (5.1.21). The steady-state voltammogram is, therefore, a wave, much like those that we saw in Chapter 1. The half-wave potential, E1/2 , where i = id,c /2 and where the surface concentration of O is ∗ , is a marker for the position of the wave. It depends on the thermodynamics and half of CO kinetics of the electrode process (Sections 5.3 and 5.4). In Figure 5.1.3, the current measurements are spaced rather far apart, just to illustrate the concept of recording a steady-state voltammogram. In practical work, the spacing would be much smaller, so that one obtains better definition of the wave. Actually, a common practice is to apply a linear potential sweep from the initial potential to the final potential [e.g., by using the linear sweep voltammetry (LSV) mode of an automated instrument]. The current is recorded essentially continuously. If the scan rate, v, is slow enough, diffusion can continuously adjust the steady state to the changing potential. For example, if a 1-μm electrode is used, the steady-state renewal time, 𝜏 ss , is on the order of 13 ms. If the potential is scanned at 50 mV/s, then E moves less than 1 mV over 𝜏 ss ; consequently. diffusion has the capacity to preserve the steady state mV-by-mV. A useful rule is based on the idea that the potential should not change over the steady-state renewal time by more than 0.1RT/F (i.e., the electron energy should not change more than 0.1 kT, or 2.6 meV at 25 ∘ C); thus, v≤

0.1RT 1 × 10−6 ≈ mV∕s F𝜏ss r02

(with r0 in cm)

(5.1.22)

For a disk with a 1-μm radius, v should be 100 mV/s or less. One can verify that the scan rate is adequately slow simply by checking for a dependence of the response on scan rate. If the chosen rate is too fast, a peak starts to appear on the plateau, because the result is not purely a steady-state current, but also includes transient effects involved in the adjustment of the steady state.

213

214

5 Steady-State Voltammetry at Ultramicroelectrodes

Figure 5.1.4 Steady-state voltammogram of TCNQ at a spherical gold UME (r0 = 3 μm) in acetonitrile with ∗ = 6.5 mM, v = 100 mV/s). The dual 0.1 M TBABF4 (CO trace arises from a forward scan from 0.6 to 0.2 V and a reverse scan back to 0.6 V. [Reprinted with permission from Demaille, Brust, Tsionsky, and Bard (4). © 1997, American Chemical Society.]

25

20

15 i/nA 10

5

0

–5 0.6

0.5

0.4 E/V vs. SCE

0.3

0.2

Figure 5.1.4 presents a steady-state voltammogram recorded by linear sweep at a spherical gold UME. The electroactive species is tetracyanoquinodimethane (TCNQ), which is reduced in a one-electron process to the anion radical in acetonitrile. The double trace in Figure 5.1.4 arises because the experiment was carried out using the cyclic voltammetry (CV) mode of an automated potentiostat. Much steady-state voltammetry is recorded in a CV mode, and the literature may refer to the experimental approach as steady-state cyclic voltammetry; however, one should not generally regard steady-state CV as a distinct methodology. The forward and reverse scans ought to produce the same result when steady-state behavior is truly maintained.6,7

5.2 Shapes and Properties of Ultramicroelectrodes The advent of UMEs opened major opportunities by extending electrochemical methodology into broad new domains of space, time, chemical medium, and methodology (5–12). UMEs were barely in view when the first edition of this book appeared, but their impact drove much of the change in the second edition, and they underlie even more change in this edition. The essential concept of a UME is that the electrode is smaller than the scale of the diffusion layer developed in readily achievable experiments. Not all applications depend on this aspect of behavior, but many do, including SSV. Other applications of UMEs rest on the low ohmic drops or small time constants characteristic of very small electrodes. We will cover those matters in Section 5.6 and in Chapter 6. 6 In some situations, steady-state behavior cannot truly be maintained, e.g., where an electrode reaction alters the surface and changes either the reaction kinetics or the availability of electroreactants at the surface. Steady-state CV can have diagnostic value in such circumstances. We are not yet ready to discuss CV, so we will postpone any further mention for Chapter 7, in which CV is a prime focus. 7 Hysteresis between the two scans in a steady-state CV can be an indicator that the scan rate is too fast to fully eliminate the transient effect. This may be the reason for the small hysteresis in Figure 5.1.4.

5.2 Shapes and Properties of Ultramicroelectrodes

A UME is defined operationally (13) as an electrode having at least one dimension (such as the radius of a disk) smaller than 25 μm. This aspect is called the characteristic dimension.8 Electrodes with a characteristic dimension as small as 10 nm can be made. Since only one dimension of an electrode must be small to establish the properties of a UME, there is latitude in other physical dimensions and, consequently, a variety of shapes. We will now review the most useful of them. 5.2.1

Spherical or Hemispherical UME

Steady-state behavior at a spherical electrode was covered in Section 5.1. The sphere was chosen for that initial treatment because it is the most ideal shape. It has the property of being uniformly accessible, meaning that the geometry of mass transfer is the same for every point on the electrode surface. In consequence, the flux of any species is the same at any point on the electrode. This is true because there are no physical discontinuities or boundaries except the electrode surface itself, and because the curvature is the same at every point on a sphere. The diffusion-limited steady-state current is given by (5.1.20) or (5.1.21). A hemispherical UME bounded by an infinite planar mantle has exactly half of the diffusion field of a spherical UME of the same r0 , so it has half of the current of the corresponding sphere. Equation 5.1.20 compensates for the difference through the proportionality with area, so it applies to the hemisphere as well as the sphere. Equation 5.1.21 is accurate only for the sphere. If the radius of the mantle is comparable to that of the hemispherical electrode, diffusion can take place to and from portions of the solution on the “back side” of the plane of the mantle. This effect results in enhanced currents (Section 18.2). An infinite mantle preserves uniform accessibility at a hemispherical electrode. For the rest of this chapter, we assume that hemispherical electrodes behave ideally in this respect. This assumption is experimentally realistic. Spherical UMEs can be made for gold (4), but are difficult to realize with other materials. Hemispherical UMEs can be formed by plating mercury onto a disk-shaped UME. Spherical electrodes can present geometric nonidealities where electrical contact is made with the sphere, because diffusion in that locale can be partially or wholly shielded. Historically, electrochemical work with spherical electrodes was carried out with mercury droplets expressed from a capillary and left in contact with the mercury thread in the capillary. For “large” droplets (with r0 of perhaps 250 μm), the shielding is hardly detectable, but for a spherical UME, the contact can take up a sizable fraction of the sphere. In a case like that in Figure 5.2.1, the area used for (5.1.20) should be corrected for the shielded part of the sphere (Problem 6.12). Equation 5.1.21 should be avoided when there is extensive shielding, because it is based on the full area of a sphere. 5.2.2

Disk UME

The disk UME, by far the most important practical shape, can be fabricated by sealing a fine wire in an insulator, such as a glass or a plastic resin, and then exposing and polishing a cross-section of the wire. The characteristic dimension is the radius, r0 , which must be 25 μm or smaller. The preparation of disk UMEs is an important art, discussed in Section 5.9.1. Disks with r0 as small 10 nm have been made regularly. The geometric area of a disk scales with the square of the radius and can be tiny. For r0 = 1 μm, the area, A, is only 3 × 10−8 cm2 , 6 orders of magnitude smaller than the geometric area of a 1-mm diameter microelectrode. 8 In the second edition, it was called the critical dimension. We now bow to the Electrochemical Dictionary (13).

215

216

5 Steady-State Voltammetry at Ultramicroelectrodes

Figure 5.2.1 Optical micrograph of a spherical gold UME with r0 = 4 μm. The conical shape extending out to the sphere is a glass micropipet through which electrical contact is made. [Reprinted with permission from Demaille, Brust, Tsionsky, and Bard (4). © 1997, American Chemical Society.]

5 μm

z axis (r = 0)

r axis (z = 0)

Flux into mantle = 0

r0

Mantle in z = 0 plane (extends beyond diffusion layer)

Inlaid UME disk

Figure 5.2.2 Geometry of diffusion at an ultramicroelectrode disk.

(a) Steady-State Diffusion at a Disk Electrode

A disk UME is complicated theoretically by the fact that diffusion occurs in two dimensions (14, 15)—radially with respect to the axis of symmetry and normal to the plane of the electrode (Figure 5.2.2). From Table 4.4.2, the corresponding Laplace equation describing steady-state diffusion of species j is ∇2 Cj (r, z) =

𝜕 2 Cj (r, z) 𝜕r2

+

2 1 𝜕Cj (r, z) 𝜕 Cj (r, z) + =0 r 𝜕r 𝜕z2

(5.2.1)

Let us consider the experiment equivalent to that treated in Section 5.1, i.e., a UME disk ∗ . The electrode held at potential E in a quiescent solution of species O at bulk concentration CO electrode reaction is O + ne ⇌ R. Species R can be generated at the electrode but is not present in the bulk solution.

5.2 Shapes and Properties of Ultramicroelectrodes

Four boundary conditions are needed for a solution to (5.2.1). Two come from the semiinfinite conditions ∗ lim CO (r, z) = CO

(5.2.2a)

∗ lim CO (r, z) = CO

(5.2.2b)

r→∞

z→∞

A third comes from recognition that there can be no flux into or out of the insulating mantle surrounding the disk, since no reaction occurs there: 𝜕CO (r, z) || =0 | | 𝜕z |z=0

(r > r0 )

(5.2.3)

The final condition is defined by the methodology and the kinetic behavior of the system. For now, let us consider the case where the surface concentration over the face of the electrode is uniform at C O (z = 0) for r ≤ r0 . Newman (16) solved an isomorphic problem relating to electrical potential profiles at an inlaid disk electrode, and we can adapt his solution to obtain the result we seek. To begin, we ∗ − C (r, z). Then, the recast our problem in terms of the “depleted concentration,” ΔCO = CO O Laplace equation becomes ∇2 ΔCO (r, z) =

𝜕 2 ΔCO (r, z) 𝜕r2

+

2 1 𝜕ΔCO (r, z) 𝜕 ΔCO (r, z) + =0 r 𝜕r 𝜕z2

(5.2.4)

It is easy to show that (5.2.4) is the same as (5.2.1) for species O. Boundary conditions (5.2.2a,b) become lim ΔCO (r, z) = 0

(5.2.5a)

lim ΔCO (r, z) = 0

(5.2.5b)

r→∞

z→∞

and boundary condition (5.2.3), written for ΔC O (r, z), remains the same. At the electrode sur∗ − C (z = 0) for r ≤ r . Our problem is now identical in form to the one face, ΔCO (z = 0) = CO O 0 that Newman solved. He proceeded by making a transformation to elliptical coordinates, 𝜉 and 𝜂, relating to r and z according to r = r0 [(1 + 𝜉 2 )(1 − 𝜂 2 )]1∕2

(5.2.6a)

z = r0 𝜉𝜂

(5.2.6b)

The coordinate 𝜉 generally expresses distance from the electrode surface (where 𝜉 = 0). The coordinate 𝜂 generally describes an angular position relative to the surface of the mantle, which corresponds to 𝜂 = 0. The rotational axis of symmetry through the center of the disk corresponds to 𝜂 = 1. The Laplace equation for these coordinates is [ ] [ ] 𝜕ΔCO (𝜉, 𝜂) 𝜕ΔCO (𝜉, 𝜂) 𝜕 𝜕 2 2 2 ∇ ΔCO (𝜉, 𝜂) = (1 + 𝜉 ) + (1 − 𝜂 ) =0 (5.2.7) 𝜕𝜉 𝜕𝜉 𝜕𝜂 𝜕𝜂

217

218

5 Steady-State Voltammetry at Ultramicroelectrodes

r

r0 0.8 0.7 0.6 0.5 0.4

z

0.3 0.2

∆CO ∆CO(z = 0)

= 0.1

Figure 5.2.3 Steady-state concentration profiles at a disk UME. Solid arcs represent surfaces of fixed values of ΔC O /ΔC O (z = 0). Dotted lines represent flux paths. [Adapted from Newman (16). Reprinted with permission of the publisher, The Electrochemical Society, Inc.]

and the boundary conditions become lim ΔCO (𝜉, 𝜂) = 0

𝜉→∞

𝜕ΔCO (𝜉, 𝜂) || =0 | | 𝜕𝜂 |𝜂=0 ΔCO (0, 𝜂) = ΔCO (z = 0)

(far from the disk)

(5.2.8)

(on the mantle)

(5.2.9)

(on the disk surface)

(5.2.10)

Note that ΔC O (z = 0), like C O (z = 0), is a constant. The solution (16) is compact, but the details need not be covered here. The result, re-expressed for our problem, is the steady-state concentration function arising by diffusion to a disk: ΔCO (𝜉, 𝜂) = 1 − (2∕𝜋)tan−1 𝜉 (5.2.11) ΔCO (z = 0) Figure 5.2.3 represents the two-dimensional concentration profile. This figure was originally published to depict the electrical potential profile in the solution surrounding a disk electrode undergoing electrolysis; however, it applies equally to the concentration functions of interest to us.9 The solid arcs represent surfaces of equal ΔC O /ΔC O (z = 0), which is a measure of the 9 In Chapter 10, we will cover this same figure in the context of Newman’s original paper.

5.2 Shapes and Properties of Ultramicroelectrodes

relative depletion of species O caused by electrolysis. For example, the outer arc, corresponding to ΔC O /ΔC O (z = 0) = 0.1, describes the surface for which the depleted concentration of O is 10% of the depleted concentration at the electrode surface. If the electrode surface is fully ∗ , and the outer depleted, as it would be at very negative potentials, then ΔC O (z = 0) would be CO ∗ (i.e., where ΔC = 10% of C ∗ ). Note arc would represent the surface where O is at 90% of CO O O that the arcs are closely spaced near the edges of the electrode, showing that the concentration profiles are steep. This phenomenon reflects a high rate of diffusive transport to the edge of the electrode. (b) Diffusion-Limited Steady-State Current at a UME Disk

The local steady-state current density, j, at any point on the disk is given by [ ] [ ] 𝜕CO (r, z) 𝜕ΔCO (r, z) j = nFDO = −nFDO 𝜕z 𝜕z z=0 z=0 [ ] [ ] 𝜕ΔCO (𝜉, 𝜂) 𝜕𝜉 1 𝜕ΔCO (𝜉, 𝜂) j = −nFDO = −nFDO 𝜕𝜉 𝜕z 𝜉=0 r0 𝜂 𝜕𝜉 𝜉=0

(5.2.12) (5.2.13)

For the result at the right of (5.2.13), one can use (5.2.6a) to evaluate r0 𝜂 in terms of r and 𝜉 and (5.2.11) to evaluate the derivative inside the brackets; thus, ( )1∕2 ⎡ 2 2ΔCO (z = 0) ⎤ 2nFDO ΔCO (z = 0) 1 + 𝜉 ⎥ j = nFDO ⎢ 2 = (5.2.14) 2 ⎢ r − r2 + r2 𝜉 2 ⎥ 𝜋(1 + 𝜉 ) 𝜋(r02 − r2 )1∕2 0 ⎣ 0 ⎦ 𝜉=0

The total current in any annulus of width dr at radius r is 2𝜋rjdr, and the total steady-state current is the integral over all annuli, i = 2𝜋

r0

∫0

jrdr = 4nFDO ΔCO (z = 0)

r0

∫0

(r02

r dr − r2 )1∕2

(5.2.15)

The value of the final integral is simply r0 , so ∗ − C (z = 0)] i = 4nFDO r0 [CO O

(5.2.16)

Recognizing A = 𝜋r02 gives i=

4nFADO 𝜋r0

∗ − C (z = 0)] [CO O

(5.2.17)

When the potential is negative enough that species O cannot coexist with the electrode, then C O (z = 0) = 0, and we have the cathodic diffusion-limited steady-state current, id,c =

∗ 4nFADO CO

𝜋r0

∗r id,c = 4nFDO CO 0

(5.2.18a) (5.2.18b)

These are important, widely used relationships. A common application of (5.2.18a,b) is to determine r0 (Section 5.9.3). The above derivation of i and id,c assumes that the disk is surrounded by an infinite mantle for exactly the reason identified in Section 5.2.1. In this book, we ordinarily assume ideality in this regard; however, Section 18.2 quantitatively addresses behavior at disks with finite mantles.

219

5 Steady-State Voltammetry at Ultramicroelectrodes

(c) Variation of Current Density on a Disk UME

An important feature of the disk geometry is that the current density varies across the face of the electrode, being much greater at the edge, which offers the nearest point of arrival to electroreactants drawn from a large surrounding volume. The effect is illustrated in Figure 5.2.4, which is drawn from (5.2.14) for the diffusion-limited case when C O (z = 0) = 0. About 40% of id,c is passed at radii between 90% and 100% of r0 . The current densities can become impressively high. In the case, presented in Figure 5.2.4, they reach toward 0.1 A/cm2 at the edge of the electrode and are everywhere above 10 mA/cm2 . The total steady-state limiting current is only 4.8 nA, but all of the action happens on just 2 × 10−7 cm2 . The nonuniformity of steady-state current density over the face of a disk complicates the interpretation of certain kinds of experiments. Modeling may be required to make the most valid interpretations of experimental results, especially when kinetic parameters are sought (Section 5.4.4). (d) Local Relationship between the Surface Concentrations of O and R

Even though the flux varies over the face of the disk, the fluxes of O and R must be equal and opposite at every point on the electrode. This is just a consequence of the conservation of mass. Thus, [ ] [ ] dC O (r, z) dC R (r, z) + DR =0 (5.2.19) DO dz dz z=0 z=0 or [ ] [ ] dC O (𝜉, 𝜂) 𝜕𝜉 dC R (𝜉, 𝜂) 𝜕𝜉 DO + DR =0 (5.2.20) d𝜉 𝜕z 𝜉=0 d𝜉 𝜕z 𝜉=0 Using the process that carried us from (5.2.12) to (5.2.14), we find 2DO ΔCO (z = 0) 2DR ΔCR (z = 0) + =0 𝜋(r02 − r2 )1∕2 𝜋(r02 − r2 )1∕2 or DR ΔCR (z = 0) = −DO ΔCO (z = 0) CR (z = 0) = CR∗ +

DO DR

∗ − C (z = 0)] [CO O

(5.2.23) Figure 5.2.4 Local diffusion-limited current density on the face of a disk UME of 2.5-μm radius ∗ = 5 mM). (n = 1; DO = 1 × 10−5 cm2 /s; CO

80 60 40 20 0 0.0

(5.2.21)

(5.2.22)

100 Local current density/mA cm−3

220

0.5 1.0 1.5 2.0 Distance from center of disk/μm

2.5

5.2 Shapes and Properties of Ultramicroelectrodes

We previously discovered this same relationship for surface concentrations at a spherical electrode, as (5.1.17). 5.2.3

Cylindrical UME

A cylindrical UME can be fabricated simply by exposing a length l of fine wire with radius r0 . The length is macroscopic, typically millimeters, so the characteristic dimension is r0 . The corresponding Laplace equation is the same as for the disk (equation 5.2.1). The coordinate r describes radial position normal to the axis of symmetry, while z is the position along the length. Since we normally assume uniformity along the length of the cylinder, 𝜕C/𝜕z = 𝜕 2 C/𝜕z2 = 0, and z drops out of the problem. The boundary conditions are exactly those used in solving the spherical case (Section 5.2.1), and the result is available in the literature (9). At a potential where the surface concentration of O is zero and in the long-time limit, when the diffusion length becomes large compared to r0 , the diffusion-limited current becomes id,c = id,c =

∗ 2nFADO CO

r0 ln(4DO t∕r02 ) ∗ 4𝜋nFlDO CO

ln(4DO t∕r02 )

(5.2.24a) (5.2.24b)

Equation 5.2.24b, derived by recognizing A = 2𝜋r0 l, shows that id,c depends rather weakly on the critical dimension. Since time appears in (5.2.24a,b), id,c is not a true steady-state limit such as we found for the sphere and the disk. However, the inverse logarithmic relationship ensures that the current declines slowly when 4DO t∕r02 ≫ 1. It can still be used experimentally in much the same way that steady-state currents are exploited at disks and spheres. This situation is called the quasi-steady state. The cylinder is uniformly accessible except at the ends, so the electrode can be expected to have a uniform current density everywhere along the surface until the ends are approached within several r0 . Since the length is macroscopic, the nonidealities at the ends affect only a small portion of the electrode and are not generally important. 5.2.4

Band UME

The band UME has as its characteristic dimension a width, w, in the range below 25 μm. The length, l, can be much larger, even in the centimeter range. Band UMEs can be fabricated by sealing metallic foil or an evaporated film between glass plates or in a plastic resin, then exposing and polishing an edge. A band can also be produced as a microfabricated metallic line on an insulating substrate using normal methods of microelectronic manufacture. By these means, electrodes with widths ranging from 25 μm to about 10 nm can be obtained. Operationally, there are many similarities between a cylinder and a band. Both differ from the disk in that the geometric area scales linearly with the characteristic dimension, rather than with the square. Electrodes with quite small values of w (or r0 , in the case of a cylinder) can possess appreciable geometric areas and can produce sizable currents. For example, a band of 1-μm width and 1-cm length has a geometric area of 10−4 cm2 , almost 4 orders of magnitude larger than that of a 1-μm disk. In the same way that a disk electrode is a two-dimensional diffusion system behaving very much like the simpler, one-dimensional, hemispherical case, a band electrode is a two-dimensional system behaving much like the simpler hemicylindrical system. The

221

222

5 Steady-State Voltammetry at Ultramicroelectrodes

z axis (x = 0)

w

Flux into mantle = 0 l x axis (z =

Inlaid UME band (l >> w)

0)

Mantle in z = 0 plane (extends beyond diffusion layer)

Figure 5.2.5 Diffusional geometry at a band electrode. Normally the length of the electrode is very much greater than the width, and the three-dimensional diffusion at the ends does not appreciably violate the assumption that diffusion occurs only along the x and z axes.

coordinate system used to treat diffusion at the band is shown in Figure 5.2.5. For the case where the applied potential drives the surface concentration of the electroreactant effectively to zero, the current–time relationship approaches a quasi-steady state when the diffusion length grows significantly larger than w (9), id,c = id,c =

∗ 2𝜋nFADO CO

w ln(64DO t∕w2 ) ∗ 2𝜋nFlDO CO ln(64DO t∕w2 )

(5.2.25a) (5.2.25b)

Thus, the band UME also does not provide a true steady-state current at long times. Band electrodes do not offer uniform accessibility. Diffusive transport is more effective at the edges than at locations on the face away from the edges, and the local current density varies accordingly over the electrode surface. 5.2.5

Summary of Steady-State Behavior at UMEs

For any of the electrode shapes considered here, the current at the working electrode approaches a steady state or a quasi-steady state when the diffusion-layer thickness becomes large compared to the characteristic dimension. One can write the cathodic limiting current in the manner developed empirically in Section 1.3.2, ∗ id,c = nFAmO CO

(5.2.26)

where mO is a mass-transfer coefficient. The functional form of mO depends on shape as given in Table 5.2.1.10 10 Real electrodes are rarely ideal spheres, hemispheres, circular disks, circular cylinders, or rectangular bands, so the mass transfer coefficients must usually be understood as approximations. A practical disk, for example, might have an elliptical shape or an irregular circumference, but still behave for practical purposes as a disk of a given r0 , interpreted as an “average” or “apparent” value.

5.2 Shapes and Properties of Ultramicroelectrodes

Table 5.2.1 Form of mO for UMEs of Common Shapes Sphere

Hemisphere

Disk

Cylinder(a)

DO

DO

4DO

2DO

2𝜋DO

r0

𝜋r0

r0 ln(4DO t∕r02 )

w ln(64DO t∕w2 )

r0

Band(a)

(a) Long-time limit is to a quasi-steady state.

We have shown in (5.1.19) and (5.2.17) that, for the sphere and disk, ∗ − C (surface)] i = nFAmO [CO O

(5.2.27)

where C O (surface) is C O (r = r0 ) for the sphere and C O (z = 0) for the disk. For these two shapes, we have discovered that the assumed basis for the simple model in Section 1.3.2 is rigorously valid, as long as the surface concentration is uniform over the electrode surface. This last condition was assumed in the treatments of steady-state diffusion at the sphere and the disk. We can also confirm other important relationships for the simple model, including ( ) i ∗ 1− (5.2.28) CO (surface) = CO id,c For the case where R is absent from the bulk, we can use this last result to obtain ( ) ∗ mO CO i CR (surface) = mR id,c

(5.2.29)

When R is present in the bulk, we understand from the symmetry of the problem that, at positive potentials, there would have to be a limiting anodic current, id,a , reflecting the diffusion-controlled oxidation of R. It would have to have the same form as (5.2.26), except for the sign given to the anodic current; thus, id,a = −nFAmR CR∗

(5.2.30)

Also by symmetry, i = nFAmR [CR (surface) − CR∗ ]

(5.2.31)

When both O and R are present, (5.2.26)–(5.2.28) remain valid; however, the surface concentration of R is given by ( ) ∗ mO CO i ∗ CR (surface) = CR + (5.2.32) mR id,c By substitution from (5.2.26) and (5.2.30), this can be converted to ( ) i ∗ CR (surface) = CR 1 − id,a

(5.2.33)

We will have many occasions to make use of these relationships as we develop treatments of different methods.

223

224

5 Steady-State Voltammetry at Ultramicroelectrodes

We have not demonstrated the validity of these relationships for the hemisphere, cylinder, or band UME. For the hemisphere and the cylinder, the geometric simplicity suggests that the relationships would all remain valid when the surface concentrations are uniform. The band is a much more complex problem overall. For all of these geometries the relationships given here are likely to be good, useful approximations, even if they are not strictly valid. For the cylinder and the band, i, id,c , and id,a values would always be quasi-steady-state currents.

5.3 Reversible Electrode Reactions The goal in this section is to examine steady-state voltammetry for systems in which the reactants and products are chemically stable, and the kinetics of all processes, including the charge-transfer kinetics, are fast, so that every chemical process is at equilibrium. These conditions define a reversible system. We continue to consider an electrode reaction O + ne ⇌ R, where both O and R are soluble. The solution is not stirred and there is a supporting electrolyte, so convection and migration are not important. The working electrode is a UME at which steady-state voltammograms are recorded. 5.3.1

Shape of the Wave

(a) R Absent from the Bulk

Because the system is reversible, the electrode potential and the surface concentrations of O and R are always in a nernstian balance (Section 3.4.5), ′ RT CO (surface) E = E0 + ln (5.3.1) nF CR (surface) where C O (surface) and C R (surface) are surface concentrations, such as C O (r = r0 ) for a sphere or C O (z = 0) for a disk. For an electrode of any shape, the potential enforces uniform surface concentrations for O and R, so we can rely on the generalized relationships validated in Section 5.2.5. In (5.3.1), C O (surface) can be replaced using (5.2.28), and C R (surface) can be replaced (for CR∗ = 0) using (5.2.29). After rearrangement, one has ( ) id,c − i m ′ RT RT R E = E0 + ln + ln (5.3.2) nF mO nF i in which mO and mR are the mass-transfer coefficients given in Table 5.2.1. This is the shape equation for a reversible wave for any UME geometry covered in Section 5.2.5. For the sphere, disk, and hemisphere, (5.3.2) becomes ( ) id,c − i DR RT RT 0′ E=E + ln + ln (5.3.3) nF DO nF i Both (5.3.2) and (5.3.3) are rigorous results for spherical and disk-shaped UMEs. Because we have verified (5.2.28) and (5.2.29) only for those two cases, these results may be approximations for the other shapes. One can also write ) ( id,c − i RT ln (5.3.4) E = E1∕2 + nF i

5.3 Reversible Electrode Reactions

where the half-wave potential, E1/2 , is generally ′

E1∕2 = E0 +

m RT ln R nF mO

(5.3.5)

but for the sphere, disk, and hemisphere E1/2 is also ′

E1∕2 = E0 +

D RT ln R nF DO

(5.3.6)

Equations 5.3.2, 5.3.4, and 5.3.5 are exactly the same as (1.3.15), (1.3.17), and (1.3.16), which we derived using the simple approach to steady-state mass transfer in Chapter 1. As shown in Figure 5.3.1, these equations describe a wave that rises from baseline to the diffusion-controlled limit over a fairly narrow potential region centered on E1/2 . Since the ratio of diffusion coefficients in (5.3.6) [or mass-transfer coefficients in (5.3.5)] is roughly unity in ′ most cases, E1/2 is usually a good approximation to E0 for a reversible couple in which all species are soluble. Note also that E vs. log[(id,c − i)/i] should be linear with a slope of 2.303RT/nF or 59.1/n mV at 25 ∘ C. This “wave slope” is often computed from experimental data to test for reversibility. A quicker test [the Tomeš criterion (17)] is that |E3/4 − E1/4 | = 56.4/n mV at 25 ∘ C. The potentials E3/4 and E1/4 are those for which i = 3id,c /4 and i = id,c /4, respectively. If the wave slope or the Tomeš criterion significantly exceeds the expected value, the system is not reversible [Section 5.4.2(b)]. (b) Both O and R Present in the Bulk

When both O and R exist in the solution, the wave-shape equation is derived from (5.3.1) by substituting for the surface concentrations according to (5.2.28) and (5.2.33). The result is ) ( id,c − i RT E = E1∕2 + ln (5.3.7) nF i − id,a where E1/2 is given by (5.3.5) or (5.3.6), depending on the geometry. The parameters id,c and id,a are the diffusion-limited steady-state (or quasi-steady-state) currents for reduction of O and oxidation of R, respectively, and are given by (5.2.26) and (5.2.30). As for the case in which R Figure 5.3.1 A reversible cathodic wave in steady-state voltammetry (n = 1, T = 298 K, and DO = 0.7DR ). Because DO ≠ DR , E 1/2 differs slightly ′

from E 0 , in this case by about +9 mV. For n > 1, the wave rises more sharply to the plateau. E1/2

i

id,c id,c/2

100

0

–100 (E – E0′)/mV

–200

225

5 Steady-State Voltammetry at Ultramicroelectrodes

1.0

Eeq

0.5 id,c

0.0 i/nA

226

E1/2

–0.5

id,a

–1.0

(id,c + id,a )/2

–1.5 –2.0 –2.5 0.4

0.3

0.2

0.1 0 –0.1 E/V vs. reference

–0.2

–0.3

–0.4

Figure 5.3.2 Steady-state voltammogram for a system in which both O and R are present in the bulk ′ ∗ = 1 mM, C ∗ = 3 mM, D = 1 × 10−5 cm2 /s, D = 2 × 10−5 cm2 /s, r = 1 μm). (E 0 = −0.1 V, n = 1, CO O R 0 R

is absent from the bulk, (5.3.7) is rigorous for the sphere and the disk, but may be approximate for the other electrode shapes. Figure 5.3.2 shows the composite anodic–cathodic wave predicted for a system in which both O and R are present. At open circuit, the working electrode is poised by the O/R couple at Eeq

∗ RT CO =E + ln ∗ nF CR 0′

(5.3.8)

Polarizing the working electrode toward more negative potentials from Eeq produces cathodic currents, while polarizing toward more positive potentials yields anodic currents. When the argument of the logarithm in (5.3.7) is unity, E = E1/2 . This occurs at i = (id,c + id,a )/2, halfway between the anodic and cathodic plateaus. According to (5.3.7), E is expected to be linear with log[(id,c − i)/(i − id,a )], having a slope of 2.303RT/nF or 59.1/n mV at 25 ∘ C. A significantly larger figure indicates a lack of reversibility.11 Actually, the shape of the waves in Figures 5.3.1 and 5.3.2 is the same. The only difference is a vertical translation of the curve in the latter case to allow for the anodic plateau current. 5.3.2

Applications of Reversible i–E Curves

This section applies equally to steady-state voltammetry (covered in this chapter) and sampledtransient voltammetry (covered in Chapter 6). (a) Information from the Wave Height

The plateau current of a simple wave (whether reversible or not) is controlled by mass transfer and can be used to determine any single system parameter that affects the limiting flux of electroreactant at the electrode surface, including n, A, D, and C ∗ . Historically, the most common application has been to employ voltammetric wave heights to determine concentrations, or to follow changes in concentration. The plateau current of a steady-state voltammogram can also provide the characteristic dimension of the electrode (e.g., r0 for a sphere, hemisphere, or disk; Section 5.9.3). 11 Alternatively, it can indicate appreciable iRu , but this is rare with UMEs, which are central to this chapter.

5.3 Reversible Electrode Reactions

(b) Information from the Wave Shape

In a reversible (nernstian) system, the electrode reaction is always at equilibrium at the electrode surface. The kinetics are so facile that the interfacial concentrations are governed solely by thermodynamic aspects. The shapes and positions of reversible waves can be exploited to provide thermodynamic properties, such as standard potentials, free energies of reaction, and various equilibrium constants, just as potentiometric measurements can be. On the other hand, reversible systems offer no kinetic information about heterogeneous electron transfer, because the kinetics are, in effect, transparent. The wave shape is most easily analyzed in terms of the wave slope (Section 5.3.1), which is expected to be 2.303RT/nF or 59.1/n mV at 25 ∘ C for a reversible system. Larger slopes are generally found for systems that do not have both nernstian heterogeneous kinetics and overall chemical reversibility [Section 5.4.2(b)]; thus, the slope can be used to diagnose reversibility. If the system is known to be reversible, the wave slope can be used to determine the value of n. Occasionally, one encounters the idea that a wave slope near 60 mV can be taken as an indicator of both reversibility and n = 1. If the electrode reaction is simple and does not implicate, for example, adsorbed species (Chapter 14), one can support both conclusions from the wave slope. However, electrode reactions are often subtly complex, and it is safer to determine reversibility by a technique that can view the reaction in both directions, such as cyclic voltammetry at a larger electrode (Chapter 7). (c) Information from the Wave Position

′

Because the half-wave potential for a reversible wave is close to E0 , steady-state or sampledtransient voltammetry is convenient for estimating the formal potential for a chemical system that has not been previously characterized. It is essential to verify reversibility, because slow electron-transfer kinetics or coupled irreversible homogeneous reactions can cause E1/2 to be ′ quite some distance from E0 (Sections 1.4.2, 5.4.2, and 6.3.5; Chapter 13). It is a common error ′ to assume blindly that E1∕2 ≈ E0 . By definition, a formal potential describes the potential of a couple at equilibrium in a system where the oxidized and reduced forms are present at unit formal concentration, even though O and R may be distributed over multiple chemical forms (e.g., as the two members of a conjugate acid–base pair). Formal potentials always manifest activity coefficients. Frequently they also reflect chemical effects, such as complexation or participation in acid–base equilibria (Section 2.1.7); thus, the formal potential can shift systematically as the medium changes. In steady-state or sampled-transient voltammetry, the half-wave potential of a reversible wave would shift correspondingly. This phenomenon provides a highly profitable route to chemical information and has been exploited elaborately (mostly using voltammetric techniques with longer histories than SSV). Fortunately, the effects and rationales are general to all forms of voltammetry. We have already encountered one relevant example in Section 1.4.1, where we treated the effect of a preceding equilibrium on the voltammetric response. The reaction scheme was A ⇌ O + qY

(5.3.9)

O + ne ⇌ R

(5.3.10)

in which A and Y might be, for example, a complex and a ligand. The treatment is applicable to any form of association described by (5.3.9), as long as the kinetics are fast enough to keep the participants always at equilibrium. The voltammogram turned out in Section 1.4.1 to have the

227

228

5 Steady-State Voltammetry at Ultramicroelectrodes

Binding of the oxidized form: negative shift

i

0

1

2

E/V vs. reference

Figure 5.3.3 Effect of adding a substance that can manifest equilibrium binding of species O in the reversible O/R couple. Wave positions: (0) in the absence of binding agent, (1) upon addition of binding agent at significant excess concentration, (2) when the binding agent concentration is further increased.

normal shape, given by (5.3.4), but to have E1/2 more complex than (5.3.5). At 25 ∘ C, ′

E1∕2 = E0 +

m 0.059 0.059 0.059 log R + log K − q log CY∗ n mA n n

(5.3.11)

The first two terms express E1/2 for the O/R voltammetric wave in the absence of the binding agent, Y. The last two terms describe the effect of the binding.12 The equilibrium constant is for (5.3.9) as written, so it is a dissociation (or instability) constant. If the binding of O is significant, then K ≪ 1 and log K would be a negative number, commonly −3 to −30; therefore, the third term in (5.3.11) would typically have a value of −180 mV to −1.8 V. In the derivation leading to (5.3.11), the molar concentration CY∗ is assumed to be much larger than the total molar concentration of O in all forms, so CY∗ typically would be 0.1–1 M. For q = 1 to 6, the last term of (5.3.11) would be 0–360 mV. Taking the two final terms together, one sees that the effect of adding Y to the solution is usually to shift the wave negatively (Figure 5.3.3), often by hundreds of mV. Indeed, the existence of such a shift is strong qualitative evidence that Y binds O. By carrying out experiments in the absence of Y and at several concentrations of Y, one can obtain values for both K and q. In Section 1.4.1, we first encountered this case in the context of an approximate approach to mass transfer; however, we later learned in Section 5.2.5 that the principal equations underlying that approach are rigorous for steady-state currents at spherical and disk-shaped UMEs. In Section 6.2.4, we will find that they are also rigorous for sampled transient currents obtained at any electrode at which semi-infinite linear diffusion applies. Thus, (5.3.11) and results for other, similar equilibrium cases are also rigorous for sampled-transient voltammetry and for steady-state voltammetry at the sphere and the disk. They are at least approximations for SSV at UMEs of other shapes. In the example just considered, the important feature was a shift in the wave position caused by selective chemical stabilization of one of the redox forms. In a reversible system, the potential axis is a free energy axis, and the magnitude of the shift is a direct measure of the free energy involved in the stabilization. These concepts are quite general and can be used to understand many chemical effects on electrochemical responses. Any equilibrium in which either redox 12 If the equilibrium constant, K, is expressed for molar concentrations, then CY∗ would be a molar concentration. In the case where K is written for activities, CY∗ would be aY /𝛾 Y , which is also the numeric value of the molar concentration.

5.3 Reversible Electrode Reactions

species participates will influence the wave position; moreover, changes in concentrations of participants in those equilibria (e.g., species Y in the example above) will cause an additional shift in the half-wave potential. This movement of wave position may seem confusing at first, but the principles are not complicated and are very valuable: 1) If the oxidized form (e.g., species O) is chemically bound in an equilibrium process [as in (5.3.9)], then the oxidized form is stabilized. It becomes energetically harder to reduce the oxidized form, and it becomes easier to produce this species by oxidation of the reduced form. Accordingly, the voltammetric wave shifts in a negative direction by a degree that depends on the equilibrium constant for the binding process (i.e., the change in standard free energy) and the concentration of the binding agent. This is the situation we encountered in the case above, and it is illustrated generally in Figure 5.3.3. 2) If the reduced form of a redox couple (e.g., species R) is chemically bound in an equilibrium process, then the reduced form has a lowered free energy relative to the situation where the binding is not present. Oxidation of the reduced form becomes energetically more difficult, and reduction of the oxidized form becomes easier. Therefore, the voltammetric wave shifts in a positive direction by an amount reflecting the equilibrium constant for the binding process and the concentration of the binding agent. This case is illustrated generally in Figure 5.3.4. 3) Increasing the concentration of the binding agent enlarges the equilibrium fraction of bound species; therefore, it reinforces the basic effect and enhances the shift in the wave from its original position. This result is illustrated in Figures 5.3.3 and 5.3.4 by the position of Curve 2 relative to Curve 1 in each case. 4) Secondary equilibria can also affect the wave position in ways that can be interpreted within these principles. Suppose, for example, that the binding agent (e.g., NH3 ) is itself involved in an acid–base equilibrium (e.g., with NH+ ), so that the availability of the binding agent is 4 affected by the pH. If the pH is changed such that the concentration of binding agent is lessened (e.g., by adding HNO3 in the case of binding by NH3 ), the effect would be to lower the fraction of binding and, consequently, to bring the wave back toward Curve 0 in Figures 5.3.3 and 5.3.4. 5) When both redox forms engage in binding equilibria, both are stabilized relative to the situation in which the binding processes are absent. The effects offset each other. If the free energy of stabilization were the same on both sides of the basic electron-transfer process, there would be no alteration of the free energy change required for either oxidation or reduction, and the wave would not shift. If the stabilization of the oxidized form is greater, then the wave shifts in the negative direction, and vice versa. Binding of the reduced form: positive shift

i

2

1

0

E/V vs. reference

Figure 5.3.4 Effect of adding a substance that can manifest equilibrium binding of species R in the reversible O/R couple. Wave positions: (0) in the absence of binding agent, (1) upon addition of binding agent at significant excess concentration, (2) when the binding agent concentration is further increased.

229

230

5 Steady-State Voltammetry at Ultramicroelectrodes

A wide variety of binding chemistry can be understood and analyzed within this framework. Obvious by the prior example is complexation of metals. Also generally important are acid–base equilibria, which affect many inorganic and organic redox species in protic media. The principles discussed here are valid in systems involving such diverse phenomena as amalgamation, dimerization, ion pairing, adsorptive binding on a surface, coulombic binding to a polyelectrolyte, and binding to enzymes, antibodies, or DNA. Detailed treatments are easily worked out for other types of electrode reactions, including O + mH+ + ne ⇌ R

(Problem 5.2)

O + ne ⇌ Radsorbed

(Chapter 14)

Similar treatments can be developed for systems which do not involve the binding phenomena emphasized here, but which differ from the simple process O + ne ⇌ R and, yet, remain reversible. Such examples include O + ne ⇌ Rinsoluble

(Problem 5.3)

O + ne ⇌ 3R

(Problem 5.4)

Details are often available in references on voltammetry of various kinds. Reversible systems have the advantage of behaving as though all chemical participants are at equilibrium; thus, they can be treated by any set of equilibrium relationships linking the species defining the oxidized and reduced states of the system. It is not important to treat the system according to an accurate mechanistic path, because the behavior is controlled entirely by free energy changes between initial and final states, and the mechanism is invisible to the experiment. (d) Information from a Change in Limiting Current

For many systems, the diffusion coefficient for species A in (5.3.9) is not very different from that of O; hence, id,c for reduction of bound O is about the same as for unbound O (as implied in Figures 5.3.3 and 5.3.4). However, if the binding agent is very large, as might occur when Y is a protein, a polymer, or DNA, then the bound O will be much larger than free O, and there will be a significant decrease in D and in id,c . Under these conditions, the change in id,c with addition of Y can be used to obtain information about K and q. An investigation of this type was based on the interaction of Co(phen)3+ with double-strand DNA (18), where phen is 3 1,10-phenanthroline. The diffusion coefficient decreased from 3.7 × 10−6 cm2 /s for the free Co species to 2.6 × 10−7 cm2 /s upon binding to DNA.

5.4 Quasireversible and Irreversible Electrode Reactions Now we turn to situations in which heterogeneous electron-transfer kinetics begin to limit the rate of the electrode reaction. Basic ideas relevant to this situation were developed in Section 3.4.6. In these situations, kinetic parameters such as k f , k b , k 0 , and 𝛼 influence the responses to potential steps and can often be evaluated from those responses. In this section, we will treat the one-step, one-electron reaction, O + e ⇄ R, usually using the general Butler–Volmer i − E characteristic (Section 3.3.2). 5.4.1

Effect of Electrode Kinetics on Steady-State Responses

The predicted steady-state voltammetric behavior depends on the kinetic model employed and whether O and R are present in the bulk. We distinguish several situations as we develop the basic ideas.

5.4 Quasireversible and Irreversible Electrode Reactions

(a) General Kinetics, Any Mix of O and R in the Bulk

For the 1e reaction being considered, the current is i = FA[kf CO (surface) − kb CR (surface)]

(5.4.1)

where C O (surface) and C R (surface) represent the surface concentrations at the chosen UME, and k f and k b are the potential-dependent rate constants for the forward and backward reactions. Substitution for the surface concentrations according to (5.2.28) and (5.2.33) gives [ ( ) ( )] i i ∗ ∗ i = FA kf CO 1 − − kb CR 1 − (5.4.2) id,c id,a where id,c and id,a are the cathodic and anodic diffusion-limited currents. Upon rearrangement, one obtains ∗ FAk f CO FAk b CR∗ i= (id,c − i) − (id,a − i) (5.4.3) id,c id,a ∗ and i ∗ Upon identifying id,c = FAmO CO d,a = −FAmR CR from (5.2.26) and (5.2.30), and defining Λf = k f /mO and Λb = k b /mR , the result becomes

i=

Λf id,c + Λb id,a 1 + Λf + Λb

(5.4.4)

This relationship is completely general. No model of kinetics has yet been applied.13 (b) Butler–Volmer Kinetics, Any Mix of O and R in the Bulk

Now, let us invoke the BV model, for which the rate constants are given by (3.3.7a,b) and can be re-expressed as kf = k 0 𝜃 −𝛼

(5.4.5a)

0 (1−𝛼)

kb = k 𝜃

(5.4.5b)

in which 𝜃 = ef (E−E

0′ )

= kb ∕kf

(5.4.6)

with f = F/RT. If we define Λ0 = k 0 /mO and 𝜉 = mO /mR , then Λf = Λ0 𝜃 −𝛼 and Λb = Λ0 𝜉𝜃 (1 − 𝛼) . Substitution and factoring then gives i=

Λ0 𝜃 −𝛼 (id,c + 𝜉𝜃id,a ) 1 + Λ0 𝜃 −𝛼 (1 + 𝜉𝜃)

(5.4.7)

This equation has a remarkable scope, for it compactly describes the steady-state current at any potential and in any kinetic regime (reversible, quasireversible, or totally irreversible) for a one-step, one-electron process at a UME. It also describes all possible steady-state voltammograms for such a process. The assumptions are only that species O and R remain stable in solution and that BV kinetics apply.14 13 In the second edition, kinetic effects in SSV were developed in terms of two parameters, 𝜅 and 𝜅 0 , which expressed the ratio of an electron-transfer rate constant to a mass-transfer coefficient. In this edition, the treatment is simpler and more general. The parameters Λf , Λb , and Λ0 are now used to express the ratios of electron-transfer and mass-transfer rates. 14 Actually, (5.4.7) has even broader applicability than to UMEs. It applies to any system that adheres to the general mass-transfer relationships, (5.2.26)–(5.2.33). In this book, we will encounter many such systems not based on UMEs.

231

232

5 Steady-State Voltammetry at Ultramicroelectrodes

The dimensionless parameter Λ0 expresses the intrinsic facility of electron transfer relative ′ to the rate of mass transfer. This is the value of both Λf and Λb /𝜉 at E0 . Since 𝜉 is usually on ′ the order of 1, Λf ∼ Λb ∼ Λ0 at E0 . When Λ0 ≪ 1, the overall rate of reaction, and, hence, the current, is limited by the electron-transfer kinetics; when Λ0 ≫ 1, kinetics are not limiting and mass transfer controls the overall rate. ′ As the potential is changed from values far positive of E0 (where 𝜃 is large) to values far negative (where 𝜃 approaches zero), i makes a transition from an asymptote at id,a to an asymptote at id,c . However, the transition can take different forms, as shown in Figure 3.4.6, which can be understood (Problem 5.8) as the set of plots of (5.4.7) for Λ0 → ∞ and Λ0 = 1, 0.1, and 0.01, ∗ = m C∗ . given mO CO R R (c) Special Cases

For commonly encountered conditions, (5.4.7) takes simpler forms. 1) BV kinetics. Species R Absent from the Bulk. When CR∗ = 0, id,a = 0; therefore, i id,c

=

Λ0 𝜃 −𝛼 1 + Λ0 𝜃 −𝛼 (1 + 𝜉𝜃)

(5.4.8)

∗ = 0, i 2) BV kinetics. Species O Absent from the Bulk. When CO d,c = 0; therefore,

i id,a

=

Λ0 𝜉𝜃 (1−𝛼) [ ] 1 + Λ0 𝜉𝜃 (1−𝛼) (𝜉𝜃)−1 + 1

(5.4.9)

3) Reversible kinetics. In Problem 5.9, the reader is invited to show that the simplified forms of (5.4.7) for Λ0 → ∞ are identical with results obtained in Section 5.3.1. Figure 5.4.1 is a display of voltammograms according to (5.4.8) for systems in which R is absent from the bulk. The leftmost curve also corresponds to (5.3.2). Progressively smaller values of k 0 cause a broadening of the wave and a displacement toward more extreme potentials, just as one would expect from the discussion in Section 3.4.6. 5.4.2

Total Irreversibility

Any displacement in potential that activates k f also suppresses k b ; hence, the backward component of the electrode reaction becomes progressively less important at more negative potentials. If k 0 is very small, a sizable activation of k f is required at any potential where appreciable cathodic current can flow; therefore, k b is essentially fully suppressed in the range of the cathodic wave. The totally irreversible regime for a reduction process is conveniently defined by the condition that k b /k f ≈ 0 (i.e., 𝜃 ≈ 0). The following relationships give the shape of the irreversible cathodic wave: i id,c i id,c

=

=

Λf 1 + Λf Λ0 𝜃 −𝛼 1 + Λ0 𝜃 −𝛼

(5.4.10a)

(5.4.10b)

Equation 5.4.10a is derived from (5.4.4) simply by recognizing that Λb ≪ Λf . Since it comes from (5.4.4), no specific kinetic model underlies this relationship. Equation 5.4.10b is derived from (5.4.7) by allowing 𝜃 to become very small or from (5.4.10a) by setting Λf = Λ0 𝜃 −𝛼 . By either approach, (5.4.10b) implicates BV kinetics.

5.4 Quasireversible and Irreversible Electrode Reactions

1.2

1.0

0.8 Quasireversible

i/id,c 0.6

Totally irreversible

0.4 Reversible 0.2

0.0 200

100

0

–100 (E –

–200

–300

–400

–500

E0′)/mV

Figure 5.4.1 Cathodic steady-state voltammograms for various kinetic regimes. Curves are calculated for a spherical or hemispherical electrode from (5.4.8) assuming Butler–Volmer kinetics (CR∗ = 0, 𝛼 = 0.5, and r0 = 5 μm, DO = DR = 1 × 10−5 cm2 /s). The reversible curve is for Λ0 → ∞. For the curves with points, the values of k0 are 2 × 10−2 (squares), 2 × 10−3 (triangles), and 2 × 10−4 cm/s (diamonds). Since mO = DO /r0 = 2 × 10−2 cm/s, these curves correspond to Λ0 values of 1, 0.1, and 0.01, respectively.

Analogous considerations apply to an anodic wave. The totally irreversible anodic regime is defined by k f /k b ≈ 0 (i.e., 𝜃 becoming very large), and the following relationships give the shape of the anodic wave: i id,a i id,a

=

=

Λb 1 + Λb Λ0 𝜉𝜃 (1−𝛼) 1 + Λ0 𝜉𝜃 (1−𝛼)

(5.4.11a)

(5.4.11b)

where (5.4.11a) comes from (5.4.4) and involves no assumption about kinetic model, while (5.4.11b) rests on BV kinetics. The presence or absence of species R in the bulk has no impact on an irreversible cathodic wave, because the anodic component reaction is always negligible in the relevant potential range. Likewise, the presence or absence of species O in the bulk is insignificant for an irreversible anodic wave. For a cathodic wave, one can rearrange (5.4.10b) to obtain ( ) id,c − i RT RT 0′ 0 E=E + ln Λ + ln (5.4.12) 𝛼F 𝛼F i which has a half-wave potential given by ′

E1∕2 = E0 +

RT ln Λ0 𝛼F

(5.4.13)

233

234

5 Steady-State Voltammetry at Ultramicroelectrodes

A plot of E vs. log[(id,c − i)/i] is expected to be linear with a slope of 2.303RT/𝛼F (i.e. 59.1/𝛼 mV at 25 ∘ C) and an intercept of E1/2 . From the slope and intercept, one can obtain 𝛼 and k 0 , ′ although E0 and mO must be known to obtain the latter. The analogous relationships for an anodic wave are readily derived from (5.4.11b): ( ) id,a − i RT RT 0′ 0 E=E − ln Λ 𝜉 − ln (5.4.14) (1 − 𝛼)F (1 − 𝛼)F i ′ RT E1∕2 = E0 − ln Λ0 𝜉 (5.4.15) (1 − 𝛼)F 5.4.3

Kinetic Regimes

One can define the boundaries between kinetic regimes in SSV by focusing on Λ0 = k 0 /mO , ′ which is the value of Λf and Λb /𝜉 at E0 . In effect, this parameter compares the intrinsic electron-transfer rate to the mass-transfer rate. Even though currents are small at UMEs, current densities can be extremely high because of the very high mass transfer rates that can apply at UMEs. This is the aspect of their nature that provides access, through Λ0 , to heterogeneous rate constants of the most facile known reactions. It has been estimated (19) that values of k 0 as large as 200 cm/s would be measurable using steady-state current measurements in scanning electrochemical microscopy (SECM; Chapter 18) at the smallest UMEs; however, the largest figures actually reported to date are in the range of 10–40 cm/s (Sections 3.3.3 and 18.4.1). ′ If a system is to appear reversible, then the rate constants at potentials near E0 must be large enough that (5.4.7) converges to the reversible result, (5.3.7). This will be true within normal limits of precision if Λ0 > 10. For a reduction process, total irreversibility applies when the cathodic wave is displaced neg′ atively to such a degree that 𝜃 ≈ 0 across the whole wave. If E1∕2 − E0 is more negative than −4.6RT/F, that condition will be satisfied. The second term in (5.4.12) must then be more negative than −4.6RT/F, implying that log Λ0 < − 2𝛼. Essentially the same conclusion is reached by considering an oxidation wave instead. The quasireversible regime is between these boundaries, in the range 10−2𝛼 ≤ Λ0 ≤ 10. The kinetic regime is defined partly by experimental conditions, and it can change if those conditions are altered. The most important experimental variable affecting the kinetic regime in SSV is the characteristic dimension of the electrode (e.g., r0 ). A good example of this effect is found in Figure 5.4.2, which shows steady-state voltammograms for the oxidation of ferrocenylmethyltrimethylammonium (FcTMA+ ; Figure 1) at roughly hemispherical electrodes of different sizes (20). For a hemisphere, Λ0 = k 0 r0 /D; therefore, Λ0 becomes proportionately smaller as the electrode size is reduced. For the largest electrode, the system is reversible, but for the smaller three electrodes it is not, progressively more clearly so as the UME is made smaller (see also Section 5.4.3 and Problem 5.6). In estimating kinetic parameters, the actual shape of the electrode can be important. For example, in making small electrodes (sub-μm radius), the metal disk is sometimes recessed inside the insulating sheath and has access to the solution only through a small aperture (Problem 5.5). Such an electrode will show a limiting current characteristic of the aperture radius, but the heterogeneous kinetics will be governed by the radius of the recessed disk (21, 22). 5.4.4

Influence of Electrode Shape

The foregoing discussion is rigorous only for spherical electrodes, which are uniformly accessible. Steady-state voltammetry can be carried out readily at other UMEs; however, the results

5.4 Quasireversible and Irreversible Electrode Reactions

1.0 35

rapp 31.1 nm

i/id,a r0 = 12.5 μm

25 0.5

i/pA

rapp 31.1 nm 8.8 nm 3.2 nm

15 8.8 nm

5 –5

3.2 nm

0.0 200

(a)

300 400 500 E/mV vs. Ag/AgCl (b)

600

Figure 5.4.2 (a) Steady-state voltammograms for the oxidation of 2 mM FcTMA+ at three Pt UMEs in aqueous 0.2 M KCl (v = 10 mV/s). Apparent r0 values, rapp , were calculated from the limiting currents, id,a , using the hemispherical relationship, id,a = 2𝜋nFDC * rapp and D = 7.5 × 10−6 cm2 /s. (b) Smoothed curves from (a) normalized to id,a . The normalized steady-state voltammogram at a 25-μm-diameter Pt UME is also shown. [Reprinted with permission from Watkins et al. (20). © 2003, American Chemical Society.]

for quasireversible and totally irreversible systems are affected by the nonuniformity in the flux at different points on the electrode surface. At a disk UME, for example, mass transfer can support a flux to points near the edge that is much higher than to points near the center [Section 5.2.2(c)]; thus, the kinetics must be activated more strongly at the edge than in the center to support the diffusion-limited current. There is a significant contrast here with Section 5.2.5, where we found that the results for reversible systems observed at spherical electrodes could be extended generally to electrodes of other shapes (23). This is true because the potential directly controls the surface concentrations of O and R for a reversible system and keeps them uniform across the surface. For quasireversible and irreversible systems, the potential controls rate constants, not surface concentrations, uniformly across the surface. The surface concentrations become defined indirectly by the local balance of interfacial electron-transfer rates and mass-transfer rates. When the electrode surface is not uniformly accessible, this balance varies over the surface in a way that is idiosyncratic to the geometry. This is a complicated situation that can be treated theoretically for an arbitrary shape by simulation. For UME disks, however, the geometric problem can be simplified by symmetry, and results exist in the literature to facilitate the quantitative analysis of voltammograms (6, 7, 24–26). 5.4.5

Applications of Irreversible i–E Curves

Immediately below, we review the kinds of information that one can obtain from irreversible steady-state voltammograms; however, it is important to understand that these comments are limited to one-step, one-electron electrode reactions, i.e., to outer-sphere elementary electrode reactions. A complex electrode process (e.g., the reduction of O2 to H2 O) can involve multiple electron-transfer steps, chemical reorganizations, adsorption or desorption steps, or irreversible homogeneous processes linked to the electrode reaction. We are not yet prepared to deal with any of those complications. Chapters 13 and 15 focus on them. (a) Information from the Wave Height

The plateau of an irreversible or quasireversible wave for a one-step, one-electron system involving chemically stable species is controlled by diffusion and can be used to determine

235

236

5 Steady-State Voltammetry at Ultramicroelectrodes

any variable that contributes to the limiting current, including A, D, C*, or the characteristic dimension of the UME. Section 5.3.2(a), which covers these ideas, also applies to irreversible and quasireversible one-step, one-electron systems in SSV. (b) Information from the Wave Shape and Position

When the wave is not reversible, the half-wave potential is not usually a good estimate of the formal potential and cannot be used readily to determine thermodynamic quantities in the manner discussed in Section 5.3.2(c). In the case of a totally irreversible system, the wave shape and position can furnish only kinetic information, but quasireversible waves can sometimes ′ provide approximate values of E0 in addition to kinetic parameters. Because the interpretation and information content of a wave’s shape and position depend on the kinetic regime, it is essential to be able to diagnose the regime confidently. For a known one-step, one-electron case, wave shape is a useful indicator toward that purpose. One can characterize reversibility either by the slope of a plot of E vs. log[(id − i)/i] (the wave slope) or by the difference |E3/4 − E1/4 | (the Tomeš criterion). Table 5.4.1 provides a summary of expectations for SSV in all three kinetic regimes. For reversible systems, these figures of merit are near 60/n mV at ambient temperatures. Significantly larger figures often signal a degree of irreversibility. For example, if the one-step, one-electron mechanism applies and 𝛼 is between 0.3 and 0.7 (commonly true), then a system with totally irreversible electron-transfer kinetics would show |E3/4 − E1/4 | between 80 and 190 mV. Such behavior would represent a clear departure from reversibility. The wave slope behaves similarly; however, it is not always easy to analyze it precisely, because wave-slope plots are slightly nonlinear for quasireversible voltammograms. The advantage of the Tomeš criterion is that it is always applicable. If n = 1 and the system shows totally irreversible behavior in SSV based on the kinetics of interfacial electron transfer, then kinetic parameters can be obtained in any of four ways: 1) Point-by-point evaluation of k f . From a recorded cathodic wave, one can measure i/id,c at various potentials in the rising portion of the wave and use (5.4.10a) to obtain corresponding values of Λf , which will yield values of k f if mO is known. For an anodic wave, the same approach, using (5.4.11a), provides values of Λb , from which corresponding values of k b can be calculated if mR is known. This analysis involves no assumption that the kinetics follow a particular model. If a model is subsequently assumed, the set of k f values can be analyzed to obtain other parameters (e.g., k 0 and 𝛼 for BV kinetics or k 0 and 𝜆 for Marcus kinetics). If BV ′ kinetics apply, for example, a linear plot of log k f vs. E − E0 will provide 𝛼 from the slope and k 0 from the intercept. Regardless of the model assumed, one can determine k 0 only if ′ E0 is known by some other means (e.g., potentiometry), because it cannot be separately determined from the wave position. Table 5.4.1 Wave Shape Characteristics for Steady-State Voltammetry at 25 ∘ C Kinetic Regime

Wave Slope (mV)

|E3/4 –E 1/4 | (mV)

Reversible (n ≥ 1)

Linear, 59.1/n

56.4/n

Quasireversible (n = 1)

Nonlinear

>56.4, < irreversible

Irreversible (cathodic, n = 1)

Linear, 59.1/𝛼

56.4/𝛼

Irreversible (anodic, n = 1)

Linear, 59.1/(1 − 𝛼)

56.4/(1 − 𝛼)

5.4 Quasireversible and Irreversible Electrode Reactions

2) Wave-slope plot. If BV kinetics apply, a totally irreversible cathodic wave gives a linear plot of E vs. log[(id,c − i)/i] in accord with (5.4.12). The slope provides 𝛼 and the intercept ′ at E0 (if known) yields k 0 . An analogous procedure applies to anodic waves, based on (5.4.14). 3) Tomeš criterion and half-wave potential. As one can see from Table 5.4.1, |E3/4 − E1/4 | for a totally irreversible system provides 𝛼 directly. That figure can then be used in conjunc′ tion with (5.4.13) or (5.4.15) to obtain k 0 . BV kinetics are implicit; E0 and mO or mR must be known. 4) Curve fitting. The most general approach to the evaluation of parameters is to employ a nonlinear least-squares algorithm to fit a whole digitized voltammogram to a theoretical function. For a totally irreversible wave, one could develop a fitting function from (5.4.10a) or (5.4.11a). A specific kinetic model would be needed to describe the potential dependence of k f or k b in terms of adjustable parameters. For Marcus kinetics, the adjustable parameters would be k 0 and 𝜆. If the BV model is assumed, one can use (5.4.10b) or (5.4.11b), and the adjustable parameters are 𝛼 and k 0 . The algorithm determines the values of the parameters that best describe the experimental results. If the steady-state voltammetry is quasireversible, one cannot use simplified descriptions of the wave shape, but must analyze results according to the appropriate expression for general kinetics. The most useful approaches are: • Method of Mirkin and Bard. One can analyze a quasireversible steady-state wave very conveniently in terms of two differences, |E1/4 − E1/2 | and |E3/4 − E1/2 |. Published tables for spherical and disk-shaped UMEs (26) link these differences to corresponding sets of k 0 and 𝛼, assuming BV kinetics; therefore, one can evaluate the kinetic parameters by a look-up process. • Curve fitting. This method proceeds essentially exactly as described for totally irreversible systems, except that the fitting function must be developed from (5.4.7), (5.4.8), or (5.4.9), depending on the bulk composition. ′

For a quasireversible wave, E1/2 is not far removed from E0 and is sometimes used as a rough estimate of the formal potential. Better estimates can be made from fundamental equations after the kinetic parameters have been evaluated from the wave shape. The tables published by ′ Mirkin and Bard for SSV actually provide E1∕2 − E0 with sets of k 0 and 𝛼 (26). 5.4.6

Evaluation of Kinetic Parameters by Varying Mass-Transfer Rates

In Section 5.4.3, we recognized that it is possible to alter the kinetic regime—sometimes all the way from reversible to totally irreversible—by changing the rate of mass transfer to the working electrode. In fact, one can obtain kinetic parameters from experiments in which the mass-transfer rate is varied as the independent variable, while the potential of the working electrode is held constant. This approach originated for rotating disk electrodes and is known as the Koutecký–Levich method (Section 10.2.5). It has been redeveloped in the context of steady-state current measurements at UMEs (27, 28), for which mass-transfer rates can be varied by using electrodes of different sizes. As shown in Table 5.2.1, the mass-transfer coefficients at UMEs of all shapes depend inversely on the critical dimension, so progressively smaller electrodes involve progressively higher rates of mass transfer. With UMEs, it is possible to vary mass-transfer rates over orders of magnitude. A simple illustration of the method is based on a steady-state cathodic process occurring at constant potential in the totally irreversible regime, so that (5.4.10a) describes the current. If

237

238

5 Steady-State Voltammetry at Ultramicroelectrodes

we multiply that equation by the diffusion-limited current, then invert, the result is 1 1 1 = + i Λf id,c id,c

(5.4.16)

∗ and Λ = k /m , substitution gives With recognition that id,c = FAmO CO f f O

1 1 1 = + ∗ ∗ i FAk f CO FAmO CO

(5.4.17)

If we now multiply by the area and identify i/A as the current density, j, we have 1 1 1 = + ∗ ∗ j Fk f CO FmO CO

(5.4.18)

This form of the Koutecký–Levich equation is applicable not only to UMEs, but also to the original context of the method, the RDE. In the case of the RDE, the mass-transfer coefficient is varied by changing the rotation rate. From (5.4.18), we see that a plot of 1/j, measured at a fixed potential, should be linear with ∗ , from which the heterogeneous rate con1/mO and should have a positive intercept, 1∕Fk f CO stant, k f , can be calculated. Alternatively, k f is the ratio of slope to intercept. In deriving (5.4.18), we used only (5.4.10a), which rests on (5.4.4). At no stage did we introduce a model of electrode kinetics, so the rate constants obtained with the method, as described so far, are independent of any assumption about kinetics. If the BV model is assumed, then ln k f should be linear with E, as predicted by (5.4.5a). ′ The slope of such a plot provides 𝛼. If E0 is known, one can also evaluate k 0 from the intercept ′ at E0 . When the electrode kinetics are quasireversible, the equations become more elaborate, but the Koutecký–Levich approach remains applicable. One still finds that 1/j should be linear with 1/mO (or 1/mR ), and one can still obtain a rate constant from the slope and intercept of the plot. For a quasireversible system, the Koutecký–Levich equation must be developed from the general kinetic relationship, (5.4.4) or (5.4.7), or from the appropriate special case [Section 5.4.1(c)]. The result consistently takes the form, S 1 = IKL + KL j m

(5.4.19)

with the slope, SKL , and the intercept, I KL , being given for particular cases in Table 5.4.2. The last column of Table 5.4.2 shows that SKL /I KL yields the available kinetic parameter directly (scaled in most cases by a factor that depends on potential). Figure 5.4.3 shows experimental Koutecký–Levich plots15 for a system with rather facile kinetics. Only the oxidized form is present in the bulk solution, so the measured currents are cathodic (corresponding to the second line in Table 5.4.2). In most experimental circumstances, this system appears reversible; however, the extraordinary mass transfer rates available at smaller UMEs made it possible to capture kinetic effects. The results were consistent with k 0 = 0.45 cm/s and 𝛼 = 0.5. The Koutecký–Levich method is a broadly useful means for measuring kinetic limitations at an electrochemical interface under steady-state conditions. In the cases we have considered 15 Koutecký–Levich plots are not always of 1/j vs. 1/m. Often, the ordinate is 1/i, and the abscissa may be the inverse of an experimental variable proportional to the mass-transfer coefficient.

5.5 Multicomponent Systems and Multistep Charge Transfers

Table 5.4.2 Slopes and Intercepts of Koutecký–Levich Plots(a) Case

SKL

IKL

SKL ÷IKL

General kinetics, general composition(b)

1 + 𝜉𝜃 ∗ )] FC ∗O [1 − 𝜃(CR∗ ∕CO

1 ∗ [1 − 𝜃(C ∗ ∕C ∗ )] Fk 0 𝜃 −𝛼 CO R O

(1 + 𝜉𝜃)k 0 𝜃 −𝛼

General kinetics, CR∗ = 0(b)

1 + 𝜉𝜃 FC ∗O

(1 + 𝜉𝜃)k 0 𝜃 −𝛼

∗ = 0(c) General kinetics, CO

−

Totally irreversible reduction(b)

1 FC ∗O

Totally irreversible oxidation(c)

−

1 ∗ Fk 0 𝜃 −𝛼 CO 1 − 0 Fk 𝜃 (1−𝛼) CR∗ 1 ∗ Fk 0 𝜃 −𝛼 CO 1 − 0 Fk 𝜃 (1−𝛼) CR∗

1 + 𝜉𝜃 F𝜉𝜃CR∗

1 FC ∗R

( ) 1 1+ k 0 𝜃 (1−𝛼) 𝜉𝜃 k 0 𝜃 −𝛼 = k f k 0 𝜃 (1 − 𝛼) = k b

(a) Describing the effect of heterogeneous kinetics for the one-step, one-electron process, O + e ⇌ R, with both O and R remaining dissolved. All cases involve the assumption of BV kinetics. (b) Plot of 1/j vs. 1/mO . (c) Plot of 1/j vs. 1/mR .

Figure 5.4.3 Koutecký–Levich plots for reduction of 5 mM Ru(NH3 )3+ in 0.1 M KNO3 at spherical Pt 6 nanoparticle UMEs with r0 = 0.7, 1.2, 1.6, and 3.3 nm (from left to right within point groups). Top ′ to bottom, data sets and lines are for E − E 0 values of −50, −70, −100, −150, and −200 mV. [Adapted with permission from Kim and Bard (28). © 2016, American Chemical Society.]

0.2 1/j (cm2/A) 0.1 Slope ≈ 1/F

0.0

0

5,000 1/mOCO* (cm2 s/mol)

10,000

here, the limitations arise from the heterogeneous electron-transfer kinetics, but in other systems, they can result from other kinds of processes, as we will see when we consider dynamics at modified electrodes (Chapter 17).

5.5 Multicomponent Systems and Multistep Charge Transfers Consider the situation in which two reducible substances, O and O′ , are present in the same ′ ′ solution, so that the electrode reactions O + ne → R and O + ne → R can occur. Suppose the first process takes place at less extreme potentials than the second, and that the second does not commence until the mass-transfer-limited region has been reached for the first (Figure 5.5.1). The reduction of O can then be studied without interference from O′ , but one must observe the current from O′ superimposed on that caused by the mass-transfer-limited flux of O. An example might involve the successive 1e reductions of ferrocenium (Fc+ ) and benzoquinone (BQ) in acetonitrile with 0.1 M TBABF4 , where Fc+ is reduced with E1/2 near +0.3 V vs. SCE, but the BQ remains inactive until the potential becomes more negative than about −1.4 V. Upon reduction, Fc+ goes to ferrocene, Fc, and BQ goes to the anion radical, BQ−∙.

239

240

5 Steady-State Voltammetry at Ultramicroelectrodes

Figure 5.5.1 SSV for a two-component system.

i (id,c)total = id,c + i′d,c id,c E

5 nA

+0.5

+0.3

+0.1

–0.1

–0.3

–0.5

Figure 5.5.2 Cyclic steady-state voltammogram for 1 mM TCNQ in deoxygenated acetonitrile with 0.1 M TBAP. Working electrode is a Pt disk UME with r0 = 12.4 μm. [Adapted with permission from Norton et al. (29). © 1991, American Chemical Society.]

–0.7

E/V vs. SSCE

In the potential region where both processes are limited by the rates of mass transfer [i.e., CO (surface) = CO′ (surface) ≈ 0], the total current is the sum of the individual diffusion currents. For cathodic SSV at a UME, ∗ + n′ m C ∗ ) (id,c )total = FA(nmO CO O′ O′

(5.5.1)

where mO and mO′ are defined in Table 5.2.1, given the shape of the UME. The diffusion current for the first wave can be subtracted from the total current of the composite wave to obtain the current attributable to O′ alone. That is, i′d,c = (id,c )total − id,c

(5.5.2)

where id,c and i′d,c are the limiting current components due to O and O′ , respectively. Similar considerations hold for a system in which a single species O is reduced in several steps, depending on potential, to more than one product. That is, O + n1 e → R1

(5.5.3)

R1 + n2 e → R2

(5.5.4)

where the second step occurs at more extreme potentials than the first. An example is provided by the steady-state voltammetry of TCNQ (Figure 1) in acetonitrile (Figure 5.5.2). The current at potentials more positive than about +0.3 V is essentially zero. The first wave, with E1/2 near +0.25 V, corresponds to the 1e reduction of TCNQ to the anion radical, TCNQ−∙. The second wave, with E1/2 close to −0.3 V, reflects an additional 1e reduction to TCNQ2− . At potentials for which the reduction of O to R2 is diffusion-controlled (more negative than −0.4 V in Figure 5.5.2), the steady-state current for the entire process [(5.5.3) and (5.5.4)] is ∗ (n + n ) id,c = FAmO CO 1 2

(5.5.5)

5.6 Additional Attributes of Ultramicroelectrodes

where mO is available in Table 5.2.1, given the UME shape. In solutions containing a low concentration of supporting electrolyte, migration contributes to the fluxes, and the limiting currents for the first and second waves are not additive (Section 5.7.4). Our focus here is on the limiting current resulting from multistep electron transfers yielding stable products. There are many interesting kinetic and mechanistic aspects of processes involving sequential electron transfers, but we defer them for consideration in Chapter 13.

5.6 Additional Attributes of Ultramicroelectrodes The large impact of UMEs is rooted in their ability to support very useful extensions of electrochemical methodology into previously inaccessible domains. UMEs allow one to investigate chemical systems: a) using new experimental modes (including SSV), b) using highly resistive media that could not previously have been employed, c) in confined (but still macroscopic) spaces (e.g., a rat brain or a single biological cell), for which one desires highly localized analytical or diagnostic information, d) for measurements of analytes at high concentrations, e) where spatial relationships are examined on a distance scale relevant to molecular events, f ) involving discrete events in time such as nucleation and single-particle collision, g) using short time scales that could not otherwise have been reached electrochemically. In this section and in Section 5.7, we will cover properties of UMEs that support applications in categories (a)–(c). Category (d) is covered in Section 5.8. Applications in categories (e) and (f ) are addressed in Chapters 18 and 19. Those in category (g) are held for Chapter 6, where transient experiments are introduced. Some of these applications do not involve SSV; however, we mention them together here, because they all relate to UMEs, which form a principal focus of this chapter. 5.6.1

Uncompensated Resistance at a UME

As current flows in solution between a working electrode and a counter electrode, one can think of it as passing along paths of roughly equal length, terminated by the faces of the two electrodes. These paths are largely contained in the portion of the electrolyte bounded by the electrodes and the closed surface representing the locus of minimum-length connections between points on the perimeters of the electrodes (Figure 5.6.1). Usually the counter electrode is much larger than the working electrode; hence, this solution volume is broadly based at the counter electrode, but narrowly based at the working electrode. The value of Ru depends on where the tip of the reference electrode intercepts the current path. Figure 5.6.1 shows the situation for a working electrode having a radius one tenth that of the counter electrode, but if the working electrode is a UME, its radius can easily be a thousandth or even a millionth of the counter electrode’s radius. In such a case, all of the current must pass through a solution volume of extremely small cross-sectional area near the working electrode, and this turns out to be the part of the current path that defines the value of Ru . The resistance to uniform current flow offered by any element of a solution is l/𝜅A, where l is the thickness of the element along the current path, A is the cross-sectional area, and 𝜅 is the conductivity. Usually, we assume that the solution is homogeneous—that 𝜅 remains constant everywhere in space, even during electrochemical activity. This will be true in electrochemical systems with excess supporting electrolyte, but not when there is little to no supporting electrolyte (Section 5.7).

241

242

5 Steady-State Voltammetry at Ultramicroelectrodes

Electrolyte solution

Working electrode

Volume containing current paths

Counter electrode

Electrolyte solution

Figure 5.6.1 Schematic representation of the volume of solution containing the current paths between disk-shaped working and counter electrodes situated on a common axis. Current paths are largely, but not strictly, confined to the volume defined by minimum-length connections between electrode perimeters.

For systems with excess electrolyte, the resistance of the disk-shaped volume of solution adjacent to the working electrode and extending out a distance r0 /4 is 1/(4𝜋𝜅r0 ). A similar relation applies for the counter electrode; but its radius is typically 103 to 106 bigger than r0 . One can readily see that the uncompensated resistance is likely to be dominated by the tiny part of the solution very near the working electrode. In a system with spherical symmetry, which would apply approximately to any working electrode that is essentially a point with respect to the counter electrode, the uncompensated resistance is given by (30), ( ) 1 x Ru = (5.6.1) 4𝜋𝜅r0 x + r0 where x is the distance from the working electrode to the tip of the reference. In a UME-based system, it is impractical to place a reference tip so that x is comparable to r0 ; thus, the parenthesized factor approaches unity, and Ru =

1 4𝜋𝜅r0

(5.6.2)

Ru rises as the electrode is made smaller. The volume of solution controlling Ru also becomes smaller, but with a length scale that shrinks proportionately with r0 and a cross-sectional area that shrinks with the square. The effect of decreasing area overrides that of decreasing thickness. 5.6.2

Effects of Conductivity on Voltammetry at a UME

The uncompensated resistance creates a control error in any potentiostatic experiment such that the actual potential at the working electrode, E, differs from the target value, Eappl , by the uncompensated ohmic drop, iRu (Sections 1.5.4 and 16.7). The actual potential is more positive than Eappl if a cathodic current is flowing, but more negative if the net current is anodic. As recorded with conventional instrumentation, voltammograms are plots of recorded current vs. Eappl ; thus, the plotted waves incorporate effects of iRu . These effects generally mimic quasireversibility; that is, they cause a displacement of the voltammogram toward more extreme apparent potentials, as well as a broadening of the voltammogram along the potential axis. They can cause misinterpretation of data, so it is

5.6 Additional Attributes of Ultramicroelectrodes

important to understand when they are significant and how to minimize or correct for them. The topic is discussed in various contexts in later chapters, especially in Section 16.7. Consider now a disk UME with radius r0 at which we desire to carry out SSV. The size of the potential-control error at E1/2 is (iRu )E

= 1∕2

id,c 2

∙

∗ nFDO CO 1 = 4𝜋𝜅r0 2𝜋𝜅

(5.6.3)

where the first factor in the middle part of (5.6.3) is the half-wave current [i.e., half of (5.2.18b)] and the second factor is Ru as given by (5.6.2). According to (5.6.3), (iRu )E does not depend on 1∕2

r0 . While the current depends linearly on r0 , the uncompensated resistance depends inversely on it; so the effects cancel. To record a steady-state voltammogram in which the half-wave potential is shifted less than 5 mV by the effect of uncompensated resistance, we require that ∗ nFDO CO

2𝜋𝜅

< 5 × 10−3 V

(5.6.4)

∗ = 1 mM, the conductivity must exceed 3 × 10−5 S/cm. With n = 1, DO = 10−5 cm2 /s, and CO This minimum would be met by all aqueous electrolytes having concentrations above that of the electroactive species, as well as by most common solvent systems of lower polarity containing weakly dissociated electrolytes. Thus, it is practical to record steady-state voltammograms at disk-shaped, hemispherical, or spherical UMEs in all such media without serious error from the effects of uncompensated resistance.16 A fascinating aspect of voltammetry at UMEs is that one can often record voltammograms in media that are far less conducting than we have just discussed. Useful data have been gathered, for example, in solvents without any added supporting electrolyte or in polymers of very high viscosity. We will address such cases further in Section 5.7. One can often simplify the instrumentation used with UMEs because both the working electrode and the current are very small. There is nothing to be gained by trying to position a separate reference electrode near the working electrode, and there is no danger of polarizing a poised reference by passing the tiny cell current through it. Consequently, two-electrode cells are often used in experiments based on UMEs, including SSV (Section 16.8.4).

5.6.3

Applications Based on Spatial Resolution

Because UMEs are physically small, they can be used to probe small spaces. Single electrodes have been employed frequently in physiological applications, such as the measurement of time-dependent concentrations of neurotransmitters near synapses of neurons (7, 10–12). Single electrodes also provide the basis for SECM (Chapter 18). Alternatively, groups of UMEs can be used in various interesting ways to provide a spatially sensitive characterization of a system (7, 31). Combinations and arrays of UMEs can be made by microlithographic techniques, and they sometimes consist of parallel bands. If the bands are also connected electrically in parallel, they behave as a single segmented electrode and follow principles outlined in Section 6.1.5. If they are independently addressable, they can be used as separate working electrodes to characterize different regions of a sample, such as a polymer overlayer (32). 16 In contrast, work at mm- or cm-sized electrodes is often strongly affected by uncompensated resistance when media of low conductivity are employed. This is a subject we must save for later chapters.

243

244

5 Steady-State Voltammetry at Ultramicroelectrodes

Redox cycling

O

Generator

R

R

O

Collector

Figure 5.6.2 Schematic representation of two microband electrodes operating in the generator–collector mode.

One can also use the elements of an array to probe chemistry occurring at neighboring elements. The simplest example is a double-band system used in the generator–collector mode (Figure 5.6.2). The two bands are spaced closely enough together to allow the diffusion fields to overlap, so that events at each electrode can be affected by the other. One of the electrodes, called the generator, is used to drive the experiment, often by having its potential scanned slowly enough to produce a steady-state voltammogram. Now suppose the double-band assembly is immersed in a solution containing only species O in the bulk. Assume further that the reaction O + ne ⇌ R is reversible and that the product R is chemically stable. In the absence of any influence from the second electrode, one would record, at either of the electrodes, a quasi-steady-state voltammogram characteristic of a single band. However, in the generator–collector mode, the second electrode is set at a potential in the base region of the reduction wave for O, so that any R arriving there is immediately reconverted to O. A current flows at this collector only when the generator is producing R; thus, a plot of the current at the collector vs. the potential of the generator should have the same shape as that recorded at the generator, but with the opposite sign. Also, currents at the collector are smaller than corresponding values at the generator, because the collector does not collect all of the generated R. This is our first mention of a reversal experiment, in which a second phase of electrochemistry is used to characterize the product of a prior phase. Reversals can happen in space, at closely spaced electrodes, as in this example. However, they can also happen in time at a single electrode. We will encounter many examples. Like other reversal experiments, generation–collection at the double band is sensitive to the chemical stability of species R. If it does not survive long enough to diffuse to the collector, no current will be recorded there, and if only a part survives, then only a part of the expected current will be seen.17 Another interesting phenomenon is that the current at the generator can be enhanced by the collector through a mechanism called redox cycling (33). Without active collection, all R produced at the generator would diffuse into solution and would have no further effect on the experiment at the generator electrode. However, if the collector reconverts a portion of the R to O, then some of the regenerated O will diffuse back to the generator, where it adds to the 17 Generation–collection experiments originated with the rotating ring-disk electrode (Section 10.3.2), with which they have a long history of application. The first mention of generation–collection arises here because of its relevance to UME arrays. Redox cycling was first observed (33) in dual-electrode thin-layer cells (Sections 12.7.3 and 19.6).

5.7 Migration in Steady-State Voltammetry

flux arriving from the bulk. Thus, the current at the generator becomes larger than it would be without activity at the collector. Generation–collection experiments can be carried out in UME arrays aside from the double band. A more elaborate approach involves an interdigitated array, which is an extensive series of parallel bands, the alternate members of which are connected in parallel. One set serves as the generator and the other as the collector. For all of these systems, the dynamics are dependent on the widths of the bands and the gaps between them (7). A striking example of redox cycling (34) was based on microfabrication to establish a pair of independent metal electrodes to each tiny volume of electrolyte solution captured in an array of sealed pores in an inert plastic matrix. Within each pore, the electrodes were separated from each other by about 100 nm. Thus, each volume became a separate generator–collector cell of only 10 aL (10−17 L) volume, but the entire array of such cells was operated in parallel, so that larger currents could be observed. With Fe(CN)3− and Fe(CN)4− in each cell, the current at the 6 6 generator electrodes could be enhanced by up to 250 times, a figure reflecting the tiny volume of each cell and the close spacing of the two electrodes. Collection and redox cycling are especially important in SECM (Chapter 18), which is a much more powerful arena for exploiting them, because the distance between the generator and collector can be varied at will on a nanometer scale.

5.7 Migration in Steady-State Voltammetry It is practical to use UMEs to record steady-state voltammograms for systems in which there is very low ionic strength, sometimes even without added supporting electrolyte. There are opportunities in such experiments, so interest in effective theory has grown (35). In any such experiment, migration must significantly influence the movement of charged electroreactants and electroproducts, even near the electrode surface, because they make up a notable portion of the total ionic population everywhere in the system. The theoretical treatment becomes much more complicated than we have so far encountered, because we must include a migration term in each mass-transfer equation, and because it is necessary to treat the mass transfer of all ionic species. Before this point, we have generally assumed conditions assuring that only diffusion would be important; thus, we acquired the luxury that only participants in the electrode reaction (e.g., O and R) needed to be included in the treatment of mass transfer, while nonparticipants could be entirely ignored. To obtain the steady-state current (Section 5.1), only one participant, the electroreactant, had to be considered. 5.7.1

Mathematical Approach to Problems Involving Migration

For the more complicated problem now before us, the starting point is with the first two terms of the Nernst–Planck equation, (4.1.9), describing the contributions of diffusion and migration, respectively, to the flux of species j at any location, Jj = −Dj 𝛁 Cj −

zj F

D C 𝛁𝜙 (5.7.1) RT j j The vector operator 𝛁 depends on the geometry of the coordinate system. In (5.7.1), every factor is particular to species j, except the local electric potential, 𝜙, which depends on the distributions of all ionic species. This is the aspect of the problem that necessitates a simultaneous solution comprehending all ionic species in the solution. If there are m ionic species, there is

245

246

5 Steady-State Voltammetry at Ultramicroelectrodes

a differential equation like (5.7.1) for each one. In addition, there is a requirement for local electroneutrality,18 m ∑

zj Cj = 0

(5.7.2)

j

In principle, these m + 1 equations can allow a solution for the m + 1 unknown functions, viz. the m different C j , plus 𝜙. Ionic species not involved in the electrode reaction have zero flux at the electrode surface. However, at least one of the ionic species must be a reactant or a product of the electrode reaction, and its flux at the surface at any point is proportional to the local current density. The integral of the current density over the area of the electrode provides the current, i. These considerations afford some of the boundary conditions for a solution. It can be challenging to solve any system of simultaneous mass-transfer and electrostatic equations for arbitrary choices of electrode shape, electrode reaction, ionic composition, and kinetics. Computational software for solving coupled differential equations, such as COMSOL Multiphysics , can accommodate much variation in the specific details of a system, but simplifications in formulating the problem can still be helpful. If an analytical solution is sought, simplifications are essential. One or more of the following is commonly selected:

®

1) A simple geometry (e.g., based on a spherical or hemispherical electrode). 2) The absence of a time dependence (i.e., a steady state applies). 3) A simplified electrode reaction (e.g., a neutral electroreactant being reduced or oxidized in a 1e step to a charged product). 4) A simplified electrolyte, (e.g., a 1:1 salt, such as KCl or TBABF4 ). 5) Uniform diffusion coefficients for all species. 6) The validity of the Nernst–Einstein relationship, uj = Dj zj F/RT. Actually, this assumption is already built into the version of the Nernst–Planck equation given in (5.7.1). 7) Nernstian electrode kinetics, requiring no activation overpotential. Some important analytical treatments have been offered (36–38) and will be discussed below. In general, however, these problems lend themselves much better to numeric treatment than to analytical solution (39). 5.7.2

Concentration Profiles in the Diffusion–Migration Layer

Oldham (36) treated the steady-state oxidation of ferrocene to ferrocenium, Fc → Fc+ + e, in solutions of varying concentrations of a supporting electrolyte for which both the electrolyte anion and cation have a unit charge (e.g., NaClO4 ). The working electrode was assumed to be a uniformly accessible hemisphere. The key parameter in this and other treatments of migration in SSV at UMEs is the dimensionless ratio, 𝛾, of the bulk concentration of supporting electrolyte relative to that of the electroactive redox species, ∗ ∗ 𝛾 = Celectrolyte ∕Credox

(5.7.3)

When 𝛾 ≫ 1, transport of the redox species is dominated by diffusion, and the SSV response is as described in Sections 5.3 and 5.4. When 𝛾 < 10, both migration and diffusion contribute significantly to transport of a charged electroreactant or electroproduct, leading to changes in the observed wave shape and limiting current in the SSV response. 18 Local electroneutrality normally applies in electrochemical systems, but it can break down at high current densities (including those found at UMEs with r0 ≤ 10 nm). In such cases, the Poisson equation must be used.

5.7 Migration in Steady-State Voltammetry

1.0 Fc 0.8

0.6 * C/CFc 0.4 Anion

Fc+

0.2

Cation 0.0

0

1

2

3

4 r/r0

5

6

∞

Figure 5.7.1 Steady-state concentration profiles near a hemispherical electrode. Fc is oxidized at 75% of the ∗ , i.e. 𝛾 = 0.1. [From Oldham (36), with permission.] transport-limited rate. Bulk electrolyte concentration is 0.1CFc

Figure 5.7.1 shows a set of steady-state concentration profiles from this analysis for the case of 𝛾 = 0.1. The curves for Fc and Fc+ look similar to those that we have come to expect for electroreactants and primary electroproducts in purely diffusive systems; however, the profiles of the supporting electrolyte ions are very different. In the purely diffusive cases, the supporting electrolyte is typically present at a concentration 20–1000 times larger than the electroreactants, so the ionic demands of the electrode reaction are trivial relative to the local supply of ions. There is no need for ionic species to adjust their concentrations to enable the electrode reaction, so the ionic profiles are essentially flat across the diffusion layer at the bulk concentrations. In the depicted situation, where the electrolyte is present at only a tenth of the concentration of the electroreactant, the oxidation of the ferrocene cannot happen unless anions arrive from the bulk or cations are expelled to the bulk, to preserve local electroneutrality. Anions are held at a much higher concentration in the diffusion–migration layer than in the bulk, and cations have been preferentially removed. Migration creates these effects. Because the electrolyte concentration is low, the electric field in solution is much larger than when there is a large excess of electrolyte. This is an example of an electrode reaction that elevates the ionic population in the diffusion–migration layer, relative to the bulk, and, therefore, increases local conductivity in the zone that effectively lowers the uncompensated resistance (Section 5.6.1). Oldham noted (36) that this effect should be common in electrode reactions. In his language, the “steady-state resistance” of the cell is typically much lower than the “static resistance” because of this pattern of ionic redistribution. This effect underlies the striking feasibility of SSV in systems of low conductivity. It has now become routinely practical to record useful steady-state voltammograms for solutes in media without any added electrolyte.19

19 In the absence of added electrolyte, SSV must, nevertheless, rest on real ions from some source. Impurity ions exist in common solvents and can migrate into the depletion layer when a faradaic process occurs, increasing the local conductivity adequately. Very clean solvents may support SSV only at very slow scan rates or not at all.

247

5 Steady-State Voltammetry at Ultramicroelectrodes

5.7.3

Wave Shape at Low Electrolyte Concentration

The kinetics of the ferrocene/ferrocenium couple are fast, so the oxidation wave has a reversible form when excess supporting electrolyte is present (leftmost curve of Figure 5.7.2). As the electrolyte concentration is lowered, the electrode reaction becomes more complicated because significant work is required to neutralize the ionic charge created upon oxidation of the ferrocene. For any given current, that work shows up on the potential axis as a shift toward a more extreme potential (in the positive direction for an oxidation). As the electrolyte concentration is made progressively lower, the wave shifts further from its position in excess electrolyte. The curves in Figure 5.7.2 resemble those of Figure 5.4.1, so one might interpret the shift and broadening in terms of slowing kinetics. However, the effects in Figure 5.7.2 are not rooted in electrode kinetics. The potential shift, ΔE, measured at any current vs. the curve for excess electrolyte, is a manifestation of the electric potential difference that must develop across the diffusion–migration layer to maintain electroneutrality as the current flows. A potential difference across a zone of solution reflects resistance to current flow, so ΔE in this instance is akin to the ohmic drop, iRu , which we have encountered in more ordinary experiments. However, this ΔE is not ohmic (not linear with current) and ΔE/i differs from the conventional estimate of Ru given in (5.6.2). As a rule, one can expect the broadening of a wave and a shift to more extreme potentials as the electrolyte concentration is lowered, even to 10 times the electroreactant’s bulk concentration, or below. 5.7.4

Effects of Migration on Wave Height in SSV

In Figure 5.7.2, the wave height remains unchanged as the electrolyte concentration is lowered, even though there are large effects on shape and position. The reason is that the electroreactant, ferrocene, is neutral, so that its mass transfer remains purely diffusive. The limiting current is still based on the diffusion-controlled arrival of ferrocene at the electrode surface. As the electrolyte concentration is reduced, a more extreme potential is required to establish 1.0

0.8 i 2𝜋FDFcC*Fcr0

248

≥10

1 0.1

0.01

0.001

0.6

0.4

0.2

0.0

–100

0

100 (E – E0)/mV

200

Figure 5.7.2 Predicted steady-state voltammograms at a hemispherical electrode for oxidation of ferrocene at various levels of added electrolyte. Anodic current is positive and is normalized by the steady-state current in ∗ ∗ excess electrolyte. The number by each curve is 𝛾 = Celectrolyte ∕Credox . For 𝛾 ≫ 1, the system is diffusion controlled; for 𝛾 < 10, migration is significant. [From Oldham (36), with permission.]

5.7 Migration in Steady-State Voltammetry

25 a 20

b c

Current/nA

15

d 10 5 0 –5 –0.6

–0.8

–1.0 E/V

–1.2

–1.4

Figure 5.7.3 Voltammograms for reduction of 0.65 mM Tl2 SO4 at a mercury film on a silver ultramicroelectrode (r0 = 15 μm) in the presence of (a) 0, (b) 0.1, (c) 1, and (d) 100 mM LiClO4 . The potential was controlled vs. a Pt wire QRE whose potential was a function of solution composition. This variability affects the wave position along the potential axis. [Reprinted with permission from Ciszkowska and Osteryoung (40). © 1995, American Chemical Society].

the limiting flux of ferrocene for reasons discussed just above, but that flux is the same for all cases, once the surface concentration of ferrocene is driven effectively to zero. If an electroreactant bears a charge, then migration eventually becomes important to its flux at the electrode surface as the electrolyte concentration is lowered; thus, the current is also affected. The limiting current becomes larger or smaller than in the presence of excess electrolyte, depending on the sign of the ionic charge. Anodic processes build positive charge in the diffusion–migration layer, so negative ions are attracted toward the electrode, and positive ions are repelled. Cathodic processes build negative charge near the electrode, so positive ions are attracted, and negative ions are repelled. If the electroreactant is cationic, migration augments a cathodic limiting current, but suppresses an anodic one. If the electroreactant is anionic, the effect is vice versa. Figure 5.7.3 shows this effect for reduction of Tl+ to the amalgam at a mercury film. The limiting current increases systematically as the LiClO 4 concentration is reduced (40). Figure 5.7.4, taken from a general treatment of migration effects on limiting currents in SSV (37), summarizes key results for electrode reactions with n = 1 occurring in a 1:1 electrolyte. The figure shows the ratio of the steady-state limiting current measured at arbitrary 𝛾, relative to the limiting current recorded in the presence of excess supporting electrolyte, il (𝛾 → ∞). One can readily see the qualitative effects identified in the previous paragraph. The greatest effect is for the case of an electroreactant with z = + 1 for a reduction or z = − 1 for an oxidation, either of which is predicted to exhibit a doubling of the wave height at low electrolyte concentration vs. a large excess of electrolyte. Figure 5.7.4 shows that when 𝛾 > 103 , contributions from migration to the limiting current are entirely eliminated, regardless of the charge on the electroactive molecule; it is under these conditions that the SSV response is purely diffusion-controlled. This treatment (37) also covered systems having other values of n, which was handled as a signed quantity, positive for reductions and negative for oxidations.20 Useful equations were 20 Elsewhere in this book, n is regarded as a positive integer.

249

250

5 Steady-State Voltammetry at Ultramicroelectrodes

2.0

+1

1.8 1.6

il (γ) il (γ → ∞)

1.4 +2 1.2

+3 +4

0 –4

1.0 0.8

–1 –6

–4

–2

0

2

4

log γ

∗ ∗ Figure 5.7.4 Effects of migration on SSV wave height for a 1e reduction. The parameter 𝛾 is Celectrolyte ∕Credox , and il (𝛾)/il (𝛾 → ∞) is the ratio of the SSV wave height at a given 𝛾 to the wave height at excess electrolyte (𝛾 → ∞). The numbers by each curve correspond to the charge on the electroactive species. The curves for charge values of −3 and −2 are not labeled but are seen between those for −4 and −1, with the curve for −3 above that for −2. For a 1e oxidation wave, this figure remains applicable, but each charge has the opposite sign. [From Amatore et al. (37), with permission.]

developed, allowing computation of the limiting current when no supporting electrolyte is added to the solution, i.e., il (𝛾 = 0). When n = z, il (𝛾 = 0) il (𝛾 → ∞)

= 1 + |n|

(5.7.4)

When n ≠ z, il (𝛾 = 0) il (𝛾 → ∞)

{ =1±z

[

1 1 + (1 + |z|)(1 − z∕n) ln 1 − (1 + |z|)(1 − z∕n)

]} (5.7.5)

The negative sign is used for n > z and the positive sign for n < z. One can clearly exploit the effect of low electrolyte concentrations on wave height to identify at least the sign of the charge on the electroactive species (Problem 5.11). There are more quantitative uses as well. An example is provided in Figure 5.7.5, which shows the oxidation in acetonitrile of tetrathiafulvalene (TTF; Figure 1) in two 1e steps (29). The first + yields the cation radical, TTF ∙, and the second gives the dication, TTF2+ . As the supporting electrolyte concentration is reduced, the two waves behave very differently. The wave height of the initial oxidation (E1/2 near +0.3 V, plateau between +0.4 and +0.6 V) hardly changes at all, showing that the electroreactant is uncharged. This result is in keeping with TTF as the active species in bulk solution. In contrast, the height of the second wave (E1/2 near +0.7 V, plateau beyond +0.8 V) is strongly suppressed at lower electrolyte concentrations. This finding clearly implicates a positive species in the corresponding electrode reaction. Different scenarios exist for the second oxidation. One possibility is comproportionation, in which the outbound TTF2+ produced at the electrode undergoes rapid electron exchange with

5.8 Analysis at High Analyte Concentrations

Figure 5.7.5 Steady-state CV for 5 mM TTF in acetonitrile at a Pt disk (r0 = 12.5 μm). Supporting electrolyte is TBAP at (a) 100, (b) 1, (c) 0 mM. [Reprinted with permission from Norton et al. (29). © 1991, American Chemical Society.]

(a) 20 nA

(b)

(c)

+1.2

+0.8 +0.4 +0.0 E/V vs. SSCE

the inbound TTF, +

TTF2+ + TTF → 2TTF ∙

(5.7.6)

The resulting cation radical can diffuse back to the electrode to pick give up a second electron and depart as the dication. If (5.7.6) is fast enough, none of the TTF would actually diffuse through the outbound flux of the dication, and the only electroreactant at the surface would be + TTF ∙. On the other hand, if (5.7.6) were ineffective, the inbound TTF could pass through the outbound flux of TTF2+ and undergo a 2e oxidation directly at the electrode. The strong suppression of the second oxidation at low electrolyte concentration is much more consistent with the first of these scenarios. Using simulation (29) or analytical methods (41), it was possible to treat the dynamics in this system, including the migration effects, and to estimate the rate constant for (5.7.6).

5.8

Analysis at High Analyte Concentrations

Faradaic electroanalysis is the application of electrochemical techniques based on the passage of current for the determination of identities and quantities of analytes. Typically, the species of interest are in solution at concentrations of 10 mM to 1 μM, together with an electrolyte considerably in excess. Many different faradaic techniques have evolved for electroanalysis and are in practical use. The important measurements are half-wave or peak potentials (confirming identities) and wave or peak heights (providing concentrations). The methods were developed over decades, mainly using conventional microelectrodes with characteristic dimensions of 0.5–10 mm—larger by orders of magnitude than the UMEs of concern in this chapter. We are not ready to deal with most aspects of electroanalysis, so we defer the topic, for the most part, until we reach Chapters 6–11. However, there is an aspect that has been entirely enabled by UMEs, and it is appropriate to cover it here. It concerns concentrated analytes.

251

252

5 Steady-State Voltammetry at Ultramicroelectrodes

Industrial processes based on electrochemistry (e.g., the recovery or electrorefining of copper) typically feature electroreactants at high concentrations (often in the molar range) in media that do not contain excess electrolyte. Direct process monitoring (carrying out analytical measurements on an operating process, preferably in situ) is an important practical arena; however, faradaic electroanalysis has been historically inapplicable to it, largely because of the inability to address the effects of migration. In the language of Section 5.7, systems of concentrated analytes typically have 𝛾 < 10, where 𝛾 is the ratio of the total bulk concentration of all other electrolytes to that of the electroactive analyte. When 𝛾 < 10, both diffusion and migration make significant contributions to the transport of charged analytes. UMEs have opened opportunities for practical electroanalysis of species at high concentration, because UMEs are effective in systems for which migration is important. The difficulty of using conventional microelectrodes in this context has several aspects. To begin, the analytical currents can become very large. Analyte concentrations are typically 103 to 106 times larger than those normally encountered in faradaic electroanalysis; therefore, currents of 0.1 μA to 10 mA, found in the normal concentration range, can become tens of mA to more than an ampere for an analyte at high concentration, depending on the area of the electrode and the details of measurement. The large currents can have several negative effects: 1) The uncompensated ohmic drop, iRu , can become excessive, creating drawn-out, poorly defined voltammetric responses.21 2) The current density can become uneven over the face of the working electrode, with higher current densities at locations closer to the counter electrode and lower ones further away. Moreover, the pattern of current density can change with time (e.g., during a voltammetric wave). This effect also leads to drawn-out, poorly defined voltammetry. 3) The large currents lead to massive chemical alteration of the diffusion layer at the working electrode. When the electrode and its diffusion layer have dimensions in the millimeter range, the density changes and limited heat transfer can lead to convective stirring near the electrode, resulting in noisy, imprecise measurements. The first two of these effects are rooted in ionic motion in solution, i.e., migration. The third effect is related to the limited mass and heat transfer into and out of the diffusion–migration layer, and, sometimes, to gravity. If a UME is used, these problems are mitigated. Even though iRu is multiplied (regardless of the size of the working electrode) in systems of high analyte concentrations vs. systems of more conventional concentration, it typically remains manageable in a UME-based cell. When 𝛾 < 10, waves may be significantly shifted due to the effect of iRu , but a good plateau can typically be reached. Also, the UME is so tiny relative to any placement of the counter electrode that the spatial effects discussed in Point 2 become negligible. Finally, the diffusion–migration layer is physically very small at a UME and features much greater rates of mass and heat transfer than with a much larger electrode. Convective mixing is far less of a problem at a UME than with a microelectrode of conventional size. The practicality of electroanalysis by SSV has been demonstrated for systems with very high analyte concentrations (42, 43). Figure 5.8.1 shows reported results for high concentrations of Cu(II) in 9 M LiCl, including a practical calibration curve for Cu(II) up to 1.5 M. 21 Because the ionic strength in one of these systems is normally larger than in the analogous conventional analyte solutions, Ru would generally be smaller, but by only a factor of 5–50 (with the ionic strength changing, say, from 0.1 M in a conventional solution to 1–5 M in a process solution). Since the current is larger by a thousand to a million-fold, iRu , should be greater in the high-concentration systems by 1–5 orders of magnitude.

5.9 Lab Note: Preparation of Ultramicroelectrodes

1.4 100

1.2

80 i/nA

Data Theoretical calculation

1.0

60

i/μA

40

0.8 0.6 0.4

20

0.2

0

0.0 0.7

0.6

0.5 0.4 0.3 E/V vs. Ag/AgCl (a)

0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 [Total Cu(II)]/M (b)

Figure 5.8.1 (a) Cyclic SSV of 71.4 mM Cu(II) in 9 M LiCl, with continuous cycling at 20 mV/s for 60 min. Electrode was a Pt disk (r0 = 12.5 μm). The electrode process was the 1e reduction of Cu(II) to Cu(I). (b) Calibration curve for Cu(II) in 9 M LiCl. [Reprinted with permission from Zhao, Chang, Boika, and Bard (43). © 2013, American Chemical Society.]

At high concentrations, the calibration curve deviates negatively from a straight line. A minor portion of the deviation was shown to be due to the increased viscosity in more concentrated solutions, which reduces diffusion coefficients and ion mobilities inversely. Most of the deviation was attributed to migrational suppression of the limiting flux of the electroreactant, as expected for a negatively charged species undergoing reduction (Section 5.7.4). Known equilibrium constants for Cu(II) suggest that it should exist in 9 M LiCl predominantly as CuCl2− 4 or CuCl− 3 . The investigators were able to model mass transfer by diffusion and migration in this system with the results shown in Figure 5.8.1b. Although the effect of viscosity on the calibration curve was small for the copper system just discussed, it is a major concern in other situations involving analytes at high concentration. An interesting example is the 1e reduction of nitrobenzene (44) in acetonitrile solutions bearing 0.2 M TBAPF6 . The nitrobenzene showed well-behaved SSV at concentrations even above 9 M. The calibration curve was fairly linear up to about 1 M, but it rolled over at higher concentrations, peaked at about 3 M, and then declined by more than 50% for concentrations up to the range of 9 M. This striking effect was shown to be caused by changes in viscosity. Over the studied concentration range, the viscosity becomes five times greater, so diffusion coefficients and ionic mobilities become five times smaller. Above 3 M, a diminished mO overwhelms ∗ , as these factors together determine the wave height in SSV; therefore, the any increase in CO calibration curve becomes inverted. In practical electroanalysis of concentrated analytes, one must be prepared for significant variations in the physical properties of the system with analyte concentration. Such effects are rarely encountered in more conventional electroanalysis. This section has been presented largely in terms of electroanalysis, but there are also opportunities for direct investigations of concentrated electroreactants having more fundamental aims (42, 44). In such work, it is often essential to employ computational models that account for migration.

5.9 Lab Note: Preparation of Ultramicroelectrodes This section introduces some practical aspects of work with UMEs, but it is not a detailed review. Research with very small UMEs is quite active, so practices evolve as new approaches are conceived and tested. The interested reader should determine the state of the art by studying recent publications and consulting experienced colleagues.

253

254

5 Steady-State Voltammetry at Ultramicroelectrodes

5.9.1

Preparation and Characterization of UMEs

UMEs are most commonly made of Pt, Au, or C, but they are also fashioned of other materials for special purposes (45). Disk-shaped Pt and Au UMEs with radii as small 5 μm are commercially available, but these are also readily prepared by gently sealing fine Pt or Au wire in a glass with a low softening temperature, using a flame or pipet-pulling methods. Hemispherical Hg electrodes can be made by plating Hg onto a Pt or C disk. UMEs of more complex geometries and arrays of UMEs are commonly fabricated by microlithography. Smaller disk-shaped UMEs are typically made by sharpening a wire chemically before sealing it in the insulating material, or by using pipet-pulling instruments to pull a metal wire inserted in glass so that it becomes dramatically reduced in size. At the same time, it becomes sealed in the insulator. Breaking, cutting, or polishing the insulating material produces a cross-section consisting of a conducting disk within an insulating mantle (Figure 5.9.1). UME disks with r0 as small as 10 nm can be made using these techniques, although it becomes challenging to reliably fabricate and characterize such small electrodes. After an electrode is fabricated, attention must usually be given to the condition of its surface. Section 6.8.1 addresses surface preparation for microelectrodes, including UMEs. As UMEs become smaller, they also become more fragile and require more caution with regard to surface preparation. When a UME is constructed, its precise geometry and dimensions are generally unknown, especially if the characteristic dimension is toward the smaller end of the achievable range ( 0)

(6.1.18)

where r0 is the radius of the electrode. (a) The Chronoamperometric Transient

The substitution, v(r, t) = rC O (r, t), converts (6.1.15) into an equation having the same form as the linear problem. The details are left to the reader (Problem 6.1). The resulting diffusion-limited current is [ ] 1 1 ∗ + (6.1.19) id,c (t) = nFADO CO (𝜋 DO t)1∕2 r0 which can be written id,c (spherical) = id,c (linear) + id,c (steady state)

(6.1.20)

The diffusion-limited current for the spherical case is just that for linear diffusion (the Cottrell current) plus the spherical diffusion-limited steady-state current, (5.1.20), with which we became quite familiar in Chapter 5. For the sphere, (6.1.19) is a full solution for the response following a potential step, including the entire transient leading to the steady state and the steady state itself. (b) Concentration Profile

The distribution of the electroactive species near the electrode also can be obtained from the solution to the diffusion equation, and it turns out to be [ )] ( r0 r − r0 ∗ (6.1.21) CO (r, t) = CO 1 − erfc r 2(DO t)1∕2 Because r − r0 is the distance from the electrode surface, this profile strongly resembles that for the linear case, (6.1.13). The difference is the factor r0 /r. If the diffusion layer remains thin compared to the electrode’s radius, this factor is essentially 1, so that the linear and spherical cases become indistinguishable. The situation is analogous to our experience living on a spherical planet. The zone of our activities above the earth’s surface is small compared to its radius of curvature; hence, we usually cannot distinguish the surface from a rough plane. At the other extreme, when the diffusion layer grows much larger than r0 (as at a UME, Chapter 5), the concentration profile near the surface becomes both independent of time and linear with 1/r. One can see this effect in (6.1.21), where the error function complement approaches unity for r − r0 ≪ 2(DO t)1/2 . In that case, ∗ [1 − r ∕r] CO (r, t) = CO 0

(6.1.22)

∗ ∕r , which gives the steady-state current, (5.1.20), from the The slope at the surface is CO 0 current–flux relationship for the spherical case, [ ] 𝜕CO (r, t) i = DO (6.1.23) nFA 𝜕r r=r 0

6.1 Chronoamperometry Under Diffusion Control

(c) Applicability of the Linear Approximation

Linear diffusion adequately describes mass transport to a sphere, provided the sphere’s radius is large enough and the time domain is small enough. More precisely, the linear treatment is adequate when the second term of (6.1.19) is small compared to the first term. For accuracy within a%, ∗ nFADO CO

r0

1∕2

∗ nFADO CO a ≤ ∙ 100 (𝜋t)1∕2

(6.1.24)

or (𝜋DO t)1∕2 r0

≤

a 100

(6.1.25)

Taking a = 10% and DO = 10−5 cm2 /s, for example, we find t 1/2 /r0 ≤ 18 s1/2 /cm. For a mercury drop of 1-mm radius, for example, the linear limit holds within 10% for about 3 s. The numerator on the left of (6.1.25) is essentially the thickness of the diffusion layer; thus, the importance of the steady-state term, which manifests spherical diffusion, depends mainly on the ratio of that thickness to the radius of the electrode. When the diffusion layer grows to a thickness that is an appreciable fraction of r0 , the steady-state term will contribute significantly to the measured current. 6.1.3

Transients at Other Ultramicroelectrodes

For the step experiment at a spherical electrode, we were able to obtain the whole diffusionlimited transient—from the rise of the potential step to steady state. In the case of a disk electrode, we have not yet developed the full transient, because the treatment in Section 6.1.1 was based on semi-infinite linear diffusion, and the disk was regarded, in effect, as an infinite plane. That approach is valid when the diffusion layer remains thin. However, we know from Section 5.2.2 that a disk will provide a steady-state current in the long-time limit, when the diffusion layer is able to grow much larger than the disk radius. For completeness, we now cover the full diffusion-limited transient at the disk and will also address behavior at the cylinder and the band. As noted in Sections 5.1.3 and 5.2, convergence to the steady state (or quasi-steady state) is practical only when disk, cylinder, and band electrodes are small enough to serve as UMEs. (a) Disk Electrode

For the disk geometry, the diffusion equation for species O is written as follows (Table 4.4.2; Figure 5.2.2): ] [ 𝜕CO (r, z, t) 𝜕 2 CO (r, z, t) 1 𝜕CO (r, z, t) 𝜕 2 CO (r, z, t) =D + ⋅ + (6.1.26) 𝜕t r 𝜕r 𝜕r2 𝜕z2 where r describes radial position normal to the axis of symmetry (r = 0), and z describes linear displacement normal to the plane of the electrode (z = 0). Five additional conditions are needed for a solution, including the initial condition and two semi-infinite conditions: ∗ CO (r, z, 0) = CO

(6.1.27)

∗ lim CO (r, z, t) = CO

(6.1.28a)

∗ lim CO (r, z, t) = CO

(6.1.28b)

r→∞

z→∞

267

268

6 Transient Methods Based on Potential Steps

A fourth condition comes from the recognition that there can be no flux of O into or out of the mantle, since O does not react there: 𝜕CO (r, z, t) || =0 (r > r0 ) (6.1.29) | | 𝜕z |z=0 The final condition defines the experimental perturbation. For a potential step that drives the surface concentration of O to zero at the electrode surface after t = 0, CO (r, 0, t) = 0

(r ≤ r0 , t > 0)

(6.1.30)

This problem was addressed (4) in terms of a dimensionless parameter, 𝜏 = 4DO t∕r02 , representing the squared ratio of the diffusion length [taken as 2(DO t)1/2 ] to the radius of the disk. The current–time curve is ∗ 4nFADO CO id,c (t) = f (𝜏) (6.1.31) 𝜋r0 where the function f (𝜏) was determined as two different series applicable in different domains of 𝜏 (4–6). At short times (𝜏 < 1), 𝜋 1∕2 𝜋 + + 0.094𝜏 1∕2 1∕2 4 2𝜏 or, with the constants evaluated f (𝜏) =

(6.1.32)

f (𝜏) = 0.88623𝜏 −1∕2 + 0.78540 + 0.094𝜏 1∕2

(6.1.33)

At long times (𝜏 > 1),6, 7 4 −1∕2 𝜏 + 0.05626𝜏 −3∕2 − 0.00646𝜏 −5∕2 · · · (6.1.34) 𝜋 3∕2 A single empirical relationship (5) covers the entire range of 𝜏 with an accuracy better than 0.6% at all points: f (𝜏) = 1 +

f (𝜏) = 0.7854 + 0.8862𝜏 −1∕2 + 0.2146e−0.7823𝜏

−1∕2

(6.1.35)

The diffusion-limited current–time relationship for a UME disk spans three regimes, as shown in Figure 6.1.3. If the experiment is on a short time scale, so that the diffusion layer remains thin compared to r0 , radial diffusion does not manifest itself appreciably, and the diffusion has a semi-infinite linear character. The early current is, therefore, the Cottrell current, (6.1.12). This point is illustrated graphically in Figure 6.1.3a, where two sets of symbols are superimposed. One can also see it mathematically as the limit of (6.1.31) and (6.1.32) when 𝜏 approaches zero. For an electrode with r0 = 5 μm and DO = 10−5 cm2 /s, the short-time region covered in Figure 6.1.3a is 60 ns to 60 μs. In this period, the diffusion layer thickness [taken as (2DO t)1/2 ] grows from 0.01 to 0.3 μm. Radial diffusion becomes important as the experiment continues into an intermediate regime where the diffusion layer thickness is appreciable vs. r0 . The current is larger than for a continuation of pure linear diffusion, for which this “edge effect” is ignored. Figure 6.1.3b illustrates 6 The two versions of f (𝜏) overlap for 0.82 < 𝜏 < 1.44. The dividing point given in the text is convenient and appropriate. 7 By analogy to the rigorous result for the spherical system, the current at the disk has sometimes been approximated 1∕2 ∗ −1∕2 −1∕2 as the simple linear combination of the Cottrell and steady-state terms: id,c (t) ≈ nFADO CO 𝜋 t + ∗ r This estimate is accurate at the short- and long-time limits and deviates from the rigorous result by a few 4nFDO CO 0 percent in the range of Figure 6.1.3b. The largest error (∼ +7%) is near 𝜏 = 1.

6.1 Chronoamperometry Under Diffusion Control

Figure 6.1.3 Diffusion-limited current–time relationships at a disk UME. Time is expressed as 𝜏, which is proportional to t. Triangles, Cottrell current. Filled squares, (6.1.31) and (6.1.33). Open squares, (6.1.31) and (6.1.34). Dashed line is steady state.

300 200 id,c /id,css 100 0 10–5

10–4

10–3 𝜏 (a) Short time regime

10–2

0.1

10

100

10000

10 8 id,c /id,css

6 4 2 0 0.01

1 𝜏 (b) Intermediate time regime

1.0 id,c /id,css

0.0 10

1000 𝜏 (c) Long time regime

the result. For r0 = 5 μm and DO = 10−5 cm2 /s, this frame corresponds to an experimental time between 60 μs and 60 ms. The diffusion layer thickness is in the range from 0.3 to 11 μm. At still longer times, when the diffusion field grows to a size much larger than r0 , the system resembles the hemispherical case and the current approaches the steady state, iss d,c (Figure 6.1.3c). The steady-state equation for the disk, (5.2.18a,b), can be seen easily as the limit of (6.1.31) and (6.1.34) when 𝜏 becomes very large.8 For the specific values of r0 and DO used as examples above, Figure 6.1.3c describes the time period from 60 ms to 60 s, when the diffusion layer thickness enlarges from 11 μm to its steady-state value, in the range of 25 μm. The experimental time ranges discussed here relate to practical values of electrode radius and diffusion coefficient, and are all readily accessible with standard commercial electrochemical instrumentation. A notable feature of a UME is the ability to operate in different mass-transfer regimes. In essence, we used the ability to approach or to achieve the steady state as the basis for our operational definition of a UME in Section 5.2. In the intermediate and late time regimes, the current density at a UME disk is intrinsically nonuniform because the edges of the electrode are more accessible geometrically to the diffusing electroreactant [Section 5.2.2(c)]. This nonuniformity affects the interpretation of 8 Here we need to distinguish between the cathodic diffusion-limited transient current, id,c (t), and the cathodic diffusion-limited steady state current, iss . In Chapter 5, we just used id,c for the latter, because all currents in that d,c chapter were steady-state currents.

269

270

6 Transient Methods Based on Potential Steps

phenomena that depend on local current density, such as heterogeneous electron-transfer kinetics or the kinetics of second-order reactions involving electroactive species in the diffusion layer (Section 5.4.4). Section 5.7 introduced the concept of steady-state measurements at UMEs in systems of low ionic strength, for which accurate treatment requires that both diffusion and migration be taken into account. The focus there was on SSV; however, transient measurements are also practical at UMEs in such systems (7). (b) Cylindrical Electrode

We return to a simpler geometry by considering a cylindrical electrode, which involves only a single dimension of diffusion. The corresponding expression of Fick’s second law (Table 4.4.2) is: ] [ 𝜕CO (r, z, t) 𝜕 2 CO (r, z, t) 1 𝜕CO (r, z, t) 𝜕 2 CO (r, z, t) =D + ⋅ + (6.1.36) 𝜕t r 𝜕r 𝜕r2 𝜕z2 where r describes radial position normal to the axis of symmetry, and z is the position along the length. Since we normally assume uniformity along the length of the cylinder, 𝜕C/𝜕z = 𝜕 2 C/𝜕z2 = 0, and z drops out of the problem. The boundary conditions are exactly those used in solving the spherical case (Section 6.1.2). A practical approximation, reported (8) to be valid within 1.3%, is ] ∗ [ nFADO CO 2 exp(−0.05𝜋 1∕2 𝜏 1∕2 ) 1 + (6.1.37) id,c (t) = r0 𝜋 1∕2 𝜏 1∕2 ln(5.2945 + 0.7493𝜏 1∕2 ) where 𝜏 = 4DO t∕r02 . In the short-time limit, when 𝜏 is small, only the first term of (6.1.37) is important and the exponential approaches unity. Thus, (6.1.37) reduces to the Cottrell equation, (6.1.12), as expected for the situation where the diffusion length is small compared to r0 . The deviation from the Cottrell current resulting from cylindrical diffusion remains below 4% until 𝜏 reaches ∼0.01, where the diffusion layer thickness has become about 10% of r0 . In the long-time limit, when 𝜏 becomes very large, the first term in (6.1.37) dies away completely, and the logarithmic function in the denominator of the second term approaches ln𝜏 1/2 . Thus, the current becomes (5.2.24a), the quasi-steady state considered in Chapter 5. (c) Band Electrode

We will not dwell on the behavior at a band electrode (8, 9). At short times, the current converges, as we now expect, to the Cottrell form, (6.1.12). At long times, the current-time relationship approaches the quasi-steady state, (5.2.25). 6.1.4

Information from Chronoamperometric Results

We have now developed descriptions of id − t curves following the application of large-amplitude potential steps to electrodes of various sizes and shapes, given the general electrode reaction O + ne → R. For all types of electrodes, the id − t relationship follows the Cottrell form, at least for short times. Depending on the shape of the electrode, there might eventually be a divergence from the Cottrell form, and, for UMEs, the system eventually reaches a steady or quasi-steady state. Useful information is available from all parts of the chronoamperometric transient. (a) Cottrell Slope

In the time regime where the Cottrell equation, (6.1.12), applies, one should see (in the cathodic 1∕2 ∗ case) a linear plot of id,c (t) vs. t −1/2 , for which the slope is nFADO CO ∕𝜋 1∕2 (often called the Cottrell slope). It is essential to omit data obtained while double-layer charging is still occurring

6.1 Chronoamperometry Under Diffusion Control

(earlier than 5Ru C d ; Section 6.8.2). From the Cottrell slope, one can obtain any one of n, A, DO , ∗ , if the other quantities are known. Experimentally, it is often easier and more precise to or CO obtain a Cottrell slope using chronocoulometry (Section 6.6), which is form of chronoamperometry in which the current is integrated as it is recorded. (b) Diffusion-Limited Steady-State Current

The steady-state (or quasi-steady-state) current obtained in the long-time chronoamperometric limit at a UME is exactly the same as the diffusion-limited steady-state (or quasi-steadystate) current discussed extensively in Chapter 5. Section 5.3.2(a) covers the chemical information available from these currents. Ordinarily, it is easier to obtain a steady-state (or quasi-steady-state) current from SSV than from chronoamperometry, if only the steady state is of interest. (c) Combination of Transient and Steady-State Data

If one is working with a UME, there is an opportunity to use the shorter-time and longer-time chronoamperometric data together in a useful way (10, 11). The approach rests on the fact that DO contributes to the Cottrell slope as a square root, but to the steady-state current as a linear factor. Because of this feature, it becomes practical to measure DO without separate ∗ . The method has proven valuable and is commonly practiced. We now knowledge of n or CO develop the details. The diffusion-controlled chronoamperometric transient a disk UME is given by (6.1.31)–(6.1.34). Suppose we normalize id,c (t) by dividing every point by the diffusion-limited steady-state current, iss , which can be obtained experimentally from the same transient at d,c long times [and is given by (5.2.18a)]. The two-part result is id,c (t) 𝜋 1∕2 𝜋 = f (𝜏) = + + 0.094𝜏 1∕2 (for 𝜏 < 1) (6.1.38) ss 1∕2 4 id,c 2𝜏 id,c (t) id,ss

=1+

4 𝜋 3∕2

𝜏 −1∕2 + 0.05626𝜏 −3∕2 − 0.00646𝜏 −5∕2 · · ·

(for 𝜏 > 1)

(6.1.39)

where 𝜏 = 4DO t∕r02 . If we just take the first two terms in each case and substitute for 𝜏, then we obtain id,c (t) 𝜋 𝜋 1∕2 r0 = + (for 𝜏 < 0.16 or t < 0.04r02 ∕DO ) (6.1.40) 1∕2 1∕2 4 iss 4D t d,c O

id,c (t) iss d,c

=1+

2r0 1∕2 𝜋 3∕2 DO t 1∕2

(for 𝜏 > 4 or t > r02 ∕DO )

(6.1.41)

These relationships are accurate within 1% for segments of the transient limited to the ranges given to the right of each equation (11).9 For either of these ranges, one can expect a linear plot of id,c (t)∕iss vs. t −1/2 , the slope of d,c which can supply DO . The electrode radius, r0 , must be known, but usually can be measured by the method discussed in Section 5.9.3. Once DO is known, one can use (5.2.18b) to find either ∗ from iss . n or CO d,c 6.1.5

Microscopic and Geometric Areas

If the electrode surface were strictly a plane with a well-defined boundary, such as an atomically smooth metal disk mounted in a glass mantle, the area in the Cottrell equation would 9 One must discard points recorded before 5Ru C d because of interference from charging current (Section 6.8.2).

271

272

6 Transient Methods Based on Potential Steps

Geometric area Projected enclosure

Rough electrode surface

Figure 6.1.4 An electrode surface and the enclosure formed by projecting the boundary outward in parallel with the surface normal. The cross-section of the enclosure is the geometric area of the electrode.

be readily understood. However, the concept of area becomes less distinct for real electrode surfaces, which are not ideally smooth. Figure 6.1.4 helps to define two different measures of area for a given electrode. First there is the microscopic area, which is computed by integrating the exposed surface over all of its undulations, crevices, and asperities, even down to the atomic level.10 An easier quantity to evaluate operationally is the geometric area (sometimes called the projected area). Mathematically, it is the cross-sectional area of the enclosure formed by projecting the boundary of the electrode outward in parallel with the mean surface normal. The microscopic area, Am , is, of course, always larger than the geometric area, Ag , and the roughness factor, 𝜌, is the ratio of the two: 𝜌 = Am ∕Ag

(6.1.42)

Routinely polished metal electrodes typically have roughness factors of 2–3, but single crystal faces of high quality can have roughness factors below 1.5. Liquid-metal electrodes (e.g., mercury) are often assumed to be atomically smooth. One can estimate the microscopic area by measuring the double-layer capacitance (Section 14.4) or the charge required to form or to strip a compact monolayer electrolytically from the surface. For example, the real area of a platinum or gold electrode is often evaluated from the charge passed upon removal of an adsorbed layer under well-defined conditions (12). For Pt, Am can be estimated from the charge needed to desorb hydrogen (Section 14.6). The widely used (but somewhat arbitrary) standard figure for conversion is 210 μC/cm2 , which corresponds to an adsorption site density equal to the density of Pt atoms in a (100) plane. For Au, Am is available from the reduction of a layer of adsorbed oxygen, with 386 μC/cm2 used as the factor for conversion. This figure also corresponds to an adsorption site density matching the density of host atoms in a (100) plane. Uncertainty arises in any measurement of Am because it can be difficult to subtract contributions from other faradaic processes and double-layer charging, and because the charge for desorption varies for different exposed planes of the metal (Figure 14.4.6). The area to be used in the Cottrell equation (or in other similar equations describing current flow) depends on the time scale of the measurements. In the derivation of the Cottrell equation (Section 6.1.1), the current is defined by the flux of species at x = 0. The total rate of reaction in moles per second, giving the total current in amperes, is the product of that flux and the effective area of the diffusion field, which is the area needed for the final result. 10 The microscopic area is alternatively called the real area or true area.

6.1 Chronoamperometry Under Diffusion Control

Solution

Solution Solution

Electrode (a)

Electrode (b)

Figure 6.1.5 Diffusion fields at (a) long and (b) short times at a rough electrode. Depicted here is an idealized electrode where the roughness is caused by parallel triangular grooves cut on lines perpendicular to the page. Dotted lines show surfaces of equal concentration in the diffusion layer. Vectors show concentration gradients driving the flux toward the electrode surface.

In most chronoamperometry, with measurement times of 1 ms to 10 s, the diffusion layer is several micrometers to even tens of micrometers thick. These distances are much larger than the scale of roughness on a reasonably polished electrode, which will have features no larger than a small fraction of a micrometer. On the scale of the diffusion layer, the electrode appears flat; the surfaces connecting equal concentrations in the diffusion layer are planes parallel to the electrode surface; and the effective area of the diffusion field is the geometric area of the electrode. When these conditions apply, as in Figure 6.1.5a, the geometric area should be used in the Cottrell equation or other similar relationship. Let us now imagine a contrasting situation involving a much shorter time scale, perhaps 100 ns, where the diffusion layer thickness is only 10 nm. In this case, depicted in Figure 6.1.5b, much of the roughness is of a scale larger than the thickness of the diffusion layer; hence, the surfaces of equal concentration in the diffusion layer tend to follow the features of the surface. They define the effective area of the diffusion field, which is generally larger than the geometric area. It approaches the microscopic area, but might not be quite as large, because features of roughness smaller in scale than the diffusion length tend to be averaged within the diffusion field. Similar considerations are needed to understand chronoamperometry at an electrode that is active only over a portion of a larger area, as in Figure 6.1.6. Such a situation can arise when arrays of electrodes are fabricated by microelectronic methods, or when the electrode is a composite based on conducting particles, e.g., graphite in an insulating phase, such as a polymer. Another important case involves an electrode covered by a blocking layer with pinholes through which the electroactive species may access the electrode surface (Section 17.6.1). At short times, when the diffusion layer thickness is small compared to the size of the active spots, each spot generates its own diffusion field (Figure 6.1.6a), and the area of the overall diffusion field is the sum of the geometric areas of the active spots. At longer times, the individual diffusion fields begin to extend outside the projected boundaries of the spots, and linear diffusion is augmented by a radial component. (Figure 6.1.6b) At still longer times, when the diffusion layer is much thicker than the distances between the active zones, the separated diffusion fields merge into a single field, again exhibiting linear diffusion and having an area equal to the geometric area of the entire array, even including the insulating zones between the active sites (Figure 6.1.6c). The individual active areas are no longer distinguishable. Molecules diffusing to the electrode come, on average, from so far away that the added distance and time required to reach an active place on the surface becomes negligible. This problem has been treated analytically for cases in which the active spots are uniform in size and situated in a regular array (13), but, in general, simulation is required.

273

274

6 Transient Methods Based on Potential Steps

Solution phase Inactive surface

Electronically conducting phase

Insulating matrix (a)

(b)

(c)

Figure 6.1.6 Evolution of the diffusion field during chronoamperometry at an electrode with active and inactive areas on its surface. In this case, the electrode is a regular array such that the active areas are of equal size and spacing; however, the same principles apply for irregular arrays. (a) Short electrolysis times, (b) intermediate times, (c) long times. Arrows indicate flux lines to the electrode.

Since charging currents are generated by events occurring within nanometer distances at an electrode surface (Section 1.6.3 and Chapter 14), they always reflect the microscopic area. For an electrode made of a polished polycrystalline metal, the area giving rise to a nonfaradaic current may be significantly larger than that characterizing the diffusion field. On the other hand, the opposite can be true if one is using an array of small, widely spaced electrodes embedded in an inert matrix. Exactly this effect was exploited (14) with composite electrodes consisting of gold-filled nanopores passing perpendicularly through a polycarbonate film. A portion of the composite membrane could be mounted for use as a working electrode, providing a surface array of gold nanodisks in polycarbonate. Background currents at such electrodes were smaller by 1–3 orders of magnitude vs. solid gold electrodes of the

6.2 Sampled-Transient Voltammetry for Reversible Electrode Reactions

same geometric area; yet, faradaic responses on a timescale of seconds were essentially the same. Analytical detection limits (Section 8.6) were, consequently, much lower at the composite electrodes.

6.2 Sampled-Transient Voltammetry for Reversible Electrode Reactions Suppose we now consider a series of step experiments using a solution of anthracene (An) in DMF, as discussed in the opening to Section 6.1. Between each experiment the solution is stirred, so that the initial conditions are always re-established. Also, the initial potential (before the step) is always at a common value where no faradaic processes occur. The change from experiment to experiment is in the final step potential, as depicted in Figure 6.2.1a. Suppose further that experiment 1 involves a step to a potential at which An is not yet electroactive; that experiments 2 and 3 involve potentials where An is reduced, but not so effectively that its surface concentration is zero; and that experiments 4 and 5 have step potentials in the mass-transfer-limited region. Experiment 1 yields no faradaic current, and experiments 4 and 5 yield the Cottrell current (Section 6.1.1). In both 4 and 5, the surface concentration is zero; hence, An arrives as fast as diffusion can bring it, and the current is limited by this factor. Once the electrode potential becomes so extreme, the potential no longer influences the current from reduction of An. In experiments 2 and 3, the story is different because the reduction process is not so dominant that An cannot coexist with the electrode. Still, its concentration is less than the bulk value, so An does diffuse to the surface, where it must be eliminated by reduction. Since the difference between the bulk and surface concentrations is smaller than in the mass-transfer-limited case, less material arrives at the surface per unit time than in experiments 4 and 5, and the currents for corresponding times are smaller. The depletion effect applies to experiments 2 and 3, so the current still decays with time in both of those cases (Figure 6.2.1b). If the current is measured at some fixed time, 𝜏, into each of these step experiments; then one can plot the sampled current, i(𝜏), vs. the potential to which the step takes place. As shown in Figure 6.2.1c, the resulting current–potential curve has a wave shape like the voltammograms that we encountered earlier in Chapters 1 and 5. This kind of experiment is sampled-transient voltammetry (STV), several forms of which are in common practice.11 The simplest, operating exactly as described above, is normal pulse voltammetry (NPV). In this chapter, we will consider STV in a general way, with the aim of establishing concepts that apply across a broad range of methods. Chapter 8 covers the details for several forms of voltammetry based on step waveforms, including NPV and its historical predecessors and successors. In principle, current sampling can occur in the time regime where a transient current flows or in the later period when a steady state or quasi-steady state might be reached. In Chapter 5, we covered the latter case fully and need not address it further. As we discuss STV here and in Chapter 8, we will always be discussing experiments with a timescale short enough that semi-infinite linear diffusion applies. The electrode always acts in these experiments as a plane of area A, to which edge diffusion is not significant. Even if the electrode is not actually planar, as in the case of an Hg drop, the diffusion length remains much smaller than the radius of curvature, so that the curvature does not influence the measurements. 11 In the second edition, we used the term sampled-current voltammetry. In this edition, sampled-transient voltammetry is being used, to make the contrast with steady-state voltammetry.

275

276

6 Transient Methods Based on Potential Steps

–1.0

20

60 5 4

E/V

3 2

–0.5

𝜏 = 150 ms

30

1 0.0 –200

0 t/ms

200

4 5

i(𝜏)/μA

i/μA

0 –200

(a)

1 0 t/ms (b)

4, 5 3 2 200

3

10

0 0.0

1

2 –0.5 E/V

–1.0

(c)

Figure 6.2.1 Sampled-transient voltammetry. (a) Step waveforms applied in a series of experiments. (b) Faradaic current–time curves following steps. Currents recorded from 10 to 200 ms after step. Curve 1 is indistinguishable from baseline. (c) Sampled-transient voltammogram. In this example, initial E is −0.1 V vs. ′ reference; steps are to −0.3, −0.47, −0.53, −0.7, and −0.8 V. Electroactive species has E 0 = −0.50 V, n = 1, ∗ −5 2 CO = 1 mM, DO = 1 × 10 cm /s. Working electrode is a disk (r0 = 1 mm, Ru C d = 30 μs). Current sampling at 𝜏 = 150 ms. Curve in (c) would be traced out if many experiments were performed having step potentials separated by a few mV.

6.2.1

A Step to an Arbitrary Potential

Let us now develop the theory for STV. We again consider an experiment like that of Section 6.1.1, but this time we allow a potential step of any magnitude. Each experiment starts at a potential at which no current flows; then, at t = 0, E is shifted instantaneously to another value. Charge-transfer kinetics are assumed to be very rapid, so that ′

E = E0 +

RT CO (0, t) ln nF CR (0, t)

(6.2.1)

Because (6.2.1) involves both O and R, we must address the diffusive motion of both. We start with the general formulation of the mass-transfer problem (Section 4.5.2), which comprehends two diffusion equations, two initial conditions, two semi-infinite conditions, and the flux balance. Species R is absent from the bulk; hence, CR∗ = 0. Extension into Laplace space (Section A.1.6) allows distillation of this formulation into just two equations, which we now adopt as starting points: C O (x, s) =

∗ CO

s

+ A(s)e−(s∕DO )

C R (x, s) = −𝜉A(s)e−(s∕DR )

1∕2 x

1∕2 x

(6.2.2) (6.2.3)

where 𝜉 = (DO /DR )1/2 and A(s) is not yet defined. The final boundary condition needed for a solution is (6.2.1), invoking reversibility. It is conveniently rewritten as 𝜃=

CO (0, t) CR (0, t)

= enf (E−E

0′ )

(6.2.4)

Laplace transformation of (6.2.4) shows that C O (0, s) = 𝜃C R (0, s). With substitution from (6.2.2) and (6.2.3), we have ∗ CO

s

+ A(s) = −𝜉𝜃A(s)

(6.2.5)

6.2 Sampled-Transient Voltammetry for Reversible Electrode Reactions ∗ ∕[s(1 + 𝜉𝜃)]. Therefore, the transformed concentration profiles become giving A(s) = −CO ∗ CO

C O (x, s) =

s

−

∗ e−(s∕DO ) CO

1∕2 x

s(1 + 𝜉𝜃)

∗ e−(s∕DR ) 𝜉CO

(6.2.6)

1∕2 x

(6.2.7) s(1 + 𝜉𝜃) Equation 6.2.6 differs from (6.1.8) only by the factor 1/(1 + 𝜉𝜃) in the second term. Since 𝜉𝜃 is independent of x and t, the current can be obtained (exactly as for the Cottrell case) by evaluating i(s) and then inverting [see (6.1.9)–(6.1.12)]: C R (x, s) =

1∕2

i(t) =

∗ nFADO CO

𝜋 1∕2 t 1∕2 (1 + 𝜉𝜃)

(6.2.8)

This is the general response function for a step experiment in a reversible system with CR∗ = 0. ′ The Cottrell equation, (6.1.12), is a special case for the diffusion-limited region, where E − E0 is very negative, so that 𝜃 → 0. It is convenient to represent the cathodic Cottrell current as id,c (t) and to rewrite (6.2.8) as i(t) =

id,c (t)

(6.2.9)

1 + 𝜉𝜃

Now we see that, for a reversible couple, every current–time curve has the same shape; however, the magnitude is scaled by 1/(1 + 𝜉𝜃), according to the potential to which the step is made. ′ For very positive potentials relative to E0 , this scale factor approaches zero; for very negative potentials, it approaches unity. Thus, i(t) has a value between zero and id,c (t), depending on E, as shown in Figure 6.2.1b. 6.2.2

Shape of the Voltammogram

In STV, our goal is to obtain an i(𝜏) − E curve by (a) performing step experiments with different final potentials E, (b) sampling the current response at a fixed time 𝜏 after the step, and (c) plotting i(𝜏) vs. E. Here we consider the shape of this plot for a reversible couple and the kinds of information one can obtain from it. Equation 6.2.9 provides all the answers for the cathodic case. For a fixed sampling time 𝜏, id,c (𝜏) i(𝜏) = (6.2.10) 1 + 𝜉𝜃 which can be rewritten as id,c (𝜏) − i(𝜏) 𝜉𝜃 = (6.2.11) i(𝜏) and expanded: ′ RT 1 RT id,c (𝜏) − i(𝜏) E = E0 + ln + ln (6.2.12) nF 𝜉 nF i(𝜏) When i(𝜏) = id,c (𝜏)/2, the third term vanishes. The corresponding potential is E1/2 , the half-wave potential: 1∕2

E1∕2

RT DR =E + ln 1∕2 nF DO 0′

(6.2.13)

277

278

6 Transient Methods Based on Potential Steps

so that E = E1∕2 +

RT id,c (𝜏) − i(𝜏) ln nF i(𝜏)

(6.2.14)

The last two equations describe the voltammogram for a reversible system in STV when semi-infinite linear diffusion holds and R is absent from the bulk. It is interesting to compare (6.2.12) and (6.2.14) with the wave shape equations derived for steady-state voltammetry in Section 5.3.1. They are identical in form; however, E1/2 depends on 𝜉 = (DO /DR )1/2 in transient voltammetry, but on 𝜉 = DO /DR in steady-state voltammetry. Figure 5.3.1, already presented as a steady-state voltammogram, also accurately represents a sampled-transient voltammogram as given by (6.2.13) and (6.2.14), but for DO = 0.5DR , rather than DO = 0.7DR (as represented in the caption for Figure 5.3.1). Since (DO /DR )1/2 is nearly unity in most cases, the second term in (6.2.13) is normally small, ′ and E1/2 for a reversible couple in STV is usually a good approximation to E0 . As in earlier cases of reversible waves, E vs. log[(id,c (𝜏) − i(𝜏))/i(𝜏)] should be linear with a “wave slope” of 2.303RT/nF or 59.1/n mV at 25 ∘ C. Also, the Tomeš criterion (15) applies: One should find that |E3/4 − E1/4 | = 56.4/n mV at 25 ∘ C, where E3/4 and E1/4 are the potentials for which i(𝜏) = 3id,c (𝜏)/4 and i(𝜏) = id,c (𝜏)/4, respectively. If the wave slope or the Tomeš criterion significantly exceeds the expected value, the system is not reversible. For the case where both O and R are present in the bulk, one can derive the expected behavior by the methods developed just above, but the details are left for the reader (Problem 6.5). Diffusion-limited currents exist for both O and R, because either can be electrolyzed under Cottrell conditions. For the reduction of O at rather negative potentials, the cathodic Cottrell current, id,c (t), is given directly by (6.1.12). For the oxidation of R in a rather positive range, the anodic Cottrell current, id,a (t), can be stated from (6.1.12) by symmetry:12 1∕2

nFADR CR∗

(6.2.15) 𝜋 1∕2 t 1∕2 The equation for the composite STV wave has the same form as (5.3.7), describing SSV for the same situation: id,a (t) = −

E = E1∕2 +

RT id,c (𝜏) − i(𝜏) ln nF i(𝜏) − id,a (𝜏)

(6.2.16)

For the case where R is initially present, but O is absent from the bulk, the wave is entirely anodic and has the following form, derived easily from (6.2.16): E = E1∕2 + 6.2.3

−i(𝜏) RT ln nF i(𝜏) − id,a (𝜏)

(6.2.17)

Concentration Profiles When R Is Initially Absent

Inverse transformation of (6.2.6) and (6.2.7) yields the concentration profiles for CR∗ = 0: ] [ ∗ CO x ∗ CO (x, t) = CO − erfc (6.2.18) 1 + 𝜉𝜃 2(DO t)1∕2 [ ] ∗ 𝜉CO x CR (x, t) = erfc (6.2.19) 1 + 𝜉𝜃 2(DR t)1∕2 12 Whether O or R is electrolyzed under Cottrell conditions, the diffusion problem is the same, so the result must be the same, except for the subscripts identifying the diffusing species. difference in current direction.

6.3 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions

from which we obtain the surface concentrations, ( ) ( ) 𝜉𝜃 1 ∗ ∗ CO (0, t) = CO 1 − = CO 1 + 𝜉𝜃 1 + 𝜉𝜃 ( ) 𝜉 ∗ CR (0, t) = CO 1 + 𝜉𝜃 Since (6.2.9) shows that i(t)/id.c (t) = 1/(1 + 𝜉𝜃), [ ] i(t) ∗ CO (0, t) = CO 1 − id,c (t) ∗ CR (0, t) = 𝜉CO

6.2.4

i(t) id,c (t)

(6.2.20) (6.2.21)

(6.2.22)

(6.2.23)

Simplified Current–Concentration Relationships

For STV, (6.2.22) and (6.2.23) can be rearranged and re-expressed in the following way for the sampling time 𝜏: 1∕2

i(𝜏) =

nFADO

𝜋 1∕2 𝜏 1∕2

∗ − C (0, 𝜏)] = nFAm [C ∗ − C (0, 𝜏)] [CO O O O O

(6.2.24)

1∕2

nFADR

C (0, 𝜏) = nFAmR CR (0, 𝜏) (6.2.25) 𝜋 1∕2 𝜏 1∕2 R Thus, we arrive, with rigor, at a set of simple relations of precisely the same form as those assumed in the naive approach to mass transport used in Section 1.3.2. To translate the relationships exactly, one need only define i(𝜏) =

1∕2

mO =

DO

𝜋 1∕2 𝜏 1∕2

(6.2.26a)

1∕2

DR

(6.2.26b) 𝜋 1∕2 𝜏 1∕2 In Section 5.2.5, we were able to prove corresponding equations for steady-state voltammetry and to identify the mO and mR functions for UMEs of different shapes. It follows easily from what we have done here that (5.2.26)–(5.2.33) are all valid for STV of reversible systems, including cases for which R is present in the bulk. We have also now confirmed that, for linear diffusion, 𝜉 = mO /mR = (DO /DR )1/2 . mR =

6.2.5

Applications of Reversible i–E Curves

For a detailed discussion of applications, the reader is referred to Section 5.3.2, which applies to both STV and SSV.

6.3 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions In this section, we will treat the one-step, one-electron reaction O + e ⇌ R using the general i − E characteristic. In contrast with the reversible cases just examined, the interfacial electron-transfer kinetics considered here are not so fast as to be transparent; thus, kinetic

279

280

6 Transient Methods Based on Potential Steps

parameters such as k f , k b , k 0 , and 𝛼 influence the current transients following potential steps and can often be evaluated from those transients. 6.3.1

Effect of Electrode Kinetics on Transient Behavior

The predicted transient behavior depends on the kinetic model employed and the presence or absence of O and R in the bulk. We distinguish several situations as we develop the basic ideas. (a) General Kinetics, Any Mix of O and R in the Bulk

To develop theory for a current transient governed by both charge-transfer kinetics and diffusion, one can begin with the general formulation (Section 4.5.2), which is embodied in Laplace space (Section A.1.6) as (A.1.62) and (A.1.63). The final boundary condition, allowing a solution, is the kinetic relationship for a one-step, one-electron reaction at the electrode surface: [ ] 𝜕CO (x, t) i = DO = kf CO (0, t) − kb CR (0, t) (t > 0) (6.3.1) FA 𝜕x x=0 Since k f and k b are constant during a potential step, the Laplace transform of (6.3.1) is [ ] 𝜕C O (x, s) DO = kf C O (0, s) − kb C R (0, s) (6.3.2) 𝜕x x=0

Substitution from (A.1.62) and (A.1.63), followed by rearrangement, then provides C O (x, s) =

∗ CO

−

s

∗ − k C ∗ )e−(s∕DO ) (kf CO b R

1∕2 x

(6.3.3)

1∕2

DO s(H + s1∕2 )

where H=

kf 1∕2

DO

+

kb

(6.3.4)

1∕2

DR

The transform of the current is given in (6.1.10), which, with evaluation from (6.3.3), provides [ ] ∗ − k C∗ ) FA(kf CO 𝜕C O (x, s) b R i(s) = FADO = (6.3.5) 𝜕x s1∕2 (H + s1∕2 ) x=0

∗ and C ∗ , Inverse transformation affords the current transient for the general case of CO R ∗ − k C ∗ ) exp(H 2 t)erfc(Ht 1∕2 ) i(t) = FA(kf CO b R

(6.3.6)

Two special cases are ∗ exp(H 2 t)erfc(Ht 1∕2 ) i(t) = FAk f CO

i(t) = −FAk b CR∗ exp(H 2 t)erfc(Ht 1∕2 )

(CR∗ = 0) ∗ = 0) (CO

(6.3.7) (6.3.8)

At any given step potential, k f , k b , and H are constants. The functional product exp(x2 ) erfc(x) is unity for x = 0, but falls monotonically toward zero as x becomes large. Thus, the current–time curve has a shape like that in Figure 6.3.1, corresponding to (6.3.7). The heterogeneous kinetics limit the current at t = 0 to a finite value from which k f can be evaluated. The same general behavior applies in any of the cases covered by (6.3.6)–(6.3.8). Since a charging current also exists in the moments after the step is applied, the faradaic component at t = 0 must be

6.3 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions

i FAkfC*O

t

Figure 6.3.1 Current decay after the application of a step to a potential where species O is reduced with quasireversible kinetics.

determined by extrapolation from data taken after the charging current has decayed [Sections 1.6.4(a) and 6.8.2]. Extrapolation is facilitated for small values of Ht 1/2 , because exp(H 2 t) erfc(Ht 1/2 ) can be linearized by taking the first two terms from the corresponding Maclaurin series, exp(H 2 t) erfc(Ht 1∕2 ) = 1 − 2Ht 1∕2 ∕𝜋 1∕2

(6.3.9)

Thus, (6.3.7) and (6.3.8) become, respectively, ∗ (1 − 2Ht 1∕2 ∕𝜋 1∕2 ) i = FAk f CO

(6.3.10a)

i = −FAk b CR∗ (1 − 2Ht 1∕2 ∕𝜋 1∕2 )

(6.3.10b)

One can apply a step to the potential region where H remains small (generally at the foot of a cathodic or anodic response), then obtain a linear plot of i vs. t 1/2 providing k f (if CR∗ = 0) or k b ∗ = 0) from the intercept. (if CO To this point, we have not invoked any model of electrode kinetics, so any value of k f or k b obtained through the results of this section would be independent of any model. (b) Butler–Volmer Kinetics, Any Mix of O and R in the Bulk

If the Butler–Volmer model is chosen, the forward and backward rate constants for the one step, one-electron process O + e ⇌ R are given by kf = k 0 e−𝛼f (E−E

0′ )

kb = k 0 e(1−𝛼)f (E−E

= k 0 𝜃 −𝛼 0′ )

(6.3.11a)

= k 0 𝜃 (1−𝛼)

(6.3.11b)

0′

where 𝜃 = ef (E−E ) and k 0 and 𝛼 are the fundamental kinetic parameters (Section 3.3). If one uses the methods described in the preceding section to obtain values of k f or k b , one has the latitude to interpret the rate constants in the context of the Butler–Volmer model. A ′ plot of ln k f or ln k b vs. E should be linear, with a slope that provides 𝛼 and an intercept at E0 equal to k 0 . When both O and R are present in the bulk, an equilibrium potential exists, and one can describe the effect of potential on the i − t curve in terms of the overpotential, 𝜂 = E − Eeq . This is a natural way to examine the results for a poised system, because Eeq must be chosen as the initial potential to avoid pre-electrolysis; hence, any potential step has an amplitude 𝜂. An alternate expression for (6.3.6) can be given for BV kinetics by noting that ∗ − k C ∗ = k 0 [C ∗ e−𝛼f (E−E kf CO b R O

0′ )

0′

− CR∗ e(1−𝛼)f (E−E ) ]

(6.3.12)

281

282

6 Transient Methods Based on Potential Steps

Substitution for k 0 in terms of i0 using (3.4.6) and rearrangement with use of (3.4.2) then provides i ∗ − k C ∗ = 0 [e−𝛼f 𝜂 − e(1−𝛼)f 𝜂 ] (6.3.13) kf CO b R FA Therefore, (6.3.6) may be written i = i0 [e−𝛼f 𝜂 − e(1−𝛼)f 𝜂 ] exp(H 2 t)erfc(Ht 1∕2 )

(6.3.14)

By similar substitutions into the definition for H, one has H=

⎤ i0 ⎡ e−𝛼f 𝜂 e(1−𝛼)f 𝜂 ⎥ ⎢ − FA ⎢ C ∗ D1∕2 C ∗ D1∕2 ⎥ ⎣ O O R R ⎦

(6.3.15)

The form of (6.3.6) and (6.3.14) is i = [i in the absence of mass-transfer effects] × [f (H, t)] where f (H, t) accounts for the effects of mass transfer. Using (6.3.9), we can linearize exp(H 2 t) erfc(Ht 1/2 ) in (6.3.14) to obtain i = i0 [e−𝛼f 𝜂 − e(1−𝛼)f 𝜂 ](1 − 2Ht 1∕2 ∕𝜋 1∕2 )

(6.3.16)

Thus, a plot of i vs. t 1/2 has as its intercept the kinetically controlled current free of mass-transfer effects. For small values of 𝜂, the linearized i − 𝜂 characteristic, (3.4.12), can be used, so that (6.3.14) becomes Fi 𝜂 (6.3.17) i = − 0 exp(H 2 t) erfc(Ht 1∕2 ) RT Then, for small 𝜂 and small Ht 1/2 , one has a “completely linearized” form: Fi 𝜂 i = − 0 (1 − 2Ht 1∕2 ∕𝜋 1∕2 ) (6.3.18) RT A plot of it = 0 vs. 𝜂 can then be used to obtain i0 . 6.3.2

Sampled-Transient Voltammetry for Reduction of O

Let us return to (6.3.7), which is the i − t expression for the case where only species O is present ′ in the bulk. Assuming BV kinetics, we have k b /k f = 𝜃 = exp[ f (E − E0 )]; therefore, H=

kf 1∕2

DO

(1 + 𝜉𝜃)

(6.3.19)

Now (6.3.7) can be rephrased as 1∕2

∗ FADO CO

[𝜋 1∕2 Ht 1∕2 exp(H 2 t)erfc(Ht 1∕2 )] (6.3.20) 𝜋 1∕2 t 1∕2 (1 + 𝜉𝜃) Since semi-infinite linear diffusion applies, the diffusion-limited current is the Cottrell current for reduction of O, which is easily recognized in the factor preceding the brackets. On the basis of (6.1.12), we can simplify (6.3.20) to i=

i=

id,c (1 + 𝜉𝜃)

F1 (𝜆)

(6.3.21)

6.3 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions

where F1 (𝜆) = 𝜋 1∕2 𝜆 exp(𝜆2 )erfc(𝜆)

(6.3.22)

and 𝜆 = Ht 1∕2 =

kf t 1∕2 1∕2

(1 + 𝜉𝜃)

(6.3.23)

DO

Equation 6.3.21 is a compact representation of the way in which the current in a step experiment depends on potential and time, and it holds for all kinetic regimes: reversible, quasireversible, and totally irreversible. It is the analogue, for potential step experiments, of (5.4.7), derived earlier for steady-state voltammetry. The function F 1 (𝜆) manifests the kinetic effects on the current in terms of the dimensionless parameter 𝜆, which compares the maximum cur∗ , to the maximum current rent supportable by the kinetics at a given step potential, FAk f CO supportable by diffusion at that potential, id,c /(1 + 𝜉𝜃). Thus, 𝜆 is an analogue, for potential step experiments, of the parameters Λf and Λb , used to make the same sort of comparison in steady-state voltammetry (Section 5.4.1). At small values of 𝜆, kinetics strongly influence the current; at large values of 𝜆, kinetics are facile and the response is controlled by diffusion. The function F 1 (𝜆) rises monotonically from a value of zero at 𝜆 = 0 toward an asymptote of unity as 𝜆 becomes large (Figure 6.3.2). Simpler forms of (6.3.21) are used for the reversible and totally irreversible limits. For example, consider (6.2.9), which describes the current–time curve following a step to an arbitrary potential in a reversible system. That same relationship is available from (6.3.21) simply by recognizing that, with reversible kinetics, 𝜆 is very large, so that F 1 (𝜆) is always unity. The totally irreversible limit will be considered separately in the next section. So far, it has been most convenient to think of (6.3.21) as describing the current–time response following a potential step; however, it also describes the current–potential curve in STV, just as we understood (6.2.9) to do for reversible systems. At a fixed sampling time, 𝜏, 1∕2 𝜆 becomes (kf 𝜏 1∕2 ∕DO )(1 + 𝜉𝜃), which is a function only of potential among the variables ′ that change during a voltammetric run. At very positive potentials relative to E0 , 𝜃 is very large, so i ≈ 0, regardless of F 1 (𝜆). At very negative potentials, 𝜃 → 0 and k f becomes very large; consequently, F 1 (𝜆) → 1 and i → id,c . From these simple considerations, we expect the 1.2 1.0 0.8 F1(𝜆) 0.6 0.4 0.2 0.0 0.01

0.1

𝜆

1

10

Figure 6.3.2 General kinetic function for chronoamperometry and STV.

283

284

6 Transient Methods Based on Potential Steps

1.2

1.0 Quasireversible 0.8 i/id,c

Reversible 0.6

Totally irreversible

0.4

0.2

0.0 300

–100

100

–300

–500

–700

–900

(E – E 0′)/mV

Figure 6.3.3 Sampled-transient voltammograms for various kinetic regimes. The reaction is O + e → R with R absent from the bulk. Curves are calculated from (6.3.21) assuming BV kinetics with 𝛼 = 0.5 and 𝜏 = 1 s, DO = DR = 1 × 10−5 cm2 /s. From left to right, k0 is 10, 1 × 10−3 , 1 × 10−5 , and 1 × 10−7 cm/s.

sampled-transient voltammogram to have a wave shape generally similar to that found in the reversible case. Figure 6.3.3, which offers several voltammograms corresponding to different kinetic regimes, bears out this expectation. For very facile kinetics, corresponding to large k 0 , the wave has the reversible shape, and the ′ ′ half-wave potential is near E0 . (In Figure 6.3.3, where DO = DR , E1∕2 = E0 exactly.) For smaller values of k 0 , the kinetics must be driven by an overpotential, so the wave is displaced toward more extreme potentials (in the negative direction for a reduction). In addition, the wave is broadened by kinetic effects, as one can see clearly in Figure 6.3.3. The displacement represents the required kinetic activation. For small k 0 , it can be hundreds of millivolts, or even volts. Still, k f is activated exponentially with potential and can become large enough at negative potentials to handle the diffusion-limited flux of electroactive species; thus, the wave eventually shows a plateau at id,c , unless the background limit of the system is reached first. 6.3.3

Sampled Transient Voltammetry for Oxidation of R

Now we turn to the case in which species O is absent from the bulk, and the electrode process being observed is the net oxidation of R. The starting point for treatment is (6.3.8), describ∗ = 0. Given Butler–Volmer kinetics, H can be ing the current transient in a system where CO restated as ( ) k 1 H= b 1+ (6.3.24) 1∕2 𝜉𝜃 D R

then (6.3.8) becomes 1∕2

FADR CR∗ i=− ( ) [𝜋 1∕2 Ht 1∕2 exp(H 2 t)erfc(Ht 1∕2 )] 1 1∕2 1∕2 1 + 𝜉𝜃 𝜋 t

(6.3.25)

6.3 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions

The Cottrell current for oxidation of R, id,a (t), is given by (6.2.15) and can be seen readily in (6.3.25). Given F 1 (𝜆) from (6.3.22), we can rewrite (6.3.25) as id,a i= ( ) F1 (𝜆) 1 1 + 𝜉𝜃

(6.3.26)

Thus, we have found that equations of very similar form, (6.3.21) and (6.3.26), describe the ∗ = 0. In consequence, much of the cathodic wave when CR∗ = 0 and the anodic wave when CO discussion of the last section applies equally to the anodic case, but there are some differences: • Because k b is the dominant kinetic parameter for an anodic wave (vs. k f for a cathodic wave), it is more useful to think about 𝜆 at sampling time 𝜏 in the following form: ( ) k 𝜏 1∕2 1 𝜆= b 1+ (6.3.27) 1∕2 𝜉𝜃 DR ′

• At very negative potentials relative to E0 , 𝜃 → 0. According to (6.3.26), i ≈ 0 in this range, regardless of F 1 (𝜆). At very positive potentials, both 𝜃 and k b become large; therefore, F 1 (𝜆) → 1 and i → id,a . Thus, there would be a sigmoidal wave rising from a baseline of zero to a plateau at id,a , but facing oppositely on the potential axis vs. the cathodic case. • Since id,a is negative, the net current is always negative. ∗ = 0, except that it would be rotated • Figure 6.3.3 applies in essence to the anodic case when CO ∘ by 180 , so that the waves rise to a negative current plateau as the potential becomes more ′ positive. The reversible system would still be situated at E1/2 = E0 for DO = DR , but progressively smaller values of k 0 would cause the wave to shift positively (i.e., toward more extreme potentials for an oxidation). • Because k b = k 0 𝜃 (1 − 𝛼) dominates the kinetics of an anodic wave, the wave shape and the shift of wave position with k 0 generally depend on 1 − 𝛼. In the cathodic case, k f = k 0 𝜃 −𝛼 dominates, so these aspects generally reflect 𝛼. This distinction will become more apparent in the next section. 6.3.4

Totally Irreversible Reactions

The irreversible regime is defined by the condition that the electrode reaction is unidirectional (either O + e → R or R → O + e) over the entire potential range of practical observation. A large overpotential is required to render either process observable, and it fully inhibits the reverse reaction. In the range where the cathodic branch can be studied, k b /k f = 𝜃 ≈ 0. Correspondingly, for the anodic branch, k f /k b = 1/𝜃 ≈ 0. The presence of R in the bulk does not affect a totally irreversible cathodic wave for species O, because the oxidation of R is fully suppressed in the region of study. Likewise, an irreversible anodic process is unaffected by the presence of species O. (a) Cathodic Currents 1∕2

In the region of irreversible cathodic response, 𝜃 = 0. From (6.3.19), we have H = kf ∕DO , so that (6.3.6) or (6.3.7) becomes ( 2 ) ⎛ k t 1∕2 ⎞ kf t ∗ ⎟ i = FAk f CO exp erfc ⎜ f (6.3.28) ⎜ D1∕2 ⎟ DO ⎝ O ⎠

285

286

6 Transient Methods Based on Potential Steps

Equation 6.3.21 remains valid, but simplifies to i = F1 (𝜆) = 𝜋 1∕2 𝜆 exp(𝜆2 )erfc(𝜆) id 1∕2

(6.3.29) 1∕2

where 𝜆 has become kf t 1∕2 ∕DO or kf 𝜏 1∕2 ∕DO , depending on whether one is using (6.3.29) to describe a current transient or a sampled-transient voltammogram. The half-wave potential for an irreversible STV wave occurs where F 1 (𝜆) = 0.5, which is where 𝜆 = 0.433. If k f = k 0 𝜃 −𝛼 and t = 𝜏, then k 0 𝜏 1∕2 1∕2 DO

′

exp[−𝛼f (E1∕2 − E0 )] = 0.433

(6.3.30)

By taking logarithms and rearranging, one obtains ′

E1∕2 = E0 +

⎛ ⎞ RT ⎜ 2.31k 0 𝜏 1∕2 ⎟ ln 𝛼F ⎜ D1∕2 ⎟ ⎝ ⎠ O

(6.3.31)

where the second term expresses the overpotential required to activate the kinetics. For an irreversible cathodic wave, the second term is negative, and the wave is shifted negatively vs. ′ E0 as illustrated in Figure 6.3.3. Equation 6.3.31 provides a simple way to evaluate k 0 when 𝛼 ′ and E0 are known. The wave-slope plot for a totally irreversible STV curve is nonlinear and is not useful; however, the Tomeš criterion is valid and can supply 𝛼. For an irreversible cathodic wave in STV, |E3∕4 − E1∕4 | = 45.0∕𝛼 mV

(6.3.32)

(b) Anodic Currents

In the potential region of an irreversible anodic wave, 1/𝜃 → 0. From (6.3.24), we see that 1∕2 H = kb ∕DR ; hence, (6.3.6) or (6.3.8) becomes ( i=

−FAk b CR∗ exp

kb2 t DR

)

⎛ k t 1∕2 ⎞ ⎟ erfc ⎜ b ⎜ D1∕2 ⎟ ⎝ R ⎠

(6.3.33) 1∕2

This can be shown to be (6.3.26), which simplifies to (6.3.29), but with 𝜆 = kb t 1∕2 ∕DR . By the logic leading to (6.3.31), and given k b = k 0 𝜃 (1 − 𝛼) and t = 𝜏, one obtains the half-wave potential for an irreversible anodic STV wave: ′

E1∕2 = E0 −

⎛ ⎞ RT 2.31k 0 𝜏 1∕2 ⎟ ln ⎜ (1 − 𝛼)F ⎜ D1∕2 ⎟ ⎝ ⎠ R

(6.3.34)

For an irreversible wave, the second term is positive overall, and the wave is shifted positively ′ vs. E0 . The Tomeš criterion for an irreversible anodic wave in STV is |E3∕4 − E1∕4 | = 45.0∕(1 − 𝛼) mV

(6.3.35)

6.3 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions

6.3.5

Kinetic Regimes

Just as we did for steady-state voltammetry in Section 5.4.3, we can distinguish conditions defin′ ing the three kinetic regimes. To do so, let us focus on the particular value of 𝜆 at E0 , designated as 𝜆0 . This parameter has essentially the same role that Λ0 had in Section 5.4.3. ′ 1∕2 At E0 , k f = k b = k 0 and 𝜃 = 1; therefore, 𝜆0 = (1 + 𝜉)k 0 𝜏 1∕2 ∕DO , which can be approxi1∕2

mated for our purpose as 2k 0 𝜏 1∕2 ∕DO . It is useful to understand 𝜆0 as a comparator of the intrinsic abilities of kinetics and diffusion to support a current. The greatest possible forward ∗ , corresponding to the absence of depletion at the electrode reaction rate at any potential is kf CO ′ ∗ and the resulting current is FAk 0 C ∗ . The greatest current supsurface. At E = E0 , this is k 0 CO O portable by diffusion at sampling time 𝜏 is the Cottrell current. The ratio of these currents is 1∕2 𝜋 1∕2 k 0 𝜏 1∕2 ∕DO , or (𝜋 1/2 /2)𝜆0 , which is essentially the same as 𝜆0 . If a system is to appear reversible, 𝜆0 must be sufficiently large that F 1 (𝜆) approaches unity ′ 1∕2 at potentials neighboring E0 . For 𝜆0 > 2 (or k 0 𝜏 1∕2 ∕DO > 1), F 1 (𝜆0 ) exceeds 0.90, a value high enough to assure reversible behavior within practical experimental limits. Smaller values of 𝜆0 will produce measurable kinetic effects in the voltammetry. Thus, we can set 𝜆0 = 2 as the boundary between the reversible and quasireversible regimes, although we also recognize that the delineation is not sharp and depends operationally on the precision of experimental measurements. Total irreversibility on the cathodic side requires that k b /k f ≈ 0 (i.e., 𝜃 ≈ 0) at all potentials ′ where the current is measurably above the baseline. Because 𝜃 is also exp[ f (E − E0 )], this con′ dition implies that the rising portion of the wave be significantly displaced from E0 in the ′ 0 negative direction. If E1/2 − E is at least as negative as −4.6RT/nF, then k b /k f will be no more than 0.01 at E1/2 , and the condition for total irreversibility will be satisfied. The implication is that the second term on the right side of (6.3.31) is more negative than −4.6RT/nF, and by rearrangement one finds that log𝜆0 < − 2𝛼 + log(2/2.31). The final term can be neglected for our purpose here, so the condition for total irreversibility becomes log𝜆0 < − 2𝛼. For 𝛼 = 0.5, 𝜆0 must be less than 0.1. One can derive essentially the same condition by considering total irreversibility on the anodic side. In the middle ground, where 10−2𝛼 ≤ 𝜆0 ≤ 2, the system is quasireversible, and one cannot simplify (6.3.21) or (6.3.26) as a descriptor of current decay or voltammetric wave shape. It is important to grasp that the kinetic regime depends not only on the intrinsic kinetic characteristics of the electrode reaction, but also on the experimental conditions. The time scale, expressed as the sampling time 𝜏 in STV, is a particularly important experimental variable and can be used to change the kinetic regime for a given system. For example, suppose one has an electrode reaction with the following (not unusual) properties: k 0 = 10−2 cm/s, 𝛼 = 0.5, and DO = DR = 10−5 cm2 /s. For sampling times longer than 100 ms, 𝜆0 > 2 and the voltammetry would be reversible. Sampling times between 100 ms and 250 μs correspond to 2 ≥ 𝜆0 ≥ 0.1 and would produce quasireversible behavior. Values of 𝜏 smaller than 250 μs would produce total irreversibility. In Section 5.4.3, we found a similar effect of experimental conditions on the observed kinetic regime in SSV, where the prime experimental variable is the critical dimension of the electrode (often r0 ). For STV, it is the sampling time, 𝜏. 6.3.6

Applications of Irreversible i–E Curves

(a) Information from the Wave Height

The reader is referred to Section 5.3.2(a), which applies to both STV and SSV. The reversibility of the electrode process in terms of electron-transfer kinetics is not a concern in the extraction of information from the wave height.

287

288

6 Transient Methods Based on Potential Steps

Table 6.3.1 Wave Shape Characteristics at 25 ∘ C in Sampled-transient Voltammetry. |E3/4 − E 1/4 | (mV)

Kinetic Regime

Wave Slope (mV)

Reversible (n ≥ 1)

Linear, 59.1/n

56.4/n

Quasireversible (n = 1)

Nonlinear

Between 56.4 and 45.0/𝛼

Irreversible (n = 1)

Nonlinear

45.0/𝛼 (cathodic), 45.0/(1 − 𝛼) (anodic)

(b) Information from the Wave Shape and Position

When the wave is not reversible, the half-wave potential is not a good estimate of the formal potential and cannot be used directly to determine thermodynamic quantities in the manner discussed in Section 5.3.2. However, information about heterogeneous electron transfer kinetics is available for one-step, one-electron cases. Section 5.4.5 provides a detailed discussion for SSV that is generally also applicable to STV. The interested reader is urged to review it now. Details particular to STV are covered in the remaining paragraphs of this section. The wave slope and the Tomeš criterion are different in STV vs. SSV. Table 6.3.1 summarizes expectations for STV in all three kinetic regimes. A large wave slope is a clear indicator that a system is not showing simple reversible behavior; however, it does not necessarily imply that one has an electrode process controlled by the kinetics of electron transfer. Electrode reactions frequently include purely chemical steps taking place in the diffusion layer. A system involving “chemical complications” can show a wave shape essentially identical to that expected for a simple electron transfer in the totally irreversible regime. For example, the reduction of nitrobenzene in aqueous solutions can lead, depending on the pH, to phenylhydroxylamine (16): H PhNO2 + 4H+ + 4e → PhNOH + H2 O

(6.3.36)

However, the first electron-transfer step PhNO2 + e ⇌ PhNO2 −∙

(6.3.37)

is intrinsically quite rapid, as found from measurements in nonaqueous solvents, such as DMF. The irreversibility observed in aqueous solutions arises because of the protonations and electron transfers following the first electron addition. If one treated the observed voltammetric curve of nitrobenzene using the totally irreversible one-step, one-electron model, kinetic parameters for the electron transfer might be obtained, but they would be of no significance. Treatment of such complex systems requires elucidation of the electrode reaction mechanism, as discussed in Chapter 13. Before one employs wave-shape parameters to diagnose the kinetic regime, one must be sure of the basic chemistry of the electrode process. It is easier to establish confidence on this point by investigating the system with a method that can provide direct observations of the reaction in both directions (e.g., cyclic voltammetry, Chapter 7). If a one-step, one-electron system shows totally irreversible behavior in STV based on the kinetics of interfacial electron transfer, then kinetic parameters can be obtained in three ways: 1) Point-by-point evaluation of k f or k b . From a recorded voltammogram, one can measure i/id at various potentials on the rising portion of the wave and then use (6.3.29) to determine the corresponding value of 𝜆 for each potential. For a cathodic wave, a value of k f can then be calculated from each value of 𝜆, given 𝜏 and DO . For an anodic wave, the procedure yields

6.4 Multicomponent Systems and Multistep Charge Transfers

values of k b , given 𝜏 and DR . This approach requires no assumption that the kinetics follow a particular model. If a model is subsequently assumed, the set of k f or k b values can be analyzed to obtain other parameters. The Butler–Volmer model implies that a linear plot of ′ log k f or log k b vs. E − E0 will provide 𝛼 from the slope and k 0 from the intercept. To yield ′ k 0 , this procedure requires that E0 be obtained by some other means (e.g., potentiometry), because it cannot be determined from the wave position. 2) Tomeš criterion and half-wave potential. As one can see from Table 6.3.1, |E3/4 − E1/4 | for a totally irreversible system provides 𝛼 directly. That figure can then be used in conjunction ′ with (6.3.31) or (6.3.34) to obtain k 0 . BV kinetics are implicit, and E0 must be known. 3) Curve fitting. The most general approach to the evaluation of parameters is to employ a nonlinear least-squares algorithm to fit a whole digitized voltammogram to a theoretical function. For a totally irreversible wave, one could develop a fitting function from (6.3.29) by using a specific kinetic model to describe the potential dependence of k f or k b in terms of adjustable parameters. If the BV model is assumed, the appropriate substitution is (6.3.11a,b), and the adjustable parameters are 𝛼 and k 0 . The algorithm then determines the values of the parameters that best describe the experimental results. If the voltammetry is quasireversible, one cannot use simplified descriptions of the wave shape, but must analyze results according to the appropriate general expression, (6.3.21) or (6.3.26). Point-by-point evaluation or curve-fitting are options.

6.4 Multicomponent Systems and Multistep Charge Transfers Section 5.5 addressed the SSV of systems with multiple electroactive species or with solutes that can undergo stepwise reduction or oxidation. The same concepts apply to STV. The reader might review Section 5.5 now. A multicomponent case might be the successive reduction at an Hg drop of Cd(II) and Zn(II) in deoxygenated aqueous KCl. The Cd(II) is reduced with an E1/2 near −0.6 V vs. SCE, but the Zn(II) remains inactive until the potential becomes more negative than about −0.9 V. The STV would resemble Figure 5.5.1. Beyond −1.0 V, the total current is simply the sum of the individual diffusion currents for the two species, which we label as O and O′ . Equation 5.5.1 would accurately describe the total current, with mO = DO 1/2 /𝜋 1/2 𝜏 1/2 and mO′ = DO′ 1∕2 ∕𝜋 1∕2 𝜏 1∕2 , as demonstrated for STV in Section 6.2.4. In effect, we have assumed that the reactions of O and O′ are independent and that the products of one electrode reaction do not interfere with the other. While this is commonly the situation, there are cases where reactions in solution can invalidate (5.5.1). A classic example is the reduction of cadmium ion and iodate at a mercury electrode in an unbuffered medium. The in the second wave occurs by the reaction IO− + 3H2 O + 6e → I− + 6OH− . reduction of IO− 3 3 The liberated hydroxide diffuses away from the electrode and reacts with Cd2+ diffusing toward the electrode, causing precipitation of Cd(OH)2 and, thus, decreasing the height of the first wave (from reduction of Cd2+ to the amalgam) at potentials where the second wave occurs. The consequence is a second plateau that is much lower than would be observed if the reactions were independent (or if the solution were buffered). Molecular oxygen is the basis of a multistep system with great biological and technological importance. It is reduced at Hg in two steps in neutral solution. Figure 6.4.1 shows the behavior in the polarographic form of STV.13 In the first reduction step, oxygen goes to hydrogen 13 Polarography is voltammetry at a dropping mercury electrode. Section 8.1 provides more detail on this historically important methodology.

289

290

6 Transient Methods Based on Potential Steps

H2O2 + 2e → 2OH–

10

O2 + 2H2O + 2e H2O2 + 2OH–

i/μA

+0.2 –0.2

–0.4

–0.6

–0.8

–1.0

–1.2

–1.4

–1.6

E/V vs. SCE

Figure 6.4.1 Polarographic form of STV for air-saturated 0.1 M KNO3 . The working electrode is a dropping mercury electrode, which produces oscillations as individual drops grow and fall over a lifetime of 2–4 s. The potential is scanned slowly in a linear fashion, so the abscissa is both a time axis and a potential axis. The top edge of the envelope is defined by the set of currents at the end of each drop’s life; thus, it is a sampled-transient voltammogram where sampling occurs at the drop lifetime. The downward spikes occur at the fall of each drop, followed by regrowth of the next drop.

peroxide with a 2e change, manifested by a wave near −0.1 V vs. SCE. A second 2e step takes hydrogen peroxide to water. At potentials less extreme than about −0.5 V, the second step does not occur to any appreciable extent; hence, one sees only a single wave corresponding to a diffusion-limited 2e process. At still more negative potentials, the second step begins to occur, and beyond −1.2 V, oxygen is reduced completely to water at the diffusion-limited rate. Equation 5.5.5 accurately describes the overall current in STV for a two-step process to stable products, if mO is taken as identified above and in (6.2.26a). The electroreductions of O2 , first to H2 O2 , then to H2 O, are extremely complex. The overall process leading to H2 O requires the breaking of a strong O—O bond and the addition of 4e and 4H+ . Several steps are likely to exist in the mechanism, and key electron transfers apparently involve close interaction with the electrode surface [i.e., they are inner-sphere reactions (Section 3.5.1)]. No attempt is made here to address the kinetics of oxygen reduction on the basis of Figure 6.4.1. The O2 reduction mechanism is treated in Section 15.3.1.

6.5 Chronoamperometric Reversal Techniques After the application of an initial potential step, one might wish to apply an additional step, or even a complex sequence of steps. The most common arrangement is the double-step

6.5 Chronoamperometric Reversal Techniques

Ef

E 0′ (–) E

Er

Ei

τ

0 t

Figure 6.5.1 General waveform for a double potential step experiment.

technique, in which the first step (the forward step) is used to generate a species of interest, and the second is used to examine it. The latter might be made to any potential within the working range, but is usually employed to reverse the effects of the initial step. An example is shown in Figure 6.5.1. Suppose an electrode is immersed in a solution of species O that is ′ reversibly reduced at E0 . If CR∗ = 0 and the initial potential, Ei , is much more positive than ′ E0 , no electrolysis occurs before t = 0, when the potential is changed abruptly to Ef , far more ′ negative than E0 . A cathodic Cottrell current results as species R is generated electrolytically for a period 𝜏, then the second step (the reversal step) shifts the electrode to the more positive value Er . Usually, Er is equal to Ei . The reduced form R then can no longer coexist with the electrode, and it is reoxidized to O. During the reversal step, a large anodic current flows initially, but decays away as R is consumed (Figure 6.5.2). This experiment, called double potential step chronoamperometry, is our first example of a transient reversal technique. Such methods comprise a large class, all featuring the initial generation of an electrolytic product, then a reversal of electrolysis, so that the first product is directly examined electrolytically. Reversal methods are powerful for studying complex electrode reactions, and we will have much to say about them. i

tf

0

τ

tr

t

Figure 6.5.2 Current response in double-step chronoamperometry.

291

292

6 Transient Methods Based on Potential Steps

6.5.1

Approaches to the Problem

To obtain a quantitative description of this experiment, one might first treat the forward step, then use the concentration profiles applicable at 𝜏 as initial conditions for the diffusion equations describing the reversal step. In the case outlined above, the effects of the forward step are well known (Section 6.1.1), and this direct approach is practical. More generally, however, a reversal experiment presents complex concentration profiles at the end of the forward step, and it is often simpler to rely on the principle of superposition (17, 18) to treat the reversal. We now introduce that approach to solve the present problem. The applied potential can be represented as the superposition of two signals: a constant component Ef for all t > 0 and a step component Er − Ef superimposed on the constant perturbation for t > 𝜏. Figure 6.5.3 illustrates this idea, which is written mathematically as E(t) = Ef + S𝜏 (t)(Er − Ef )

(t > 0)

(6.5.1)

where the step function, S𝜏 (t), is zero for t ≤ 𝜏 and unity for t > 𝜏. Similarly, the concentrations of O and R can be expressed as a superposition of two concentrations that may be regarded as responsive to the separate potential components: I II (x, t) + S𝜏 (t)CO (x, t − 𝜏) CO (x, t) = CO

(6.5.2)

CR (x, t) = CRI (x, t) + S𝜏 (t)CRII (x, t − 𝜏)

(6.5.3)

The boundary conditions and initial conditions for this problem are most easily formulated in terms of the actual concentrations C O (x, t) and C R (x, t). The initial situation is ∗ CO (x, 0) = CO

(6.5.4a)

CR (x, 0) = 0

(6.5.4b)

During the forward step, we have ′ CO (0, t) = CO

CR (0, t) =

(6.5.5a)

CR′

(6.5.5b)

We will treat only situations where the O/R couple is nernstian; thus, ′ CO = 𝜃 ′ CR′

(6.5.6)

where ′

𝜃 ′ = exp[nf (Ef − E0 )]

(6.5.7)

The reversal step is defined by ′′ CO (0, t) = CO

(6.5.8a)

CR′′

(6.5.8b)

CR (0, t) = E

E

Ef

E

Ef Er

0

τ

t

0

τ t

0

τ

(Er – Ef) Component I

Component II

Composite

Figure 6.5.3 A double-step waveform as a superposition of two components.

t

6.5 Chronoamperometric Reversal Techniques

and ′′ CO = 𝜃 ′′ CR′′

(6.5.9)

where ′

𝜃 ′′ = exp[nf (Er − E0 )]

(6.5.10)

At all times, the semi-infinite conditions, ∗ lim CO (x, t) = CO

(6.5.11a)

lim CR (x, t) = 0

(6.5.11b)

x→∞ x→∞

and the flux balance, (6.5.12)

JO (0, t) = −JR (0, t)

are applicable. All of these conditions, as well as the diffusion equations for O and R, are linear. An imporI , C II , C I , and C II can all be carried tant consequence is that the component concentrations CO O R R through the problem separately. Each makes a separable contribution to every condition. We can, therefore, solve individually for each component, then combine them through (6.5.2) and (6.5.3) to obtain the real concentration profiles, from which we derive the current–time relationship. These steps, which are detailed in the first edition,14 are left for the reader now as Problem 6.7 The method of superposition can succeed when linearity exists and separability of the component concentrations can be assured. Unfortunately, many electrochemical situations do not satisfy this requirement. In such cases, other methods, such as simulation, must be applied. Quasireversible electron transfer in a system with chemically stable O and R has been addressed for double-step chronoamperometry, initially on the basis of a special case (19), and subsequently in a general way yielding a series solution (20) that allows extraction of kinetic parameters from experimental data under a wide variety of conditions. 6.5.2

Current–Time Responses

Since the experiment for 0 < t ≤ 𝜏 is identical to that treated in Section 6.2.1, the current is given by (6.2.8), which is restated for the present context as 1∕2

∗ nFADO CO

(6.5.13) 𝜋 1∕2 t 1∕2 (1 + 𝜉𝜃 ′ ) From the approach outlined in the previous section, the current during the reversal step turns out to be 1∕2 ∗ {( )[ ] } nFADO CO 1 1 1 1 −ir (t) = − − (6.5.14) 1 + 𝜉𝜃 ′ 1 + 𝜉𝜃 ′′ 𝜋 1∕2 (t − 𝜏)1∕2 (1 + 𝜉𝜃 ′ )t 1∕2 A special case involves stepping in the forward phase to a potential on the diffusion plateau ′ ′ ≈ 0), then reversing to a potential on the diffusion plateau for of the reduction wave (𝜃 ≈ 0, CO ′′ ′′ reoxidation (𝜃 → ∞, CR ≈ 0). Then, (6.5.14) simplifies to if (t) =

−ir (t) =

1∕2 ∗ [ nFADO CO

𝜋 1∕2

14 First edition, pp. 178–180.

1 1 − (t − 𝜏)1∕2 t 1∕2

] (6.5.15)

293

294

6 Transient Methods Based on Potential Steps

1.0

–ir if

0.0 1.0

2.0 tr /τ

3.0

Figure 6.5.4 Working curve of −ir (tr )/if (tf ) vs. tr for tr = tf + 𝜏 as defined by (6.5.17). The system is O + ne ⇌ R, with both O and R being stable on the time scale of observation. Responses during both potential steps are diffusion-limited. ′ ≈ 0 and C ′′ ≈ 0 This relation could also have been derived under the conditions CO R without requiring nernstian behavior. It, therefore, holds also for systems with sluggish electron-transfer kinetics, provided that the potential steps are large enough to drive the kinetics to the diffusion-controlled limit. Figure 6.5.2 shows the kind of current response predicted by (6.5.13) and (6.5.14). In comparing a real experiment to the prediction, it is useful to divide a sampled value of the reversal current, −ir , by a sampled value of the forward current, if . If t f and t r are the times at which the current measurements are made, then for the purely diffusion-limited case described by (6.5.15), ( )1∕2 ( )1∕2 tf t −ir = − f (6.5.16) if tr − 𝜏 tr

If t f and t r values are selected in pairs so that t r − 𝜏 = t f , then −ir if

( )1∕2 𝜏 =1− 1− tr

(6.5.17)

When one calculates these ratios for several different values of t r , they ought to fall on the working curve shown in Figure 6.5.4. A quick reference for a stable system is that −ir (2𝜏)/if (𝜏) = 0.293. Deviations from the working curve indicate kinetic complications in the electrode reaction. For example, if species R decays to an electroinactive species, then |ir | would be smaller than predicted by (6.5.15) and the current ratio −ir /if would fall below the prediction from (6.5.17) and Figure 6.5.4. Chapter 13 covers in more detail the ways in which reversal methods can be used to characterize electrode processes coupled to homogeneous chemistry.

6.6 Chronocoulometry To this point, this chapter has concerned either i − t transients stimulated by potential steps or voltammograms constructed by sampling the transients. A very useful alternative is to integrate the current, so that one obtains the charge passed vs. time, Q(t). This chronocoulometric mode (21, 22) is often favored in place of chronoamperometry because it offers important experimental advantages: • The measured signal normally grows with time; hence, the later parts of the transient, which are most accessible experimentally and are least distorted by nonideal potential rise, offer

6.6 Chronocoulometry

better signal-to-noise ratios than the early time results. The opposite is true for chronoamperometry. • The act of integration smooths random noise on the current transients; hence, the chronocoulometric records are inherently cleaner. • Contributions to Q(t) from double-layer charging and from electrode reactions of adsorbed species can be distinguished from those due to diffusing electroreactants. An analogous separation of the components of a current transient is not generally feasible. This advantage of chronocoulometry is especially valuable for the study of surface processes. 6.6.1

Large-Amplitude Potential Step

The simplest chronocoulometric experiment is the Cottrell case discussed in Section 6.1.1. One begins with a quiescent, homogeneous solution of species O, in which a planar working electrode is held at an initial potential, Ei , where insignificant electrolysis takes place. At t = 0, the potential is shifted to Ef , which enforces a diffusion-limited current. The Cottrell equation, (6.1.12), describes the chronoamperometric response, and its integral from t = 0 gives the cumulative charge passed in reducing the diffusing reactant: 1∕2

t

Qd =

∫0

id,c (t)dt =

∗ t 1∕2 2nFADO CO

(6.6.1)

𝜋 1∕2

As shown in Figure 6.6.1, Qd rises with time, and a plot of its value vs. t 1/2 is linear. The slope ∗ , given knowledge of the others. of this plot is useful for evaluating any one of n, A, DO , or CO Equation 6.6.1 shows that the diffusional component to the charge is zero at t = 0; yet, a plot of the total charge Q vs. t 1/2 generally does not pass through the origin, because additional components of Q arise from double-layer charging and from the electroreduction of any O molecules that might be adsorbed at Ei . The charges devoted to these processes are passed very quickly compared to the slower accumulation of the diffusional component; hence, they may

6

4 Qd

Q/μC 2

Qdl + nFAΓO 0 0.0

0.1

0.2

0.3

0.4

0.5

t1/2/s1/2

Figure 6.6.1 Linear chronocoulometric plot for reduction at a Pt disk of 0.95 mM 1,4-dicyanobenzene ′ (DCB; Figure 1) in benzonitrile + 0.1 M TBABF4 . T = 25 ∘ C, A = 0.018 cm2 . E 0 for DCB + e ⇄ DCB−∙ is −1.63 V vs. QRE. The actual chronocoulometric trace is the part of Figure 6.6.2 corresponding to t < 250 ms. [Data courtesy of R. S. Glass.]

295

296

6 Transient Methods Based on Potential Steps

be included by adding two time-independent terms: 1∕2

Q=

∗ t 1∕2 2nFADO CO

𝜋 1∕2

+ Qd1 + nFAΓO

(6.6.2)

where Qdl , is the capacitive charge and nFAΓO is the faradaic component given to the reduction of the surface excess, ΓO (mol/cm2 ), of adsorbed O. The intercept of Q vs. t 1/2 is Qdl + nFAΓO . A common application of chronocoulometry is to evaluate surface excesses of electroactive species; hence, it is of interest to separate these two interfacial components. Doing so reliably usually requires other experiments, such as those described in the next section. An approximate value of nFAΓO can be had by comparing the intercept of the Q − t 1/2 plot obtained for a solution containing O, with the “instantaneous” charge passed in the same experiment performed with supporting electrolyte only. The latter quantity is Qdl for the background solution, and it may approximate Qdl for the complete system. However, these two capacitive components will not be identical if O is adsorbed, because adsorption influences the interfacial capacitance (Chapter 14). 6.6.2

Reversal Experiments Under Diffusion Control

Chronocoulometric reversal experiments are nearly always designed with step magnitudes large enough to ensure that any electroreactant diffuses to the electrode at its maximum rate. A typical experiment begins exactly like the one described just above. At t = 0, the potential is shifted from Ei to Ef , where O is reduced at the diffusion-limited rate. That potential is held for a fixed period, 𝜏, then the electrode is returned to Ei , where R is reconverted to O, again at the diffusion-limited rate. This sequence is a special case of the general reversal experiment considered in Section 6.5, and we have already found, in (6.5.15), the chronoamperometric response for t > 𝜏: 1∕2 ∗ [ ] −nFADO CO 1 1 − (6.6.3) ir = 𝜋 1∕2 (t − 𝜏)1∕2 t 1∕2 Before 𝜏, the experiment is the same as that treated just above; hence, the cumulative charge devoted to the diffusional component after 𝜏 is 1∕2

Qd (t > 𝜏) =

∗ 𝜏 1∕2 2nFADO CO

𝜋 1∕2

t

+

∫𝜏

ir dt

(6.6.4)

or 1∕2

∗ 2nFADO CO

[t 1∕2 − (t − 𝜏)1∕2 ] (6.6.5) 𝜋 1∕2 This function declines with increasing t, because the second step withdraws charge injected in the forward step. The overall Q − t curve would resemble that of Figure 6.6.2, and one could expect a linear plot of Q(t > 𝜏) vs. [t 1/2 − (t − 𝜏)1/2 ]. There is no net capacitive component in the total charge after time 𝜏, because the net potential change is zero. Although Qdl was injected with the rise of the forward step, it was withdrawn upon reversal. Now consider the quantity of charge removed in the reversal, Qr (t > 𝜏), which experimentally is the difference Q(𝜏) − Q(t > 𝜏), as depicted in Figure 6.6.2. Qd (t > 𝜏) =

1∕2

Qr (t > 𝜏) = Qdl +

∗ 2nFADO CO

𝜋 1∕2

[𝜏 1∕2 + (t − 𝜏)1∕2 − t 1∕2 ]

(6.6.6)

6.6 Chronocoulometry

6

Q(𝜏)

Qr(t > 𝜏)

4 Q/μC)

𝜏= 250 ms

2

0

0

100

200

t/ms

300

400

500

Figure 6.6.2 Chronocoulometric response for a double-step experiment performed on the system of Figure 6.6.1. Steps were to potentials enforcing diffusion-limited electrolysis. [Data courtesy of R. S. Glass.]

where the bracketed factor is usually denoted as 𝜃. For simplicity, we consider the case in which R is not adsorbed. A plot of Qr (t > 𝜏) vs. 𝜃 should be linear and possess the same slope magnitude seen in the forward chronocoulometric plot. Its intercept is Qdl . The pair of graphs depicting Q(t ≤ 𝜏) vs. t 1/2 and Qr (t > 𝜏) vs. 𝜃 in the manner of Figure 6.6.3 (often called an Anson plot) is extremely useful for quantifying electrode reactions of adsorbed species. In the case we considered above, where O is adsorbed and R is not, the difference between the intercepts is simply nFAΓO . Ideally, this difference cancels Qdl and leaves only the net faradaic charge devoted to the adsorbed O. The cancellation should be precise if the second step quickly (relative to 𝜏) restores the interfacial conditions that existed before the first step was made. For the situation under discussion, that requirement would include full restoration of the layer of adsorbed O early in the second step. In practice, one does not always observe precise cancellation of Qdl in cases where adsorption of O or R is not suspected (as in Figure 6.6.3). Perhaps there is indeed some adsorption (despite expectation to the contrary), or perhaps the experiment produces a residual change in the interfacial structure of the electrode or its microscopic area. If O is adsorbed at the initial potential and R is adsorbed at the step potential, then the difference of intercepts would be nFA(ΓO − ΓR ), assuming precise cancellation of Qdl . For details of interpretation concerning various possible situations, the original literature should be consulted (21, 23, 24). Equations 6.6.3, 6.6.5, and 6.6.6 are all based on the assumption that the concentration profiles at the start of the second step are exactly those that would be produced by an uncomplicated Cottrell experiment. In other words, we have regarded those profiles as being unperturbed by the additions or subtractions of diffusing material that are implied by adsorption and desorption. This assumption obviously cannot hold strictly. Rigorous treatments are available (23) showing how chronocoulometric data can be corrected for such effects (Section 17.3.1). Reversal chronocoulometry can also be useful for characterizing the homogeneous chemistry of O and R. The diffusive faradaic component Qd (t) is especially sensitive to solution-phase reactions (24, 25), and it can be conveniently separated from the overall charge Q(t) as described above. If both O and R are stable and are not adsorbed, then Qd (t) is fully described by (6.6.1) and (6.6.5). If we divide Qd (t) by the Cottrell charge passed in the forward step, Qd (𝜏), the resulting

297

298

6 Transient Methods Based on Potential Steps

6

4

Q(t < 𝜏) vs. t1/2

2 Q/μC 0

0.0

0.1

0.2

0.3

0.4

0.5

(t1/2 or 𝜃)/s1/2

2 Qr(t > 𝜏) vs. 𝜃 4

Figure 6.6.3 Linear chronocoulometric plots for data from the trace shown in Figure 6.6.2. For Q(t ≤ 𝜏) vs. t1/2 , the slope is 9.89 μC/s1/2 and the intercept is 0.79 μC. For Qr (t > 𝜏) vs. 𝜃, the slope is 9.45 μC/s1/2 and the intercept is 0.66 μC. [Data courtesy of R. S. Glass.]

ratio takes a simple form: Qd (t ≤ 𝜏) ( t )1∕2 = Qd (𝜏) 𝜏 ]1∕2 Qd (t > 𝜏) ( t )1∕2 [( t ) −1 = − , Qd (𝜏) 𝜏 𝜏

(6.6.7) (6.6.8)

∗ . For a given which is independent of the specific experimental parameters n, A, DO , and CO value of t/𝜏, the charge ratio is even independent of 𝜏. Equations 6.6.7 and 6.6.8 describe the essential shape of the chronocoulometric response for a stable system. If the experimental results for any real system do not adhere to this shape function, then chemical complications are indicated. For a quick examination of chemical stability, one can evaluate the charge ratio Qd (2𝜏)/Qd (𝜏) or, alternatively, the ratio [Qd (𝜏) − Qd (2𝜏)]/Qd (𝜏). For a stable system, they are 0.414 and 0.586, respectively. In contrast, consider the O/R couple in which R rapidly decays in solution to electroinactive X. In the forward step, O is reduced at the diffusion-controlled rate and (6.6.7) is obeyed. However, (6.6.8) is not followed, because species R is partially lost in the solution-phase decay. The ratio Qd (t > 𝜏)/Qd (𝜏) falls less rapidly than for a stable system, and in the limit of completely effective conversion of R to X, no reoxidation is seen at all. Then Qd (t > 𝜏)/Qd (𝜏) = 1 for all t > 𝜏. Various other kinds of departure from (6.6.7) and (6.6.8) can be observed. See Chapter 13 for a discussion concerning the diagnosis of prominent homogeneous reaction mechanisms. The large body of chronoamperometric theory for systems with coupled chemistry can be used

6.6 Chronocoulometry

directly to describe chronocoulometric experiments, because there are no differences in fundamental assumptions. The only differences are that the response is integrated in chronocoulometry and that the chronocoulometric experiment manifests more visibly the contributions from double-layer capacitance and electrode processes of adsorbates. 6.6.3

Effects of Heterogeneous Kinetics

In the foregoing discussion, we examined only situations in which electroreactants arrive at the electrode at the diffusion-limited rate. At the potentials required to enforce that condition, the heterogeneous rate parameters are experimentally inaccessible. On the other hand, if one wished to evaluate those parameters, it would be useful to obtain a chronocoulometric response governed wholly or partially by the interfacial charge-transfer kinetics. That goal can be reached by using a step potential in the rising portion of the sampled-transient voltammogram corresponding to the time scale of interest, which must be sufficiently short that electrode kinetics influence current flow for a significant period. The appropriate experiment involves a step at t = 0 from an initial potential where electrolysis does not occur, to potential E, where it does. Let us consider the special case in which species O ∗ and species R is initially absent (26, 27). In Section 6.3.1, is initially present at concentration CO we found that the current transient for quasireversible electrode kinetics was given by (6.3.7). Integration from t = 0 provides the chronocoulometric response: ] ∗ [ nFAkf CO 2Ht 1∕2 2 1∕2 Q(t) = exp(H t)erfc(Ht ) + −1 (6.6.9) H2 𝜋 1∕2 1∕2

1∕2

where H = (kf ∕DO ) + (kb ∕DR ). For Ht 1/2 > 5, the first term in the brackets is negligible compared to the others; hence, (6.6.9) takes the limiting form: ( 1∕2 ) 2t 1 ∗ Q(t) = nFAkf CO − 2 (6.6.10) H𝜋 1∕2 H A plot of the faradaic charge vs. t 1/2 should, therefore, be linear and display a negative intercept on the Q axis and a positive intercept on the t 1/2 axis. The latter involves a shorter extrapola1∕2 tion, as shown in Figure 6.6.4; hence, it can be evaluated more precisely. Designating it as ti , we find H to be H=

𝜋 1∕2 1∕2

(6.6.11)

2ti

∗ ∕H𝜋 1∕2 . With H in hand, k f is found from the linear slope, 2nFAkf CO Equations 6.6.9 and 6.6.10 do not include contributions from adsorbed species or double-layer charging. For accurate application of this treatment, one must correct for those terms or render them negligibly small compared to the diffusive component of the charge. In practice, it is difficult to measure kinetic parameters in this way, so the method is not widely practiced. The principal value in considering the problem is in the insight that it provides to the origin of negative intercepts in chronocoulometry, which are rather common, especially with modified electrodes (Chapter 17). The lesson here is that a rate limitation on the delivery of charge to a diffusing or migrating species produces an intercept smaller than predicted in Sections 6.6.1 and 6.6.2. A negative intercept clearly indicates such a rate limitation. It may be due to sluggish interfacial kinetics, as treated here, but it may also be from other sources, including slow establishment of the potential because of uncompensated resistance. Using a more extreme step potential can ameliorate this behavior if it is not the object of study.

299

300

6 Transient Methods Based on Potential Steps

Figure 6.6.4 Chronocoulometric response for 10 mM Cd2+ in 1 M Na2 SO4 . The working electrode was a hanging mercury drop with A = 2.30 × 10−2 cm2 . The initial potential was −0.470 V vs. SCE, and the step potential was −0.620 V. The slope of the plot is 1∕2 3.52 μC/ms1/2 and ti = 5.1 ms1/2 . [Christie, Lauer, and Osteryoung (27). © 1964, Elsevier.]

90 80 70 60 50 Q/μC 40 30 20 10 0

5

10

15

20

25

30

t1/2/ms1/2

6.7 Cell Time Constants at Microelectrodes In Section 1.6.4, we found that changing the potential (hence, changing the double-layer charge) involves the uncompensated resistance, Ru , and the double-layer capacitance, C d . The cell time constant, Ru C d , controls the time required to charge the electrode to the desired potential (Section 6.8.2), so it sets the lower limit of the experimental time scale. The size of the electrode is the most important factor determining the cell time constant. Let us consider a disk-shaped working electrode at which the interfacial capacitance per unit area,Cd0 , is in the typical range of 10–50 μF/cm2 . Then, Cd = 𝜋r02 Cd0

(6.7.1)

With a radius, r0 , of 1 mm, C d is 0.3–1.5 μF, but for r0 = 1 μm, C d is 6 orders of magnitude smaller, only 0.3–1.5 pF. The uncompensated resistance also depends on the electrode size, although in a less transparent way (Section 5.6.1). From (6.7.1) and (5.6.2), one can express the cell time constant as Ru Cd =

r0 Cd0 4𝜅

(6.7.2)

where 𝜅 is the conductivity of the electrolyte. Even though Ru rises inversely as r0 becomes smaller, C d decreases with the square; hence, Ru C d scales with r0 . This is an important result indicating that smaller electrodes can provide access to shorter time domains. Consider, for example, the effect of electrode size in a system with Cd0 = 20 μF/cm2 and 𝜅 = 0.013 Ω−1 cm−1 (characteristic of 0.1 M aqueous KCl at ambient temperature). With r0 = 1 mm, the cell time constant is about 30 μs, and the lower limit of time scale in step experiments, including time for logging data, would be on the order of 1 ms. This conclusion is consistent with the general experience that experiments with electrodes of conventional (millimeter) size need to be limited to the millisecond time domain or longer.

6.7 Cell Time Constants at Microelectrodes

However, with r0 = 5 μm, the cell time constant becomes about 170 ns, so that the lower limit of time scale drops to a few μs. Before UMEs were understood and readily available, the microsecond regime was very difficult to reach in electrochemical studies. However, UMEs have opened it to relatively convenient investigation (28–31), not only by potential step methods, but also by other experimental approaches covered later in this book. UMEs even allow access to the nanosecond domain, although not with routine ease or convenience. To reach it, one must reduce the electrode size further and employ highly conducting electrolytes. Specialized instrumentation is also needed (32–34), because commercial instruments generally do not have adequate bandwidth (Section 16.9). In addition, one must contend with stray capacitance associated with the working electrode. In Section 2.2, we learned that the potential of a conducting phase is determined by surface charges on that phase and on adjacent phases. In Section 1.6, we assumed implicitly that the double-layer charge at the electrode/solution interface alone determines the potential of the working electrode. Actually, it is the totality of charge, qtotal , on all surfaces of the working electrode that determines the potential. All must be charged when the potential of the working electrode is varied, and the associated capacitances limit the speed with which the potential can be altered at the electrode/solution interface. We can think about qtotal in two parts: (a) the double-layer charge at the electrode/solution interface, q, and (b) the “stray” charge, qstray , at all other surfaces of conductors making up the working electrode or connections to it. Normally, these “other surfaces” would at least include • The sides of the working electrode material (e.g., Pt wire) as it passes through the insulator at the tip of the assembly (typically involving an interface with glass or a polymer). • The sides of that material as it might continue into the interior of the working electrode assembly (often involving an interface with air). • The surfaces of any material (e.g., solder) used to join the working electrode material to an electrical lead (e.g., a Cu wire). A lead like this is usually included in the working electrode assembly to allow for convenient external connection to the potentiostat. • The surfaces of the electrical connection from the working electrode assembly to the potentiostat (e.g., surfaces inside the potentiostat cable). • The surfaces of conducting elements in cables and switches inside the electronics. Ideally, all surfaces except that at the electrochemical interface itself are in contact with unpolarizable insulators (including air or vacuum). By analogy to (1.6.11), qtotal = q + qstray = Cd (E − Ez ) + Cstray (E − Ez )

(6.7.3)

Thus, the functional capacitance of the working electrode is C d + C stray . For ordinary electrochemical situations, the high polarizability of the electrode/solution interface causes q to be much larger than qstray ; therefore, qtotal ≈ q and C stray ≪ C d . In such cases, (1.6.11) is valid, and the way we have generally thought about charging at the working electrode is appropriate. However, the picture changes as the size of the electrode/electrolyte interface becomes much smaller. As C d shrinks with r02 , it can readily fall to the point where C stray can no longer be neglected. Then, the cell time constant is no longer given by (6.7.2), but becomes Ru C =

r0 Cd0 4𝜅

+

Cstray 4𝜋𝜅r0

(6.7.4)

where the first term is Ru C d , given by (6.7.2), and the second is Ru C stray . While Ru C d falls with r0 , Ru C stray rises. The cell time constant shows a minimum at the radius where C d = C stray , as shown in Figure 6.7.1.

301

6 Transient Methods Based on Potential Steps

–6 log (time constant/s)

302

–7 Ru(Cd + Cstray)

–8

–9

RuCd

–10 –2

RuCstray –1

0 log r0 /μm

1

2

Figure 6.7.1 Time constants vs. radius for a disk microelectrode. The overall cell time constant (solid curve) is the sum of Ru C d and Ru C stray . Here, 𝜅 = 0.75 Ω−1 cm−1 ; Cd0 = 150 μF/cm2 ; and C stray = 1 pF. The conductivity is characteristic of 5 M HClO4 . The minimum cell time constant for these conditions is about 5 ns, found in the range of r0 = 0.5 μm.

The origins and effects of stray capacitance have been examined in detail (28, 32–34). In one study (33), cell time constants for Pt disk UMEs in 5 M HClO4 remained well behaved (proportional to r0 with zero intercept) down to about 12 ns for r0 = 2.5 μm. The results were consistent with the model system depicted in Figure 6.7.1. With the ability to obtain cell time constants smaller than a few tens of nanoseconds, it should be possible to carry out step experiments on time scales of a few hundred nanoseconds, and perhaps shorter. However, experimental work in this range is complicated by difficult measurement problems and by fundamental issues related to the availability of molecules when the diffusion layer thickness becomes small (35–37). For a potential step lasting 100 ns, (2Dt)1/2 is only about 12 nm; hence, relatively few solute molecules in a conventional electrolyte solution are close enough to the electrode to react if they must reach it by diffusion. Obtaining an adequate signal can be an issue (Problem 6.13). Detection of electrochemical processes by conversion to photons has been useful (38). Better prospects exist for systems in which the electroreactant species are attached to the electrode, so that they are present in large numbers at the surface, and diffusion is not required (33, 39). Experiments with step widths in the range of 500 ns have been conducted with such systems. An example is shown in Figure 6.7.2, which relates to the electrode reaction, 2,6-AQDS + 2H+ + 2e ⇌ 2,6-DHADS

(6.7.5)

where 2,6-AQDS is 2,6-anthraquinonedisulfonate, and 2,6-DHADS is the 2,6-disulfonate of 9,10-dihydroxyanthracene (structures in Figure 1). Both 2,6-AQDS and 2,6-DHADS produce well-behaved adsorbate layers on an Hg surface, and they can be interconverted electrolytically. The double-step experiment of Figure 6.7.2 first converts the monolayer of 2,6-AQDS into a monolayer of 2,6-DHADS, then reverses the initial conversion. Because both substances are bound and do not diffuse, the current decays exponentially in each step (after the charging current falls off ). The rate constants depend on the potentials to which the steps are made and on the proton concentration in solution. From kinetic measurements taken as functions of those variables, a detailed kinetic picture was developed (33). Chapters 14 and 17 will cover electrochemical activity of surface layers in greater detail.

6.8 Lab Note: Practical Concerns with Potential Step Methods

Figure 6.7.2 (a) Current transient for 2,6-AQDS adsorbed on a Pt-based Hg UME (r0 = 1 μm). Solution was 0.1 mM 2,6-AQDS in 1.8 M H2 SO4 . A 600-ns forward step was made from 0.270 to −0.122 V vs. Ag/AgCl, then the potential was returned to 0.270 V. (b) log|i| vs. time. The linear part of the forward-step log plot corresponds to a rate constant of 8.5 × 107 s−1 for the reduction process and a lifetime of 113 ns for 2,6-AQDS. The cell time constant was 24 ns. [From Xu (33).]

200 100 i/μA 0 –100 –200 0.0

1.0 t/μs (a)

2.0

1.0 t/μs

2.0

–3.52 –4.32 log |i|/A –5.12 –5.92 –6.72 –7.52 0.0

(b)

Most fast electrochemical studies of dissolved solutes have been carried out by cyclic voltammetry, rather than by chronoamperometry, so we will pick up this subject again in the next chapter.

6.8 Lab Note: Practical Concerns with Potential Step Methods 6.8.1

Preparation of the Electrode Surface at a Microelectrode

Working electrode surfaces need attention, not only while fabricating a new electrode, but also prior to a new set of experiments, and, sometimes, even during an experimental sequence. Surface roughness, chemical state, and contamination are the important considerations. To obtain meaningful and reproducible data, one must assure that the physical and chemical state of the surface is known and remains consistently the same in a series of measurements. This is especially true if the focus is on heterogeneous kinetics, where small changes in interfacial structure can greatly alter the rate of an electron-transfer reaction. The preparation of an electrode surface is very specific to the material being used, as well as to the intended application of the electrode. For instance, a gold microelectrode surface may simply require polishing, while the preparation of a CdTe photoelectrode requires very specific chemical treatments. Similarly, the preparation of a Pt electrode for investigating a dissolved redox molecule often requires less attention to surface condition than preparing the same Pt electrode for studying an inner-sphere reaction involving an adsorbed intermediate, where surface structure plays a critical role in the overall reaction. In general, when beginning

303

304

6 Transient Methods Based on Potential Steps

any new investigation, it is important to carefully read the literature describing electrode surface preparations and to be able to identify the electrochemical characteristics of a well-prepared surface. Surface roughness (Section 6.1.5) is affected by polishing and by any electrochemical process that causes electrodissolution or electrodeposition of the electrode material (e.g., the dissolution or plating of Ag at an Ag electrode). Oxidation and reduction of a metallic species at the surface of a bulk metal usually happen at favored sites, so either process generally increases roughness. Physical polishing by traditional means can reduce roughness, but will normally leave polishing striations comparable in size with the particle diameter used in polishing. Physical polishing can be avoided altogether when using metal layers that are vapor deposited onto a smooth surface, but these electrodes are generally for one-time applications. The term “chemical state” relates to the chemical form of the electrode material at the active surface, as well as the controllable deposition of species unrelated to the electrode material, perhaps by specific adsorption (Chapter 14) or designed covalent binding (Chapter 17). Many metals, including Pt, undergo the anodic formation of oxide layers at positive potentials (Chapter 14). Often these layers can be re-reduced at a suitably negative potential. Adsorption can involve an electroreactant, an electroproduct, an intermediate in the electrode reaction, a component of the supporting electrolyte, a species deliberately added to modify the surface, or a contaminant. The surfaces of carbon electrodes are susceptible to oxidation processes that introduce functional sites, e.g., acid groups, which may alter the reactivity of the electrode during the experiment. In general, “contamination” is meant here as any process that modifies the electrode surface undesirably and uncontrollably, usually by unknown species. Electrodes are often fouled by substances present in the solvent–electrolyte system (e.g., proteins in clinical samples) or by side-products (sometimes insoluble polymers) produced in complex electrode reactions. Airborne contaminants strongly adsorb on metal surfaces; thus, it is generally wise to avoid prolonged exposure of a polished and precleaned electrode to laboratory air prior to immersing it in the electrolyte solution. Contamination of the electrode surface is reduced by using highly purified solvents and electrolytes and by ensuring that species produced at the counter and reference electrodes do not find their way to the working electrode. Trace quantities of metal ions from dissolution of reference electrode materials (e.g., AgCl− 2 from an Ag/AgCl electrode) can deposit on the working electrode and cause large unintended changes in the kinetics of electron-transfer reactions. Larger UMEs (characteristic dimension ≥ 50 nm) can be mechanically polished. This is normally done during fabrication and as experience dictates (often before each day of experiments, sometimes less frequently). Coarse polishing during fabrication may be done with emery paper, but later stages are accomplished with alumna or diamond paste. Finishing is usually with a process based on abrasive particles 50 nm–1 μm in diameter. After the multistep polishing phase that often completes the fabrication process, routine maintenance of the electrode normally involves only the final polishing step used in fabrication. Smaller UMEs, especially toward the lower end of the practical range, are generally not robust enough to be mechanically polished. Besides, the utility of polishing electrodes with particles larger than the electrode radius is questionable. More elaborate surface preparations, such as focused ion beam milling, can be employed to prepare a very smooth and clean metal UME. Often, very small UMEs are made for one-time use, because re-cleaning the surface is impractical. Historically, Pt electrodes have been conditioned by applying two or three cycles of alternating 1-s steps between the positive and negative background limits in an aqueous electrolyte, ending on the negative limit. This procedure has two main effects: First, it repeatedly grows and

6.8 Lab Note: Practical Concerns with Potential Step Methods

re-reduces layers of PtO and PtO2 , creating a fresh Pt surface as the process ends on the negative side. Second, the background processes generate OH∙, which is an active chemical scavenger of contaminants. This procedure is usable for larger UMEs (with a critical dimension above 1 μm), but smaller UMEs can be irreparably damaged. As an alternative, it is practical to condition a Pt working electrode simply by holding it for several seconds at a potential where O2 is reduced in aqueous solution. This step reduces surface oxides and generates OH∙ near the surface, which acts to clean the surface. Tiny UMEs, at the lower limit of practical size, can hold up to this procedure, which also has the advantage of being compatible with a wide range of electrode materials. 6.8.2

Interference from Charging Current

Transient techniques normally involve changes in the working electrode’s potential with time. Since any potential change requires that the charge on the double layer be altered, a charging current, ic , must flow (Section 1.6.4). This current is added to any faradaic current that might be passed at the same time, so it represents an interference to the measurement normally sought. For potential step techniques, the charging current passes just after each step edge in Eappl (the voltage difference between working and reference electrodes, as applied by the potentiostat). Let us examine Figure 6.8.1a, which shows events just after a step that initiates a Cottrell experiment. The electrode is a fairly large disk, 2.5 mm in radius, at which the cell time constant, Ru C d , is 39 μs. Section 1.6.4(a) covers double-layer charging for this situation and shows that ic falls away exponentially, as given by (1.6.17). This current is represented by the dashed curve in Figure 6.8.1a. The faradaic response is complicated by the charging process. The ideal result (the Cottrell current, id ) is given by the dotted curve. However, the measured response is not the ideal, because the actual potential of the working electrode does not change instantaneously at the step edge. The charging process establishes the potential of the working electrode, and time is required for it to reach the intended step value, as expressed in (1.6.18). For the system in Figure 6.8.1a, there is no faradaic activity in the first 20 μs, because the potential has not shifted ′ far enough negative to initiate it. By 28 μs, the potential has arrived at E0 (−0.45 V), which drives electrolysis, but not at the diffusion-controlled rate. By 65 μs, the potential has reached −0.60 V, which is negative enough to enforce diffusion control. A total of 180 μs (nearly 4Ru C d ) is required to approach the desired step potential of −0.7 V. The delayed faradaic response follows the curve shown by the crosses in Figure 6.8.1a. It lags the ideal at the shortest times for reasons just discussed, but it exceeds the ideal soon after electrolysis begins, because the delay in start-up has left more electroactive species near the electrode than would be available in the same time range in the ideal case. As the elapsed time becomes longer, the effects of the early events dissipate, and the delayed faradaic current converges on the ideal. The observed total current is the sum of ic and the delayed faradaic response, as shown in the solid curve of Figure 6.8.1a. For the first 80 μs, ic dominates the total current, and it contributes appreciably for about 100 μs more. At about 5Ru C d , ic has faded away to the point of being negligible. Data logging in chronoamperometry or STV must be deferred until double-layer charging no longer interferes, so the step width must be much be much larger than 5Ru C d . Commonly, it is at least 50Ru C d , but it can arbitrarily longer. Figure 6.8.1b is a logarithmic display of the total current over a 200-ms time range, such as might be used experimentally (more than 5000Ru C d for the system being discussed). Data might be taken at equal intervals (e.g., every 1 ms) from 1–200 ms, or might only be sampled over a small time period near the end of the step (e.g., in

305

6 Transient Methods Based on Potential Steps

60 50

i/mA

40 30 20 10 0

0

50

100 t/μs

150

200

150

200

(a) 5.0 4.5 log i/μA

306

4.0 3.5 3.0 2.5 2.0 1.5 1.0

0

50

100 t/ms (b)

′

Figure 6.8.1 Current after a step in E appl from −0.2 to −0.7 V in a system with E 0 = − 0.45. The working ∗ = 1 mM, C ∗ = 0, A = 0.196 cm2 , R = 10 Ω, electrode is a disk with r0 = 0.25 cm. n = 1, DO = 10−5 cm2 /s, CO u R C d = 3.9 μF, Ru C d = 39 μs. (a) Components of the current at times shorter than 5Ru C d . Solid curve, total current. Dashed curve, ic . Dotted curve, Cottrell current (ideal faradaic transient). Crosses, delayed faradaic transient. (b) Total current over a typical experimental timescale. Large dot on vertical axis marks the current maximum at t = 0.

the last few ms). The charging-current spike at the beginning exceeds the currents of interest by 1–3 orders of magnitude, clearly demonstrating why current-sampling methods are needed to exclude it. As we proceed through the next several chapters, interference from double-layer charging and means for defeating it will be a recurring theme.

6.9 References F. G. Cottrell, Z. Physik, Chem., 42, 385 (1902). H. A. Laitinen and I. M. Kolthoff, J. Am. Chem. Soc., 61, 3344 (1939). H. A. Laitinen, Trans. Electrochem. Soc., 82, 289 (1942). K. Aoki and J. Osteryoung, J. Electroanal. Chem., 122, 19 (1981). D. Shoup and A. Szabo, J. Electroanal. Chem., 140, 237 (1982). K. Aoki and J. Osteryoung, J. Electroanal. Chem., 160, 335 (1984). (a) W. Hyk, M. Palys, and Z. Stojek, J. Electroanal. Chem., 415, 13 (1996); (b) W. Hyk and Z. Stojek, J. Electroanal. Chem., 422, 179 (1997). 8 A. Szabo, D. K. Cope, D. E. Tallman. P. M. Kovach, and R. M. Wightman, J. Electroanal. Chem., 217, 417 (1987). 9 K. Aoki, K. Tokuda, and H. Matsuda, J. Electroanal. Chem., 225, 19 (1987). 1 2 3 4 5 6 7

6.10 Problems

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

C. P. Winlove, K. H. Parker, and R. K. C. Oxenham, J. Electroanal. Chem., 170, 293 (1984). G. Denuault, M. Mirkin, and A. J. Bard, J. Electroanal. Chem., 308, 27 (1991). R. Woods, Electroanal. Chem., 9, 1 (1976). T. Gueshi, K. Tokuda, and H. Matsuda, J. Electroanal. Chem., 89, 247 (1978). V. P. Menon and C. R. Martin, Anal. Chem., 67, 1920 (1995). J. Tomeš, Coll. Czech. Chem. Commun., 9, 12, 81, 150, (1937). C. K. Mann and K. K. Barnes, “Electrochemical Reactions in Nonaqueous Solvents,” Marcel Dekker, New York, 1970, Chap. 11. T. Kambara, Bull. Chem. Soc. Jpn., 27, 523 (1954). D. D. Macdonald, “Transient Techniques in Electrochemistry,” Plenum, New York, 1977. W. M. Smit and M. D. Wijnen, Rec. Trav. Chim., 79, 5 (1960). D. H. Evans and M. J. Kelly, Anal. Chem., 54, 1727 (1982). F. C. Anson, Anal. Chem., 38, 54 (1966). G. Inzelt in “Electroanalytical Methods,” 2nd ed., F. Scholz, Springer, Berlin, 2010, Chap. II.4. J. H. Christie, R. A. Osteryoung, and F. C. Anson, J. Electroanal. Chem., 13, 236 (1967). J. H. Christie, J. Electroanal. Chem., 13, 79 (1967). M. K. Hanafey, R. L. Scott, T. H. Ridgway, and C. N. Reilley, Anal. Chem., 50, 116 (1978). J. H. Christie, G. Lauer, R. A. Osteryoung, and F. C. Anson, Anal. Chem., 35, 1979 (1963). J. H. Christie, G. Lauer, and R. A. Osteryoung, J. Electroanal. Chem., 7, 60 (1964). R. M. Wightman and D. O. Wipf, Acc. Chem. Res., 23, 64 (1990). J. Heinze, Angew. Chem. Int. Ed. Engl., 32, 1268 (1993). R. J. Forster, Chem. Soc. Rev., 23, 289 (1994). C. Amatore in “Physical Electrochemistry.” I. Rubenstein, Ed., Marcel Dekker, New York, 1995, Chap. 4. C. Amatore, C. Lefron, and F. Pfluger, J. Electroanal. Chem., 270, 43 (1989). C. Xu, “Fast Electrochemistry of Surface Monolayers on Ultramicroelectrodes,” Ph. D. Thesis, University of Illinois at Urbana-Champaign, 1992. C. Amatore and E. Maisonhaute, Anal. Chem., 77, 303A (2005). R. Morris, D. J. Franta, and H. S. White, J. Phys. Chem., 91, 3559 (1987). J. D. Norton, H. S. White, and S. W. Feldberg, J. Phys. Chem., 94, 6772 (1990). C. P. Smith and H. S. White, Anal. Chem., 65, 3343 (1993). M. M. Collinson and R. M. Wightman, Science, 268, 1883 (1995). R. J. Forster and L. R. Faulkner, J. Am. Chem. Soc., 116, 5444, 5453 (1994).

6.10 Problems 6.1

Fick’s law for diffusion to a spherical electrode of radius r0 is given by (6.1.15). Solve that equation using the boundary conditions given in (6.1.16)–(6.1.18), showing that the current follows (6.1.19). [Hint: By making the substitution v(r, t) = rC(r, t) in Fick’s equation and in the boundary conditions, the problem becomes essentially the same as that for linear diffusion.]

6.2 Given n = 1, C * = 1.00 mM, A = 0.02 cm2 , and D = 10−5 cm2 /s, calculate the current for diffusion-controlled electrolysis at (a) a planar electrode and (b) a spherical electrode of the same area at t = 0.1, 0.5, 1, 2, 3, 5, and 10 s, and as t → ∞. Plot both i vs. t curves on the same graph. How long can the electrolysis proceed before the current at the spherical electrode exceeds that at the planar electrode by 10%?

307

308

6 Transient Methods Based on Potential Steps

Integrate the Cottrell equation to obtain the total charge consumed in electrolysis at any time, then calculate the value for t = 10 s. Use Faraday’s law to obtain the number of moles reacted by that time. If the total volume of the solution is 10 mL, what fraction of the sample has been altered by electrolysis? 6.3 Consider a diffusion-controlled electrolysis at a hemispherical mercury electrode protruding from a glass mantle. The radius of the mercury surface is 5 μm, and the diameter of the glass mantle is 5 mm. The electroactive species is 1 mM thianthrene (TH; Figure 1) in acetonitrile + 0.1 M TBABF4 , and the electrolysis produces the cation radical. The diffusion coefficient is 2.7 × 10−5 cm2 /s. Calculate the current at t = 0.1, 0.2, 0.5, 1, 2, 3, 5, and 10 ms, and also at 0.1, 0.2, 0.5, 1, 2, 3, 5, and 10 s. Do the same for the system under the approximation that linear diffusion applies. Plot the pairs of curves for the shortand long-time regimes. How long is the linear approximation valid within 10%? 6.4 The following measurements were made at 25 ∘ C on the reversible sampled-transient voltammogram for the reduction of a metallic complex ion to metal amalgam (n = 2):

NaX Concentration/M

0.10

0.50

1.00

E1/2 /V vs. SCE

−0.448

−0.531

−0.566

(a) Calculate the number of ligands X− associated with the metal M2+ in the complex. (b) Calculate the stability constant of the complex if E1/2 for the reversible reduction of the simple metal ion is +0.081 V vs. SCE. Assume that the diffusion coefficients for the complex ion and the metal atom are equal, and that all activity coefficients are unity. 6.5 Consider the reversible system O + ne ⇄ R in which both O and R are present initially. From Fick’s laws, derive the current–time curve for a step experiment in which the initial potential is the equilibrium potential and the final potential is any arbitrary value E. Assume that a planar electrode is used and that semi-infinite linear diffusion applies. Derive the shape of the current–potential curve that would be recorded in a sampled-transient experiment performed in the manner described here. What is the value of E1/2 ? Does it depend on concentration? 6.6 Derive the Tomeš criteria for (a) a reversible sampled-transient voltammogram, (b) a totally irreversible cathodic sampled-transient voltammogram, (c) a totally irreversible anodic sampled-transient voltammogram, and (d) a totally irreversible cathodic steady-state voltammogram. 6.7 Derive (6.5.14) and (6.5.15) from (6.5.1)–(6.5.12). 6.8 Calculate k f for the reduction of Cd2+ to the amalgam from the data in Figure 6.6.4. 6.9 Devise a chronocoulometric experiment for measuring the diffusion coefficient of Tl in mercury.

6.10 Problems

6.10 Consider the data in Figures 6.6.1–6.6.3. Calculate the diffusion coefficient of DCB. How well do the slopes of the two lines in Figure 6.6.3 bear out the expectations for a completely stable, reversible system? These data are typical for a solid planar electrode in nonaqueous media. Offer at least two possible explanations for the slight inequalities in the magnitudes of the slopes and intercepts in Figure 6.6.3. 6.11 Denuault et al. (11) [Section 6.1.4(c)] suggested that by normalizing the diffusionlimited transient current, id (t), obtained at a UME at short times, by the steady-state current, iss , one can determine the diffusion coefficient, D, without knowledge of the d,c number electrons involved in the electrode reaction, n, or the bulk concentration of the reactant, C * . (a) Why would this procedure not be suitable for a large electrode? (b) A disk UME of radius 13.1 μm is used to measure D for Ru(NH3 )3+ in an aqueous 6 polymer gel into which the working and reference electrodes can be inserted. The gel also contains KNO3 well in excess of the Ru(NH3 )3+ . The slope of id (t)/iss vs. t −1/2 6 d,c for the 1e reduction of Ru(NH3 )3+ is found to be 0.427 s1/2 . The intercept is 0.780. 6 Calculate D. (c) In the experiment in part (b), iss = 6.3 nA. What is the concentration of Ru(NH3 )3+ 6 d,c in the gel?

6.12 Suppose you have a spherical gold UME like that in Figure 5.2.1, at which you can use both steady-state and sampled-transient voltammetry to record the 1e reduction wave for 5 mM Ru(NH3 )3+ in 0.09 M phosphate buffer at pH 7.4. For this system, 6 DO = 5.3 × 10−6 cm2 /s (Table C.4). (a) Devise a method for using the limiting currents in SSV and STV (in the normal pulse voltammetric mode) to determine r0 and A. (b) The limiting currents recorded for the system identified above are 16.5 nA in SSV and 402 nA in STV. The sampling time in STV is 5 ms. Determine r0 . (c) Determine A and the fractional shielding of the total spherical area. (d) What is the diffusion length in the STV experiment in (b). Is semi-infinite linear diffusion reasonably applicable to that experiment? 6.13 Consider a 200-ns potential step at a disk UME having r0 = 1 μm. The electroactive species is at 1 mM in an aqueous electrolyte and has D = 1 × 10−5 cm2 /s. The cell time constant is 10 ns. The experiment is carried out under Cottrell conditions. (a) Assuming that at t = 0, Eappl rises to the step value in less than Ru C d , how early could data corresponding to the faradaic transient be accurately recorded? (b) What is the diffusion layer thickness corresponding to the end of the forward step? (c) Using your answer in part (b), justify the use of the Cottrell equation for planar diffusion at a 1 μm radius electrode. (d) At t = 0, how many solute molecules would be in the volume of solution extending out one diffusion layer thickness from the face of the working electrode? (e) Suppose, instead, that the electroactive species exists as a monolayer in which each molecule takes up an area of 0.8 nm × 1.2 nm. How many molecules are on the electrode at t = 0?

309

311

7 Linear Sweep and Cyclic Voltammetry We now turn to the use of a potential sweep for generating a transient response. Although potential sweeps were used in Chapter 5 to record steady-state voltammograms, care was always taken that the sweep rate was low enough to maintain a steady state (Section 5.1.3). In this chapter, we consider much faster sweeps. At conventional microelectrodes (e.g., disks with r0 between 0.1 and 10 mm), sweep rates might range from 10 mV/s to about 1000 V/s. At UMEs, the sweep rate might exceed 106 V/s. In sweep experiments, one usually records the current as a function of potential, which is equivalent to recording current vs. time. The formal name for this method is linear potential sweep chronoamperometry, but most workers call it linear sweep voltammetry (LSV).1 The corresponding reversal experiment is cyclic voltammetry (CV), a powerful diagnostic method—perhaps the most widely practiced of all electrochemical techniques.

7.1 Transient Responses to a Potential Sweep In principle, one can obtain the complete electrochemical behavior of a system through a series of steps to different potentials, with the corresponding i − t curves being recorded as described in Chapter 6. The results might be plotted as a three-dimensional i − t − E surface, as in Figure 7.1.1a. However, the accumulation and analysis of the data would be tedious, because closely spaced potential steps (perhaps 1 mV apart) would be needed for full definition. Also, it is not easy to recognize the presence of different species (e.g., to observe waves) from the set of recorded i − t curves alone. Information can be gained more efficiently in a single experiment by sweeping the potential with time and recording the i − E curve directly. This approach amounts conceptually to traversing the three-dimensional i − t − E realm (Figure 7.1.1b). One can readily grasp the essentials of the transient response. As a concrete example, consider the anthracene (An) system previously discussed in Section 6.1. The solution contains An in an aprotic solvent, such as MeCN, together with a supporting electrolyte. It is stirred before the experiment, but quiescent during the potential sweep. Figure 7.1.2a shows how the potential is changed with time, and Figure 7.1.2b illustrates the i − E response. The scan begins at an ′ initial potential, Ei , well positive of E0 for the reduction of An to its anion radical, An ∙. Only ′ a charging current [Section 1.6.4(b)] flows for a while, but when E reaches the vicinity of E0 , reduction begins and faradaic current starts to flow. As the potential becomes steadily more negative, the surface concentration of An must drop; hence, the flux to the surface increases, as ′ does the current. As the potential moves past E0 , the surface concentration drops essentially 1 This method was also once called stationary electrode polarography; however, we follow the recommended practice of reserving the term polarography for STV at a renewable mercury electrode (Chapter 8). Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

7 Linear Sweep and Cyclic Voltammetry

i

i

t

t

E

E

(a)

(b)

Figure 7.1.1 (a) A portion of the i − t − E surface for a nernstian reaction. Potential axis is in units of 60/n mV. (b) Linear potential sweep across this surface. [Reinmuth (1). © 1960, American Chemical Society.] C

i

E (–) →

312

An + e → An–∙ C*An An–∙

Ei

0

0

Ei

An

E 0′

t→

E (–) →

(a)

(b)

0

x→ (c)

Figure 7.1.2 (a) Linear potential sweep starting at E i . (b) Resulting i − E curve. (c) Concentration profiles of An and An ∙ at potentials beyond the peak.

to zero. Mass transfer of An to the surface reaches a maximum rate and then declines as the depletion of An becomes progressively greater near the electrode. The result is a peaked i − E curve like that depicted. At this point in the experiment, the concentration profiles near the electrode would be as shown in Figure 7.1.2c. Suppose the potential scan is now reversed, as in Figure 7.1.3a. Near the electrode, there is a ′ large concentration of An ∙. As the potential reapproaches, then passes, E0 , the electrochemical balance at the surface progressively favors the neutral anthracene species; thus, the An ∙ becomes reoxidized, and an anodic current flows. The reversal transient (Figure 7.1.3b) has a shape much like that of the forward transient, for essentially the same reasons. This reversal experiment, called CV, is the potential-scan analogue of double potential step chronoamperometry (Section 6.5). CV is very widely used for characterization and mechanistic diagnosis of electrode reactions (Chapter 13). In Sections 7.2–7.4, we develop the theory for LSV and CV for electrode reactions with reversible, quasireversible, and totally irreversible electrode kinetics. An analytical approach is provided, because it shows directly how the current is affected by different experimental variables, especially scan rate and concentration. However, digital simulation [Section 4.5.4(b)

7.2 Nernstian (Reversible) Systems

E (–) →

i E𝜆

An + e → An–∙

Switching potential Ei E 0′

Switching time

Ei

E𝜆

An–∙ – e → An 𝜆

0 t→

E (–) →

(a)

(b)

Figure 7.1.3 (a) Cyclic potential sweep. (b) Resulting cyclic voltammogram.

and Appendix B] is commonly used to calculate voltammograms, particularly when the overall electrode process is complicated by multistep electron transfers or coupled homogeneous reactions (Chapter 13).

7.2 Nernstian (Reversible) Systems Consider now the reaction O + ne ⇄ R taking place under conditions of semi-infinite linear diffusion in a solution initially containing only species O and a supporting electrolyte. The electrode kinetics are facile; hence, a nernstian balance always exists between the potential of the working electrode and the surface concentrations of O and R: CO (0, t) CR (0, t)

= enf [E(t)−E

0′ ]

= 𝜃(t)

(7.2.1)

where f = F/RT and the time dependence of E is explicitly recognized. The electrode is held initially at a potential Ei , where no electrode reaction occurs. At t = 0, the potential begins to follow a linear sweep at scan rate v (V/s). Treatment of this system begins with the general formulation (Section 4.5.2), encompassing the diffusion equations, initial conditions, and semi-infinite conditions for both O and R, plus the flux balance. In Laplace space, these seven equations distill into two expressions (Section A.1.6) that we now use as points of departure. For CR∗ = 0, they are ∗ CO

+ A(s)e−(s∕DO ) s 1∕2 C R (x, s) = −𝜉A(s)e−(s∕DR ) x

C O (x, s) =

1∕2 x

(7.2.2) (7.2.3)

where A(s) is independent of x, and 𝜉 = (DO /DR )1/2 . The one remaining boundary condition, allowing a solution, is the nernstian balance, (7.2.1). 7.2.1

Linear Sweep Voltammetry

LSV was first treated successfully by Randles (2) and Ševˇcík (3), but the development and notation below follow the later work of Nicholson and Shain (4).

313

314

7 Linear Sweep and Cyclic Voltammetry

(a) Solution of the Mass-Transfer Problem

We begin with (7.2.2) and (7.2.3), without need for the last boundary condition, (7.2.1). From (4.4.21), the Laplace transform of the current can be written as [ ] 𝜕C O (x, s) i(s) = nFADO (7.2.4) 𝜕x x=0

From (7.2.2), one can evaluate the derivative in (7.2.4), leading, after rearrangement, to A(s) = 1∕2 −i(s)∕nFADO s1∕2 . Thus, (7.2.2) becomes, for x = 0, C O (0, s) =

∗ CO

s

−

i(s)

(7.2.5)

1∕2

nFADO s1∕2

which one can invert using the convolution theorem (Section A.1.3) to obtain2 ∗ − [nFA(𝜋D )1∕2 ]−1 CO (0, t) = CO O

t

∫0

i(𝜏)(t − 𝜏)−1∕2 d𝜏

(7.2.6)

By letting i(𝜏) nFA (7.2.6) becomes

(7.2.7)

f (𝜏) =

∗ − (𝜋D )−1∕2 CO (0, t) = CO O

t

∫0

f (𝜏)(t − 𝜏)−1∕2 d𝜏

(7.2.8)

From (7.2.3), one can obtain a similar expression for C R (0, t): CR (0, t) = (𝜋DR )−1∕2

t

∫0

f (𝜏)(t − 𝜏)−1∕2 d𝜏

(7.2.9)

Equations 7.2.8 and 7.2.9 flow directly from the general formulation without any assumption about electrode kinetics or technique; hence, they are generally applicable to any system involving stable, soluble O and R, semi-infinite linear diffusion, and CR∗ = 0. Now we express the time-dependent potential as E(t) = Ei − v t

(t > 0)

(7.2.10)

thus, the last boundary condition, (7.2.1), can be rewritten as CO (0, t) CR (0, t)

= 𝜃i e−𝜎t = 𝜃i S(t) 0′

(7.2.11)

where 𝜃i = enf (Ei −E ) , S(t) = e–𝜎t , and 𝜎 = nfv. In the treatment of step experiments in Chapter 6, we were able to resolve problems in Laplace space, reaching simple, explicit expressions for the transform of the current, i(s), which could be inverted to obtain the i − t responses. In those cases, the last boundary condition was C O (0, t) = 𝜃C R (0, t), which could be readily transformed to C O (0, s) = 𝜃C R (0, s) because 𝜃, being independent of time, could be factored from the Laplace integral. We are not so fortunate in the treatment of sweeps, where 𝜃 depends on time. Instead, we must apply the final boundary condition without conversion to Laplace space. 2 In the integral, 𝜏 is not a specific time, but a dummy variable, lost when the definite integral is evaluated.

7.2 Nernstian (Reversible) Systems

Using (7.2.8) and (7.2.9) to substitute for the surface concentrations in (7.2.11), we obtain ∗ t CO f (𝜏)(t − 𝜏)−1∕2 d𝜏 = (7.2.12) ∫0 [𝜃i S(t)(𝜋DR )−1∕2 + (𝜋DO )−1∕2 ] t

∫0

1∕2

i(𝜏)(t − 𝜏)−1∕2 d𝜏 =

∗ nFA𝜋 1∕2 DO CO

[𝜃i S(t)𝜉 + 1]

(7.2.13)

This last relationship is an integral equation that can be solved for i(t), which is the desired i − t curve (or, since potential is linearly related to time, the desired i − E curve). A numerical method must be employed, because a closed-form solution cannot be obtained. Before undertaking the solution, it is worthwhile to make two changes: • To convert from i(t) to i(E), since data are usually presented as i − E curves. • To put the equation in a dimensionless form, so that a single numerical solution will describe all combinations of experimental conditions. First, we make the following substitution: 𝜎t = nfvt = nf (Ei − E)

(7.2.14)

then, we let f (𝜏) = g(𝜎𝜏). Also, we let z = 𝜎𝜏, so that 𝜏 = z/𝜎, d𝜏 = dz/𝜎, z = 0 at 𝜏 = 0, and z = 𝜎t at 𝜏 = t. Then, we have ( ) t 𝜎t z −1∕2 dz f (𝜏)(t − 𝜏)−1∕2 d𝜏 = g(z) t − (7.2.15) ∫0 ∫0 𝜎 𝜎 Now, (7.2.13) can be written 𝜎t

∫0

g(z)(𝜎t − z)−1∕2 𝜎 −1∕2 dz =

∗ (𝜋D )1∕2 CO O

1 + 𝜉𝜃i S(𝜎t)

(7.2.16)

∗ (𝜋D )1∕2 , we obtain Finally, dividing by CO O 𝜎t

∫0

𝜒(z)dz (𝜎t − z)1∕2

=

1 1 + 𝜉𝜃i S(𝜎t)

(7.2.17)

where 𝜒(𝜎t) =

g(z) ∗ (𝜋D 𝜎)1∕2 CO O

=

i(𝜎t) nFAC∗O (𝜋DO 𝜎)1∕2

(7.2.18)

Equation (7.2.17) is the desired re-expression in terms of the dimensionless variables 𝜒(z), 𝜉, 𝜃 i , S(𝜎t), and 𝜎t. At any value of S(𝜎t), which expresses E, 𝜒(𝜎t) can be obtained by solution of (7.2.17), and, from it, the current can be obtained by rearrangement of (7.2.18): i = nFAC∗O (𝜋DO 𝜎)1∕2 𝜒(𝜎t)

(7.2.19)

At any value of 𝜎t, 𝜒(𝜎t) is a pure number; therefore, (7.2.19) explicitly states the functional relationship between the current at any point on the LSV curve and the experimental variables ∗ , and v. Notably, i is proportional to C ∗ and v1/2 . n, A, DO , CO O The solution of (7.2.17) has been carried out numerically (4), by a series solution (3, 5), analytically in terms of an integral that must be evaluated numerically (6, 7), and by related methods (8, 9). The general result is a set of values of 𝜒(𝜎t) [or 𝜋 1/2 𝜒(𝜎t), as in Figure 7.2.1 and Table 7.2.1] as a function of 𝜎t or n(E – E1/2 ).3 ′

3 ln 𝜉𝜃 i S(𝜎t) = nf (E – E1/2 ), where E1/2 ≡ E0 + (RT/nF) ln (DR /DO )1/2 .

315

7 Linear Sweep and Cyclic Voltammetry

Epc

ipc

E1/2

0.4 Dimensionless current, 𝜋1/2𝜒(𝜎t)

316

0.3

Epc/2

0.2

0.1

0.0

+100

–100

0

n(E – E1/2) = n(Ei – E ) – (RT/F)𝜎t + (RT/F) ln 𝜉 (mV) 0′

Figure 7.2.1 Linear potential sweep voltammogram in terms of dimensionless current. Values on the potential axis are for 25 ∘ C.

(b) Peak Current and Potential

At its maximum, 𝜋 1/2 𝜒(𝜎t) = 0.4463; thus, from (7.2.19), the cathodic peak current, ipc , is given by the Randles–Ševˇcík equation, (

ipc

F3 = 0.4463 RT

)1∕2

1∕2

∗ v1∕2 n3∕2 ADO CO

(7.2.20)

At 25 ∘ C, 1∕2

∗ v1∕2 ipc = (2.69 × 105 )n3∕2 ADO CO

(7.2.21)

∗ in mol/cm3 , and v in V/s. where ipc is in amperes for A in cm2 ; DO , in cm2 /s, CO The corresponding cathodic peak potential, Epc , is found from Table 7.2.1 to be

Epc = E1∕2 − 1.109

RT 28.5 = E1∕2 − mV at 25∘ C nF n

(7.2.22)

It is sometimes convenient to report the half-peak potential, Epc/2 , where i = ipc /2, Epc∕2 = E1∕2 + 1.09

RT 28.0 = E1∕2 + mV at 25∘ C nF n

(7.2.23)

Thus, E1/2 is just about midway between Epc and Epc/2 . A convenient diagnostic for a nernstian wave is |Epc − Epc∕2 | = 2.20

RT 56.5 = mV at 25∘ C nF n

(7.2.24)

For reversible kinetics, Epc is independent of scan rate, and ipc (as well as the current at any other point on the wave) is proportional to v1/2 . The latter property indicates diffusion control and is analogous to the variation of id,c (t) with t −1/2 in chronoamperometry. A convenient

7.2 Nernstian (Reversible) Systems

Table 7.2.1 Current Functions for Reversible Charge Transfer(a),(b) nf (E − E 1/2 )

n(E – E 1/2 )(c) 𝝅 1/2 𝝌(𝝈t)

𝝓(𝝈t)

nf (E − E1/2 )

4.67

120

0.008

−0.19

−5

0.400

0.548

3.89

100

0.020

0.019

−0.39

−10

0.418

0.596

3.11

80

0.042

0.041

−0.58

−15

0.432

0.641

2.34

60

0.084

0.087

−0.78

−20

0.441

0.685

1.95

50

0.117

0.124

−0.97

−25

0.445

0.725

0.009

n(E – E 1/2 )(c) 𝝅 1/2 𝝌(𝝈t)

𝝓(𝝈t)

1.75

45

0.138

0.146

−1.109

−28.50

0.4463

0.7516

1.56

40

0.160

0.173

−1.17

−30

0.446

0.763

1.36

35

0.185

0.208

−1.36

−35

0.443

0.796

1.17

30

0.211

0.236

−1.56

−40

0.438

0.826

0.97

25

0.240

0.273

−1.95

−50

0.421

0.875

0.78

20

0.269

0.314

−2.34

−60

0.399

0.912

0.58

15

0.298

0.357

−3.11

−80

0.353

0.957

0.39

10

0.328

0.403

−3.89

−100

0.312

0.980

0.19

5

0.355

0.451

−4.67

−120

0.280

0.991

0.00

0

0.380

0.499

−5.84

−150

0.245

0.997

(a) To calculate the current: (1) i = i(plane) + i(spherical correction). 1∕2 ∗ 1∕2 1∕2 ∗ (1∕r )𝜙(𝜎t). (2) i = nFADO CO 𝜎 𝜋 𝜒(𝜎t) + nFADO CO 0 1∕2

∗ v1∕2 𝜋 1∕2 𝜒(𝜎t) + (9.65 × 104 )nAD C ∗ 𝜙(𝜎t)∕r at 25 ∘ C with i in A, (3) i = (6.02 × 105 )n3∕2 ADO CO O O 0 ∗ in mol/cm3 , v in V/s, and r in cm. A in cm2 , DO in cm2 /s, CO 0 ′

(b) E1/2 ≡ E0 + (RT/nF) ln(DR /DO )1/2 . (c) mV at 25 ∘ C. Modified from Nicholson and Shain (4). ∗ (sometimes called the current function), which depends on n3/2 constant in LSV is ipc ∕v1∕2 CO 1∕2

and DO . The current function can be used to estimate n for a reversible electrode reaction when DO is known or can be estimated, for example, from the LSV of a compound of similar size that undergoes a reversible electrode reaction with a known n value. We have now worked out the principal features of LSV for the reduction of O to R. Analogous features apply for oxidation of R to O. In either case, • E follows (occurs later in the scan than) E (at 25 ∘ C, 28.5/n mV more negative for a p

1/2

cathodic wave and 28.5/n mV more positive for an anodic wave). • Ep/2 precedes (occurs earlier in the scan than) E1/2 (at 25 ∘ C, 28.0/n mV more positive for a cathodic wave and 28.0/n mV more negative for an anodic wave). • E1/2 is very near the average of Ep and Ep/2 . • ip is given quantitatively by (7.2.21), but has a negative sign for an anodic peak. (c) Spherical Electrodes and UMEs

For LSV with a spherical electrode [e.g., a hanging mercury drop electrode (HMDE)], a similar treatment can be presented (5). The resulting current is i = i(plane) +

∗ 𝜙(𝜎t) nFADO CO

r0

(7.2.25)

317

318

7 Linear Sweep and Cyclic Voltammetry

where r0 is the radius of the electrode, and 𝜙(𝜎t) is a tabulated function (Table 7.2.1). For large values of v and with electrodes of conventional size, i(plane) is much larger than the spherical correction term, and the electrode can be considered planar (i.e., semi-infinite linear diffusion applies). At fast scan rates, hemispherical electrodes and ultramicroelectrodes also behave as though diffusion is linear. However, for a UME, the second term will dominate at low scan rates. One can show from (7.2.25) that this is true when v ≪ RTD∕nFr20

(7.2.26)

so that the voltammogram becomes an SSV wave, independent of v, as discussed in Chapter 5.4 For r0 = 5 μm, n = 1, D = 10−5 cm2 /s, and T = 298 K, the right side of (7.2.26) has a value of 1000 mV/s; thus, a scan made at ∼100 mV/s or slower should permit the accurate recording of steady-state currents. Criterion (7.2.26) depends on the square of the radius, so it becomes impractical to record steady-state voltammograms with electrodes much larger than those normally considered to be UMEs. Conversely, with very small UMEs, one requires a high sweep rate to see behavior other than the steady state. For example, at a UME disk with r0 = 0.5 μm and with n, D, and T as given above, steady-state behavior would apply up to 10 V/s. In Figure 7.2.2, the forward sweep shows the transition from a peak-shaped voltammogram in the linear-diffusion regime at high sweep rates to a wave-like steady-state voltammogram at small v, as described by the treatment in Section 5.3.1. Ultramicroelectrodes are almost always employed in the limiting regions—the linear-diffusion region, when v1/2 /r0 is large, or the steady-state region, when v1/2 /r0 is small. There is nothing additional to be gained from working in the intermediate zone. 12 10

1 V/s

8 i/nA

0.1 V/s

6 0.01 V/s 4 2 0 –2 –4 0.3

0.2

0.1

0.0 (E – E 0′)/V

–0.1

–0.2

–0.3

Figure 7.2.2 Effect of scan rate on CV for a hemispherical UME with r0 = 10 μm. Simulated curves for a ′ ∗ = 1.0 mM, and T = 25 ∘ C. At 1 V/s, the nernstian reaction with n = 1, E 0 = 0.0 V, DO = DR = 1 × 10−5 cm2 /s, CO forward response (i.e., the LSV response) begins to show the peak expected for linear diffusion, but the high current at the switching potential and the small ipa /ipc show that the steady-state component is still dominant. 4 Relationship 7.2.26 involves a comparison of a diffusion length to the radius of the electrode [Section 6.1.2(c)]. The diffusion length is [DO /(nfv)]1/2 , which corresponds to the time 1/nfv. This is the period required for the scan to cover an energy kT/n along the potential axis (25.7/n meV at 25 ∘ C). It is often regarded as the characteristic time of an LSV or CV experiment (Section 13.2.2).

7.2 Nernstian (Reversible) Systems

(d) Effect of Double-Layer Capacitance

As shown in Section 1.6.4(b), a charging current, ic , always flows during a potential sweep, because the potential is continuously changing. During most of the sweep, it is given by5 ic = ±AC d v

(+ for negative scan, − for positive scan)

(7.2.27)

The faradaic current must always be measured from this baseline (Figure 7.2.3). Figure 7.2.3 Effect of double-layer charging at different sweep rates in LSV for C d independent of E. The magnitudes of the charging current, ic , and the faradaic peak current, ip , are shown. The current scales in (c) and (d) are 10× and 100×, respectively, that in (a) and (b).

𝑣 = 900 a ip

(d)

100

ic 0 𝑣 = 100 a

10

ip

(c) ic 0

(b)

1 ip 𝑣=9a

ic 0 (a)

ip

𝑣=a

1 ic

0 E

5 See (1.6.23). It is normal in the electrochemical literature to treat scan rate, v, as an unsigned quantity. Charging current has a sign depending on the direction of the scan. For negative scans, the charging current looks “cathodic”; for positive ones, it looks “anodic.”

319

320

7 Linear Sweep and Cyclic Voltammetry

While ip varies with v1/2 for linear diffusion, ic varies with v. Thus, ic becomes relatively more important at faster scan rates. From (7.2.21) and (7.2.27), |i | Cd v1∕2 (10−5 ) | c| (7.2.28) | |= | ip | 2.69n3∕2 D1∕2 C ∗ | | O O or for DO ≈ 10−5 cm2 /s and C d ≈ 20 μF/cm2 , | i | (2.4 × 10−8 )v1∕2 | c| (7.2.29) | |≈ ∗ | ip | n3∕2 CO | | ∗ , the faradaic component of the LSV response becomes small in compariAt high v and low CO son to the background charging current, and this effect often sets the upper limit of useful scan rate and the lower limit of useful concentration. (e) Effect of Uncompensated Resistance

A potentiostat controls Eappl = E − iRu , rather than the actual potential, E, of the working electrode [Sections 1.5.4, 1.6.4(d), and 16.7.1]. Since i varies as the peak is traversed, the control error vs. E varies correspondingly. If ip Ru is appreciable compared to the precision of measurement, the sweep will not be truly linear and the condition given by (7.2.10) is compromised. The practical effect of Ru is to flatten the wave and to shift the peak toward more extreme potentials. Since the current increases with v1/2 , the larger the scan rate, the more Ep will be shifted. Thus, appreciable Ru causes Ep to become a function of v. It moves negatively with increasing v for a reduction, but positively for an oxidation. In this way, significant uncompensated resistance mimics shifts in Ep resulting from quasireversible heterogeneous kinetics (Section 7.3). (f) Correction for Charging Current

A procedure for correcting LSV for charging current has been presented (10, 11). For a blank experiment in the absence of electroactive species, the charging current, ibc , is ibc = −Cdb (dE∕dt)

(7.2.30)

where Cdb is the double-layer capacitance (μF) for the blank system, and E is the actual potential vs. the reference. For a negative-going scan, E = Eappl + ibc Ru = Ei − vt + ibc Ru

(7.2.31)

Thus, at any given value of E, Cdb =

ibc v − Ru (dibc ∕dt)

(7.2.32)

If Ru is known [e.g., from one of the methods discussed in Section 16.7.3(a)], the measured values of ibc and dibc ∕dt allow the determination of Cdb as a function of E. When substance O is introduced into solution, the total current is i = if + ic

(7.2.33)

where if is the faradaic component. In general, the charging current, ic , differs from ibc because the presence of if causes dE/dt to differ in the potential range of the wave for the test solution vs. the blank. However, the interfacial capacitance, C d for the test solution is still given by ic = −Cd (dE∕dt)

(7.2.34)

7.2 Nernstian (Reversible) Systems

with (7.2.35)

E = Ei − vt + iRu Therefore, if = i − Cd v + Cd Ru

di dt

(7.2.36)

If one assumes that C d is the same as in the blank (i.e., that Cd = Cdb for all potentials), then (7.2.36) provides if vs. E. This method is applicable to LSV and CV of dilute, nonadsorbing, freely diffusing species; however, any change of the electrode surface induced by a component of the test solution absent from blank will invalidate the assumption that Cd = Cdb . Many electroreactants, intermediates, or electroproducts adsorb on the electrode and can cause big changes in C d vs. a blank. 7.2.2

Cyclic Voltammetry

The reversal experiment, CV, is carried out by changing the direction of an LSV scan at a switching time, t = 𝜆 (or, equivalently, at a switching potential, E𝜆 ). Thus, the potential is given at any time by E = Ei − vt

(0 < t ≤ 𝜆)

(7.2.37)

E = Ei − 2v𝜆 + vt

(t > 𝜆)

(7.2.38)

While it is possible to use a different scan rate on reversal (12), this is rarely done. Only the case of a symmetrical triangular wave is considered here. The forward phase is just an LSV scan, already treated in Section 7.2.1. Here, it is necessary only to address the reverse scan. (a) Theory

The application of (7.2.38) to the nernstian balance of surface concentrations, (7.2.1), for t > 𝜆 also yields (7.2.11), but with S(t) given by S(t) = e𝜎t−2𝜎𝜆

(t > 𝜆)

(7.2.39) CR∗

In Section 7.2.1(a), we addressed the case in which = 0 and the forward scan corresponds to the reduction of O to R. We now treat a reverse scan in the positive direction. The treatment proceeds via the approach of Section 7.2.1(a). The shape of the curve depends on the switching potential, E𝜆 (i.e., on how far beyond the cathodic peak the scan proceeds before reversal). However, if E𝜆 is at least 35/n mV past the cathodic peak,6 the reversal peaks all have the same general shape—a curve like the forward i − E curve, but plotted in the opposite direction on the current axis, with the decaying current of the cathodic wave used as a baseline. Typical i − t curves for different switching potentials are shown in Figure 7.2.4. This type of presentation would result if the curves were recorded on a time base. The more usual i − E presentation is shown in Figure 7.2.5. Two parameters of interest on these cyclic voltammograms are the ratio of peak currents, |ipa /ipc |, and the separation of peak potentials, Epa – Epc .

6 This condition applies if the potentiostat responds ideally and the effects of Ru are negligible (Section 16.7). A larger margin between the peak and the switching potential would be needed in less ideal circumstances.

321

7 Linear Sweep and Cyclic Voltammetry

(b) Peak Current Ratio

For a nernstian wave with a stable, soluble product (i.e., O + ne ⇌ R), |ipa /ipc | = 1 when ipa is measured from the decaying cathodic current as a baseline (see Figures 7.2.4 and 7.2.5). The ratio |ipa /ipc | = 1 applies regardless of v, DO , DR , or E𝜆 (for E𝜆 > 35/n mV past Epc ). If the cathodic sweep is stopped and the current is allowed to decay to zero (Figure 7.2.5, curve 4), the resulting anodic i − E curve is identical in shape to the cathodic one, but is plotted in the opposite directions on both the i and E axes. Allowing the cathodic current to decay to zero results in a thick diffusion layer, depleted of O and populated with R at a concentration ∗ , so that the anodic CV scan is virtually the same as that for anodic LSV in a solution near CO containing only R. Departure of |ipa /ipc | from unity indicates mechanistic complications in the electrode process (13), as discussed extensively in Chapter 13. Nicholson (14) suggested that the ratio can be calculated from (a) the uncorrected anodic peak current, (ipa )0 , measured with respect to the zero-current baseline7 (see Figure 7.2.5, curve 3) and (b) the current at E𝜆 , (isp )0 : | ipa | | (ipa )0 | 0.485(isp )0 | | | | + 0.086 | |=| |+ | ipc | | ipc | ipc | | | |

(7.2.40)

(c) Peak Separation

The difference Epa − Epc , often symbolized by ΔEp , is a useful diagnostic for a nernstian reaction. Although ΔEp for a reversible system is slightly a function of E𝜆 (Table 7.2.2), it is always 0.5

Epc

0.4

Dimensionless current, 𝜋1/2𝜒(𝜎t)

322

E𝜆1 E 𝜆2

0.3 0.2

ipc

E𝜆3

0.1 0.0 ipa

–0.1 –0.2 1

2

–0.3 –0.4

𝜆1

𝜆2

Time →

3

Epa

𝜆3

Figure 7.2.4 Cyclic voltammograms for reversal at different E 𝜆 , with presentation on a time base. Noted potentials correspond to the indicated values of t. 7 Here, “zero-current baseline” means the level of zero faradaic current. In experimental work, this would be the baseline defined by the charging current [Section 7.2.2( f )].

7.2 Nernstian (Reversible) Systems

0.5

Epc

0.4 ipc

Dimensionless current, 𝜋1/2𝜒(𝜎t)

0.3

E𝜆1 1

E𝜆2 2

0.2 1′ 0.1

E𝜆3 E𝜆4

3

2′ 3′

(isp)0

0.0

E𝜆∞

ipa

–0.1

(ipa)0 –0.2 –0.3

Epa

4

–0.4 –0.5 +200

+100

0

–100

–200

–300

–400

n(E – E1/2)/mV

Figure 7.2.5 Cyclic voltammograms under the same conditions as in Figure 7.2.4, but in an i − E format. E 𝜆 : (1) E 1/2 – 90/n; (2) E 1/2 – 130/n; (3) E 1/2 – 200/n mV; (4) for potential held at E 𝜆4 until the cathodic current decays to zero. [Curve 4 results from reflection of the cathodic i − E curve through the E axis and then through the vertical line at n(E – E 1/2 ) = 0. Curves 1, 2, and 3 result by addition of this curve to the decaying current of the cathodic i − E curve (1′ , 2′ , or 3′ ).]

close to 2.303RT/nF (59.1/n mV at 25 ∘ C). Repeated cycling will establish a steady response with ΔEp = 58/n mV at 25 ∘ C (6). (d) Estimation of E1/2

For a reversible system, the average of Epa and Epc is very close to E1/2 . When the effects of uncompensated resistance are negligible (with ipc Ru and ipa Ru being small and offsetting), there is only an error determined by the switching potential, but it is less than 1.8/n mV for any of the switching conditions in Table 7.2.2. (e) Effect of Uncompensated Resistance

When iRu is significant, (7.2.37) and (7.2.38) do not accurately describe the potential sweep, for it is Eappl , the voltage applied between the working and reference electrodes, that is swept linearly. Since E = Eappl + iRu (Section 1.5.4), the actual sweep condition for CV becomes E = Ei − vt + iRu

(0 < t ≤ 𝜆)

(7.2.41)

E = Ei − 2v𝜆 + vt + iRu

(t > 𝜆)

(7.2.42)

Table 7.2.2 Variation of ΔE p with E 𝜆 for a Nernstian System at 25 ∘ C. n(Epc – E𝜆 ) (mV)

71.5

121.5

171.5

271.5

n(Epa − Epc ) (mV)

60.5

59.2

58.3

57.8

Based on Nicholson and Shain (4).

∞ 57.0

323

324

7 Linear Sweep and Cyclic Voltammetry

Because i varies across a wave, iRu compromises both the linearity of E and the theory for LSV or CV. As mentioned in Section 7.2.1(e), the effect of uncompensated resistance is to shift cathodic waves more negatively and anodic waves more positively; consequently, the peak separation in CV, ΔEp , widens with increasing Ru and v. If the magnitude of iRu remains small relative to the peak separation in the absence of uncompensated resistance, ΔEp0 , then the separation in the presence of Ru will be approximately ΔEp0 + 2ip Ru . For a reversible system at 25 ∘ C, this is about 60/n mV + 2ip Ru . As iRu becomes appreciable compared to ΔEp0 , its compromising effect grows more serious, and it can no longer be treated as a linear perturbation. Close comparison of theory and experiment must then be based on simulation. (f) Charging-Current Background

In real cyclic voltammograms, the faradaic response is usually superimposed on a roughly constant charging current. The situation for the forward scan was discussed in Section 7.2.1(d). Upon reversal, the magnitude of dE/dt remains constant, but the sign changes. It forms a baseline for the faradaic reversal response, just as for the forward scan, and both ipc and ipa must be corrected correspondingly. The correction technique described in Section 7.2.1( f ) is applicable to CV. In the reversal phase, the actual potential of the working electrode is given by (7.2.42). In routine practice, the measurement of peak currents in CV is inherently imprecise because the correction for charging current is uncertain. For the reversal peak, the imprecision is further increased because one cannot ideally define the folded faradaic response for the forward process (e.g., curve 1′ , 2′ , or 3′ in Figure 7.2.5) to use as a reference for the measurement. In general, CV is not ideal for quantitative evaluation of system properties that must be derived from peak heights (e.g., the concentration of an electroactive species). The method’s power lies in its diagnostic strength, which is derived from the ease of interpreting qualitative and semiquantitative behavior. Once a system is understood mechanistically, other methods are often better suited for the precise evaluation of parameters. (g) Information from Reversible Cyclic Voltammograms

The information content in reversible cyclic voltammograms is essentially the same as in reversible SSV and STV, so one can obtain stoichiometric and thermodynamic data much as we have already discussed (Section 5.3.2). Figure 7.2.6 provides an example in which chemical information is derived from wave potentials, rather than the peak heights. The depicted voltammetry is for the stable radical tetramethylpiperidine-1-oxyl (TEMPO∙; Figure 1), which undergoes a reversible 1e oxidation to TEMPO+ in MeCN (15): TEMPO∙ ⇌ TEMPO+ + e

E1∕2 = 0.25 V vs. Fc+ ∕Fc

(7.2.43)

When tetra-n-butylammonium azide (TBAN3 ) is added, the CV (Figure 7.2.6a) remains reversible, but shifts negatively with added TBAN3 . The observed behavior suggests that the oxidized form, TEMPO+ , is reversibly stabilized by addition of TBAN3 [Section 5.3.2(c)]. The investigators proposed an equilibrium involving formation of a 1:1 charge-transfer complex, + − TEMPO+ + N− 3 ⇌ (TEMPO N3 )

(7.2.44)

the stoichiometry of which was supported by the plot in Figure 7.2.6b (Problem 7.12). Spectroscopic evidence confirmed other aspects of the hypothesis. This system is of interest because it can facilitate valuable electrosynthetic steps, perhaps because the charge-transfer complex can furnish the azidyl radical, N3∙ , to a suitable coreactant.

7.3 Quasireversible Systems

220 ½(Epa + Epc) / mV vs. Fc+/Fc

40 30 20 i/μA

10 0 –10 –20 –30 –0.2

0.0

0.2

0.4

200

180

160

140 0.0

E/V vs. Fc+/Fc

0.2

0.4 0.6 log (Cazide/mM)

(a)

(b)

0.8

1.0

Figure 7.2.6 (a) Cyclic voltammetry at a glassy carbon disk (r0 = 1.5 mm) of 1 mM TEMPO∙ in the presence of N− in MeCN with 0.1 M LiClO4 . v = 100 mV/s. Azide added as the tetra-n-butylammonium salt. Curves are for 3 1 mM (rightmost) to 10 mM (leftmost) azide in increments of 1 mM. Scans start at −0.2 V and first move positively. Anodic currents are up. (b) Slope, −62 mV; intercept at 1 mM azide, 210 mV. [Adapted from Siu et al. (15). © 2018, American Chemical Society.]

(h) More General Notation for LSV and CV

So far, we have developed equations for LSV and CV in the specific context of a negative-going forward sweep, causing a reduction. Voltammetric features are labeled as cathodic or anodic (e.g., ipc or Epa ) to preserve clarity for the reader. The same will be done in Sections 7.3 and 7.4. However, it should be evident by now that the interpretation of CV is based on distinguishing features of the forward scan from those of the reverse scan. Thus, everything labeled in this section with subscripts c and a, for “cathodic” and “anodic” could have been labeled with subscripts f and r, for “forward” and “reverse,” respectively. Then, we would have had Epf , Epr , ipf and ipr , as well as |ipr /ipf | and ΔEp = |Epf − Epr |. Since the half-peak potential always relates to the forward scan, one can simply use Ep/2 . This notation has the advantage of labeling features accurately, regardless of the direction of the forward scan. In later chapters, we will favor it. One must recognize that anodic and cathodic peaks are on opposite sides of E1/2 [e.g., as discussed at the end of Section 7.2.1(b)] and behave differently with respect to 𝛼 (Sections 7.3 and 7.4).

7.3 Quasireversible Systems Matsuda and Ayabe (6) coined the term quasireversible for reactions that show electron-transfer kinetic limitations such that both the forward and reverse processes must be considered. They also provided the first treatment of such systems for sweep voltammetry. For the one-step, one-electron case, kf

O + e⇌ R kb

(7.3.1)

the problem begins with the general formulation (Section 4.5.2) for the case of CR∗ = 0. A solution requires one additional boundary condition embodying the electrode kinetics and the

325

326

7 Linear Sweep and Cyclic Voltammetry

potential program. For this problem, that condition is ( ) 𝜕CO (x, t) 0′ 0′ DO = k 0 e−𝛼f [E(t)−E ] {CO (0, t) − CR (0, t) e f [E(t)−E ] } 𝜕x x=0

(7.3.2)

where E(t) is given by (7.2.10). 7.3.1

Linear Sweep Voltammetry

The shape of the LSV peak and the various peak characteristics are functions of 𝛼 and of a kinetic parameter, Λ, defined (6) as Λ=

k0 (D1−𝛼 D𝛼R O

(7.3.3)

fv)1∕2

For DO = DR = D, Λ=

k0 (D fv)1∕2

(7.3.4)

The current is given by 1∕2

∗ f 1∕2 v1∕2 Ψ(E) i = FADO CO

(7.3.5)

where Ψ(E) is a dimensionless expression of the i − E curve. Examples are illustrated in Figure 7.3.1 for various values of Λ and 𝛼. When Λ > 15, the behavior approaches that of a reversible system. For smaller values of Λ (i.e., for smaller k 0 or greater v), the wave broadens and shifts toward more extreme (more negative) potentials. The values of ip , Ep , and Ep/2 depend on Λ and 𝛼 according to published functions (6).8 For a quasireversible reaction, ip is not proportional to v1/2 , because the value of Λ changes with v, causing alterations in the wave shape. ∗ = 0 and a positive-going sweep), the behavior For an anodic electrode process (i.e., with CO follows essentially the same principles. If Λ > 15, the response is reversible, but for smaller Λ, the wave broadens and moves to more extreme potentials (more positive potentials for the anodic case). A given system may show nernstian, quasireversible, or totally irreversible behavior, depending on the time scale of the experiment, which can be regarded as the time needed to traverse the LSV wave. At small v (or long time scale), the system might yield a reversible wave, while at large v (or short time scale), irreversible behavior might even be observed. In Section 6.3, we encountered similar effects in the context of potential step experiments. Matsuda and Ayabe (6) suggested the following zone boundaries for LSV9 : • Reversible (nernstian) • Quasireversible • Totally irreversible 7.3.2

Λ > 15 15 ≥ Λ ≥ 10–2(1 + 𝛼) Λ < 10–2(1 + 𝛼)

k 0 > 0.3v1/2 cm/s 0.3v1/2 ≥ k 0 ≥ 2 × 10–5 v1/2 cm/s k 0 < 2 × 10–5 v1/2 cm/s

Cyclic Voltammetry

For the quasireversible one-step, one-electron process, one can derive cyclic voltammetric i − E curves using the approach described in Section 7.3.1 and the potential program given by (7.2.37) 8 Section 6.4 in previous editions presents these functions graphically and includes a brief discussion. 9 The k 0 values are based on n = 1, 𝛼 = 1, T = 25 ∘ C, and D = 10−5 cm2 /s. For v in V/s, Λ ≈ k 0 /(39Dv)1/2 .

7.3 Quasireversible Systems

0.5

(a) 𝛼 = 0.7

0.4 III

II

0.3

IV

Ψ(E ) 0.2 0.1 0 0.5 (b) 𝛼 = 0.5 0.4 I

0.3

II

Ψ(E )

III

IV

0.2 0.1 0 0.5

(c) 𝛼 = 0.3

0.4

I

0.3 Ψ(E ) 0.2 III

II

IV

0.1 0 128

0

–128 –257 (E – E1/2)/mV at 25 °C

–385

Figure 7.3.1 Dimensionless current, Ψ(E), vs. potential for a quasireversible system. For each graph: 𝛼 is as specified; values of Λ are (I) 10; (II) 1; (III) 0.1; (IV) 10−2 ; dashed curve is for reversible kinetics. Major divisions on the potential axis are separated by 5kT. [Adapted from Matsuda and Ayabe (6).]

and (7.2.38). The wave shape and ΔEp depend on v, k 0 , 𝛼, and E𝜆 ; however, the effect of E𝜆 is small when E𝜆 is at least 90 mV beyond the cathodic peak. The quasireversible voltammograms are expressed in terms of 𝛼 and either the dimensionless function Λ [defined in (7.3.3)] or an equivalent function (14), 𝜓 defined by10 𝜓 = Λ𝜋 −1∕2 =

(DO ∕DR )𝛼∕2 k 0 (𝜋DO fv)1∕2

(7.3.6)

Typical behavior is shown in Figure 7.3.2. As we saw for LSV of quasireversible systems, the effect of lowered Λ or 𝜓 (reflecting lowered k 0 or elevated v) is to broaden a wave and to shift its peak to more extreme potentials. In CV, these effects are manifested in both the forward and reverse responses; therefore, the peak splitting, ΔEp , widens. The cathodic peak is more negative than in the reversible case, and the anodic peak is more positive. 10 The function 𝜓 is not the same as Ψ(E) in (7.3.5).

327

7 Linear Sweep and Cyclic Voltammetry

0.5

0.5

3

1 0.4

Dimensionless current, 𝜋1/2𝜒(𝜎t)

Dimensionless current, 𝜋1/2𝜒(𝜎t)

328

2

0.3 0.2 0.1 0.0 –0.1 –0.2

0.4 4

0.3 0.2 0.1 0.0 –0.1 –0.2

–0.3 180 120 60 0 –60 –120 –180 (E – E1/2)/mV

–0.3 180 120 60 0 –60 –120 –180 (E – E1/2)/mV

Figure 7.3.2 Theoretical cyclic voltammograms showing effect of 𝜓 and 𝛼 on curve shape for a one-step, one-electron reaction. Curve 1: 𝜓 = 0.5, 𝛼 = 0.7. Curve 2: 𝜓 = 0.5, 𝛼 = 0.3. Curve 3: 𝜓 = 7.0, 𝛼 = 0.5. Curve 4: 𝜓 = 0.25, 𝛼 = 0.5. [Adapted from Nicholson (14). © 1965, American Chemical Society.] Table 7.3.1 Variation of ΔE p with 𝜓 at 25 ∘ C(a) 𝜓

20

7

6

5

4

3

2

1

0.75

0.50

0.35

0.25

0.10

(Epa − Epc ) (mV)

61

63

64

65

66

68

72

84

92

105

121

141

212

(a) For a one-step, one-electron process with E𝜆 = Ep – 112.5 mV and 𝛼 = 0.5. Nicholson (14). © 1965, American Chemical Society.

(a) Estimation of E1/2

One may be tempted to use the average of Epc and Epa as an estimate of E1/2 for a quasireversible process, just as one can do for a reversible case. This practice is valid when ΔEp does not greatly exceed 60 mV (i.e., when the electrode reaction is almost reversible). However, if ΔEp is larger, a significant error can arise, because quasireversible cathodic and anodic peaks may be unsymmetrically shifted from the reversible positions. In general, the negative shift in Epc caused by heterogeneous kinetics is roughly proportional to 𝛼 −1 , but the positive shift in Epa is approximately proportional to (1 − 𝛼)−1 . For 𝛼 > 0.5, the average of Epc and Epa will be more positive than E1/2 and vice versa for 𝛼 < 0.5. From the different dependences on 𝛼, one can develop the following approximation: E ≈ (E + E )∕2 − (𝛼 − 0.5)(ΔE − 57 mV) (at 25∘ C) (7.3.7) 1∕2

pa

pc

p

(b) Estimation of Kinetic Parameters

For 0.3 < 𝛼 < 0.7, ΔEp is nearly independent of 𝛼 and depends only on 𝜓. Table 7.3.1, linking 𝜓 to k 0 for this range of 𝛼, is the basis for the method of Nicholson (14), widely used for estimating k 0 in quasireversible systems. For a set of scan rates giving values of ΔEp covered by Table 7.3.1, one can look up or interpolate the corresponding values of 𝜓. According to (7.3.6), a plot of 𝜓 vs. v−1/2 should be linear with a zero intercept and a slope of (DO /DR )𝛼/2 k 0 /(𝜋DO f )1/2 . One can often assume that DO = DR = D, so that the slope of 𝜓 vs. v−1/2 simplifies to k 0 /(𝜋D f )1/2 , from which k 0 can be calculated, if D is known.

7.4 Totally Irreversible Systems

The uncompensated resistance, Ru , must be sufficiently small that the corresponding voltage drops (of the order of ip Ru ) are negligible compared to the part of ΔEp attributable to kinetic effects. In fact, Nicholson (14) showed that resistive effects cannot be readily detected in the ΔEp − v behavior, because the effect of uncompensated resistance on the CV is so similar to that of heterogeneous kinetics. Error due to Ru is most important when the currents are large and when k 0 approaches the reversible limit (where ΔEp differs only slightly from the reversible value). There are two practical ways to address the effects of uncompensated resistance on kinetic measurements based on ΔEp : • One can employ electronic resistance compensation to reduce the effective value of Ru . Contemporary instruments have this capability, which is covered in detail in Section 16.7.3. It may be possible to reduce the effective ohmic drop to a negligible value using this approach; however, electronic compensation of resistance needs to be approached carefully, for it is possible to overcompensate unknowingly. • If the value of Ru is known, one can correct ΔEp by subtracting the contribution from ohmic drop, which is Ru (ipc − ipa ). Some instruments can measure and report the value of Ru , which then can be used in this calculation. Correction of ΔEp can be used together with electronic compensation, when the latter cannot reduce the effective Ru to insignificance. Given the general availability of electrochemical simulators capable of comparing experimental results with simulated curves [Section 4.5.4(b)], iterative modeling and comparison has become a practical alternative for extracting kinetic parameters from CV data. In any such work, it is essential to include realistic modeling of the charging current in the simulations or to correct the CV data for the underlying charging current [Section 7.2.1( f )].

7.4 Totally Irreversible Systems kf

For a totally irreversible one-step, one-electron reduction, O + e −−−→ R, the mass-transfer problem begins with the general formulation (Section 4.5.2) with CR∗ = 0. The final boundary condition, defining the kinetics and enabling a solution, is [ ] 𝜕CO (x, t) i = DO = kf (t)CO (0, t) (7.4.1) FA 𝜕x x=0 For BV kinetics, kf (t) = k 0 e−𝛼 f [E(t)−E

0′ ]

(7.4.2)

To solve this mathematical problem, one must know the functional form of k f (t), which requires knowledge of k f (E). It is not possible to treat LSV for an irreversible system without a specific kinetic model, even though we were able to do that for SSV and STV (Sections 5.4.1 and 6.3.1). 7.4.1

Linear Sweep Voltammetry

(a) Current–Potential Curves

Introducing E(t) from (7.2.10) into (7.4.2) yields 0′

kf (t) = k 0 e−𝛼f (Ei −E ) ebt

(7.4.3)

329

330

7 Linear Sweep and Cyclic Voltammetry

where b = 𝛼fv. The solution follows in a manner analogous to that described in Section 7.2.1(a) and, again, requires a numerical solution of an integral Equation (4, 6). The current is given by i = FAC ∗O (𝜋DO b)1∕2 𝜒(bt) 1∕2

i = FAC ∗O DO v1∕2

(

𝛼F RT

(7.4.4)

)1∕2

𝜋 1∕2 𝜒(bt)

(7.4.5)

where 𝜒(bt) is a function [different from 𝜒(𝜎t)] tabulated in Table 7.4.1. At any point on the ∗. wave, i is linear with v1/2 and CO The LSV wave resembles the examples for Case IV in Figure 7.3.1. While that case does not lie outside the quasireversible range defined by Matsuda and Ayabe (Section 7.3.1), it approaches the boundary between quasireversible and totally irreversible behavior. For spherical electrodes, a procedure analogous to that employed at planar electrodes has been proposed. Table 7.4.1 contains values of the spherical correction factor, 𝜙(bt) employed in the equation i = i(plane) +

∗ 𝜙(bt) FADO CO

(7.4.6)

r0

Table 7.4.1 Current Functions for Irreversible Charge Transfer(a) Potential(b) Potential(c) 𝝅 1/2 𝝌(bt)

𝝓(bt)

Potential(b) Potential(c) 𝝅 1/2 𝝌(bt)

𝝓(bt)

6.23

160

0.003

0.58

15

0.437

0.323

5.45

140

0.008

0.39

10

0.462

0.396

4.67

120

0.016

0.19

5

0.480

0.482

4.28

110

0.024

0.00

0

0.492

0.600

3.89

100

0.035

−0.19

−5

0.496

0.685

3.50

90

0.050

−0.21

−5.34

0.4958

0.694

3.11

80

0.073

0.004

−0.39

−10

0.493

0.755

2.72

70

0.104

0.010

−0.58

−15

0.485

0.823

2.34

60

0.145

0.021

−0.78

−20

0.472

0.895

1.95

50

0.199

0.042

−0.97

−25

0.457

0.952

1.56

40

0.264

0.083

−1.17

−30

0.441

0.992

1.36

35

0.300

0.115

−1.36

−35

0.423

1.000

1.17

30

0.337

0.154

−1.56

−40

0.406

0.97

25

0.372

0.199

−1.95

−50

0.374

0.78

20

0.406

0.253

−2.72

−70

0.323

(a) To calculate the current: (1) i = i(plane) + i(spherical correction). 1∕2 ∗ 1∕2 1∕2 ∗ 𝜙(bt)∕r . (2) i = FADO CO b 𝜋 𝜒(bt) + FADO CO 0 1∕2

∗ 𝛼 1∕2 v1∕2 𝜋 1∕2 𝜒 (bt) + (9.65 × 104 )AD C ∗ 𝜙(bt)∕r at 25 ∘ C with (3) i = (6.02 × 105 )ADO CO O O 0 ∗ , mol/cm3 ; v, quantities in the following units: i, amperes; A, cm2 ; DO , cm2 /s; CO V/s; r0 , cm. ′

(b) Dimensionless potential = (𝛼F/RT)(E – E0 ) + ln[(𝜋DO b)1/2 /k 0 ]. ′ (c) Potential scale in mV for 25 ∘ C = 𝛼(E – E0 ) + 59.1 log [(𝜋D b)1/2 /k 0 ]. Modified from Nicholson and Shain (4).

O

7.4 Totally Irreversible Systems

(b) Peak Current and Potential

The product 𝜋 1/2 𝜒(bt) has a maximum value of 0.4958. Introduction of this value into (7.4.5) yields the cathodic peak current: 1∕2

ipc = (2.99 × 105 )𝛼 1∕2 AC ∗O DO v1∕2

(7.4.7)

where the units are as for (7.2.21). As for the reversible case, the peak current for an irreversible wave is proportional to v1/2 . From Table 7.4.1, the cathodic peak potential, Epc , is given by ] [ 1∕2 b) (𝜋D ′ RT RT O 𝛼(Epc − E0 ) + ln = −0.21 = −5.34 mV at 25∘ C (7.4.8) 0 F F k or ′

Epc = E0 −

⎡ ⎛ D1∕2 ⎞ ( ) ⎤ RT ⎢ 𝛼Fv 1∕2 ⎥ 0.780 + ln ⎜ O0 ⎟ + ln ⎥ ⎜ k ⎟ 𝛼F ⎢ RT ⎦ ⎣ ⎠ ⎝

(7.4.9)

We also find from Table 7.4.1 that |Epc − Epc∕2 | =

1.857RT 47.7 = mV at 25∘ C 𝛼F 𝛼

(7.4.10)

where Epc/2 is the potential preceding Epc where the current is ipc /2. In any totally irreversible system, k 0 is very small; hence, the second term in brackets of (7.4.9) ′ causes Epc to lie far negative relative to E0 , manifesting a large activation overpotential. The negative displacement of the peak would resemble that in Case IV in Figure 7.3.1, but would be even larger, because any totally irreversible system has less facile kinetics than Case IV. From the third bracketed term in (7.4.9), we see that Epc depends on scan rate, shifting negatively by 1.15RT/𝛼F (or 30/𝛼 mV at 25 ∘ C) for each tenfold increase in v. An expression for ipc in terms of Epc can be obtained by (a) rearranging (7.4.9) to give 1∕2

DO (𝛼Fv∕RT)1∕2 , (b) substituting into (7.4.5), (c) setting 𝜒(bt) = 0.4958 for the peak, and (d) evaluating constants. The result is (4, 7): ′

ipc = 0.227FAC ∗O k 0 exp[−𝛼f (Epc − E0 )]

(7.4.11)

′

A plot of ln ipc vs. Epc − E0 determined at different scan rates should have a slope of –𝛼F/RT ′

and an intercept proportional to k 0 . One must know E0 by some other method to use this ′ means for evaluation of k 0 , but, even if E0 remains unknown, the slope can still provide 𝛼. For an irreversible process more complicated than the one-step, one-electron reaction, such as a multistep heterogeneous reaction or a heterogeneous electron transfer followed by solution-phase chemistry (Chapters 13 and 15), it is usually not feasible to derive equations describing the current–potential relationship. The practical approach is to compare experimental behavior with the predictions from simulations. (c) Anodic Case

The treatment given just above applies to an irreversible cathodic peak observed in a negative-going scan for a solution containing species O. In an irreversible system, the cathodic and anodic branches are fully separated on the potential scale, so the initial presence or absence of species R is irrelevant to the response. A parallel treatment can be developed for an irreversible anodic peak observed in a positive-going scan for a solution containing species R.

331

332

7 Linear Sweep and Cyclic Voltammetry

In the anodic branch of an electrode process, 1 − 𝛼 has the mathematical role that −𝛼 plays for the cathodic branch. Recognizing that principle, it is easy to summarize the key results: With units as for (7.4.7), the anodic peak current is 1∕2

ipa = −(2.99 × 105 )(1 − 𝛼)1∕2 AC ∗R DR v1∕2

(7.4.12)

The anodic peak potential is Epa

( )1∕2 ⎤ ⎡ ⎛ D1∕2 ⎞ (1 − 𝛼)Fv RT ⎢ ⎥ =E + 0.780 + ln ⎜ R0 ⎟ + ln ⎥ ⎜ k ⎟ (1 − 𝛼)F ⎢ RT ⎦ ⎣ ⎠ ⎝ 0′

(7.4.13)

The relationship between Epa and Epa/2 is |Epa − Epa∕2 | =

1.857RT 47.7 = mV at 25∘ C (1 − 𝛼)F (1 − 𝛼)

(7.4.14)

The analogue of (7.4.11) is ′

ip,a = −0.227FAC ∗R k 0 exp[(1 − 𝛼)f (Epa − E0 )]

(7.4.15)

′

A plot of ln ipa vs. Epa − E0 determined at different scan rates should have a slope of (1 – 𝛼)F/RT and an intercept proportional to k 0 . 7.4.2

Cyclic Voltammetry

By definition, the backward reaction for a totally irreversible process cannot occur in the potential region where the forward peak is observed. Accordingly, a positive-going reverse scan in CV of an irreversible system with a cathodic forward peak consists only of the continued decay of that peak, folded back along the potential axis, as discussed for reversible systems in connection with Figures 7.4.4 and 7.4.5. When the scan returns to potentials more positive than Epc the electrode kinetics become progressively inhibited, and the current drops toward the baseline near the foot of the forward wave. There is no anodic process at all in this potential region. If the reverse scan is continued into a far more positive region, it might be possible to observe a totally irreversible oxidation wave for R, but it would be far separated from the reduction wave, because both kinetic branches require large activation overpotentials. In most cases of totally irreversible heterogeneous kinetics, one cannot see waves for both oxidation and reduction. If one is visible, the other usually is obscured by a limiting background process. In cases where both processes are visible, the CV is essentially the sum of individual anodic and cathodic LSV scans.

7.5 Multicomponent Systems and Multistep Charge Transfers 7.5.1

Multicomponent Systems

The consecutive reduction of two substances O and O′ in a potential scan experiment is more difficult to analyze for LSV and CV than for SSV or STV (Sections 5.5 and 6.4) (16, 17). As for ′ ′ the earlier methods, we consider two reactions, O + ne → R and O + ne → R , with the second occurring at more extreme potentials. If O and O′ diffuse independently, their fluxes are additive and the i − E curve for the mixture is the sum of the individual i − E curves of O and O′ (Figure 7.5.1). However, the measurement of i′p in LSV must be made using the decaying current

7.5 Multicomponent Systems and Multistep Charge Transfers

i 3

i′p O + ne → R

1

2

0

100

0

–100

E/mV

O′ + n′e → R′

–200

–300

–400 mV ′

Figure 7.5.1 Voltammograms for solutions of (1) O alone; (2) O′ alone and, (3) both O and O′ , with n = n , ∗ = C ∗ , and D = D . CO O O′ O′

of the first wave as the baseline. Usually, this baseline is estimated by assuming that the current past the peak potential behaves like that for the large-amplitude potential step and decays as t −1/2 . A better fit based on an equation with two adjustable parameters has been suggested (17), but the procedure depends on the reversibility of the reactions and is cumbersome. Given the capabilities of electrochemical workstations, a better general approach may be to compare simulated and experimental results. It is practical to include all relevant elements, including a nonfaradaic baseline, in the simulations. This issue of defining the baseline for a second peak reinforces a prior message [Section 7.2.2( f )] that LSV and CV do not lend themselves to precise quantification. While they are remarkably powerful for behavioral diagnosis, other methods may be better when precision is a goal, especially when peaks overlap. Fortunately, overlap is not always a problem. Multicomponent systems sometimes provide CV responses with well-separated peaks, as in Figure 7.5.2. Problem 7.5 offers the reader an opportunity to interpret the details of this voltammogram. 7.5.2

Multistep Charge Transfers

The situation is more complicated for the stepwise reduction of a single substance O, e.g., ′

O + n1 e → R1

E10

R1 + n2 e → R2

0′

E2

(7.5.1) (7.5.2)

The nature of the i − E curve depends on (a) the difference between the two formal potentials, ′ ′ ′ ΔE0 = E20 − E10 , (b) the reversibility of each step, and (c) n1 and n2 . Just to illustrate the range ′ of behavior, Figure 7.5.3 provides calculated cyclic voltammograms for different values of ΔE0 in a system with two 1e reversible steps. Only Figure 7.5.3a resembles the behavior considered in Chapters 5 and 6 for multistep charge transfers in SSV and STV. Because of the complexity of these multistep charge transfers, which may also involve homogeneous electron-transfer reactions among the participants, we defer detailed discussion for Chapter 13.

333

7 Linear Sweep and Cyclic Voltammetry

10

5 i/μA 0

–5

–10 1.0

0.0

–1.0 E/V vs. QRE

–2.0

Figure 7.5.2 Cyclic voltammogram of 1 mM benzophenone (BP, Figure 1) and 1 mM tri-p-tolylamine (TPTA, Figure 1) in acetonitrile with 0.1 M TBABF4 . Scan begins at 0.0 V vs. QRE and first moves positively. QRE is near −0.03 V vs. SCE. v = 100 mV/s. [Based on Michael (18).]

1.0

(a)

(c)

(b)

(d)

Figure 7.5.3 Cyclic voltammograms for a reversible two-step system at 25 ∘ C with n2 /n1 = 1. Dimensionless current is analogous to 𝜋 1/2 𝜒(𝜎t), ′ defined in (7.2.18). Values of ΔE 0 : (a) −180 mV, (b) −90 mV, (c) 0 mV, (d) 180 mV. [Polcyn and Shain (17). © 1966, American Chemical Society.]

0.5 0.0 Dimensionless current

334

–0.5

1.0 0.5 0.0 –0.5

0

–200 200 n(E – E10′)/mV

0

7.6 Fast Cyclic Voltammetry Routine CV at conventional microelectrodes typically involves scan rates of 10 mV/s to 1 V/s, which is a range sufficiently fast and broad for convenient investigation of many systems, but also slow enough to limit the experimental complications attending higher v. There has always been interest in extending the practice of CV to higher scan rates, both to gain access to shorter

7.6 Fast Cyclic Voltammetry

timescales and to achieve briefer measurement cycles. However, the idea of “fast cyclic voltammetry,” often expressed in the literature, can mean rather different things in different contexts, depending on the size of the electrode and the complexity of the medium. • With conventional microelectrodes in simple solutions, “fast CV” involves scan rates of perhaps 5–100 V/s and total sweep times on the order of 0.2 s to 10 ms. The experimental limitations are usually imposed by iRu and Ru C d . • For investigators interested in CV for in vivo electroanalysis at an implanted UME (e.g., in the brain of a rat), scan rates are often in the range of 10–1000 V/s, and the main goal is to observe changes over a time period of seconds using background-corrected CVs taken several times per second (19–22) (Section 17.8.4). Time requirements for appropriate sampling by CV and the kinetics of the electrode process determine the optimal scan rate and its practical limits. • Investigators using CV to examine the kinetics of very fast electrode processes (21–26) employ UMEs and use scan rates that may exceed 1 MV/s, providing access to events in the high nanosecond range. The upper limit of scan rate is imposed by (a) the realizable cell time constant (including effects of stray capacitance; Section 6.7), (b) instrumental bandwidth (Section 16.9), and (c) the ability to tolerate a very large capacitive background (Figure 7.6.1). The time scale of a potential sweep experiment, 𝜏 obs , can be understood as the period required for the potential to traverse voltammetric peak. Although there is no precise definition of peak width, 𝜏 obs is often taken as RT/Fv (Section 13.2.2), which, at 25 ∘ C, is the sweep time for 25.7 mV (covering kT for a 1e process). As (7.2.24) shows, |Epc − Epc/2 | is ∼60 mV at 25 ∘ C for a 1e reversible process and ∼30 mV for a 2e process, so this definition of 𝜏 obs is of the right order. Table 7.6.1 provides a few values of 𝜏 obs vs. v. If one is to obtain experimental results that remain undistorted by inability of the working electrode to follow the potential program, the cell time constant must be significantly shorter than 𝜏 obs (ideally by an order of magnitude or more). Section 6.7 covers cell time constants in detail and shows that a UME is essential for CV at the fastest sweep rates in Table 7.6.1. The shortest well-documented cell time constants are in the range of 10 ns (Section 6.7); therefore, the fastest meaningful sweep experiments would feature 𝜏 obs ∼ 100 ns, or v ∼ 3 MV/s. Sweep rates approaching this figure have been used successfully (25, 27). Special instrumentation with 10

i/μA

2

0

0

–2 –10 –2.0

–1.5 (a)

–1.0

–2.0 E/V vs. QRE

–1.5

–1.0

(b)

Figure 7.6.1 CV at v = 1.96 MV/s for 14.3 mM anthracene (An) in MeCN with 0.9 M TEABF4 at a gold disk (r0 = 2.5 ± 0.1 μm). (a) As recorded in the absence (dashed curve) and presence (thick solid curve) of An. Difference is the lighter curve in the middle. (b) Difference voltammogram from (a) for the region of response from An. The experimental result (thick solid curve) is compared with a theoretical voltammogram (dashed) for ′ E 0 = 1.61 V vs. QRE, 𝛼 = 0.45, k0 = 5.1 cm/s, and D = 1.6 × 10−5 cm2 /s. Positive potentials to the right; anodic currents are positive (up). The QRE is near −0.25 V vs. Ag/AgCl. Ringing in the voltammograms is caused by the high level of resistance compensation [Section 16.7.3(a)]. [Amatore et al. (25, 27).]

335

336

7 Linear Sweep and Cyclic Voltammetry

Table 7.6.1 Experimental Time Scales vs. Scan Rate for Potential Sweep Experiments. v

25 mV/s

100 mV/s

1 V/s

10 V/s

1000 V/s

100 kV/s

1 MV/s

𝜏 obs (a)

1.0 s

260 ms

26 ms

2.6 ms

260 μs

2.6 μs

260 ns

(a) 𝜏 obs = RT/Fv at 25 ∘ C.

extraordinary bandwidth and very careful design of electrodes, cells, and electrical connections are required for such work (Section 16.9). Because measured currents at a UME are small, the uncompensated ohmic drop does not perturb the response to the same extent as with larger electrodes. However, even with the UME, (7.2.29) applies, so the faradaic wave lies on top of a capacitive current that increases linearly with v. To extract the desired information from the voltammogram, the total response (capacitive plus faradaic) can be simulated (28) or perturbations caused by C d and Ru can be subtracted. Alternatively, positive feedback circuitry with a fast response can be used to compensate for distortions otherwise caused by Ru (27). An important practical limitation of very fast voltammetry is the sensitivity of the method to adsorption of even small amounts of electroactive substance or to faradaic changes involving the electrode surface (e.g., formation of an oxide layer). For surface processes like these, the current varies directly with v (Section 17.2); therefore, surface effects of minor importance at small v can dominate at high scan rates.

7.7 Convolutive Transformation Voltammograms obtained by LSV or CV can be mathematically transformed to wave-like forms resembling SSV responses, and this treatment sometimes leads to easier interpretation. The transformation makes use of the convolution principle, (A.1.21). In Section 7.2.1(a), we showed that the general formulation for semi-infinite linear diffusion yields (7.2.8) for any electrochemical technique. It can be re-expressed, using (7.2.7), as ∗ − [nFA(𝜋D )1∕2 ]−1 CO (0, t) = CO O

t

∫0

i(𝜏)(t − 𝜏)−1∕2 d𝜏

(7.7.1)

Let us define I(t) as the following convolutive transformation of i(t): I(t) =

t

1

𝜋 1∕2 ∫0

i(𝜏)(t − 𝜏)1∕2 d𝜏

(7.7.2)

Thus, (7.7.1) becomes (29) ∗ − CO (0, t) = CO

I(t) 1∕2

(7.7.3)

nFADO

Under diffusion-controlled conditions, where C O (0, t) = 0, I(t) reaches a limiting value, I l : 1∕2

∗ Il = nFADO CO

(7.7.4)

Thus, CO (0, t) =

Il − I(t) 1∕2

nFADO

(7.7.5)

7.7 Convolutive Transformation

Similarly, for species R, assumed absent initially, the corresponding expression, derived from (7.2.9), is CR (0, t) =

I(t)

(7.7.6)

1∕2

nFADR

These relationships hold for any form of signal excitation in any electrochemical technique applied under conditions in which the general formulation (Section 4.5.2) applies. No assumptions have been made concerning the reversibility of the charge-transfer reaction. Equations 7.7.5 and 7.7.6 have the same forms as the corresponding generalized relationships 1∕2 1∕2 governing SSV (Section 5.2.5). In this case, mO = DO and mR = DR (30, 31).11 If the electron-transfer reaction is nernstian, the application of (7.7.5) and (7.7.6) yields E = E1∕2 +

RT Il − I(t) ln nF I(t)

(7.7.7)

′

Dimensionless current, 𝜋1/2𝜒(𝜎t)

where E1/2 = E0 + (nF/RT) ln(mR /mO ). This relationship is identical in form with the analogous results for SSV and STV [(5.3.4) and (6.2.14), respectively]. Thus, we see that the convolutive transformation of an LSV response converts the peaked voltammogram to a sigmoidal wave (Figure 7.7.1). Convolution is normally a digital post-treatment of CV results acquired in the conventional manner. Several different algorithms have been proposed (32, 33). The second edition presents options.12

1.0

1.0 (b)

0.5

0.5 (a)

I(t) Il

0.0

0.0 + 5RT/nF

0

– 5RT/nF

E – E1/2

Figure 7.7.1 Comparison of (a) LSV and (b) convolutive LSV. Curve a is the essentially as in Figure 7.2.1. For curve b, Il is defined in (7.7.4). [Adapted from Imbeaux and Savéant (29). © 1973, Elsevier.] 11 Following the definition of Riemann–Liouville operators, I(t) can be considered as the semi-integral of i(t), generated by the operator d–1/2 /dt –1/2 , so that (30, 31) d−1∕2 i(t) = m(t) = I(t) dt −1∕2 Both m(t) and I(t), which represent the expression on the right in (7.7.2), have been used in presentations of this technique. The convolutive and semi-integral approaches are equivalent. 12 Second edition, Section 6.7.2.

337

338

7 Linear Sweep and Cyclic Voltammetry

i/mA

0.4

25

0.2

20

0.0

15 I(t)/μC s–1/2 10

–0.2

5

–0.4

0 –1.0

–1.1

–1.2 –1.3 E/V vs. SCE (a)

–1.4

Il Ii = 0 5 × 10–5

5 × 10–6

i

0

0

I

Ei = 0 –0.9

–1.1 –1.3 –1.5 E/V vs. Ag/AgI

–1.7

(b)

Figure 7.7.2 Experimental cyclic voltammogram and convolution of (a) 1.84 mM p-nitrotoluene in MeCN containing 0.2 M TEAP at an HMDE, v = 50 V/s. [Whitson, Vanden Born, and Evans (33). © 1973, American Chemical Society.] (b) 0.5 mM tert-nitrobutane in DMF containing 0.1 M TBAI, v = 17.9 V/s. E 1/2 determined for quasireversible system from E 1/2 = E i = 0 − (RT/F) ln[(Il – Ii = 0 )/Ii = 0 ]. [Savéant and Tessier (34). © 1975, Elsevier.]

The convolutive approach has been applied in studies of electrode kinetics because one need not assume that any particular kinetic model applies; thus, there is freedom to explore for variances from classic Butler–Volmer expectations. Convolutive CV was used to examine the irreversible electroreductions of many aliphatic nitro compounds in aprotic solvents, and significant departures from linearity (34, 35) were observed in plots of ln k f (E) vs. E. In general, 𝛼 appeared to be potential-dependent in the manner predicted by Marcus kinetics [Section 3.5.4(c)]. For a reversible reaction, the forward and backward convolutive CV scans should superimpose, with I(t) returning to zero at sufficiently positive potentials, where C R (0, t) = 0. This behavior has been verified experimentally (10, 29, 33) (Figure 7.7.2a). However, for a quasireversible reaction, the forward and backward I(t) curves should not coincide (Figure 7.7.2b). One can regard this effect as an alternative manifestation of the shifting

7.8 Voltammetry at Liquid–Liquid Interfaces

of the Epc and Epa values from their reversible values in CV.13 The procedure used to obtain the “reversible” E1/2 value for such a system is shown in Figure 7.7.2b. Rigorous correction for the uncompensated resistance, Ru , is much more straightforward for I(t) − E curves (10, 29, 33) than for the i(t) − E curve. As we have seen [Sections 7.2.1(e) and 7.2.2(e)], Ru compromises the linearity of the sweep; hence, it becomes difficult to accurately convert i(t) to i(E). The linkage is easier for convoluted currents, because I(E) does not depend on details of the excitation function, E(t). To correct for the effects of Ru , one first computes I(t) in the normal manner on a time base. It is then converted to a potential base simply by identifying each data point with a potential calculated from (7.2.41) and (7.2.42). In this process, iRu must be computed for each time by multiplying the experimentally determined value of i at that time with a value for Ru obtained separately (Section 16.7.3). Correction for charging current can be accomplished by the method of Section 7.2.1(f ) before the convolution is performed.

7.8 Voltammetry at Liquid–Liquid Interfaces In Section 2.3.6, we considered ion transfer at an interface between two immiscible electrolyte solutions (ITIES), where we found that a potential difference can arise because the transfers of different ions involve different energy changes. In the earlier context, we considered only the junction potential formed spontaneously at the ITIES. However, ion movement across the interface can also be actively driven by applying an external potential, and the rate of ion transfer can be detected as a current flow. In this way, one can study an ITIES with voltammetric methods (36–42). According to (2.3.46), the junction potential between phases 𝛼 and 𝛽 is 𝛽

𝛽

𝜙𝛽 − 𝜙𝛼 = Δ𝛼 𝜙 = (−1∕zj F)[ΔGtransfer,j + RT ln(aj ∕a𝛼j )] 0a→𝛽

(7.8.1)

0𝛼→𝛽

where ΔGtransfer,j is the standard Gibbs free energy required to transfer ionic species j, hav0𝛼→𝛽

ing charge zj , between the two phases. It is convenient to re-express ΔGtransfer,j as a standard 𝛽

potential difference, Δ𝛼 𝜙0j , the standard Galvani potential of ion transfer for species j from 𝛼 to 𝛽: 𝛽

0𝛼→𝛽

Δ𝛼 𝜙0j ≡ (−1∕zj F)ΔGtransfer,j

(7.8.2)

which allows (7.8.1) to be written in a nernstian form: 𝛽

𝛽

𝛽

Δ𝛼 𝜙 = Δ𝛼 𝜙0j + (RT∕zj F) ln(a𝛼j ∕aj )

(7.8.3)

13 This effect might cause one to wonder about the comment above that the principal results from the treatment of SSV also apply to convolutive LSV and CV. The forward and backward curves do overlap for a quasireversible system in SSV. Quasireversibility is determined in SSV by the critical dimension of the electrode, and the response is independent of scan rate or scan direction, as long as the scan is slow enough for steady state to be maintained. In convolutive CV, quasireversibility is determined by the timescale, which is set by the scan rate. The forward and reverse responses are displaced from each other to a degree that depends on scan rate. In convolutive CV, this information is encoded in I(t). Although equations of the same form describe SSV and convolutive LSV and CV, those for SSV are built on a response that does not depend on timescale, while those for CV are constructed on a response that does.

339

340

7 Linear Sweep and Cyclic Voltammetry 𝛽

0𝛼→𝛽

As shown in Section 2.3.6, ΔGtransfer,j , and, hence, Δ𝛼 𝜙0 , can be obtained from thermochemical, solubility, or potentiometric measurements (with an extrathermodynamic assumption). 7.8.1

Experimental Approach to Voltammetry 𝛽

In a suitable cell, one can vary the potential difference across the ITIES, Δ𝛼 𝜙, causing the activities of species in the two phases to change according to (7.8.3) and producing a measurable current as ions cross the interface. An experimental system for such experiments is shown in Figure 7.8.1. It features a four-electrode cell with working and counter electrodes on opposite sides of the ITIES and two reference electrodes, Ref 𝛼 and Ref 𝛽, also on opposite sides. The variable voltage source, V , allows alteration of the potential difference across the ITIES. Any current flowing between the working and counter electrodes and can be measured as indicated. The two reference electrodes and the high impedance voltmeter constitute a measurement path for potential, which can be represented as Ref 𝛼∕(H2 O) Li+ , Cl− ∕(nitrobenzene) TBA+ , TPB− ∕Ref 𝛽 𝛽

Δ𝛼 𝜙

Ej,𝛼

(7.8.4)

Ej,𝛽

where TBA+ is tetra-n-butylammonium, and TPB– is tetraphenylborate. If the two reference electrodes are identical, the measured potential difference is 𝛽

ΔEref = ERef,𝛽 − ERef,𝛼 = Δ𝛼 𝜙 − Ej,𝛼 + Ej,𝛽

(7.8.5)

where the last two terms are junction potentials at the reference electrode tips, measured from the interior electrolyte of the reference electrode to the outside solution.14 One can develop a voltammogram by varying the voltage V , measuring i and ΔEref , and plotting i vs. ΔEref . Since Ej, 𝛽 and Ej, 𝛼 can be expected to remain invariant during an experiment, 𝛽

𝛽

ΔEref is Δ𝛼 𝜙 plus a constant; therefore, changes in ΔEref reflect changes in Δ𝛼 𝜙. Figure 7.8.2 provides examples of ion-transfer cyclic voltammograms recorded essentially in this manner. They will be discussed below, after we develop some additional ideas. Counter

Figure 7.8.1 Schematic diagram of apparatus for voltammetry at an ITIES between water and nitrobenzene. TBA+ is tetra-n-butylammonium, TPB– is tetraphenylborate.

Ref 𝛼

i

ITIES

V

Phase 𝛼, H2O 0.01 M Li+ Cl–

ΔEref

Phase 𝛽, nitrobenzene 0.01 M TBA+ TBP–

Working

Ref 𝛽

14 The measured value would also include any ohmic drop between the tips of each reference electrode arising from current in the cell. By careful positioning of the reference electrodes, the ohmic component can often be rendered insignificant. We ignore this term here.

7.8 Voltammetry at Liquid–Liquid Interfaces

100 TMA+(aq) → TMA+(nb)

(b)

50 i/μA

(a) 0

–50

TMA+(nb) → TMA+(aq)

–100 100

200

300

400

500

Δϕ/mV

Figure 7.8.2 CV at an ITIES. (a) i − E curve for 0.1 M LiCl in an aqueous phase and 0.1 M TBATPB in nitrobenzene. (b) With addition of 0.47 mM TMA+ to the aqueous phase. v = 20 mV/s. Δ𝜙 = − ΔE ref , as defined in (7.8.5). It 𝛽

includes junction potentials at both reference electrode tips such that Δ𝜙 ≈ 300 − Δ𝛼 𝜙 mV. The aqueous phase becomes relatively more positive as the scan extends to the right, and a positive current represents transfer of TMA+ from aqueous phase to nitrobenzene. [Vanýsek (40). © 1995, Elsevier.]

The process for recording a voltammetric curve has just been described in manual terms, so the elements of the measurement can be easily followed; however, one can construct a four-electrode potentiostat that can control ΔEref and carry it through a program (such as a cyclic sweep). Such instrumentation is described in Section 16.4.5 and is routinely used in this field. 7.8.2

Effect of Interfacial Potential on Composition

Let us now turn to the way in which the equilibrium composition on either side of an ITIES 𝛽 0𝛼→𝛽 𝛽 depends on Δ𝛼 𝜙. Table 7.8.1 provides values of ΔGtransfer,j and Δ𝛼 𝜙0j for the species in the 𝛽

system of Figure 7.8.1. One can use these data to calculate a𝛼j ∕aj for each ion in the two phases 𝛽

for any value of Δ𝛼 𝜙. By assuming that each ion undergoes equilibrium partitioning in very thin zones on either side, one can develop a distribution diagram for the interfacial region like that 𝛽 in Figure 7.8.3a (Problem 7.11). With low potential applied across the interface (−250 < Δ𝛼 𝜙 < 150 mV), the hydrophilic salt, LiCl, resides almost totally in the aqueous phase, while TBATPB, being hydrophobic, remains in the nitrobenzene. 7.8.3

Voltammetric Behavior 𝛽

If the potential across the ITIES, Δ𝛼 𝜙, is varied by applying a voltage between the working and counter electrodes, as discussed for Figure 7.8.1, ions may begin to move across the interface.

341

342

7 Linear Sweep and Cyclic Voltammetry

Table 7.8.1 Gibbs Energy of Transfer and Standard Interfacial Potential Differences for Ion Transfer Between Water (𝛼) and Nitrobenzene (𝛽)(a) . Ion(b)

𝚫G

0𝜶→𝜷 transfer, j

(kJ mol−1 )

𝜷

𝚫𝜶 𝝓0 (mV) j

Li+

38.2

−396

Cl−

43.9

455

TBA+

−24.7

256

TPB–

−35.9

−372

4.0

−41

TMA+

(a) A positive Gibbs free energy change implies that work must be performed to move the species from the aqueous phase to the nitrobenzene. For cations, a negative Δ𝛽𝛼 𝜙0j goes with a greater tendency to remain in the

aqueous phase, while for anions a positive Δ𝛽𝛼 𝜙0j shows greater tendency to remain in the aqueous phase. (b) TBA+ , tetra-n-butylammonium; TPB+ , tetraphenylborate; TMA+ , tetramethylammonium. From Samec, Mareˇcek, Koryta, and Khalil (36). 𝛽

Making the aqueous phase relatively more positive (a more negative Δ𝛼 𝜙) tends to drive Li+ into the nitrobenzene and TPB– into the aqueous phase. A relatively more negative potential 𝛽 in the aqueous phase (producing a more positive Δ𝛼 𝜙) can drive Cl− into the nitrobenzene or TBA+ into the aqueous phase. Equilibrium can be achieved and maintained in the immediate vicinity of the ITIES if the transfer kinetics are fast enough (i.e., if the system is reversible). Assuming reversibility, one can calculate the relative interfacial activities of the ions using 𝛽 (7.8.3) for any Δ𝛼 𝜙. For the system of Figure 7.8.1, one can see in Figure 7.8.3 that no appreciable ionic transfer 𝛽 occurs in the potential range of about −0.2 to 0.15 V, because the ratios a𝛼j ∕aj for the different ions are such that Li+ and Cl− are negligibly in the nitrobenzene, and TBA+ and TPB– are negligibly in the water. When the potential is swept beyond these limits, ion transfer starts to 𝛽 happen. At the positive extreme of Δ𝛼 𝜙, TBA+ moves into the aqueous phase, and Cl− moves into the nitrobenzene. At the negative extreme, TPB– moves into the water, and Li+ moves into the nitrobenzene. Since these ionic movements represent passage of net charge across the interface, they produce a current in the external circuit. A voltammogram for this system is shown in Figure 7.8.2a. The limits shown are controlled 𝛽 by the movement of TBA+ into the water at Δ𝛼 𝜙 > 100 mV and the movement of TPB– into 𝛽 the water at Δ𝛼 𝜙 < 200 mV. In effect, the ITIES behaves between those limits as an ideally 𝛽 polarizable interface (Section 1.6.1). In any system, this window is set by the values of Δ𝛼 𝜙0j for 0𝛼→𝛽

the ions present in the two phases. Given typical maximum values of ΔGtransfer,j , it is generally narrower than 0.6–0.7 V (36). 0𝛼→𝛽 If an ion with a ΔGtransfer,j of smaller magnitude is introduced into one of the phases, it can 𝛽

transfer at values of Δ𝛼 𝜙 within the potential window allowed by the other ions. For example, when tetramethylammonium ion (TMA+ ) is added to the aqueous phase, it transfers more readily than Li+ from water to nitrobenzene (Figure 7.8.3b). If the concentration of this ion is small (typically 0.1–1 mM), the rate of its transfer across the interface is usually limited by the 𝛽 rate of mass transfer of the ion to the interface, and a recording of i vs. Δ𝛼 𝜙 resembles a typical cyclic voltammogram (Figure 7.8.2b).

7.8 Voltammetry at Liquid–Liquid Interfaces

(a) 0.01 TBA+ Cl–

Concentration in H2O(M)

TPB– Li+ 0.00 –800

–400

0 Δϕ𝛽𝛼 /mV

400

800

400

800

(b) 0.01 TMA+

0.00 –800

–400

0 Δϕ𝛽𝛼 /mV

Figure 7.8.3 (a) Distribution of ions near the ITIES in Figure 7.8.1 vs. interfacial Galvani potential difference, 𝛽 Δ𝛼 𝜙. In the middle range, LiCl is largely in the aqueous phase, and TBATPB is in the nitrobenzene. (b) Distribution for 0.01 M TMA+ in the same system.

Ion-transfer voltammograms show the same characteristics as nernstian faradaic LSV and CV responses—Ep independent of v, ip proportional to v1/2 , ΔEp ≈ 59/|zj | mV, and ip ∝ Cj∗ . Measurements are, however, often complicated by uncompensated resistance because of the low conductivity of the organic phase. In almost all cases investigated, the voltammograms have proven reversible, showing no effect of slow ion-transfer kinetics across the interface. Accordingly, voltammetric measurements can be used to find Gibbs energies of transfer, diffusion coefficients, and solution concentrations. By exploiting ITIES, voltammetric electrodes can be prepared for ions that are not faradaically electroactive. For example, a voltammetric lithium ion electrode can be fabricated based on Li+ transfer between water and o-nitrophenylphenylether containing a crown ether to improve the specificity of ion transfer (43). Ionic transfer at the ITIES also affects the electron-transfer voltammetry of three-phase systems in which an electrode is brought into contact with an ITIES. An illustration is provided in Figure 7.8.4, featuring a cylindrical UME placed in different locations relative to an ITIES between 1,2-dichloroethane (DCE) and H2 O (44). The DCE phase contained ferrocene and TBAPF6 ; the H2 O, only NaCl. Figure 7.8.4a shows results when the wire is placed in the interior of either phase. There is a reversible voltammogram for Fc in the DCE, but only a residual current in the aqueous phase, where Fc is practically insoluble. The behavior is very different when the wire UME is placed through the ITIES. The forward scan is the same as in the DCE phase alone, but the reversal scan and the second forward scan show a response for Fc+ /Fc in the aqueous phase. During the first forward scan, some of the Fc+ generated electrolytically crosses the interface, where it can be interrogated voltammetrically. This kind

343

344

7 Linear Sweep and Cyclic Voltammetry

10

In DCE phase In H2O phase

20 15

8 6

10 i/μA 5

i/μA

0

1st cycle 2nd cycle

4 2 0

–5

5 μA

–10 –15 0.0

5 μA

–2 –4

0.2

0.4

0.6

0.8

1.0 0.0 0.2 E/V vs. Ag/AgCl

(a)

0.4

0.6

0.8

1.0

(b)

Figure 7.8.4 Voltammetry of ferrocene (Fc) in a system with an ITIES between water and 1,2-dichloroethane (DCE). Working electrode was a Pt–Ir wire of 0.25 μm dia., 2–3 mm long. (a) Pt–Ir wire in the individual phases. (b) Pt–Ir wire penetrating the ITIES. In DCE: 5 mM Fc, 0.1 M TBAPF6 ; in H2 O: 0.1 M NaCl, v = 100 mV/s. [Adapted from Weatherly et al. (44). © 2019, American Chemical Society.]

of work can illuminate dynamics at a complex interface, including effects caused by the ion populations on either side of the ITIES. Electrochemistry at three-phase interfaces has received significant attention, both in fundamental research and with respect to applications, especially in electrosynthesis and electroanalysis (44, 45).

7.9 Lab Note: Practical Aspects of Cyclic Voltammetry This note focuses on appropriate experimental conditions for cyclic voltammetry, which is often the introductory laboratory activity for students and new investigators. At the bench, one faces tactical choices—e.g., of working electrode, system composition, scan limits, and experimental timescale. Using CV as a context, Sections 7.9.1–7.9.3 address relevant considerations, most of which apply broadly in electrochemical work. A beginner’s guide to CV is available elsewhere (46). 7.9.1

Basic Experimental Conditions

Lab Notes in earlier chapters have already provided practical information about microelectrodes, including UMEs. For voltammetry, one needs a reproducibly prepared electrode surface compatible with the needs of the investigation. The electrode material must provide the desired working range, as well as appropriate chemical properties toward the species in solution. One usually prefers that the electrode be chemically inert, but sometimes specific chemical interactions involving the electrode are at the focus of an investigation (as in electrocatalysis, Chapter 15). Ordinarily, the electrode should be no larger than necessary to secure the chosen mass-transfer conditions, typically either semi-infinite linear diffusion with minimal edge effect or steady-state diffusion/migration. A larger electrode than needed passes more current than needed, which leads to excessive ohmic drop. It also creates a superfluous cell time constant. These effects complicate measurement and interpretation, so it is good practice to minimize them with a properly sized electrode. For an ordinary voltammetric investigation, the concentration of the electroactive species can be as high as 10 mM, but a wiser choice is usually in the range of 0.5–5 mM. A concentration below 0.5 mM will degrade both signal to noise and signal to background. A higher

7.9 Lab Note: Practical Aspects of Cyclic Voltammetry

concentration than 5 mM brings no advantage. It does not appreciably improve signal quality, but requires added supporting electrolyte, which is wasteful, sometimes expensively so. If the concentration of the electroactive species exceeds 10 mM, the electrode reactions can appreciably change the solution density near the electrode, sometimes leading to convective disruption of the diffusion layer and erratic results. Of course, concentrations much higher than 10 mM or much lower than 0.1 mM can be used if the investigation is focused on those concentration ranges (e.g., as in Sections 5.8 and 8.6). In most voltammetric investigations, one desires that diffusion be the only mode of mass transfer; therefore, the solution must be quiescent before and during a voltammetric scan, and an excess electrolyte must be present. The rule of thumb for suppression of migration is that the concentration of the supporting electrolyte be at least 100 times that of the electroactive species. If the solute of interest is at 1 mM, for example, then 0.1 M supporting electrolyte is usually adequate. One might choose a higher concentration if some additional chemical purpose is served, such as participation by the supporting electrolyte in complexation or control of pH. The scan rate for CV or LSV should be “conveniently fast,” meaning that the scan should not be so slow or so fast that it violates mass-transfer requirements, wastes the investigator’s time, or needlessly complicates measurement or interpretation. • With a conventional microelectrode, a common scan rate is 100 mV/s. Much faster scans (by an order of magnitude or more) do not take appreciably less experimental time, but involve larger overall currents, larger background currents, and more ohmic drop. Much slower scans (again, by an order of magnitude or more) certainly have a cost in time, but also risk disruption of mass transfer by natural convection (Section 9.3.4). • With a UME operating in a steady-state mode, “conveniently fast” is a term that depends on the size of the electrode. For a radius of 1–10 μm, it usually means a scan rate as low as 10–20 mV/s. The critical concern is the steady-state renewal time of the electrode (Section 5.1.3). If the scan is too fast, the desired mass-transfer conditions are not fulfilled, and interpretive errors arise. If the scan is too slow, time is wasted. • With a UME operating in the mode of semi-infinite linear diffusion, one must avoid scan rates so slow that edge diffusion becomes significant. In general, the experimental duration (total potential span in both directions for CV divided by scan rate) must be less than about 4 × 10−4 𝜏 ss , where 𝜏 ss is the steady-state renewal time (Section 5.1.3). For a UME disk with r0 = 10 μm, the value of 𝜏 ss is 1.3 s (Table 5.1.1), so the experimental duration must not exceed 0.5 ms. If the total span is 1 V, for example, the scan rate must be at least 2000 V/s. Faster scans can be tolerated until ohmic drop, background correction, or the cell time constant complicate interpretation unreasonably. 7.9.2

Choice of Initial and Final Potentials

Investigators often ask themselves how to choose the initial and final potentials (or the switching potential, E𝜆 , in the case of CV) for a given experiment. These potentials define the processes to which the electrode and adjacent solution are subjected, so the choices should be made intelligently. When a potential is first applied to the working electrode,15 one normally desires that the system remain electrochemically inactive, passing no faradaic current and making no chemical alterations of the region near the working electrode. There are good reasons: 15 When the potentiostat actually applies a potential, not when the electrical leads from the potentiostat are physically connected to the cell.

345

346

7 Linear Sweep and Cyclic Voltammetry

• Most theory used for interpretation of results is built on the assumption that the bulk composition applies at the electrode when the experiment starts. Current flow at the initial potential can invalidate this condition. • Extensive current flow before an experimental run can contaminate the diffusion layer with deleterious species. • Current flow at the initial potential can damage the electrode surface by deposition of a passive layer or by reconstruction of the surface. Investigators often default to 0.0 V as their initial potential. This is not a sound practice, because 0.0 V is just another point on an arbitrarily chosen potential scale. Depending on the details of the system, significant current can flow at that potential. In some cases, it can even lie outside the available potential window. The only completely safe approach is to set the initial potential at the zero-current potential (the open-circuit potential) for the system. If both redox forms of a facile couple are present, the zero-current potential is also the equilibrium potential and will be well defined. If the system is not poised by a redox couple, the zero-current position is typically vaguely located within a range of low background current. In either case, one can measure the zero-current potential vs. the reference electrode by using a high-impedance voltmeter. Automated potentiostats can often make this measurement upon command. In some systems, no initial potential exists where all electroactive species of interest remain inactive, and one has no choice but to interpret results in the light of the electroactivity at the initial potential. Figure 8.6.2 provides examples in the context of pulse voltammetry. The final (or switching) potential also needs to be chosen with care. Investigators often develop the bad habit of driving the experiment into a background limit. The urge to do this should be resisted. Background reactions tend to contaminate the zone near the electrode, often generating substances that react with the species of primary interest or that damage the electrode surface. Figure 7.9.1 shows an example in which ferrocene is oxidized to ferrocenium at a Pt disk. The electrode reaction is reversible and the participants are stable, so the CV is essentially ideal, as long as the scan is limited to the wave for Fc+ /Fc (Figure 7.9.1a). If, instead, one drives the forward scan into the anodic limit (Figure 7.9.1b), the reversal scan is significantly altered by whatever happens at the electrode at very positive potentials. Unless there is a clear reason to drive into the background limit, one should keep the final or switching potential safely short of either limit. Just choose a final or switching potential that is sufficient to obtain the information of interest, e.g., a set of waves in CV. Figure 7.9.1 CV for ferrocene at a Pt disk (r0 = 0.5 mm), demonstrating the deleterious effect of scanning into the positive background limit. (a) E 𝜆 = 0.7 V and (b) E 𝜆 = 1.85 V. Scans begin at 0.2 V and first move positively. Curves (a) and (b) overlap on the forward scan. 5 mM Fc and 0.1 M TBAPF6 in 1,2-dichloroethane (DCE); v = 100 mV/s. [Courtesy of C. K. T. Weatherly, University of Utah. 2020.]

40 20 (b) (a)

i/μA 0 (b)

(a)

–20 –40 –60 2.0

1.6

1.2 0.8 0.4 E/V vs. Ag/AgCl/NaCl (sat′d)

0.0

7.10 References

Likewise, one should avoid repeated cyclic scanning, unless there is a clear experimental reason to undertake it. Repeated scanning risks fouling of the electrode surface and contamination of the solution near the electrode, even when the range of scan is limited to the features of interest. Changes in voltammetric behavior during repeated cycles may reflect no more than fouling or contamination and, consequently, have limited scientific value. In sum, be efficient with your electrochemical measurements. Do what you must, but not more. 7.9.3

Deaeration

Atmospheric oxygen interferes with many electrochemical measurements because O2 dissolves at appreciable concentrations in most solvents. In water, the air-saturated concentration is about 0.25 mM at room temperature. Molecular oxygen is electrolytically reduced in water to H2 O2 and H2 O, and in many aprotic solvents to superoxide, O2 −∙ ; thus, it can give rise to interfering voltammetric features. It also reacts homogeneously with many reduced species, often destabilizing the products of electrode reactions. For most electrochemical work, it is important to remove O2 and to maintain an inert atmosphere over the cell. Only when the entire focus is on the positive side of the potential scale—well positive of the potential where O2 would be reduced—might operation with exposure to the air be acceptable. The simplest approach to deaeration is simply to bubble the electrolyte in the cell with an inert gas (typically N2 or Ar) for 2–5 min. The cell must be fitted with a cap to define a controlled space over the solution. During bubbling, the inert gas flushes the air from that space. After bubbling, the flow of inert gas is fully diverted there to maintain an oxygen-free atmosphere. Figure 1.5.4a illustrates a common arrangement. A nitrogen inlet is shown there in an “up” position, allowing continuous refreshment of the headspace over the solution. Bubbling is accomplished just by pushing the inlet down into the electrolyte. A fuller discussion of deaeration by bubbling is available (47). While bubbling is adequate for most work with aqueous media, it is often inadequate for systems based on volatile solvents (e.g., MeCN or CH2 Cl2 ) or for very demanding situations (e.g., when one desires to preserve a reactive species for spectroscopic studies). In these cases, one must resort to a vacuum line or a glove box. A cell built for use with a vacuum line (e.g., Figure 1.5.4b) features a valved arm for connection of the headspace inside the cell to the vacuum. The deaeration procedure typically involves freeze-pump-thaw cycles. The electrolyte is frozen by immersion of the lower part of the cell into liquid nitrogen; then the valve is opened to the vacuum, and the cell is pumped to background pressure; finally, the valve is closed, and the cell is allowed to thaw. After a few cycles (typically three), the cell is disconnected from the vacuum line, and measurements can be made. Discussions of vacuum-line techniques in electrochemistry have been presented (47, 48). With a glove box, one attempts to maintain an oxygen-free, water-free environment on a sizable scale and to place all materials and apparatus inside. No separate deaeration of the cell is required. This is an effective strategy for work with air-sensitive materials. It has also been discussed in the literature (47, 49)

7.10 References 1 W. H. Reinmuth. Anal. Chem., 32, 1509 (1960). 2 J. E. B. Randles, Trans. Faraday Soc., 44, 327 (1948). 3 A. Ševˇcík, Collect. Czech. Chem. Commun., 13, 349 (1948).

347

348

7 Linear Sweep and Cyclic Voltammetry

4 5 6 7 8 9 10 11 12 13 14 15 16

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

R. S. Nicholson and I. Shain, Anal. Chem., 36, 706 (1964). W. H. Reinmuth, J. Am. Chem. Soc., 79, 6358 (1957). H. Matsuda and Y. Ayabe, Z. Electrochem., 59, 494 (1955). Y. P. Gokhshtein, Dokl. Akad. Nauk. SSSR, 126, 598 (1959). J. C. Myland and K. B. Oldham, J. Electroanal. Chem., 153, 43 (1983). A. C. Ramamurthy and S. K. Rangarajan, Electrochim. Acta, 26, 111 (1981). L. Nadjo, J.-M. Savéant, and D. Tessier, J. Electroanal. Chem., 52, 403 (1974). C. P. Andrieux, D. Garreau, P. Hapiot, J. Pinson, and J.-M. Savéant, J. Electroanal. Chem., 243, 321 (1988). J.-M. Savéant, Electrochim. Acta, 12, 999 (1967). W. M. Schwarz and I. Shain, J. Phys. Chem., 70, 845 (1966). R. S. Nicholson, Anal. Chem., 37, 1351 (1965). J. C. Siu, G. S. Sauer, A. Saha, R. L. Macey, N. Fu, T. Chauviré, K. M. Lancaster, and S. Lin, J. Am. Chem. Soc., 140, 12511 (2018). Y. P. Gokhshtein and A. Y. Gokhshtein, in “Advances in Polarography,” Vol. 2, I. S. Longmuir, Ed., Pergamon Press, New York, 1960, p. 465; Dokl. Akad. Nauk SSSR, 128, 985 (1959). D. S. Polcyn and I. Shain, Anal. Chem., 38, 370 (1966). P. R. Michael, PhD Thesis, University of Illinois at Urbana-Champaign, 1977. M. J. Ferris, E. S. Calipari, J. T. Yorgason, and S. R. Jones, ACS Chem. Neurosci., 4, 693 (2013). D. L. Robinson, A. Hermans, A. T. Seipel, and R. M. Wightman, Chem. Rev., 108, 2554 (2008). R. M. Wightman and D. O. Wipf, Acc. Chem. Res., 23, 64, 1990. R. M. Wightman and D. O. Wipf, Electroanal. Chem., 15, 267 (1989). X.-S. Zhou, B. W. Mao, C. Amatore, R. G. Compton, J.-L. Marignier, M. Mostafavi, J.-F. Nierengarten, and E. Maisonhaute, Chem. Commun., 52, 251 (2016). C. Amatore and E. Maisonhaute, Anal. Chem., 77, 303A (2005). C. Amatore, Y. Bouret, E. Maisonhaute, H. D. Abruña, and J. I. Goldsmith, C.R. Chim., 6, 99 (2003). C. Amatore in “Physical Electrochemistry—Principles, Methods, and Applications,” I. Rubinstein, Ed., Marcel Dekker, New York, 1995, p. 191. C. Amatore, E. Maisonhaute, and G. Simonneau, J. Electroanal. Chem., 486, 141 (2000). D. O. Wipf and R. M. Wightman, Anal. Chem., 60, 2460 (1988). J. C. Imbeaux and J.-M. Savéant, J. Electroanal. Chem., 44, 1969 (1973). K. B. Oldham and J. Spanier, J. Electroanal. Chem., 26, 331 (1970). K. B. Oldham, Anal. Chem., 44, 196 (1972). R. J. Lawson and J. T. Maloy, Anal. Chem., 46, 559 (1974). P. E. Whitson, H. W. VandenBorn, and D. H. Evans, Anal. Chem., 45, 1298 (1973). J. M. Savéant and D. Tessier, J. Electroanal. Chem., 65, 57 (1975). J. M. Savéant and D. Tessier, Faraday Discuss. Chem. Soc., 74, 57 (1982). Z. Samec, V. Mareˇcek, J. Koryta, and M. W. Khalil, J. Electroanal. Chem., 83, 393 (1977). D. W. M. Arrigan, G. Herzog, M. D. Scanlon, and J. Strutwolf, Electroanal. Chem., 25, 105 (2014). B. Liu and M. V. Mirkin, Anal. Chem., 73, 670A (2001). P. Vanýsek in “Modern Techniques in Electroanalysis,” P. Vanýsek, Ed., Wiley, New York, 1996, pp. 337–364. P. Vanýsek, Electrochim. Acta, 40, 2841 (1995). H. H. Girault, Mod. Asp. Electrochem., 25, 1 (1993). H. H. J. Girault and D. J. Schiffrin, Electroanal. Chem., 15, 1 (1989). S. Sawada, T. Osakai, and M. Senda, Anal. Sci., 11, 733 (1995).

7.11 Problems

44 C. K. T. Weatherly, H. Ren, M. A. Edwards, L. Wang, and H. S. White, J. Am. Chem. Soc.,

141, 18091 (2019). 45 F. Marken and J. D. Wadhawan, Acc. Chem. Res., 52, 3325 (2019). 46 N. Elgrishi, K. J. Rountree, B. D. McCarthy, E. S. Rountree, T. T. Eisenhart, and J. L.

Dempsey, J. Chem. Educ., 95, 197 (2018). 47 D. T. Sawyer, A. Sobkowiak, and J. L. Roberts, Jr., “Electrochemistry for Chemists,” 2nd ed.,

Wiley-Interscience, New York, 1995, pp. 264. 48 V. Katovic, M. A. May, and C. P. Keszthelyi in “Laboratory Techniques in Electroanalytical

49 50 51 52 53 54 55

Chemistry,” 2nd ed., P T. Kissinger and W. R. Heineman, Eds., Marcel Dekker, New York, 1996, Chap. 18. S. N. Frank and S.-M. Park in “Laboratory Techniques in Electroanalytical Chemistry,” 2nd ed., P T. Kissinger and W. R. Heineman, Eds., Marcel Dekker, New York, 1996, Chap. 19. K. M. Kadish, L. A. Bottomley, and J. S. Cheng, J. Am. Chem. Soc., 100, 2731 (1978). M. E. Peover and B. S. White, Electrochim. Acta, 11, 1061 (1966). J. L. Sadler and A. J. Bard, J. Am. Chem. Soc., 90, 1979 (1968). R. W. Johnson, J. Am. Chem. Soc., 99, 1461 (1977). M. V. Mirkin, T. C. Richards, and A. J. Bard, J. Phys. Chem., 97, 7672 (1993). I. Noviandri, K. N. Brown, D. S. Fleming, P. T. Gulyas, P. A. Lay, A. F. Masters, and L. Phillips, J. Phys. Chem., 103, 6713 (1999).

7.11 Problems 7.1

Show that (7.2.8) and (7.2.9) lead to 1∕2

1∕2

1∕2

∗D DO CO (0, t) + DR CR (0, t) = CO O

(7.11.1)

7.2

From the data in Table 7.4.1 plot the linear sweep voltammograms, i.e., 𝜋 1/2 𝜒(bt) vs. potential, for a one-step, one-electron process with k 0 = 10−5 , 10−7 , and 10−9 cm/s, given 𝛼 = 0.5, T = 25 ∘ C, v = 100 mV/s, and DO = 10−5 cm2 /s. Compare these results with those for a nernstian reaction shown in Figure 7.2.1.

7.3

Mueller and Adams (R. N. Adams, “Electrochemistry at Solid Electrodes,” Marcel Dekker, New York, 1969, p. 128) suggested that by measuring ip /v1/2 for a nernstian system by LSV, then carrying out a potential step experiment in the same solution at the same electrode to obtain the limiting value of it 1/2 , one can determine n without the ∗ , or D . Demonstrate that this is the case. Why would this method need to know A, CO O be unsuitable for an irreversible reaction?

7.4 The oxidation of o-dianisidine (o-DIA) occurs in a nernstian 2e reaction. For 2.27 mM o-DIA in 2 M H2 SO4 , LSV gives ip = 8.19 μA at a carbon paste electrode (A = 2.73 mm2 ) at v = 0.500 V/min. Calculate D for o-DIA. What ip is expected for v = 100 mV/s? For v = 50 mV/s and 8.2 mM o-DIA? 7.5

Figure 7.5.2 shows a CV for a solution containing benzophenone (BP) and tri-ptolylamine (TPTA), both at 1 mM in acetonitrile. BP can be reduced and TPTA can be oxidized inside the working range of acetonitrile; however, BP cannot be oxidized, and TPTA cannot be reduced. Account for the shape of the voltammogram. a) Assign the voltammetric features between +0.5 and 1.0 V and between −1.5 and −2.0 V to appropriate electrode reactions. Comment on the heterogeneous and homogeneous kinetics. Estimate the formal potentials.

349

350

7 Linear Sweep and Cyclic Voltammetry

b) Why does the falloff in current appear between 0.7 and 1.0 V? c) What causes the current in the negative-going and positive-going scans at −1.0 V? 7.6 Kadish et al. presented results concerning the interactions between Fe(II) phthalocyanine (FePc; see Figure 1, MPc) and various nitrogen bases, such as imidazole. The work was carried out in DMSO containing 0.1 M TEAP. Results are shown in Figure 7.11.1. In (a), both couples I and II show peak potentials and current functions that are invariant with scan rate. Interpret the voltammetric properties of the system before and after addition of imidazole.

Figure 7.11.1 Cyclic voltammograms of 1.18 mM FePc in DMSO/imidazole solutions containing 0.1 M TEAP. v = 100 mV/s. Imidazole concentrations: (a) 0.00 M; (b) 0.01 M; (c) 0.95 M. [Kadish, Bottomley, and Cheng (50). © 1978, American Chemical Society.]

5 μA II (a) I i

II (b)

I

III (c)

–0.4

–0.6

–0.8

–1.0

–1.2

–1.4

E/V vs. SCE

7.7

For O2 in an aprotic solvent such as pyridine or acetonitrile, a CV like that in Figure 7.11.2 is generally obtained. STV on a 4-s timescale at a mercury electrode gives a linear plot of E vs. log[(id,c − i)/i] with a slope of 63 mV. The reduction product at −1.0 V vs. SCE gives an ESR signal. If methanol is added in small quantities, the cyclic voltammogram shifts toward positive potentials, the forward peak rises in magnitude, and the reverse peak disappears. These trends continue with increasing methanol concentration until a limit is reached with reduction near −0.4 V vs. SCE. The STV under these limiting conditions is approximately twice as high as in methanol-free solution, and the wave slope is 78 mV. a) Identify the reduction product in methanol-free solution. b) Identify the reduction product under limiting conditions in methanolic solution. c) Comment on the charge-transfer kinetics in methanol-free solution. d) Explain the voltammetric responses.

7.8

Sadler and Bard carried out CV for a DMF solution containing 0.68 mM azotoluene (AzT; Figure 1) and 0.10 M TBAP at 25 ∘ C. The working electrode was a Pt disk with A = 1.54 mm2 . A typical voltammogram is shown in Figure 7.11.3, and other data are tabulated below. Coulometry shows that the first reduction step involves 1e. Work up these data and discuss what information is obtained about the reversibility of the reactions,

7.11 Problems

Figure 7.11.2 CV at an HMDE of O2 in pyridine with 0.2 M TBAP. Frequency 0.1 Hz. [Peover and White (51).]

i

↑ Cathodic

0

Anodic ↓

–0.6

–0.8 –1.0 E/V vs. SCE

–1.2

stability of products, diffusion coefficients, etc. (This is a set of actual data, so don’t expect numbers to conform exactly to theory.) First Wave(a) (mV s−1 )

ipc (𝛍A)

ipa (𝛍A)

−E pc

430

8.0

8.0

1.42

298

6.7

6.7

203

5.2

5.2

91

3.4

73

3.0

v

Second Wave ipc (𝛍A)

−E pc (V)(b) −E p/2 (V)(b)

1.36

7.0

2.10

2.00

1.42

1.36

6.5

2.09

2.00

1.42

1.36

4.7

2.08

2.00

3.4

1.42

1.36

3.0

2.07

1.99

2.9

1.42

1.36

2.8

2.06

1.98

(V)(b)

−E pa

(V)(b)

(a) For scan reversed 100 mV past Epc . (b) Potentials vs. SCE.

i

10 μA

0

–0.8 –1.0 –1.2 –1.4 –1.6 –1.8 –2.0 –2.2 –2.4 E/V vs. SCE

Figure 7.11.3 CV of azotoluene at a Pt disk in DMF.

351

352

7 Linear Sweep and Cyclic Voltammetry

7.9

Johnson described the electrochemical behavior of 1,3,5-tri-tert-butylpentalene (ttBP; Figure 1). Solutions of ttBP in MeCN with 0.1 M TBAP were subjected to CV, STV, and bulk electrolysis. The CV is illustrated in Figure 7.11.4. STV on a 4-s timescale at a mercury electrode showed one cathodic wave at E1/2 = − 1.46 V vs. SCE with a wave slope of 59 mV. Bulk electrolysis at +1.0 V produced a green solution giving a well-resolved ESR spectrum, and bulk electrolysis at −1.6 V gave a magenta solution that also produced a well-resolved ESR spectrum. Both bulk transformations were carried out in CH2 Cl2 . a) Describe the chemistry of the system. b) Account for the shape of the CV. Identify all peaks. c) Interpret the STV and relate the CV to it. d) Make sketches showing the expected variations with v of forward peak current and ΔEp for the couple responsible for the green solution. Do the same for the couple responsible for the magenta solution. Figure 7.11.4 CV of ttBP in MeCN with 0.1 M TBAP at a Pt disk (A = 0.25 cm2 ; v = 500 mV/s). Scan starts at 0.0 V vs. SCE and first moves positively. [Adapted from Johnson (53). © 1977, American Chemical Society.]

0.8

0.0

–0.8 E/V vs. SCE

–1.6

–2.4

7.10 Sketch the interfacial distribution diagram analogous to Figure 7.8.3 for a water/ 1,2-dichloroethane (H2 O/DCE) system containing Li+ , Cl− , TPAs+ (tetraphenylarsonium), and TPB– (tetraphenylborate). Assume that the two phases equilibrate in zones of thickness d on either side of the ITIES and that the aqueous phase is made up with 0.01 M LiCl, while the DCE phase initially contains 0.01 M TPAsTPB. Also assume ideality (that activity coefficients are unity). Plot the concentration of each ion 𝛽 0𝛼→𝛽 in the aqueous interfacial zone vs. Δ𝜙𝛼 . The following are relevant values of ΔGtransfer.j (kJ/mol): Li+ , 48.2; Cl− , 46.4; TPAs+ , −35.1; TPB– , −35.1. Use this plot to predict the current–potential behavior, such as that shown in Figure 7.8.2a.

7.11 Consider the heterogeneous electron-transfer rate for oxidation of ferrocene (Fc): a) Mirkin et al. (54) obtained k 0 = 3.7 cm/s and DR = 1.70 × 10−5 cm2 /s using MeCN with 0.5 M TBABF4 . Calculate 𝜓 and ΔEp for the CV of the Fc+ /Fc couple at 25 ∘ C, assuming DO = DR , at scan rates of 3, 30, 100, 200, 300, and 600 V/s. b) Noviandri et al. (55) reported the results tabulated below for 2 mM Fc in MeCN with 0.1 M TBABF4 at a 25-μm-diameter Au disk. Account for these results based on the calculations in (a). v (V/s)

3.2

32

102

204

297

320

640

ΔEp (V)

77

94

96

120

134

158

300

7.11 Problems

7.12

Consider the experiment represented in Figure 7.2.6. In the absence of N− , the 1e 3 reversible electrode process is given by (7.2.43). When N− is added, the CV behaves as 3 + though the formal potential for TEMPO /TEMPO∙ depends on the concentration of N− , with N− stabilizing TEMPO+ reversibly, perhaps by the association equilibrium 3 3 (7.2.44), having a dimensionless activity-based equilibrium constant K. The formal potential for TEMPO+ /TEMPO∙ is defined through the following nernstian relationship (Section 2.1.7): ′ RT [O] RT aO Eeq = E0 + ln = E0 + ln nF [R] nF aR where O and R are used for notational simplicity to denote TEMPO+ and TEMPO∙. Each bracketed term indicates a total molar concentration (without units) for all chemical forms in equilibrium. On the right, the activities are aO = 𝛾 O C O /C 0 and aR = 𝛾 R C R /C 0 , where C O and C R are molar concentrations of uncomplexed O and R, 𝛾 O and 𝛾 R are their activity coefficients, and C 0 is the standard-state concentration, defined as 1 M for this case. Each bracketed term is unitless because the total molar concentration is divided by C 0 (Section 2.1.7). ′ a) Derive a general equation for E0 in the presence of N− using (7.2.43) and (7.2.44) as 3 − 16 a chemical basis. Use [N3 ] to symbolize added azide in all chemical forms in equilibrium. For simplicity, denote the charge-transfer complex as OA and uncomplexed azide as A. Retain all factors, including activity coefficients and C 0 . ′ b) Find the limiting form of your equation for [N− ] = 0. Designate this value as E00 and 3 re-express your general equation using this quantity. c) Now assume ideality (unit activity coefficients) and that [N− ] ≫ COA ∕C 0 . Simplify 3 your general equation. d) Find the limiting form for K[N− ] ≫ 1. 3 ′ ′ e) Explain how the ordinate of Figure 7.2.6b relates to E0 . Estimate E00 . f ) Are the experimental results in Figure 7.2.6b consistent with a description of the system in terms of (7.2.43) and (7.2.44)? g) What would be the intercept of the plot in Figure 7.2.6b at [N− ] = 1? Can you draw a 3 quantitative result from it? h) Optional: Test the assumptions made in (c) and (d).

16 This problem rests on the same phenomena and principles used to treat coupled equilibria for SSV and STV in Sections 1.4.1 and 5.3.2(c); however, the approach is adapted to the context of CV.

353

355

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry In this chapter, we return to voltammetry based on potential step waveforms. Over a period of decades, progressively more elaborate schemes have been devised for applying potential steps and sampling currents. Diverse methods have evolved. Some are mainly used for electroanalysis, others, for fundamental studies. Because all are rooted in the polarographic tradition, we begin with a discussion of classic phenomena at renewable mercury electrodes, then proceed into various forms of pulse voltammetry, ending with the most powerful, square-wave voltammetry.

8.1 Polarography Electroanalytical chemistry literally developed out of the dropping mercury electrode (DME), invented by Heyrovský (1) for measurements of surface tension (Section 14.2.1). He serendipitously discovered a powerful form of voltammetry, which he named polarography, and which became the root of most methods discussed in this book (2). Heyrovský was recognized with the 1959 Nobel Prize in Chemistry for his achievement. The term “polarography” has since become a general name for voltammetry at a renewable mercury electrode. In this book, we refer to the historic form as dc polarography or conventional polarography. While that method has now been supplanted, it gave rise from the 1920s through the 1960s to a large literature, which continues to provide useful descriptive information about organic and inorganic electrochemistry, as well as valuable quantitative data, such as formal potentials, diffusion coefficients, and equilibrium constants. To preserve its value for new users, the authors retain the following summary of polarographic essentials.1 8.1.1

The Dropping Mercury Electrode

Figure 8.1.1 depicts a classical DME (3–6). A capillary with an internal diameter of ∼5 × 10−3 cm is fed by a head of mercury 20–100 cm high. Mercury issues through the capillary to form a nearly spherical drop, which grows until its weight can no longer be supported by the surface tension. A mature drop typically has a diameter on the order of 1 mm. If electrolysis occurs while the drop is growing, the current has a time dependence reflecting both the expansion of the spherical electrode and the depletion effects of electrolysis. During its fall, a drop stirs the solution and largely erases the depletion effects, so that each drop is born into renewed solution. If the potential does not change much during the lifetime of a drop (2–6 s), the process 1 Coverage of polarography is sharply abridged in this edition. Much fuller treatment, including appropriate references, is available in Section 5.3 of the first edition and Sections 7.1 and 7.2 of the second edition. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

356

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

Contact

Stand tube

Vertical support with scale

Leveling bulb Hg

Flexible tubing

Capillary Hg drop

Figure 8.1.1 A dropping mercury electrode.

is indistinguishable from a step experiment in which the potential transition coincides with the birth of a new drop. Each drop’s lifetime is a new experiment. 8.1.2

The Ilkoviˇc Equation

Let us now consider the current during the life of a single drop when a DME is held at a potential in the mass-transfer-controlled region for electrolysis. We would like to obtain an analogue of the Cottrell equation, (6.1.12), describing current flow at a planar electrode [Section 6.1.1(a)]; however, the problem at the DME is complicated by the physical growth of the electrode as the electrode reaction proceeds. Ilkoviˇc was first to provide an effective treatment (7, 8), and the result, bearing his name, is 1∕2

∗ m2∕3 t 1∕6 id,c = 708nDO CO

(8.1.1)

∗ in mol/cm3 , t in s, and m (the mass flow rate of where id,c is in amperes, DO in cm2 /s, CO ∗ in mM. mercury from the capillary) in mg/s. Alternatively, id,c can be taken in μA, and CO The Ilkoviˇc equation can indeed be understood as an extension of the Cottrell equation. Only two substitutions are needed to obtain (8.1.1) from (6.1.12):

1) The time-dependent electrode area, A(t), must be expressed explicitly, because the changing area progressively enlarges the diffusion field as the drop grows. One can show (Problem 8.2) that ( )2∕3 3mt A(t) = 4𝜋 (8.1.2) 4𝜋dHg where dHg is the density of mercury. Typically, m = 1 − 2 mg/s.

8.1 Polarography

i Drop fall

2–4 s t

Figure 8.1.2 Growth of current during three successive drops of a DME.

2) The effective diffusion coefficient becomes (7/3)DO . Steady expansion of the drop causes the existing diffusion layer to stretch over a still larger sphere, much like the membrane of an expanding balloon. At any time, the layer is thinner than it otherwise would be, so that the concentration gradient at the electrode surface is enhanced and larger currents flow. Thus, we discover that the Ilkoviˇc treatment is based on linear diffusion at a sphere. In fact, typical values of drop lifetime and drop diameter at maturity ensure that linear diffusion holds at a DME to a good approximation [Section 6.1.2(c)]. Koutecký (9, 10) later provided a rigorous treatment based on spherical diffusion, but the Ilkoviˇc equation remains the primary interpretive relationship for dc polarography. Figure 8.1.2 shows predicted current–time curves for several drops as given by (8.1.1). The current is a monotonically increasing function of time, in sharp contrast with the Cottrell decay found at a stationary planar electrode; thus, the effects of drop expansion more than counteract the depletion of the electroactive substance near the electrode. 8.1.3

Polarographic Waves

Conventional polarograms are records of the current flow at a DME as the potential is scanned linearly with time, but sufficiently slowly (1–3 mV/s) that the potential remains essentially constant during the lifetime of each drop. This constancy of potential is the basis for the descriptor “dc” in one of the names for the method. The current oscillations arising from the growth and fall of the individual drops are ordinarily quite apparent if the current is recorded continuously. An example is available in Figure 6.4.1. The most easily measured current is that which flows just before drop fall. Within the linear approximation, it is given by 1∕2

1∕6

∗ m2∕3 t (id )max = 708nDO CO max

(8.1.3)

where t max is the lifetime of a drop (usually called the drop time).2 In effect, a conventional polarogram is a form of sampled-transient voltammetry (STV) in which sampling is carried out visually, just by observing the locus of maximum currents on the current–potential curve. Sections 6.2–6.4, covering the interpretation of STV, apply fully to 2 The older polarographic literature often featured measurements of the average current, id , during a drop’s lifetime. 1∕2

1∕6

∗t From the Ilkoviˇc equation, one can readily find id to be six–sevenths of the maximum current: id = 607nDO CO max The first edition (pp. 150–152) covers the use of average currents in more detail.

357

358

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

conventional polarographic waves. Moreover, the simplified mass-transfer equations of Section 5.2.5 also apply, with the time-dependent area given by (8.1.2) and mO = [(7∕3)DO ∕𝜋tmax ]1∕2 mR = [(7∕3)DR ∕𝜋tmax ] 8.1.4

1∕2

(8.1.4a) (8.1.4b)

Practical Advantages of the DME

The DME offers valuable attributes for experimental measurements (3–6): • The dropping action is very reproducible, and the surface is continuously renewed. These factors enable high-precision measurements (11). • The electrode is not permanently modified by material that deposits in or on the electrode (e.g., by adsorption of species from solution or during electrodeposition of metals). • The repeated dropping and stirring action enables, in effect, a succession of step experiments with a consistently renewed solution and with a constant or slowly varying potential applied to the electrode, exactly as required for STV. • The current–time curve at the DME, which features a minimal rate of current change as the maximum current is approached, as well as a maximum current identified with the end of drop life, is well adapted to STV. • In turn, STV is convenient for multicomponent analysis, because current plateaus are obtained in the mass-transfer-limited region of each wave; hence, flat (or at least linear) baselines apply to each of several successive waves (Section 6.4). Besides, the DME offers the exceptionally high overpotential for hydrogen discharge characteristic of mercury surfaces. In many aqueous media, hydrogen evolution determines the cathodic background limit. The high overpotential at the DME means that the background limit is pushed to more negative potentials, so that it becomes possible to observe electrode reactions that might otherwise be lost in the background. An example is the reduction of Na+ to Na(Hg) in basic aqueous media, which is observable at the DME as a clean wave before the background limit is reached. Actually, this particular case is aided by another nice feature of the DME, viz. the ability to form amalgams. Since sodium amalgam forms spontaneously, Na + Hg → Na(Hg)

(ΔG0 < 0)

(8.1.5)

the free energy for Na+ reduction to Na(Hg) is less than for reduction to the metal, and the standard potential is correspondingly more positive. This is a general feature of electrode processes that involve reductions to amalgams, and it widens the range of processes that can be studied at the DME. A principal disadvantage of all Hg electrodes, including the DME, is the inability to operate at potentials more positive than about 0 V vs. SCE. The anodic limit, which arises from Hg oxidation, is always near that potential, although it depends somewhat on the medium. 8.1.5

Polarographic Analysis

Quantitative polarographic analysis is based on the linear linkage between the diffusion-limited current and the bulk concentration of the electroactive species. The most precise measurements of concentration are carried out by constructing a calibration curve using standard solutions. In routine work, ±1% precision can be obtained (3–6); however, Lingane (11) showed that ±0.1% is possible with careful precautions. Most sources of imprecision involve temperature effects,

8.1 Polarography

especially on mass transport, with values of D typically increasing by 1–2% per degree. The standard-addition and internal-standard methods can also be used for concentration measurements, typically with precision of a few percent. Unique to dc polarographic analysis, and to the DME, is the “absolute” method for evaluating concentrations (12). A rearrangement of (8.1.3) with the experimental variables (id , t max , m, ∗ ) placed on one side gives and CO (I)max =

(id )max 1∕6 ∗ m2∕3 tmax CO

1∕2

= 708nDO

(8.1.6)

where the diffusion current constant, (I)max , is independent of the specific values of m, t max , ∗ used in the measurement. Since it depends only on n and D , it is a constant of the and CO O electroactive substance and the medium in much the same way that molar absorptivity is a constant of the system for optical measurements. Given (I)max for the case at hand, one can ∗ simply by measuring i , t evaluate CO d max , and m. No standards are needed. The method is approximate, because (8.1.3) is itself an approximation. Many workers have reported diffusion current constants, and large tabulations exist (3, 5, 13–15). Because the DME has largely fallen out of use, diffusion current constants are rarely reported in new literature; however, the data in older literature remain useful for estimating n or D.3 In general, 1e, 2e, and 3e reactions have constants in the ranges of 1.5–2.5, 3.0–4.5, and 4.5–7.0, respectively, whenever media with water-like viscosities (∼1 cP) are employed. Those ranges rest upon the fact that diffusion coefficients for most ions and small molecules have similar values in a given medium. Exceptions include H+ and OH− in aqueous media, oxygen generally, and polymers and large biomolecules. Conventional polarographic analysis is carried out most readily in the range from 0.1 to 10 mM. Comments above about precision generally apply to this region. Above 10 mM, electrode processes tend to cause such large alterations in solution composition near the electrode that convection arises and currents become erratic. The lower end of the working range is discussed next. 8.1.6

Residual Current and Detection Limits

In the absence of an added electroactive substance, a residual current flows between the anodic and cathodic background limits in a dc polarographic experiment (3–6). It is the sum of a nonfaradaic current due to double-layer charging and faradaic currents arising from • Trace impurities (e.g., heavy metals, electroactive organics, or oxygen). • The electrode material (which often undergoes slow, potential-dependent faradaic reactions). • The solvent and supporting electrolyte (which can produce small currents over a wide potential span via reactions that greatly accelerate at more extreme potentials and eventually determine the background limits). Since the DME is always expanding, new surface appears continuously. Because it must be charged to reflect the potential of the electrode, a charging current, ic , always exists. The excess charge on the electrode is given by [Section 1.6.4(a)] qM = Ci A(E − Ez )

(8.1.7)

3 Since older work was often based on average limiting currents, reported diffusion current constants are sometimes defined from the version of the Ilkoviˇc equation for average currents, which gives: I = I = (6∕7)(I)max .

id 1∕6 ∗ m2∕3 tmax CO

1∕2

= 607nDO Thus,

359

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

0.4

Current i/μA

360

0

–0.4 0

–0.4

–0.8

–1.2

E/V vs. SCE

Figure 8.1.3 Residual current at a DME for 0.1 M HCl. The sharply increasing currents at potentials more positive than 0 V and more negative than −1.1 V arise from oxidation of mercury and reduction of H+ , respectively. The current between 0 and −1.1 V is largely capacitive. The PZC is near −0.6 V vs. SCE. [Meites (5). © 1965, Interscience Publishers].

where C i is the integral capacitance of the double layer (Section 14.2.2), A is the electrode area, and Ez is the potential of zero charge (PZC; Section 14.2.2). Since both C i and E are effectively constant during a drop’s lifetime, differentiation of (8.1.7) yields dq dA = Ci (Ez − E) dt dt From (8.1.2), dA/dt can be derived, and one finds that ic = −

ic = 0.00567Ci (Ez − E)m2∕3 t −1∕3

(8.1.8)

(8.1.9)

where ic is in μA if C i is in μF/cm2 (typically, 10–20 μF/cm2 ) and m is in mg/s. If C i and t max are not strongly varying functions of potential, ic will be linear with E, as borne out experimentally in Figure 8.1.3. During the life of a single drop, the faradaic current and the charging current have very different time profiles. Just after each drop’s birth, when dA/dt is large, a spike in charging current dominates the total current, but it dies off with t −1/3 as the drop enlarges. In contrast, the

8.2 Normal Pulse Voltammetry

0.1 μA

“cathodic” charging spikes

0.2 μA id,c

i

i

“anodic” charging spikes –0.4

–0.6 (a)

–0.8

–0.4

–0.6 (b)

E/V vs. SCE

–0.8

Figure 8.1.4 Polarograms at a DME for 10−5 M Cd2+ in 0.01 M HCl. (a) Conventional dc mode. The PZC is near −0.57 V in this experiment; hence, the charging current spikes in appear “anodic” at more positive potentials and “cathodic” at more negative ones. (b) Normal pulse mode. This method eliminates the sharp nonfaradaic spikes appearing at the birth of each drop.

faradaic current grows steadily with t 1/6 from the birth of the drop and may predominate at the end of each drop’s life. The relative contributions of these two components vary considerably ∗. with the bulk concentration of the electroactive species, CO ∗ = 10−5 M and the average chargFigure 8.1.4a is a real polarogram for a system in which CO ing and faradaic currents are roughly equal over the life of a drop. While the faradaic polarogram ∗ were reduced to is discernable, the charging spikes complicate any estimation of id,c . If CO 10−6 M, quantification of the faradaic wave would become impossible. It would be only a tenth the size in Figure 8.1.4a and would be lost among the charging current spikes, which (being ∗ ) would remain just as prominent. Accordingly, the detection limit for Cd2+ independent of CO in conventional polarography must lie between 10−5 and 10−6 M. Figure 6.4.1 provides a counter-example for a system of much higher concentration—the ∗ ≈ 0.25 mM). The faradaic current is reduction of O2 in an air-saturated aqueous solution (CO 2

about 100 times larger than in Figure 8.1.4a and dwarfs the charging-current spikes. The current oscillations in Figure 6.4.1 are almost entirely due to the loss of faradaic current at the end of drop life, whereas those in Figure 8.1.4a arise almost entirely from charging current. These cases illustrate the determining role of charging current with respect to analytical sensitivity in conventional polarography. To improve detection limits, one must develop a measurement scheme that suppresses the contribution of charging current or enhances the measured faradaic current. Normal pulse voltammetry (NPV) and differential pulse voltammetry (DPV) were conceived to improve electroanalytical sensitivity at the DME by addressing these very issues. The practice of both has now broadened far beyond the DME, as we will see next for NPV and then in Section 8.4 for DPV.

8.2 Normal Pulse Voltammetry We have already encountered the idea of NPV, because it was introduced in Section 6.2 as the paradigm for STV. The relevant theory was extensively developed in Sections 6.2–6.4. Here, the focus is on experimental aspects.

361

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

8.2.1

Implementation

Barker and Gardner (16) originated NPV (6, 17–21) essentially as illustrated conceptually in Figure 8.2.1. The method features a train of many measurement cycles. Early in each cycle, the conditions at the surface of the electrode must be renewed, so that the effects of the prior cycle are erased. During the renewal period, the electrode is held at a base potential, Eb , normally chosen so that the species of interest remains electroinactive. At time 𝜏 ′ , measured from the beginning of each cycle, the potential is stepped to value E for 1–100 ms, then is reset to the base value, Eb . The reset ends the measurement cycle and starts the next renewal period. In each cycle, the current is sampled at time 𝜏, near the end of the pulse, and the sampled value is presented as a constant output until the sample taken in the next cycle replaces it. With each additional cycle, the step potential is made a few mV more extreme. The output is a plot of sampled current vs. step potential, E, and it takes the wave shape shown in Figure 8.1.4b. A block diagram of implementation is shown in Figure 8.2.2. NPV is applicable to all sorts of microelectrodes, whether planar or not, having dimensions in the millimeter-to-low-micrometer range. Usually, the pulse width is short enough that semi-infinite linear diffusion applies, even if the electrode surface has curvature (Section 6.1.2). The renewal process exists to assure that the concentration profile of the primary species ∗ , when the pulse is applied at time 𝜏 ′ . Since (taken here as O) is uniformly at the bulk value, CO electrolysis of O is negligible before the pulse, the current follows the Cottrell equation, (6.1.12), whenever the pulse potential, E, reaches into the diffusion-limited region. Thus, the sampled faradaic current on the voltammetric plateau is 1∕2

(id,c )NP =

∗ nFADO CO

(8.2.1)

𝜋 1∕2 (𝜏 − 𝜏 ′ )1∕2

where (𝜏 − 𝜏 ′ ) is the sampling time measured from the pulse rise. Because NPV is the model for STV, the theory needed for analysis of positions, shapes, and heights of voltammetric waves is well developed (Sections 6.2 and 6.3) and is fully applicable,

Cycle j –1

Cycle j

Cycle j +1

Pulse width (10–100 ms) E(–)→

362

Renewal period (500–5000 ms)

Renewal begins

Cottrell current–time decay

Eb Current sampled 𝜏′

𝜏′

𝜏

𝜏

𝜏′

𝜏

Time

Figure 8.2.1 Potential program and sampling scheme for three cycles in NPV. A faradaic current transient like that shown for Cycle j would apply to all cycles in which the species of interest is electroactive. The zero-current axis for the transient is the level corresponding to E b for the plot of potential vs. time. The times 𝜏 and 𝜏 ′ are measured from the beginning of each cycle.

8.2 Normal Pulse Voltammetry

Waveform generation

Event sequencing

Cell Potentiostat

Renewal device (if used) Current sample i(𝜏)

i/V converter

Data recording i(𝜏) vs. E

Figure 8.2.2 Schematic experimental arrangement for NPV. Usually, an automated potentiostat is used, and tasks denoted by shaded boxes are handled digitally (Section 16.6). To improve precision, current sampling often involves repeated measurements over a brief period, which are then averaged to produce i(𝜏). The renewal device is discussed in the text.

as long as renewal of the diffusion layer is effective (Section 8.2.2). It is impractical to carry out detailed analysis of NPV results for systems where renewal cannot be achieved. The characteristic NPV time scale of milliseconds is much shorter than the ∼3-s time scale of conventional polarography; thus, it is possible for a chemical system to behave reversibly in a conventional polarographic experiment and quasireversibly or irreversibly in NPV. Many systems that show sluggish electrode kinetics behave in just this way. The opposite behavior can be seen if a system shows fast electrode kinetics, but the product of the electrode reaction decays on a 1-s time scale. In that case, NPV can show reversibility, because little product decay will occur during the measurement; yet, the conventional polarogram will show the characteristic effects of homogeneous reactions following charge transfer (Chapter 13). With the NPV waveform (Figure 8.2.1), there is a charging-current spike at each step edge, but it decays away exponentially according to the cell time constant Ru C d [Section 1.6.4(a)]. In general, the sampled current will not include a capacitive contribution if (𝜏 − 𝜏 ′ ) > 5Ru C d . In many media, the cell time constant is a few tens of microseconds to a few milliseconds (Section 6.7); hence, this condition is normally easily fulfilled for NPV. 8.2.2

Renewal at Stationary Electrodes

The cyclic renewal of the diffusion layer is essential for NPV. It happens automatically at a DME, but at a stationary electrode, such as a Pt or C disk, it may become incomplete as the electrode and diffusion layer are taken through cycle after cycle of pulsing and sampling. The electroreactant can become depleted, and products can build up, either in the diffusion layer or on the surface of the electrode. The result is a degraded voltammetric response. There are two general means for achieving cyclic renewal at stationary electrodes: 1) Electrochemical renewal. Frequently, the electrode process carried out during the pulses is reversed when the potential returns to Eb . Consider, for example, the NPV of Cu(NH3 )2+ 6 in 1 M NH3 + 1 M NH4 Cl. When the pulse potential reaches values where this complex is reduced, the electrode reaction is Cu(NH3 )2+ + e → Cu(NH3 )+ , but when the electrode 6 6

363

364

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

is restored to Eb , that process reverses: Cu(NH3 )+ → Cu(NH3 )2+ + e. Renewal can be 6 6 achieved through this reversal, especially when the holding time at Eb is much longer than the pulse duration. It is not important that the electrode kinetics be fast enough to be called “reversible,” only that the chemistry be efficiently reversed. 2) Diffusive renewal. Even without electrolytic reversal, one can obtain cyclic renewal of the diffusion layer simply by waiting long enough at Eb for diffusion to replace the consumed electroreactant (22). Section 5.1.3 covers the time required and shows that diffusive renewal is especially practical at a UME (23, 24). Even the largest UMEs offer full renewal in seconds, and those with a characteristic dimension in the 1-μm range can renew in a few milliseconds. Moreover, UMEs offer small cell time constants, especially in aqueous media. This combination of properties allows NPV to be carried out much more quickly (10–100 times faster than at conventional electrodes) and with much shorter pulses (down to 10 ms with conventional equipment, but in the high-μs range with special instrumentation). There is still another path to renewal when an electrode operates in a convective system. At a rotating Pt disk, for example, one can rely on stirring to renew the diffusion layer while the potential is held at Eb . The continuing convection may affect the current sampled in each pulse, so that the theoretical expectation based on diffusion theory is exceeded; however, the error is often irrelevant (as in analytical applications where calibration is possible) or fairly small (because a pulse of short duration creates a diffusion layer that remains confined to a relatively stagnant layer of solution). As noted in Figure 8.2.2, a special device might be employed for renewal and might require synchronization with the start of a new measurement cycle. Examples include a motor for rotation of the electrode, a microfluidic pump used to change the solution facing the electrode during each renewal period, or a drop knocker at for a DME (Section 8.2.3). If the diffusion layer cannot be effectively renewed, the polarographic wave will not show a plateau, but instead will pass through a peak, then droop at more extreme potentials as cumulative depletion of the electroreactant is manifested. The curve resembles a linear sweep voltammogram for essentially the reasons governing LSV responses (Section 7.1).

8.2.3

Normal Pulse Polarography

NPV was conceived for use with the DME as an improved form of polarography (16, 17) and, indeed, was originally called normal pulse polarography (NPP). Although the polarographic mode remains in use, the static mercury drop electrode (SMDE) has almost wholly supplanted the DME. We next examine differences in voltammetry at these two renewable Hg electrodes. (a) Behavior at a DME

At a DME, NPV follows the representation in Figures 8.2.1 and 8.2.2, with cyclic renewal happening automatically as each drop falls, leaving the new drop to grow in fresh solution. However, an electromechanical drop knocker controlled by the measurement system is used to synchronize the birth of a new drop with the start of a new measurement cycle. On the diffusion-limited plateau in conventional polarography, faradaic current flows throughout each drop’s lifetime, but is measured only just before the end. The faradaic reaction before the sampling period serves no useful purpose; indeed, it works to the detriment of sensitivity, because it depletes the region near the electrode of the electroactive species and reduces its flux to the surface at the time of actual measurement. The pulse method forestalls the early electrolysis and, thereby, increases the measured currents from the species of interest.

8.2 Normal Pulse Voltammetry

As shown in Section 8.1.2, the dc polarographic current, (id,c )DC can be rewritten as 1∕2

∗ nFA(7∕3)1∕2 DO CO

(8.2.2) 𝜋 1∕2 𝜏 1∕2 One can compare NPP and dc polarography experiments having the sampling time, 𝜏, by dividing (8.2.1) with (8.2.2) to obtain (id,c )NP ( 3 )1∕2 ( 𝜏 )1∕2 = (8.2.3) (id,c )DC 7 𝜏 − 𝜏′ (id,c )DC =

For typical values of 𝜏 = 4 s, and (𝜏 − 𝜏 ′ ) = 50 ms, this ratio is about 6; thus, the expected increase in faradaic current for NPP is substantial. In Figure 8.1.4, which compares results from NPP and conventional polarography at a DME, the larger sampled currents obtained with the pulse method are obvious. NPP also improves polarographic sensitivity by reducing the charging-current background. At a DME, dA/dt is never zero, so the charging current always contributes. From (8.1.9), we see that the charging current included in an NPP current sample is ic (𝜏) = 0.00567Ci (Ez − E)m2∕3 𝜏 −1∕3

(8.2.4)

Fortunately, this is the minimum value reached during the drop lifetime, but even more helpful is that the current sampling used in NPP wholly screens out the charging current spike at the birth of each new drop. Figure 8.1.4 confirms the reduced and much quieter background in NPP by comparison to dc polarography under the same conditions. (b) Behavior at a Static Mercury Drop Electrode

The DME has operational disadvantages with its constantly changing area and restricted timescale. The former complicates the treatment of diffusion and creates a continuous background from double-layer charging. The latter, determined by the lifetime of the drop, cannot be varied conveniently outside the range of 0.5–5 s. The SMDE (Figure 8.2.3) was invented and commercialized to remedy these drawbacks (25, 26). The SMDE is an automated device in which the mercury flow is controlled by a valve.4 A head of only about 10 cm drives mercury through a wide-bore capillary when the valve is opened in response to an electrical signal. A drop is extruded in less than 100 ms, then growth is stopped by closure of the valve. The drop remains in place until an electromechanical drop knocker dislodges it upon receiving another electronic signal. The SMDE can serve as a hanging mercury drop electrode (HMDE) or, in a repetitive mode, as a replacement for the DME. In the latter role, it retains all of the important positives of the DME (Section 8.1.4), but has the added feature that the area remains invariant after drop formation. Since dA/dt = 0 at the time of measurement, the charging current due to drop expansion is zero. In most NPP at an SMDE, the residual current is entirely faradaic and is often controlled by the purity of the solvent–supporting electrolyte system (Section 8.1.6). Virtually all contemporary polarographic work is now carried out with an SMDE, because it offers so many practical advantages vs. a DME: • The complete elimination of sampled capacitive current samples produces a lower background and improved detection limits. • For the same reason, the slope in the background found at a DME (equation 8.2.4 and Figure 8.1.4b) is removed, improving precision through better-defined wave height. 4 A newer device with similar properties is marketed as the controlled growth mercury electrode (CGME).

365

366

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

Hg reservoir

Drop knocker Electronically controlled valve

Large-bore capillary

Open Closed Time

Drop mass Time

Counter electrode Hg drop Solution

Reference electrode

Figure 8.2.3 A static mercury drop electrode. The unit includes a cell stand and facilities for stirring and deaeration of the solution by bubbling with an inert gas. These functions are normally controlled by an automated potentiostat, which also manages the issuance and dislodgment of the mercury drops and applies the potential program that they experience. When a new drop is needed, the old one is dislodged by a command to the drop knocker, then the electronically controlled valve is opened for 30–100 ms (graph on left). The drop is formed during this period (graph on right), then remains indefinitely stable in size.

• Short drop times (1 s or smaller) are practical at an SMDE; hence, one can record voltammograms with a time saving of 75% vs. practice at a DME. • Designed for automation, the SMDE interfaces readily with electrochemical workstations. • The SMDE is much more compact. 8.2.4

Practical Application

NPV is readily usable with a wide variety of microelectrodes and offers good quantitative precision over at least 2 orders of magnitude in time scale. In addition, a full body of theory exists to support interpretation and analysis of results, as detailed in Sections 6.2–6.4. Consequently, NPV lends itself well to fundamental experimental work aimed at the quantitative evaluation of parameters. NPV also has a history of application to the measurement of low concentrations of heavy metals and organics, particularly in environmental samples (6, 19–21). Much of that work has been in the polarographic mode. Detection limits in NPP are typically between 10−6 and 10−7 M, about an order of magnitude better than for dc polarography. Sensitivity is optimal at an SMDE, where the capacitive background is essentially fully eliminated. Detection limits are typically poorer at stationary electrodes for reasons discussed in Section 8.4.4. Section 8.6 deals more fully with the applications of pulse voltammetry in practical analysis.

8.3 Reverse Pulse Voltammetry

8.3 Reverse Pulse Voltammetry As we have seen, the usual practice in NPV is to select a base potential, Eb , in a region where the principal electroreactant is inactive at the electrode. The scan is made by allowing pulses in ′ successive cycles to reach first into the potential range surrounding E0 , then eventually into the diffusion-limited region. For the usual reversible case of O + ne ⇌ R, with O present in the bulk ′ and R absent, Eb would be set perhaps 200 mV more positive than E0 , and the pulses would be made in a negative direction (Figure 8.3.1a). In the time before each pulse is applied, negligible faradaic reduction of O occurs and a uniform concentration profile, extending from the bulk to the surface, prevails. In reverse pulse voltammetry (RPV) (18, 19, 27),5 the potential waveform and sampling scheme are identical with those of NPV (Figure 8.2.1). However, there are important differences in implementation (Figure 8.3.1a): • The base potential is placed in the diffusion-limited region for electrolysis of the species present in the bulk. ′ • The pulses are made “backward” through the region of E0 and then into range where that species is not electroactive. Figure 8.3.1 RPV vs. NPV in a simple reversible 1e system. (a) Waveforms and placement of E b ; NPV at left, RPV at right. Renewal is required early in the measurement cycle (Section 8.2.2). (b) Voltammograms of i(𝜏) vs. pulse potential. Time

𝜏′

Eb (NPV) 250

150

𝜏

Eb (RPV) –50

50

–150

–250

(a)

i(𝜏)

(id,c)NP

NPV

0 (id,a)RP

250

150

RPV

(id,c)DC

50 –50 (E – E1/2)/mV (b)

5 Or reverse pulse polarography (RPP), if a DME or SMDE is used.

–150

–250

367

368

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

For RPV in the case mentioned above, the base potential would be set 200 mV or more on the ′ negative side of E0 and the pulses would be made in a positive direction. During each long period at Eb , species O is electrolytically converted at the diffusion-controlled rate; hence, its concentration profile is drawn down to zero at the electrode surface, while R is produced at the electrode, and a layer of it extends outward. The pulses work on this concentration profile, which is dominated by R, not O, near the electrode. As the pulses reach more positive potentials, they become capable of oxidizing the R produced during the pre-electrolysis period at Eb ; therefore, ′ anodic current samples are obtained at 𝜏. When the pulses become more positive than E0 by 100 mV or more, the electrolysis of R proceeds at the diffusion-limited rate and does not change further with step potential; hence, an anodic plateau is established (Figure 8.3.1b). This is clearly a reversal experiment, because the focus is on detection of the product from a prior phase of electrolysis. In the case at hand, NPV involves practically a zero faradaic current at pulse potentials near Eb (Figure 8.3.1b), because O does not react at the electrode until the pulses reach the region ′ of E0 . The analogous situation is quite different in RPV, where a significant cathodic current is sampled for pulses near the base potential. This current arises because O is being consumed electrolytically at the diffusion-controlled rate. Pulses of small amplitude, not reaching into the ′ region of E0 , do not change the rate at which O is electrolyzed; hence, the same current sample is obtained for all such pulses. If semi-infinite linear diffusion applies, the cathodic plateau current (at the “base” of the RPV wave), shown as (id,c )DC in Figure 8.3.1b, is given by the Cottrell equation, (6.1.12), as 1∕2

∗ nFADO CO

(8.3.1) 𝜋 1∕2 𝜏 1∕2 The anodic plateau current (id,a )RP can be predicted from the results of Section 6.5.2, which dealt with reversal experiments involving diffusion-controlled forward electrolysis and diffusion-controlled collection of the product in a reversal step. This is exactly the situation in RPV when the steps reach the main plateau of the wave. The current during the pulse is described by (6.5.15), which can be re-expressed in terms of the time parameters of RPV as 1∕2 ∗ [ ] nFADO CO 1 1 (−id,a )RP = − (8.3.2) 𝜋 1∕2 (𝜏 − 𝜏 ′ )1∕2 𝜏 1∕2 The first term of this equation is recognizable from (8.2.1) as the diffusion-limited current for the NPV experiment, (id,c )NP , while the second term is (id,c )DC from (8.3.1). Upon rearrangement (id,c )DC =

(id,c )DC − (id,a )RP = (id,c )NP

(8.3.3)

The left side of (8.3.3) is the height of the whole reverse pulse voltammogram, which is found now to be the same as the height of the normal pulse voltammogram taken with the same timing characteristics. These principles are valid regardless of the electrode employed, as long as semi-infinite linear diffusion applies and renewal of the concentration profile can be accomplished in each cycle. For a stationary planar electrode, the relationships worked out above apply directly. For an SMDE, they apply to the extent that (id,c )DC is the Cottrell current for an electrolysis of duration 𝜏, undisturbed by the convection associated with establishment of the drop. For a DME, the picture is complicated by the steady expansion of area, but it turns out (27, 28) that (8.3.3) is still a good approximation when (id,c )DC is understood as the Ilkoviˇc current for time 𝜏, (8.2.2), and the pulse width is short compared to the pre-electrolysis time [i.e., (𝜏 − 𝜏 ′ )/𝜏 ′ < 0.05].

8.4 Differential Pulse Voltammetry

For a reversible system, the shape of the RPV wave can be derived from the general doublestep response given in (6.5.14). We confine our view to the case where Eb is in the diffusion′ limited region, so that 𝜃 ′ = exp[nf (Eb – E0 )] ≈ 0 in (6.5.14). Then, for a reverse pulse to any value of E, the sampled current would be 1∕2 ∗ [( )( ) ] nFADO CO 1 1 1 −(i)RP = 1− − (8.3.4) 1 + 𝜉𝜃 ′′ 𝜋 1∕2 (𝜏 − 𝜏 ′ )1∕2 𝜏 1∕2 ′

where 𝜃 ′′ = exp[nf (E – E0 )]. Of the three terms in (8.3.4), the first and the third together are −(id,a )RP , as defined in (8.3.2), and the second is −(id,c )NP /(1 + 𝜉𝜃 ′′ ); thus, (i)RP = (id,a )RP +

(id,c )NP

1 + 𝜉𝜃 ′′ Substitution for (id,c )NP according to (8.3.3) and rearrangement gives 𝜉𝜃 ′′ =

(id,c )DC − (i)RP

(8.3.6)

(i)RP − (id,a )RP ′

(8.3.5)

1∕2

1∕2

Defining E1∕2 = E0 + (RT∕nF) ln(DR ∕DO ), we obtain ] [ (id,c )DC − (i)RP RT E = E1∕2 + ln nF (i)RP − (id,a )RP

(8.3.7)

where (i)RP denotes the RPV current sample at potential E. This equation is identical to the shape function for a reversible composite wave, as worked out in Section 1.3.2(b). We have now established, as one suspects by a glance at Figure 8.3.1b, that the half-wave potential, the total height, and the wave slope of a reversible RPV wave are all exactly as for the corresponding NPV wave. These results were derived here for a system where simple semi-infinite linear diffusion applies; however, they apply at a DME, too (27). The principal use of RPV is to characterize the product of an electrode reaction, especially with respect to stability. If species R decays appreciably during the period of the experiment, particularly on the time scale of the pulse, it cannot be fully available to be reoxidized during the pulse. Consequently, (id,a )RP must be smaller in magnitude than expected from (8.3.2). If the decay is very fast, R will be completely unavailable, and the anodic plateau current will be zero. The ratio of plateau heights in RPV and NPV quantifies the stability, and with proper theory, one can obtain the rate constant for the following chemistry. Chapter 13 covers this kind of issue for many different mechanisms and methods. As in the application just discussed, the focus in RPV is often on the magnitude of the wave heights, rather than wave shapes and positions. One can think of RPV as a way to present double-potential-step chronoamperometric data conveniently on a potential axis, because the features of interest in RPV are rooted in chronoamperometric theory, as we have already seen for the derivations done in this section. Thus, one can make direct and confident use of the extensive published results for double-step chronoamperometry to treat data from RPV in various chemical situations.

8.4 Differential Pulse Voltammetry Analytical sensitivities better than those of NPV can be obtained with a small-amplitude pulse scheme abbreviated DPV (6, 17–21), devised originally for the DME. Practice has greatly broadened as DPV has become extensively applied with many kinds of electrodes.

369

370

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

8.4.1

Concept of the Method

Figure 8.4.1 illustrates the basis for DPV, which resembles NPV, but involves several major differences: • The base potential is not constant from cycle to cycle, but changes each cycle by a small increment. • The pulse height is only 10–100 mV and remains constant for all measurement cycles. • As for NPV, the value of the base potential (denoted as E for DPV) is set initially in a region where all species of interest are electroinactive. However, as the train of cycles proceeds, the base potential gradually approaches, then passes the characteristic potentials for those ′ species (usually their E0 values). In these regions, “pre-electrolysis” occurs at the base potential before application of the pulse. This action prepares the diffusion layer for the measurement step in a manner to be explained below. In NPV, the period before the pulse is only for renewal of the diffusion layer. • Two current samples are taken in DPV during each measurement cycle—the first at time 𝜏 ′ , immediately before the pulse, and the second at time 𝜏, late in the pulse, just before the end of the cycle. • The record of the DPV experiment is a plot of the current difference, 𝛿i = i(𝜏) − i(𝜏 ′ ), vs. the base potential. The name of the method is based on this differential current measurement. Both the pulse width (typically ∼50 ms) and the period before the pulse (0.5–4 s) are similar to the analogous periods in NPV. Figure 8.4.2 is a block diagram of the experimental system.

Cycle j –1

E

Cycle j

10 ms –1 s

Cycle j +1

1–50 mV

0.5–5 s (a) 0

Time Second current sample

E Pre-electrolysis Pulse Start of cycle

End of cycle

First current sample

(b) 0

𝜏′ Time from start of measurement cycle

𝜏

Figure 8.4.1 (a) Potential program for several measurement cycles in a DPV experiment. The potential step late in each cycle has the fixed pulse height, ΔE. The duration of the step is the pulse width, approximately 𝜏 − 𝜏 ′ . The pre-electrolysis time is from the start of a cycle until the rise of the pulse. (b) Events for a single cycle. Sampling times are usually close to the step edges, but are spaced back in this diagram for clarity.

8.4 Differential Pulse Voltammetry

Cell Waveform generation

Event sequencing

Potentiostat

i/V converter

Renewal device (if used)

Current sample i(𝜏) δi = i(𝜏) –i(𝜏′)

Data recording (δi vs. E)

Current sample i(𝜏′)

Figure 8.4.2 Schematic experimental arrangement for DPV. Usually, an automated potentiostat is used, and tasks denoted by shaded boxes are handled digitally (Section 16.6). To improve precision, current sampling often involves repeated measurements over a brief period, which are then averaged to produce i(𝜏) or i(𝜏 ′ ). Renewal is discussed in the text.

The differential measurement of current leads to a peaked output, rather than a wave-like response, as shown in Figure 8.4.3a for 10−6 M Cd2+ in 0.01 M HCl. The underlying reason for the shape is easily understood. Early in the experiment, when the base potential is much more ′ positive than E0 for Cd2+ , no faradaic current flows during the time before the pulse, and the change in potential during the pulse is too small to stimulate the faradaic process. Thus, 𝛿i is virtually zero. Late in the experiment, when the base potential is in the diffusion-limited region, Cd2+ is reduced at the maximum possible rate. The pulse cannot increase the rate further; ′ hence, the difference 𝛿i is again small. Only in the region of E0 (for this reversible system) is an appreciable faradaic difference current observed. There, the base potential is such that Cd2+ is reduced during the pre-electrolysis period at some rate less than the maximum, and the surface concentration C O (0, t) remains greater than zero. Application of the pulse forces C O (0, t) to a lower value; hence, both the flux of O to the surface and the faradaic current are enhanced, giving a significant 𝛿i. Only in potential regions where a small potential difference can make a sizable difference in current does the DPV show a response. 8.4.2

Theory

Each measurement cycle in DPV is a double-step experiment. From the start of the cycle at t = 0 until the application of the pulse at t = 𝜏 ′ , the base potential, E, is enforced. During the pulse, the potential is E + ΔE, where ΔE is the pulse height. Pre-electrolysis generally occurs before 𝜏 ′ ; hence, the pulse operates on concentration profiles that pre-electrolysis has created. This situation is analogous to that considered in Section 6.5, and it can be treated by the rigorous techniques developed there. However, we will use a more intuitive approach that preserves the essential simplicity of the problem. Because the initial period is typically 20–100 times longer than the pulse duration, any pre-electrolysis establishes a thick diffusion layer, of which the pulse can modify only a thin

371

372

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

δi

i

50 nA

Baseline

10 nA

–0.4

–0.6

–0.8

–0.4

E/V vs. SCE

(a)

–0.6

–0.8

(b)

Figure 8.4.3 Polarograms at a DME for 10−6 M Cd2+ in 0.01 M HCl. (a) DPP, ΔE = − 50 mV. (b) NPP.

zone near the electrode. One can approximate the experiment by assuming that each pulse operates on a semi-infinite homogeneous solution having bulk concentrations equal to the surface concentrations enforced before the pulse. The role of pre-electrolysis is to set up “effective bulk concentrations,” varying during successive measurement cycles from pure O to pure R (or vice versa), as the scan is made. For a given pulse, we take the differential faradaic current as the current that would flow in a system with those bulk concentrations at time 𝜏 − 𝜏 ′ after a potential step from E to E + ΔE. Now let us focus on a nernstian system in which R is initially absent. From (6.2.20) and (6.2.21), we obtain the surface concentrations during pre-electrolysis at potential E, which we ∗ ) and (C ∗ ) , for the subsequent pulse: now identify as effective bulk concentrations, (CO eff R eff ( ) 𝜉𝜃 ∗) ∗ (CO (8.4.1a) eff = CO 1 + 𝜉𝜃 ( ) 𝜉 ∗ (CR∗ )eff = CO (8.4.1b) 1 + 𝜉𝜃 ′

where 𝜃 = exp[nf (E – E0 )] and 𝜉 = (DO /DR )1/2 . Since the system is nernstian, these concentrations are in equilibrium with the electrode at potential E. The problem is now simply to find the faradaic current after a step from equilibrium to E + ΔE in a homogeneous system with bulk ∗ ) and (C ∗ ) . concentrations (CO eff R eff Through the approach of Section 6.2 (and Problem 6.5), that current turns out to be 1∕2

i=

nFADO

𝜋 1∕2 t 1∕2

⋅

∗ ) − 𝜃 ′ (C ∗ ) ] [(CO eff R eff

(1 + 𝜉𝜃 ′ )

(8.4.2)

′

where 𝜃 ′ = exp[nf (E + ΔE – E0 )] and t is time from the beginning of the pulse. Substitution according to (8.4.1a,b) gives 1∕2

∗ nFADO CO

(𝜉𝜃 − 𝜉𝜃 ′ ) (1 + 𝜉𝜃)(1 + 𝜉𝜃 ′ ) 𝜋 1∕2 t 1∕2 It is convenient (17) to introduce the parameters PA and 𝜎, where i=

⋅

′

PA = 𝜉 exp[nf (E + ΔE∕2 − E0 )]

(8.4.3)

(8.4.4)

8.4 Differential Pulse Voltammetry

and 𝜎 = exp(nf ΔE∕2)

(8.4.5)

In this notation, 𝜉𝜃 = PA /𝜎 and 𝜉𝜃 ′ = PA 𝜎; thus, ] 1∕2 ∗ [ nFADO CO PA (1 − 𝜎 2 ) i= (𝜎 + PA )(1 + PA 𝜎) 𝜋 1∕2 t 1∕2

(8.4.6)

and we take the differential faradaic current, 𝛿i = i(𝜏) − i(𝜏 ′ ), as ] 1∕2 ∗ [ nFADO CO PA (1 − 𝜎 2 ) 𝛿i = 𝜋 1∕2 (𝜏 − 𝜏 ′ )1∕2 (𝜎 + PA )(1 + PA 𝜎)

(8.4.7)

The bracketed factor describes 𝛿i as a function of potential. When E is far more positive than ′ ′ E0 , PA is large and 𝛿i is virtually zero. When E is much more negative than E0 , PA approaches zero, and so does 𝛿i. Through the derivative d(𝛿i)/dPA , one can easily show (17) that 𝛿i is maximized at PA = 1, which implies that ( ) DR 1∕2 ΔE RT ΔE 0′ Emax = E + ln − = E1∕2 − (8.4.8) nF DO 2 2 Since ΔE is small, the potential of maximum current lies close to E1/2 . Also, given that ΔE is negative in this experiment, we see that the peak anticipates E1/2 by ΔE/2. The height of the peak is 1∕2

(𝛿i)max =

∗ nFADO CO

𝜋 1∕2 (𝜏 − 𝜏 ′ )1∕2

⋅

(

1−𝜎 1+𝜎

) (8.4.9)

where (1 − 𝜎)/(1 + 𝜎) decreases monotonically with diminishing |ΔE| and reaches zero for ΔE = 0. When ΔE is negative, 𝛿i is positive (or cathodic), and vice versa. The maximum magnitude of (1 − 𝜎)/(1 + 𝜎), which applies at large pulse amplitudes, is unity. In that limit, (𝛿i)max is equal to the faradaic current sampled on top of the NPV wave obtained under the 1∕2

∗ ∕𝜋 1∕2 (𝜏 − 𝜏 ′ )1∕2 . Under same timing conditions. As (8.2.1) shows, that current is nFADO CO usual conditions, ΔE is not large enough to realize this greatest possible (𝛿i)max . Table 8.4.1 shows the influence of |ΔE| on (1 − 𝜎)/(1 + 𝜎), which is the ratio of the peak height to the limiting value. For a typical case, ΔE = 50 mV, and the peak current is 45–90% of the limiting value, depending on n. The width of the peak at half height, W 1/2 , increases as the pulse height grows larger, because differential behavior can be seen over a greater range of base potential. Normally, one refrains

Table 8.4.1 Effect of Pulse Amplitude on Peak Height (1 − 𝝈)/(1 + 𝝈) 𝚫E (mV)

n=1

n=2

n=3

−10

0.0971

0.193

0.285

−50

0.453

0.750

0.899

−100

0.750

0.960

0.995

−150

0.899

0.995

—

−200

0.960

—

—

Parry and Osteryoung (17).

373

374

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

from increasing |ΔE| much past 100 mV, because resolution is degraded unacceptably. The limiting width as ΔE approaches zero can be shown to be (17) lim W1∕2 = 3.52RT∕nF

ΔE→0

(8.4.10)

At 25 ∘ C, the limiting widths for n = 1, 2, and 3 are 90.4, 45.2, and 30.1 mV, respectively. Since a peak height in DPV is never larger than the corresponding wave height in NPV, the sensitivity gain in the differential method does not come from enhanced faradaic response, but, rather, from reduced background current. If the residual current does not change much from the first current sample to the second, then the subtractive process producing 𝛿i tends to cancel the background contribution. We will not treat the application of DPV to irreversible systems. Instead, we simply note that, as |ΔE| tends toward zero, the response in any differential scan approaches the derivative of the corresponding normal pulse voltammogram. This fact is easily demonstrated for the reversible system (Problem 8.7). If a system shows irreversibility because of slow heterogeneous kinetics, ′ one can still expect to see a differential response, but the peak will be shifted from E0 toward more extreme potentials by an activation overpotential (i.e., toward the negative for a cathodic process and toward the positive for an anodic one). Also, the peak width will be larger than for a reversible system, because the rising portion of an irreversible wave extends over a larger potential range. Since the maximum slope on the rising portion is smaller than in the corresponding reversible case, (𝛿i)max will be smaller than predicted by (8.4.9). If the irreversibility is caused by following chemistry, the peak will also be broad and low, but will be less extreme ′ than E0 for reasons discussed in Chapter 13. The range of time scales for DPV is the same as for NPV; hence, a given system ordinarily shows the same degree of reversibility toward either approach. However, the degree of reversibility toward pulse methods may differ from that shown toward methods based on longer time scales (Section 8.2.1). 8.4.3

Renewal vs. Pre-Electrolysis

In DPV, the pre-pulse stage in a measurement cycle can be called a pre-electrolysis period (Figure 8.4.1), while the analogous stage in NPV is a renewal period (Figure 8.2.1). This distinction is made here in recognition of different experimental purposes. For NPV, effective renewal in each cycle is essential for valid, quantitative results. Achieving it is the most important purpose of the pre-pulse period. For DPV, the key purpose is to condition the diffusion layer through pre-electrolysis, establishing effective bulk concentrations for interrogation by the pulse. Actual renewal is often unnecessary. If the system is kinetically reversible on the timescale of the measurement cycle, pre-electrolysis can readily establish effective bulk concentrations, even if the effects of prior cycles are not fully erased from the diffusion layer. In fact, the cumulative effect of successive cycles is gradually to thicken the diffusion layer in a manner that supports the assumptions used in the treatment of wave shape and peak height in DPV (Section 8.4.2). For this reason, DPV can generally be carried out quite successfully at a stationary electrode, such as a carbon disk or an HMDE. If the electrode reaction of interest is not chemically reversible (perhaps because one of the participants decays), then the concentrations in the diffusion layer will gradually degrade during a DPV scan at a stationary electrode. Renewal might prove helpful, but still might not be essential because DPV is mainly used for analytical measurements based on calibration. Even in the face of irreversibility, one might still obtain a reproducible response that can be practically calibrated.

8.4 Differential Pulse Voltammetry

If renewal seems important for the system at hand, the strategies discussed in Section 8.2.2 can be considered. 8.4.4

Residual Currents

In most DPV, the background current contains essentially no capacitive contribution, because dE/dt and dA/dt are zero at both moments of sampling (Section 8.2.1). However, residual currents still arise from electrolysis of impurities in solution or from slow faradaic reactions of major system components (Section 8.1.6). The rates of these processes often do not change greatly with the potential shift from E to E + ΔE and with the elapse of time from 𝜏 ′ to 𝜏; thus, the subtraction of current samples typically suppresses the faradaic background, but does not eliminate it altogether, as we will see in Section 8.6. In practical analysis by DPV, the faradaic background is usually the main factor limiting sensitivity. The improvements manifested in DPV yield sensitivities that are often an order of magnitude better than for NPV. Direct detection limits as low as 10−8 M can be achieved at Hg electrodes, but doing so requires close attention to selection of the medium. Section 8.6 provides more detail. Much lower detection limits can be achieved if preconcentration is employed, as in differential pulse stripping voltammetry (Section 12.7). Detection limits are poorer at solid electrodes, which are generally afflicted by residual currents from slow faradaic processes associated with the electrode surface itself.6 DPV moderates these contributions through its differential measurement of current, but the residual background normally remains higher than at mercury. 8.4.5

Differential Pulse Polarography

DPV continues to be performed with polarographic electrodes, in which case it is known as differential pulse polarography (DPP). Because of the SMDE’s operational advantages [Section 8.2.3(b)], it is now the strongly preferred electrode. Because of the rapid formation of the drop, the pre-electrolysis time at an SMDE can be as short as 500 ms. This time controls the duration of the overall scan, so the use of a short pre-electrolysis period can save much time in practical analysis, often 80% of that required at a DME. If a DME is used instead of an SMDE, dA/dt is never zero; therefore, a charging current contributes to the background. It is easily shown that7 𝛿ic ≈ −0.00567Ci ΔEm2∕3 𝜏 −1∕3

(8.4.11)

For a negative scan 𝛿ic is positive, and vice versa. This is a smaller contribution by an order of magnitude or more than one finds in NPP at a DME; moreover, the capacitive background in differential pulse polarography is flat, insofar as C i remains constant over a potential range. In contrast, NPP at a DME features a sloping background because of the dependence on Ez − E (equation 8.2.4). This difference is apparent in Figure 8.4.3, and the greater ease in evaluating the differential faradaic response is obvious. 6 The surfaces of electrodes often undergo faradaic transformations, such as the formation or reduction of oxides on metals or the electrochemical conversion of oxygen-containing functional groups on the edges of graphite planes. Many of these processes occur slowly and over sizable potential ranges; consequently, they give rise to background currents that can last a long time after the potential or the medium is changed. There can also be a slowly decaying nonfaradaic background if the electrode is subject to potential-dependent adsorption of a species of low concentration in the electrolyte. Background currents like these are often said to arise from “surface processes.” In general, such currents are much larger at solid electrodes than at mercury, unless the solid electrode is held for a long time (even several minutes or an hour) at a fixed potential in an unchanging medium. 7 Second edition, p. 291.

375

376

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

There is a sensitivity advantage at the SMDE in any situation where the charging current background at the DME is appreciable. Otherwise the SMDE and DME provide comparable performance with respect to sensitivity.

8.5 Square-Wave Voltammetry Exceptional versatility is found in a method called square-wave voltammetry (SWV), rooted in early work by Barker (29, 30), then transformed (31) and redeveloped extensively as automated potentiostats made the method much more practical (19, 20, 32, 33). One can view SWV as combining the best aspects of several pulse voltammetric methods, including the background suppression and sensitivity of DPV, the diagnostic value of NPV, and the ability to interrogate products directly in the manner of RPV. It also offers access to a wider range of time scales than can be achieved by any of the other pulse voltammetric techniques. The method has become widely used, and extensive reviews are available (33–38), providing many details beyond the introduction given here. 8.5.1

Experimental Concept and Practice

Square-wave voltammetry is carried out at a stationary electrode, such as a Pt disk or an HMDE, using the waveform and measurement scheme shown in Figure 8.5.1. As in other forms of pulse voltammetry, the electrode is taken through a series of measurement cycles; however, there is no renewal of the diffusion layer between cycles. In contrast to NPV, RPV, and DPV, the form of square-wave voltammetry discussed here has no polarographic mode.8 The waveform can be viewed as a special case of that used for DPV (Figure 8.4.1), in which the pre-electrolysis period and the pulse are of equal duration, and the pulse is opposite the scan direction. However, the interpretation of results is facilitated by considering the waveform as consisting of a staircase scan, each tread of which is superimposed by a symmetrical double pulse, one in the forward direction and one in the reverse. Over many cycles, the waveform is a bipolar square wave superimposed on the staircase. Figure 8.5.1 helps to define the principal parameters. The square wave is characterized by a pulse height, ΔEp , measured with respect to the corresponding tread of the staircase, and a pulse width t p , which is often expressed in terms of the square-wave frequency, f = 1/2t p . The staircase shifts by ΔEs at the beginning of each cycle; thus, the scan rate is v = ΔEs /2t p = f ΔEs . The scan begins at an initial potential, Ei , which can be applied for an arbitrary time. Current samples are taken twice per cycle, at the end of each pulse. The forward current sample, if , arises from the first pulse per cycle, which is in the direction of the staircase scan. The reverse current sample, ir , is taken during the second pulse, which is in the opposite direction. A difference current, Δi, is calculated as if − ir . There is diagnostic value in the forward and reverse currents, so they are preserved separately. Thus, the result of a single SWV run is a set of three voltammograms showing if , ir , and Δi vs. the potential on the corresponding staircase tread. 8 Barker (29, 30) invented a method that he called “square wave polarography,” in which a quite different experimental strategy is used. A small-amplitude, high-frequency square wave is superimposed on the slowly changing ramp or staircase typical of polarography, and a current-sampling scheme is employed to detect the averaged response to many cycles of the square wave for each drop at the DME. In the second edition, this method was named Barker square wave voltammetry (BSWV), while the method presented here was called Osteryoung square wave voltammetry (OSWV). In this edition, the authors refer to OSWV simply as square wave voltammetry (SWV), in keeping with contemporary practice.

8.5 Square-Wave Voltammetry

Potential

Cycle 1

Cycle 2

Cycle 3

Forward sample ΔEp tp

ΔEs

Ei

Reverse sample 0

Time

Figure 8.5.1 Waveform and measurement scheme for SWV. Shown in bold is the actual waveform applied to the working electrode. The light intervening lines indicate the underlying staircase onto which the square wave can be regarded as having been superimposed. In each cycle, a forward current sample is taken at the time indicated by the solid dot, and a reverse current sample is taken at the time marked by the shaded dot.

Square-wave voltammetry is always performed using an automated potentiostat (Section 16.6) with functional elements organized essentially as in Figure 8.4.2. In general, t p (or f ) defines the experimental time scale; ΔEs fixes the spacing of data points along the potential axis; and these parameters together determine the time required for a full scan. In normal practice, ΔEs is significantly less than ΔEp , which defines the span of interrogation in each cycle and, therefore, determines the resolution of voltammetric features along the potential axis. Only t p (or f ) is varied over a wide range, typically 0.25–100 ms (f = 5–2000 Hz). Values of ΔEs = 10/n mV and ΔEp = 50/n mV have been recommended (33) and suffice generally. With ΔEs = 10 mV and t p = 0.25 − 100 ms, the scan proceeds at 100 to 50 mV/s; thus, the time for recording a full voltammogram is shorter than for other pulse methods, but similar to a typical scan time in cyclic voltammetry. 8.5.2

Theoretical Prediction of Response

Since the diffusion layer is not renewed at the beginning of each measurement cycle, it is not possible to treat a cycle in isolation, and theoretical treatments of SWV are intrinsically much more complex than for other forms of pulse voltammetry. The initial condition for each cycle is the complex diffusion layer that has evolved from all prior pulses, and it is a function, not only of the details of the waveform, but also of the kinetics and mechanisms of the chemistry linked to the electrode process. The considerations resemble those encountered in Section 6.5 as we treated double-step responses, and the mathematical device of superposition can be applied to the extended step waveforms of SWV in simpler cases. Let us now consider the prototypical case, in which the electrode reaction O + ne ⇌ R exhibits reversible kinetics, and the solution contains O, but not R, in the bulk. The solution has been ′ homogenized, and the initial potential, Ei , is chosen well positive of E0 , so that the concentration profiles are uniform as the SWV scan begins. The experiment is fast enough to confine behavior to semi-infinite linear diffusion at most electrodes, and we assume its applicability here. These circumstances adhere to the general formulation (Section 4.5.2). The final boundary condition needed to solve the problem comes from the potential waveform, which is linked to the concentration profile through the nernstian balance at the electrode.

377

378

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

It is convenient to consider the waveform as consisting of a series of half-cycles with index m, beginning from the first forward pulse, which has m = 1. Then, [ ( ) ] m+1 Em = Ei − Int − 1 ΔEs + (−1)m ΔEp (m ≥ 1) (8.5.1) 2 where Int[(m + 1)/2] denotes truncation of the ratio to the highest integer. The nernstian balance at the surface can be expressed for each half-cycle in the following way: 𝜃m =

CO (0, t) CR (0, t)

′

= exp[nf (Em − E0 )]

(8.5.2)

The solution can be obtained analytically (31, 33, 39), and the sampled current for the mth half cycle turns out to be 1∕2

im =

∗ ∑ m nFADO CO 1∕2

𝜋 1∕2 tp

j=1

Qj−1 − Qj (m − j + 1)1∕2

(8.5.3)

where Q0 = 0 Qj =

(8.5.4a) 𝜉𝜃j

( j > 0)

1 + 𝜉𝜃j

(8.5.4b)

and 𝜉 = (DO /DR )1/2 . The sum in (8.5.3), which runs over all half-cycles preceding and including the one of interest, manifests the prior history of electrolysis. Odd values of m correspond to forward current samples and even values denote reverse current samples. In much of the theory of SWV, currents are represented dimensionlessly by normalizing with the factor preceding the sum in (8.5.3). This factor is the Cottrell current for time t p , which is the plateau current sampled in an NPV experiment with pulse width t p , (8.2.1). Designating this current as id , we can define the dimensionless current sample for the mth half-cycle, 𝜓 m , as 𝜓m =

im id

=

m ∑

Qj−1 − Qj

j=1

(m − j + 1)1∕2

(8.5.5)

The dimensionless difference current, Δ𝜓 m , is given by subtraction of pairs of samples, with the odd m taken first: Δi Δ𝜓m = m = 𝜓m − 𝜓m+1 (only odd values of m) (8.5.6) id Figure 8.5.2 shows dimensionless current transients and current samples for the experiment that we have been discussing. The currents are small in the early cycles, because the staircase ′ potential is too far positive of E0 for the forward pulse to reach the region where electrolysis ′ can occur. In the middle of the figure, the staircase has moved into the region of E0 , so that the rate of electrolysis is a strong function of potential. The forward pulse significantly amplifies the rate of reduction of O, and the reverse pulse actually reverses that reduction, so that the current becomes anodic. The right side of the figure corresponds to cycles in which the stair′ case potential has become considerably negative of E0 , so that electrolysis begins to occur at the diffusion-controlled rate regardless of potential. Then, neither the forward pulse nor the reverse affects the current much, and the samples become similar. The sampled current in the forward pulses is smaller than in the middle of the diagram because the cumulative effect of

8.5 Square-Wave Voltammetry

2.0 1.5 1.0 𝜓 0.5 0.0 –0.5 –1.0

0

5

10 15 20 Time (half-cycle index, m)

25

30

Figure 8.5.2 Dimensionless current response throughout an SWV experiment for the reversible O/R system ′ with CR∗ = 0 and with the scan beginning well positive of E 0 . Cathodic currents are upward. The staircase ′

potential reaches E 0 near m = 15. Sampled currents are shown as points. nΔE p = 50 mV and nΔE s = 30 mV. [Osteryoung and O’Dea (33), by courtesy of Marcel Dekker, Inc.]

electrolysis through many cycles is to deplete the diffusion layer and to slow the rate at which O arrives. The continued falloff of sampled current at the right extreme of the figure is caused ′ by this effect. The difference current reaches a peak near E0 and is small on either side. Figure 8.5.3 is a dimensionless representation of the voltammograms that would be derived from an experiment like that just described. The forward and reverse currents resemble a cyclic 1.0 Δ𝜓

0.5 𝜓f 𝜓

0.0 𝜓r

–0.5 0.2

0.0

–0.2

–0.4

n(E – E1/2)/V

Figure 8.5.3 Dimensionless square-wave voltammograms for the reversible O/R case with CR∗ = 0. nΔE p = 50 mV and nΔE s = 10 mV. Forward currents (𝜓 f ), reverse currents (𝜓 r ), and difference currents (Δ𝜓) vs. a potential axis referred to E1∕2 . Note that n(E m − E 1/2 ) = (RT/F) ln 𝜉𝜃 m . [Osteryoung and O’Dea (33), by courtesy of Marcel Dekker, Inc.]

379

380

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

Table 8.5.1 Dimensionless Peak Current, 𝜓 p , vs. SWV Operating Parameters n𝚫E s (mV) n𝚫E p (mV)

1

5

10

20

0(a)

0.0053

0.0238

0.0437

0.0774

10

0.2376

0.2549

0.2726

0.2998

20

0.4531

0.4686

0.4845

0.5077

50

0.9098

0.9186

0.9281

0.9432

100

1.1619

1.1643

1.1675

1.1745

(a) ΔEp = 0 corresponds to staircase voltammetry (second edition, Section 7.3.1). From Osteryoung and O’Dea (33).

voltammogram and have much of the same diagnostic value, while the difference current resembles the response from DPV and has similar sensitivity. ′ The difference current voltammogram reaches a peak at E1∕2 = E0 + (RT∕nF) ln (DR ∕DO )1∕2 and has a dimensionless peak current, Δ𝜓 p , that depends on n, ΔEp , and ΔEs as presented in Table 8.5.1. The actual peak current, Δip , is therefore, 1∕2

Δip =

∗ nFADO CO 1∕2

𝜋 1∕2 tp

Δ𝜓p

(8.5.7)

Since the Cottrell factor is the plateau current in an NPV experiment having the same pulse width, Δ𝜓 p , gauges the peak height in SWV relative to the limiting response in NPV, just as the ratio (1 − 𝜎)/(1 + 𝜎) does for DPV. For normal operating conditions of ΔEs = 10/n mV and ΔEp = 50/n mV, the SWV peak is 93% of the corresponding NPV plateau height. For DPV, the comparable figure is only about 45% (Table 8.4.1), so the SWV method is slightly more sensitive ′ than DPV. This is true because the reverse pulses near E0 produce an anodic current, which enlarges Δi. Systems involving slow heterogeneous kinetics, coupled homogeneous reactions or equilibria (as in Chapter 13), or more complex forms of mass transfer (e.g., at a UME, Sections 5.2, 5.7, and 6.1.3) are most easily treated by digital modeling. The reviews (33–38) discuss the application of SWV to a wide range of such phenomena. in a positiveFigure 8.5.4 contains data for a system involving the oxidation of Fe(CN)4− 6 going scan at a Pt disk UME. The results were analyzed by assuming reversibility at all fre′ quencies and by adjusting two parameters, the radius of the disk, r0 , and E1∕2 = E0 + (RT∕nF) ( ) 1∕2 1∕2 ln DR ∕DO , to provide the best fit. The change in behavior with frequency reflects the fact that the diffusion pattern at a disk UME can deviate from the semi-infinite linear case (Sections 5.2 and 6.1.3). The validity of the model is supported by the consistency of the fitted parameters for runs at different frequencies and by the quality of the fits. This example illustrates the typical manner of comparing SWV results with theory. 8.5.3

Background Currents

In SWV, background currents can be understood exactly as for DPV. If t p > 5Ru C d , there is no appreciable charging current contribution, either to the individual current samples or to the

8.5 Square-Wave Voltammetry

Figure 8.5.4 SWV at a Pt disk UME in a solution of 20 mM Fe(CN)4− also containing 2 M KNO3 . Each 6 scan was made from 0.0 to 0.50 V with ΔE p = 50 mV and ΔE s = 10 mV. (a) f = 5, (b) f = 60, (c) f = 500 Hz. Points are experimental; curves are fitted to give r0 and E 1/2 , respectively, as: (a) 11.9 μm, 0.2142 V, (b) 12.4 μm, 0.2137 V, (c) 12.2 μm, 0.2147 V. [Whelan, O’Dea, Osteryoung, and Aoki (40).]

100 (a) 50

0

200 i/nA

(b)

100 0

400

(c)

0 0.00

0.25 E/V vs. SCE

0.50

differences. Faradaic background processes do contribute and typically control detection limits for SWV. At solid electrodes or near background limits, the effects on the forward and reverse currents can be sizable. While they are often suppressed significantly in the difference currents, their contributions are not entirely eliminated. 8.5.4

Applications

SWV can be used diagnostically much in the manner of cyclic voltammetry (Chapters 7 and 13). The method has a high information content, especially when one considers the trio of voltammograms from a single scan; and it has the power to interrogate electrode processes over a wide potential span in a reasonable time. Its strengths with respect to CV are especially rooted in its ability to suppress the background. In general, systems can be examined at substantially lower concentrations than with CV. Moreover, there is normally much less distortion of the response by the background, so that fitting of data to theoretical models can be done with greater accuracy. On the whole, SWV is better than CV for evaluating quantitative parameters for systems that are understood mechanistically. SWV also has relative weaknesses: For most practitioners, CV is more intuitively interpretable in chemical terms; moreover, it offers a considerably wider range of time scales. Because the reversal in CV can cover a large span of potentials, it can highlight linkages between processes occurring at widely separated potentials. The reversal interrogation in SWV is limited to the pulse height, so a single SWV scan does not match CV’s ability in this regard. To remedy that deficiency, cyclic square-wave voltammetry (CSWV) has evolved. This method is simply the use of the SWV mode of data acquisition over a pair of forward and reverse sweeps, such as would apply in cyclic voltammetry. The forward scan develops as we have described, with a staircase increment ΔEs applied until the switching potential, E𝜆 , is reached. At that point, the staircase increment is changed to −ΔEs , so that the scan reverses. When the staircase returns to the initial potential, Ei , the experiment ends. Six different voltammograms are

381

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

Figure 8.5.5 Voltammetry at a Pt disk of cobalt tetraphenylporphyrin (CoTPP); Figure 1 (M = Co) in CH2 Cl2 with 0.1 M TBAP. Scans begin at 0.4 V vs. SCE and first move positively. (a) CSWV, Δi vs. E. (b) CV, i vs. E. [Helfrick and Bottomley (41) © 2009, American Chemical Society.]

10 μA (a)

Current

382

5 μA (b)

1.6

1.4

1.2

1.0 0.8 E/V vs. SCE

0.6

0.4

produced—forward samples, reverse samples, and difference currents vs. potential for the scans in both directions. The simplest mode is to use the difference currents plotted as for a normal CV. Figure 8.5.5 provides an example (41), which also highlights the superior background characteristics of SWV vs. CV. The review literature (33–38) provides entrée to many specific cases in which SWV and CSWV have been employed for mechanistic diagnosis and evaluation of kinetic parameters. Because SWV has a firm theoretical basis and operational versatility, the method is also being developed to quantify interfacial electron-transfer kinetics (42) and nonfaradaic processes at electrodes (43). The review literature just cited also references many applications of SWV for practical analysis. Figure 8.5.6 shows results from a study aimed in that direction. For analytical work, SWV is generally the best choice among all pulse methods, because it offers background suppression with the effectiveness of DPV, sensitivity slightly greater than that of DPV, and 1 30

3

i/μA 2 20

10

0 0.2

0.4

0.6 E/V vs. Ag/AgCl

0.8

1.0

Figure 8.5.6 SWV (Δi vs. E) at a glassy carbon disk for epigallocatechin gallate (I, solid curve), epigallocatechin (II, dotted curve), and gallic acid (III, dashed curve), all at 1 × 10−4 M in pH 2 buffer. f = 100 Hz; ΔE s = 2 mV; ΔE p = 50 mV; E i = 0.1 V. I and II are flavanols commonly extracted from green tea. The curves here for II and III allow assignment of electroactive moieties (1, 2, and 3) in I. [Nowak, Šeruga, and Komorsky-Lovri´c (44).]

8.6 Analysis by Pulse Voltammetry

much faster scan times. The most reproducible behavior and lowest detection limits are generally found at mercury surfaces, so an SMDE working as an HMDE is quite effective with SWV.

8.6 Analysis by Pulse Voltammetry DPV and SWV are among the most sensitive means for the direct evaluation of concentrations, and they find wide use for practical analysis (6, 19–21, 26, 32–38, 45). When they can be applied, they are often far more sensitive than competing nonelectrochemical methods. In addition, they often can provide information about the chemical form in which an analyte appears. Oxidation states can be defined; complexation can often be detected; and acid–base chemistry can be characterized. This information is frequently overlooked or unavailable with competing methods. The chief weakness of pulse analysis, common to most electroanalytical techniques, is a limited ability to resolve complex systems. DPV and SWV are frequently employed in anodic stripping (Section 12.7). They are also widely employed using modified electrodes (Chapter 17). These strategies can greatly enhance both sensitivity and selectivity (45). The pulse methods are sufficiently sensitive that one must pay special attention to impurity levels in solvents and supporting electrolytes. Contamination from the electrolyte can be reduced by lowering its concentration from the usual range of 0.1–1 to 0.01 M or even to 0.001 M. The lower limit is fixed by the maximum cell resistance that can be tolerated, if it is not set first by chemical considerations, such as the role of the supporting electrolyte for complexation or buffering. In most analytical work, aqueous media are used, both for convenience and for compatibility with the chemistry of sample preparation; however, other solvents can provide superior working ranges and merit consideration for new applications. The working range for any medium is much narrower for trace analysis by DPP or SWV than for conventional voltammetry, simply because the residual faradaic background becomes intolerably high at less extreme potentials. This point is clear in Figure 8.6.1.

With As(III) 50 nA

Background 60 mV

–120 mV vs. SCE

Figure 8.6.1 DPP for 4.84 × 10−7 As(III) in 1 M HCl containing 0.001% Triton X-100. tmax = 2 s, ΔE = − 100 mV. [Adapted from Osteryoung and Osteryoung (46)].

383

384

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

i 2 μA

0 –0.2

–0.4

–0.6 E/V vs. SCE (a)

–0.8

–0.4

–0.6 E/V vs. SCE (b)

–0.8

i 5 μA

0 –0.2

δi

0

1 μA

–0.2

–0.4

–0.6 E/V vs. SCE (c)

–0.8

Figure 8.6.2 Voltammograms at a DME of 10−3 M Fe3+ and 10−4 M Cd2+ in 0.1 M HCl. (a) Conventional polarography. (b) NPP, E b = 0.0 V vs. SCE. (c) DPP, ΔE = − 50 mV.

Under some circumstances, pulse techniques can mislead with respect to a sample’s composition. Examples are provided in Figure 8.6.2 for a system with 1 mM Fe(III) and 0.1 mM Cd(II). The Fe(III) is reduced at every potential in the working range of the Hg electrode, as shown by the dc polarogram in Figure 8.6.2a, which accurately reflects the 5 times larger value of nC * for Fe(III) vs. Cd(II). The NPP curve in Figure 8.6.2b shows a plateau for reduction of Fe(III), but suggests that the concentration of Cd(II) is much larger than that of Fe(III). This mis impression is caused by pre-electrolysis of Fe(III) at the base potential during the intended renewal period for NPP (22). For this system, there is no choice of base potential where pre-electrolysis does not occur. In DPP, a peak is seen only for the Cd2+ reduction (Figure 8.6.2c), because the observed potential range is entirely on the negative side of the Fe3+ wave. Since DPV approximates the derivative of the corresponding NPV, peaks will not be seen in DPV (or in SWV) unless distinct waves appear in NPV. Aside from this type of problem, DPV and SWV are well suited to the analysis of multicomponent systems because they usually allow the separation of signals from individual components along a common baseline. This point is illustrated in Figure 8.6.3. Note also from that figure (and Figure 8.5.6) that pulse methods are applicable to a much richer variety of analytes than heavy metal species.

8.7 References

δi

4.20 ppm Tetracycline ∙ HCl

0.2 μA

2.40 ppm Chloramphenicol

0

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9 –1.0 –1.1 –1.2 –1.3 –1.4 E/V vs. SCE

Figure 8.6.3 DPP for a mixture of tetracycline and chloramphenicol in 0.1 M acetate buffer, pH 4. ΔE = − 25 mV. [Courtesy of Princeton Applied Research].

8.7 References 1 J. Heyrovský, Chem. Listy, 16, 256 (1922). 2 A. J. Bard, J. Chem. Educ., 84, 644 (2007). 3 I. M. Kolthoff and J. J. Lingane, “Polarography,” 2nd ed., Wiley-Interscience, New York,

1952. 4 J. J. Lingane, “Electroanalytical Chemistry,” 2nd ed., Wiley-Interscience, New York, 1958. 5 L. Meites, “Polarographic Techniques,” 2nd ed., Wiley-Interscience, New York, 1958. 6 A. Bond, “Modern Polarographic Methods in Analytical Chemistry,” Marcel Dekker, New 7 8 9 10 11 12 13 14

York, 1980. D. Ilkoviˇc, Collect. Czech. Chem. Commun., 6, 498 (1934). D. Ilkoviˇc, J. Chim. Phys., 35, 129 (1938). J. Koutecký, Czech. Cas. Fys., 2, 50 (1953). J. Koutecký and M. von Stackelberg in “Progress in Polarography,” Vol. 1, P. Zuman and I. M. Koltoff, Eds., Wiley-Interscience, New York, 1962. J. J. Lingane, Anal. Chim. Acta, 44, 411 (1969). J. J. Lingane, Ind. Eng. Chem. Anal. Ed., 15, 588 (1943). L. Meites, Ed., “Handbook of Analytical Chemistry,” McGraw-Hill, New York, 1963, pp. 4–43 to 5–103. A. J. Bard and H. Lund, Eds., “Encyclopedia of the Electrochemistry of the Elements,” Marcel Dekker, New York, 1973–1986.

385

386

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

L. Meites and P. Zuman, “Electrochemical Data,” Wiley, New York, 1974. G. C. Barker and A. W. Gardner, Z. Anal. Chem., 173, 79 (1960). E. P. Parry and R. A. Osteryoung, Anal. Chem., 37, 1634 (1965). Z. Stojek in “Electroanalytical Methods,” 2nd ed., F. Scholz, Ed., Springer, Berlin, 2010, Chapter II.2. J. Osteryoung, Acc. Chem. Res., 26, 77, (1993). R. A. Osteryoung and J. Osteryoung, Phil. Trans. Roy. Soc. London, Ser. A, 302, 315 (1981). J. B. Flato, Anal. Chem., 44 (11), 75A (1972). J. L. Morris, Jr., and L. R. Faulkner, Anal. Chem., 49, 489 (1977). A. G. Ewing, M. A. Dayton, and R. M. Wightman, Anal. Chem., 53, 1842 (1981). R. M. Wightman and D. O. Wipf, Electroanal. Chem., 15, 267 (1989). W. M. Peterson, Am. Lab., 11 (12), 69 (1979). Z. Kowalski, K. H. Wong, R. A. Osteryoung, and J. Osteryoung, Anal. Chem., 59, 2216 (1987). J. Osteryoung and E. Kirowa-Eisner, Anal. Chem., 52, 62 (1980). K. B. Oldham and E. P. Parry, Anal. Chem., 42, 229 (1970). G. C. Barker and J. L. Jenkins, Analyst, 77, 685 (1952). G. C. Barker in “Proceedings of the Congress on Modern Analytical Chemistry in Industry,” St. Andrews, Scotland, 1957, pp. 209–216. L. Ramaley and M. S. Krause, Jr., Anal. Chem., 41, 1362 (1969). J. G. Osteryoung and R. A. Osteryoung, Anal. Chem., 57, 101A (1985). J. Osteryoung and J. J. O’Dea, Electroanal. Chem., 14, 209 (1986). V. Mirˇceski, R. Gulaboski, M. Lovri´c, I. Bogeski, R. Kappl, and M. Hoth, Electroanalysis, 25, 2411 (2013). A. Chen and B. Shah, Anal. Methods, 5, 2158 (2013). M. Lovri´c in “Electroanalytical Methods,” 2nd ed., F. Scholz, Ed., Springer, Berlin, 2010, Chapter II.3. V. Mirˇceski, Š. Komorsky-Lovri´c, and M. Lovri´c, “Square-Wave Voltammetry,” Springer, Berlin (2007). G. N. Eccles, Crit. Rev. Anal. Chem., 22, 345 (1991). J. H. Christie, J. A. Turner, and R. A. Osteryoung, Anal. Chem., 49, 1899 (1977). D. Whelan, J. J. O’Dea, J. Osteryoung, and K. Aoki. J. Electroanal. Chem., 202, 23 (1986). J. C. Helfrick, Jr., and L. A. Bottomley, Anal. Chem., 81, 9041 (2009). P. Dauphin-Ducharme, N. Arroyo-Currás, M. Kurnik, G. Ortega, H. Li, and K. W. Plaxco, Langmuir, 33, 4407 (2017). S. J. Cobb and J. V. Macpherson, Anal. Chem., 91, 7935 (2019). I. Nowak, M. Šeruga, and Š. Komorsky-Lovri´c, Electroanalysis, 21, 1019 (2009). Y. Lu, X. Liang, C. Niyungeko, J. Zhou, J. Xu, and G. Tian, Talanta, 178, 324 (2018). J. G. Osteryoung and R. A. Osteryoung, Am. Lab., 4 (7), 8 (1972).

8.8 Problems 8.1

The following measurements were made at 25 ∘ C on an NPV wave for which id,c = 3.24 μA. The process can be written O + ne ⇌ R. E (V) vs. SCE

−0.395

−0.406

−0.415

−0.422

−0.431

–0.445

i (μA)

0.48

0.97

1.46

1.94

2.43

2.92

8.8 Problems

Calculate (a) the number of electrons involved in the electrode reaction, and (b) the formal potential vs. NHE of the O/R couple, assuming DO = DR . 8.2

Derive the Ilkoviˇc equation, (8.1.1), from the Cottrell equation, (6.1.12), based on the information in Section 8.1.2. In the process, derive (8.1.2).

8.3

Consider a species A that can be reduced at a DME to species B. A 1-mM solution of A in acetonitrile shows a wave with E1/2 at −1.90 V vs. SCE. The wave slope is 60.5 mV at 25 ∘ C and (I)max = 2.15 in the usual units. When dibenzo-15-crown-5 is added to the solution, the polarographic behavior changes. O

O

O

O O

Dibenzo-15-crown-5 (C)

The following observations were made: Conc. of C (M)

E 1/2 (V)

Wave slope (mV)

(I)max

10−3 10−2

−2.15

60.3

2.03

10−1

–2.21

59.8

2.02

–2.27

59.8

2.04

Interpret these results. Can any thermodynamic data be derived from the data? Can you suggest the identity of species A? 8.4

A polarogram of molecular oxygen in air-saturated 0.1-M KNO3 is like that of Figure 6.4.1. The concentration of O2 is about 0.25 mM. At E = − 0.4 V vs. SCE, (id )max = 3.9 μA, t max = 3.8 s, and m = 1.85 mg/s. At E = − 1.7 V vs. SCE, (id )max = 6.5 μA, t max = 3.0 s, and m = 1.85 mg/s. Calculate (I)max at each potential. Is the ratio of the two values what you expect? Explain any discrepancy in chemical terms. Calculate the diffusion coefficient for O2 using the more appropriate constant. Defend your choice of the two.

8.5

Consider an analysis for the toxic ion Tl(I) in wastewater that also contains Pb(II) and Zn(II) in 10- to 100-fold excesses. Outline any obstacles that would impede a DPV determination and suggest means for circumventing them without implementing separation techniques. For 0.1 M KCl, E1/2 (Tl+ /Tl) = − 0.46 V, E1/2 (Pb2+ /Pb) = − 0.40 V, and E1/2 (Zn2+ /Zn) = − 0.995 V vs. SCE at a mercury electrode.

8.6

Sketch the NPV expected for a substance undergoing an irreversible electrode reaction at a gold disk (e.g., O2 → H2 O2 in 1 M KCl). Assume that the reduced form is initially absent and that no electrolysis occurs at the base potential in the starting solution. Explain the shape of the trace. (The location of the trace on the potential axis is not of interest here.) Do the same for the case in which the electrode reaction is reversible.

387

388

8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry

How would the curves differ if the disk were rotated during the recording of the polarograms? 8.7

(a) Show that the derivative of a reversible STV wave is n2 F 2 AC ∗O 𝜉𝜃 di = 1∕2 1∕2 dE (1 + 𝜉𝜃)2 RT𝜋 𝜏 (b) Show that (8.4.7) approaches this form for 𝛿i/ΔE ≈ di/dE as ΔE → 0.

(8.8.1)

389

9 Controlled-Current Techniques Chapters 5–8 covered methods in which the potential of an electrode was controlled while the current was determined as a function of time. In this chapter, we consider the opposite case—the current is controlled, and the potential is measured vs. time. Other assumptions commonly made in Chapters 6–8, such as small A/V conditions, the presence of an excess electrolyte, and semi-infinite linear diffusion, are also taken here. We do not treat UMEs separately, because the steady-state behavior under controlled current is the same as under controlled potential. The methods discussed here are called chronopotentiometric techniques, because E is determined as a function of time, or galvanostatic techniques, because the current at the working electrode is the controlled function.

9.1 Introduction to Chronopotentiometry A chronopotentiometric experiment is carried out by applying a controlled current between the working and counter electrodes of a cell, and then recording the potential between the working and reference electrodes (Figure 9.1.1). A device for supplying a controlled current is called a galvanostat. Many electrochemical workstations can function in a galvanostatic mode. Constant-current chronopotentiometry, illustrated in Figure 9.1.2a, is the most common controlled-current microelectrode method. Let us introduce it using the anthracene (An) system discussed as an example in Section 6.1. The working electrode is a Pt disk immersed in a quiescent solution where the bulk concentration of An applies everywhere. At t = 0, a steady cathodic current, i, is applied, causing reduction of An to the anion radical, An−∙, at a constant rate. The potential of the electrode moves to the range characteristic of An∕An−∙, but gradually shifts negatively as the concentration ratio changes in favor of An−∙ at the electrode surface. One can think of the process as a titration of An in the vicinity of the electrode by the continuous flux of electrons, producing an E − t curve like that obtained for a potentiometric titration (E as a function of titrant added, i × t). Eventually, the concentration of An falls to zero at the electrode surface, and the flux of An can no longer accept all electrons being forced across the electrode/solution interface. The potential of the working electrode must rapidly shift negatively, so that a second reduction process can begin. The moment when this shift occurs is called the transition time, 𝜏. It depends on the concentration and diffusion coefficient of the electroreactant and is the chronopotentiometric analogue of the peak or limiting current in controlled-potential experiments. The shape and location of the E − t curve is governed by the reversibility of the electrode kinetics.

Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

390

9 Controlled-Current Techniques

i Counter Source of controlled i (galvanostat)

i

Reference Working

i

Recording system (Ewk vs. ref)

Figure 9.1.1 Block diagram of apparatus for chronopotentiometric measurements. Excitation

Figure 9.1.2 Chronopotentiometric modes. (a) Constant-current chronopotentiometry. (b) Chronopotentiometry with linearly increasing current. (c) Current reversal chronopotentiometry. (d) Cyclic chronopotentiometry.

Response

i

E

τ 0

t

(a)

i

0

t

0

t

E

0

t

(b)

i

E 0

t

τf (c)

i

τr t

0 E

0

t

τ1 τ2 τ3 τ4 (d)

0

t

As an alternative to a constant current, one can apply a current that follows a known function of time (e.g., a current ramp, i = 𝛽t; Figure 9.1.2b). This approach is called programmed current chronopotentiometry. A constant current can also be reversed after some initial period (current reversal chronopotentiometry, Figure 9.1.2c). For example, if in the example considered above, the current is suddenly changed at or before the transition time to an anodic current of equal magnitude, the An−∙ formed during the forward step will begin to oxidize. The potential will then move positively as the An∕An−∙ ratio becomes richer in An at the electrode surface. When the An−∙ concentration falls to zero at the electrode surface, a potential transition toward positive potentials must occur, and a reverse transition time can be measured. In an extension of this technique, the current can be repeatedly reversed at each transition, resulting in cyclic chronopotentiometry (Figure 9.1.2d).

9.2 Theory of Controlled-Current Methods

As in the treatment of controlled-potential data, the derivatives of the E − t curves can be obtained, or differential methods can be employed.1 Controlled-current techniques differ from controlled-potential methods in notable ways: • The instrumentation for galvanostatic experiments is simpler, because there is no need for feedback from the reference electrode to the control device. Although constant-current sources constructed from operational amplifiers are frequently employed (Section 16.5), a simple circuit employing a high-voltage power supply and a large resistor can be adequate. • In the treatment of controlled-current experiments, the surface boundary condition is based on a known current or flux at the electrode surface (i.e., on a known concentration gradient), while for controlled-potential methods the focus is on the surface concentrations themselves. Solutions of diffusion equations in galvanostatic problems are generally simpler, and closed-form results are usually obtained. • Galvanostatic methods can be advantageous when one desires to control the rate or extent of a background process, such as the generation of solvated electrons in liquid ammonia or the reduction of quaternary ammonium ion in an aprotic solvent. Working with background processes is often difficult in a potentiostatic mode because the rate varies so sharply with potential. A simple method for determining the thickness of a metal film is by anodic stripping at constant current. • A disadvantage of controlled-current techniques is that the effects of double-layer charging are frequently larger and occur throughout the experiment. Correction is not straightforward. • Treating data from multicomponent systems and stepwise electrode reactions is also more complicated in controlled-current methods, partly because waves in E − t transients are usually less well defined than those in voltammetry.

9.2 Theory of Controlled-Current Methods 9.2.1

General Treatment for Linear Diffusion

We again consider the reaction O + ne → R, assuming a planar working electrode, an excess ∗ and C ∗ . The general formulation electrolyte, an unstirred solution, and bulk concentrations CO R (Section 4.5.2) describes this system using seven equations—the diffusion equations, the initial conditions, and the semi-infinite conditions for both O and R, plus the flux balance. After Laplace transformation, these relationships distill into two results (Section A.1.6), which we take as a point of departure: C O (x, s) = C R (x, s) =

∗ CO

s CR∗ s

+ A(s) e−(s∕DO )

1∕2 x

− 𝜉A(s) e−(s∕DR )

1∕2 x

(9.2.1) (9.2.2)

where 𝜉 = (DO /DR )1/2 and A(s) is a function still to be evaluated. Since the applied current, i(t), is controlled, the flux at the electrode surface is given at any time by (4.4.21): [ ] 𝜕CO (x, t) i(t) DO = (9.2.3) 𝜕x nFA x=0 1 These methods are uncommon, but are discussed in more depth in the first edition, Section 7.6.

391

392

9 Controlled-Current Techniques

This is the final boundary condition, defining the concentration gradient at the electrode surface and allowing the diffusion problem to be solved. The transform of (9.2.3) is [ ] 𝜕C O (x, s) i(s) DO = (9.2.4) 𝜕x nFA x=0

One can take the indicated derivative of (9.2.1) to obtain A(s), so that (9.2.1) and (9.2.2) become C∗ 1∕2 i(s) C O (x, s) = O − e−(s∕DO ) x (9.2.5) 1∕2 1∕2 s nFADO s C∗ 1∕2 i(s) C R (x, s) = R + e−(s∕DR ) x (9.2.6) 1∕2 1∕2 s nFADR s In most galvanostatic experiments the applied current is constant; however, (9.2.5) and (9.2.6) accommodate any form of applied current, i(t), including reversal. Moreover, no reference has yet been made to the electron-transfer kinetics, so (9.2.5) and (9.2.6) also apply to all kinetic regimes. The requirements are only that O and R be chemically stable solutes whose mass-transfer is controlled by semi-infinite linear diffusion.2 9.2.2

Constant-Current Electrolysis—The Sand Equation

If i(t) is constant, then i(s) = i∕s and (9.2.5) becomes ∗ CO 1∕2 i C O (x, s) = − e−(s∕DO ) x 1∕2 3∕2 s nFADO s

(9.2.7)

The inverse transform of this equation yields the concentration profile: { ( ]} [ )1∕2 ( ) 2 D t i x x O ∗ − CO (x, t) = CO 2 exp − − x ⋅ erfc nFADO 𝜋 4DO t 2(DO t)1∕2 (9.2.8) Likewise, i CR (x, t) = CR∗ + nFADR

{ ( ]} [ ) ( ) DR t 1∕2 x2 x 2 exp − − x ⋅ erfc (9.2.9) 𝜋 4DR t 2(DR t)1∕2

Figure 9.2.1 presents profiles at various times during a constant-current electrolysis for the case of CR∗ = 0. Although C O (0, t) decreases continuously, the slope at the electrode surface, [𝜕C O (x, t)/𝜕x]x = 0 , remains constant after the onset of electrolysis. The surface concentration, C O (0, t), can be obtained from (9.2.8) as ∗ − CO (0, t) = CO

2it 1∕2 1∕2

nFADO 𝜋 1∕2

(9.2.10)

At the transition time, 𝜏, C O (0, t) drops to zero, and (9.2.10) becomes the Sand equation (1): 1∕2

nFADO 𝜋 1∕2 ) i𝜏 1∕2 mA s1∕2 ( 1∕2 = = 85.5nDO A with A in cm2 ∗ 2 mM CO

(9.2.11)

2 Inversion of (9.2.5, 9.2.6) using the convolution property leads to (7.2.8, 7.2.9), which are also convenient for solving controlled-current problems.

9.2 Theory of Controlled-Current Methods

1.0

0.1 0.5 0.8

CO(x, t)

1.0

0.5

CO*

0

0

0.5

1.0

1.5

1.0

1.5

x/2(DOτ)1/2 (a) 1.0

CR (x, t)

0.5

ξCO*

1.0 0.8 0.5

0

0.1 0

0.5 x/2(DR

τ)1/2

(b)

Figure 9.2.1 Dimensionless concentration profiles of O and R at values of t/𝜏 indicated on the curves. Problem 9.1 invites the reader to derive the equations represented here from (9.2.8) and (9.2.9).

(E – Eτ/4)/mV

–240

–120

0

+120 0

0.5 t/τ

1.0

Figure 9.2.2 Theoretical chronopotentiogram for a nernstian electrode process.

Beyond the transition time, the flux of O at the surface is not large enough to satisfy the applied current; hence, the potential must shift to a more extreme value where another electrode process can occur (Figure 9.2.2). The shape of the E − t curve is discussed in Sections 9.3.1–9.3.3. The measured value of 𝜏 at known i (or, better, the values of i𝜏 1/2 obtained in experiments ∗ , or D , given knowledge at various currents) can be used to determine any one of n, A, CO O ∗ , is independent of the others. For a well-behaved case, the transition time constant, i𝜏 1∕2 ∕CO

393

394

9 Controlled-Current Techniques ∗ . A lack of constancy indicates complications from coupled homogeneous chemical of i or CO reactions (Chapter 13), adsorption (Chapter 14), or measurement artifacts, such as double-layer charging or the onset of convection (Section 9.3.4).

9.2.3

Programmed Current Chronopotentiometry

One may choose to apply a current that is programmed to vary with time, rather than remaining constant (2, 3). The treatment of programmed currents is often relatively simple, in that one begins with (9.2.5) and (9.2.6) and inserts the transform of the current program. For example, in the linear case (Figure 9.1.2b), i(t) = 𝛽t; hence, i(s) = 𝛽∕s2 and the concentration profiles and the transition time follow easily. A particularly interesting applied current is linear with t 1/2 . In Problem 9.4, the reader is invited to derive the Sand equation for this case, which is advantageous for stepwise electron-transfer reactions and multicomponent systems (Section 9.5).

9.3 Potential–Time Curves in Constant-Current Electrolysis 9.3.1

Reversible (Nernstian) Waves

For rapid electron transfer kinetics, a nernstian relationship links the potential with the surface concentrations of O and R (Sections 3.4.5 and 3.7.4). Using the Sand equation, (9.2.11), one can re-express (9.2.10) as ( ) 1∕2 ∗ 1− t CO (0, t) = CO (9.3.1) 𝜏 1∕2 For the case of CR∗ = 0, one similarly obtains ( 1∕2 ) t ∗ CR (0, t) = 𝜉CO 𝜏 1∕2

(9.3.2)

Substitution of these expressions into the nernstian balance applicable at the electrode surface, (3.7.12) yields (4) E = E𝜏∕4 +

RT 𝜏 1∕2 − t 1∕2 ln nF t 1∕2

(9.3.3)

where, at t = 𝜏/4, one has the quarter-wave potential, 1∕2

E𝜏∕4 =

′ E0

RT DR + ln 1∕2 nF DO

(9.3.4)

which is the same as the voltammetric E1/2 . The test for reversibility of an E − t curve is a linear plot of E vs. log[(𝜏 1/2 − t 1/2 )/t 1/2 ] having a slope of 59.1/n mV, or a value of |E3𝜏/4 − E𝜏/4 | = 47.9/n mV at 25 ∘ C. 9.3.2

Totally Irreversible Waves

For a totally irreversible one-step, one-electron reaction kf

O + e −−−→ R

(9.3.5)

9.3 Potential–Time Curves in Constant-Current Electrolysis

the current is related to the potential by i = nFAk0 CO (0, t)e−𝛼f (E−E

0′ )

Upon substitution of C O (0, t) from (9.3.1), one has (5) ) ( ( ) ∗ FAk 0 CO RT RT t 1∕2 0′ + E=E + ln ln 1 − 𝛼F i 𝛼F 𝜏 1∕2

(9.3.6)

(9.3.7)

An equivalent expression is obtained by using the Sand equation to substitute for 𝜏 1/2 : ′

E = E0 +

) ( RT 2k 0 RT ln + ln 𝜏 1∕2 − t 1∕2 𝛼F (𝜋DO )1∕2 𝛼F

(9.3.8)

For a totally irreversible reduction, the whole E − t wave is displaced toward more negative potentials upon increasing the current, with a tenfold increase causing a shift of 2.3RT/𝛼nF (or 59/𝛼 mV at 25 ∘ C). Uncompensated resistance produces a similar effect, so one must guard against or correct for this effect, if one desires to evaluate k 0 from the kinetic displacement of the wave. For a totally irreversible wave, |E3𝜏/4 − E𝜏/4 | = 33.8/𝛼 mV at 25 ∘ C. 9.3.3

Quasireversible Waves

For the quasireversible one-step, one-electron process, kf

O+e ⇌ R

(9.3.9)

kb

a general E − t relationship is found by combining the current-overpotential equation, (3.4.10), with equations for C O (0, t) and C R (0, t) derived from (9.2.8) and (9.2.9). Both species O and R must be present in the bulk, so that an equilibrium potential is defined (6, 7). The overall result is [ [ ( )1∕2 ] ( )1∕2 ] i 2i t 2i t = 1− e−𝛼f 𝜂 − 1 + e(1−𝛼)f 𝜂 (9.3.10) i0 FAC ∗O 𝜋DO FAC ∗R 𝜋DR Kinetic studies of quasireversible electrode reactions can be carried out by the galvanostatic or current step method, which involves small applied currents, so that the potential change from the equilibrium position remains small. One can then employ a linearized form of (9.3.10), obtained from a Taylor expansion (Section A.2):

−𝜂 =

⎡ RT ⎢ 2t 1∕2 i F ⎢ FA𝜋 1∕2 ⎣

⎛ ⎞ ⎤ 1 1 ⎟ 1⎥ ⎜ + + ⎜ C ∗ D1∕2 C ∗ D1∕2 ⎟ i0 ⎥ ⎦ ⎝ O O R R ⎠

(9.3.11)

A plot of 𝜂 vs. t 1/2 should be linear for small values of 𝜂, with i0 available from the intercept.3 Equation 9.3.11 is the constant-current analogue of (6.3.18), applicable to small-amplitude potential steps in quasireversible systems. 3 A correction must be made for the fraction of current dedicated to double-layer charging. See the second edition, Section 8.3.4 for details.

395

396

9 Controlled-Current Techniques

9.3.4

Practical Issues in the Measurement of Transition Time

Because the potential changes continuously during chronopotentiometry, a nonfaradaic current, ic , is always needed for charging of the double-layer capacitance: (9.3.12)

ic = −AC d (d𝜂∕dt) = −AC d (dE∕dt) Of the total applied constant current, i, only a portion, if , goes to the faradaic reaction:

(9.3.13)

if = i − ic

Since dE/dt is a function of time, both ic and if vary with time, even when i is constant. The balance between ic and if changes most rapidly just after application of the current and near the transition (the two time periods where dE/dt is large; Figure 9.3.1). This shifting balance alters the overall shape of the E − t curve and makes measurement of 𝜏 both difficult and inaccurate. In general, a transition time is harder to measure accurately than a voltammetric peak height or wave height. Problems with distorted E − t curves and the difficulty of obtaining corrected values of 𝜏 are significant in chronopotentiometry; hence, mitigating techniques have been proposed. In the simplest approach, ic is assumed to be constant for 0 < t < 𝜏. This is not strictly so, of course, since dE/dt and C d (a function of E) change throughout the E − t curve (9, 10); however, the approximation leads to i = if + ic

(9.3.14)

i 𝜏 if 𝜏 1∕2 i𝜏 1∕2 = + ∗c ∗ ∗ CO CO CO 𝜏 1∕2

(9.3.15)

∗ = a, is the “true chronopotentiometric constant,” equal to where the first term, if 𝜏 1∕2 ∕CO 1∕2

nFADO 𝜋 1∕2 ∕2. In the second term, Qc = ic 𝜏 = b is the total number of coulombs needed to charge the average double-layer capacitance over the ΔE from the initial potential to the potential at which 𝜏 is measured; thus, ic 𝜏 ≈ (C d )avg ΔE. With a and b so defined, (9.3.15) can be rearranged and rewritten as i𝜏 = aC ∗O 𝜏 1∕2 + b

(9.3.16)

1.00

1

0.75

2 3 4

if /i 0.50

5

0.25

0.00 0.0

0.5

1.0 t/τ

1.5

2.0

Figure 9.3.1 Fraction of total current contributing to the faradaic process (if /i) vs. time for a nernstian electrode process. Curves 1–5 show the effect of a progressively greater overall share of charge going to ∗ is the double-layer capacitance (e.g., because CO progressively lowered). The effect of double-layer charging is most important at small 𝜏. [De Vries (8). Elsevier.]

9.3 Potential–Time Curves in Constant-Current Electrolysis

A plot of i𝜏 vs. 𝜏 1/2 for a set of chronopotentiograms obtained at different applied currents or concentrations should yield a slope aC ∗O and an intercept b. An equation of this form can also be used to correct for formation of an oxide film (e.g., on a platinum electrode during an electrochemical oxidation) or for electrolysis of adsorbed material in addition to diffusing species. Including all such effects, (9.3.14) becomes (10): (9.3.17)

i = if + ic + iox + iads

where iox is the current going to formation (or reduction) of the oxide film, and iads is the current required for the adsorbed material. A treatment similar to that given above again yields (9.3.16), but b includes added terms for Qox = iox 𝜏 and Qads = nFAΓ, where Γ is the number of moles of adsorbed species per unit area (Section 17.3.1). Although these approximations are rough, treatments of actual experimental data by (9.3.16) yield fairly good results, even at rather low concentrations and short transition times, where the surface effects are most important (11). A more rigorous approach involves only the assumption that C d is independent of E (8, 12, 13). In this case, one must solve the diffusion problem as in Section 9.2.1, but with the flux condition, (9.2.3), replaced by4 ( ) ( ) 𝜕CO dE i = nFADO − AC d (9.3.18) 𝜕x x=0 dt In addition, one needs the appropriate i − E characteristic (i.e., for reversible, totally irreversible, or quasireversible kinetics). The resulting nonlinear integral equation must be evaluated numerically. Alternatively, the problem can be addressed by digital simulation. A different set of problems may develop at long experimental times from convection and nonlinear diffusion. Convective effects, caused by motion of the solution with respect to the electrode, arise by accidental vibrations transmitted to the cell (e.g., from hood fans, vacuum pumps, passing traffic) or as a result of density gradients arising at the electrode surface from differences in density between reactants and products. Convective effects can be minimized by using shielded electrodes (Figure 9.3.2) and by orienting the electrode so that the denser species is always below the less dense one (14, 15). Vertically oriented electrodes (e.g., foils or wires) often suffer from convective effects, even at not very long times (e.g., 60–80 s). A shielded electrode also has the virtue of constraining diffusion to lines normal to the electrode surface, so that true linear diffusion conditions are approached (Figure 4.4.5). An unshielded electrode, e.g., a Pt disk in a planar glass mantle, manifests “edge effects” when the diffusion layer thickness becomes comparable to the smallest dimension, so that the electroreactant begins to arrive appreciably from the sides. In chronopotentiometry this effect increases the Figure 9.3.2 (a) Shielded electrode for maintaining linear diffusion and suppressing convection. (b) Tubes to which the shielded electrode is attached to provide: (1) horizontal electrode, diffusion upward; (2) horizontal electrode, diffusion downward; (3) vertical electrode. [Bard (14). © 1961, American Chemical Society.]

7 mm

14 mm (a)

1

2 (b)

4 Equations (9.3.12) and (9.3.18) imply that C d is capacitance per unit area, in F/cm2 for this discussion.

3

397

398

9 Controlled-Current Techniques

transition time. With properly oriented shielded electrodes, linear diffusion can be maintained for 300 s or longer.

9.4 Reversal Techniques 9.4.1

Response Function Principle

A useful technique for treating current reversal in chronopotentiometry (and other electrochemical problems) is based on the response function principle (2, 16), which is also used extensively to describe electrical circuits. The idea is to discover the system’s response to an excitation by exploiting a linear linkage in Laplace transform space. We express it as (9.4.1)

R(s) = Ψ(s)S(s)

where Ψ(s) is the transform of the excitation function, R(s) is the transform of the response, and S(s) is the system transform, which connects the excitation and the response. For excitation by an applied current i(t), we discovered such a linkage in Section 9.2.1. From (9.2.5) at x = 0, 1∕2

∗ ∕s − (nFAD C O (0, s) = CO s1∕2 )−1 i(s) O

(9.4.2)

or 1∕2

C ∗O − C O (0, s) = (nFADO s1∕2 )−1 i(s)

(9.4.3) 1∕2

Thus, we can identify, Ψ(s) = i(s), R(s) = C ∗O − C O (0, s), and S(s) = (nFADO s1∕2 )−1 . The system transform, S(s), embodies behavioral principles for the electrochemical system being studied, which, for the present case, include stable participants and semi-infinite linear diffusion (i.e., adherence to the general formulation). For controlled-current problems involving other kinds of systems (e.g., spherical or cylindrical diffusion, first-order kinetic complications), the system transform would be different.5 An analogous relation for species R is available from (9.2.6): 1∕2

C ∗R − C R (0, s) = −(nFADR s1∕2 )−1 i(s)

(9.4.4) 1∕2

from which we identify Ψ(s) = i(s), R(s) = C ∗R − C R (0, s), and S(s) = (nFADR s1∕2 )−1 . 9.4.2

Current Reversal

Consider an O/R system where only O is initially present, semi-infinite linear diffusion prevails, and a constant cathodic current i is applied for a time t 1 (where t 1 ≤ 𝜏 1 , with 𝜏 1 being the transition time for reduction of O). At t 1 , the current is reversed from cathodic to anodic, so that R formed during the forward step begins to oxidize back to O. The electrolysis proceeds until time 𝜏 2 (measured from t 1 ), when C R drops to zero at the electrode surface. At this reverse transition time, the potential shows a rapid change toward positive values. We desire an expression for 𝜏 2 (17, 18). Using step function notation (Section A.1.7), the current can be expressed as i(t) = i + St (t)(−2i) 1

(9.4.5)

5 This approach, and transform methods in general, are useful only for linear problems. Second-order reactions and other nonlinear complications cannot be addressed.

9.4 Reversal Techniques

where St (t) = 0 for t ≤ t 1 and St (t) = 1 for t > t 1 . Laplace transformation of (9.4.5) gives 1

1

)i ( i ) i (2e (1 − 2e−t1 s ) i(s) = − = s s s Since (9.4.3) and (9.4.4) continue to apply, we write for CR∗ = 0 that ( ) i 1∕2 (1 − 2e−t1 s )(nFADR s1∕2 )−1 C R (0, s) = s Inverse transformation yields −t1 s

CR (0, t) =

2i 1∕2

nFADR 𝜋 1∕2

(9.4.6)

(9.4.7)

[t 1∕2 − 2St (t)(t − t1 )1∕2 ]

(9.4.8)

1

We define t = (t 1 + 𝜏 2 ) as the moment when the surface concentration of species R reaches zero. 1∕2 From (9.4.8), one finds that C R (0, t 1 + 𝜏 2 ) = 0 when (t1 + 𝜏2 )1∕2 − 2𝜏2 = 0; therefore, 𝜏2 = t1 ∕3

(9.4.9)

Thus, the reverse transition time for stable R is always 1/3 that of the forward switching time, ∗ , or t 1 (Figure 9.4.1), up to and including 𝜏 1 . This behavior does not depend on n, DO , DR , CO the kinetics of electron transfer (assuming they are sufficiently facile to show a reverse transition). The factor of 1/3 means that only one third of the R generated during the forward step (it 1 /nF mol) returns to the electrode during the reversal by the time of the transition at 𝜏 2 . The remainder diffuses into the bulk solution. At first thought, one might wonder at the independence of 𝜏 2 /t 1 on DR . If DR is very large, then a larger amount might be expected to diffuse away. However, a large DR also implies that a larger amount will diffuse back during the reverse step, and the mathematics demonstrates that the effects are exactly compensating. The 𝜏 2 /t 1 ratio of 1/3 in chronopotentiometry is the analogue of −ir (2𝜏)/if (𝜏) = 0.293 in potential step reversal (Section 6.5) and |ipr /ipf | = 1 in cyclic voltammetry [Section 7.2.2(b)]. Expressions can be written for the E − t curve by combining the appropriate kinetic relationship with the equations for C O (0, t) and C R (0, t), just as in Section 9.3. For a nernstian reversal wave, E0.215𝜏 = E𝜏∕4 . For a quasireversible system the separation between E𝜏/4 and E0.215𝜏 can 2

2

be used to determine k 0 (20). Figure 9.4.1 Experimental chronopotentiogram with current reversal. Oxidation of 1.04 mM diphenylpicrylhydrazyl (DPPH) in CH3 CN with 0.1 M NaClO4 , followed by reduction of the stable + radical cation, DPPH ∙ . A shielded Pt electrode (A = 1.2 cm2 ) was employed with i = 100 μA. [Solon and Bard (19). © 1964, American Chemical Society.] E/V vs. SCE

0.6

0.7 1 0.8 10 s t→

2

399

400

9 Controlled-Current Techniques

9.5 Multicomponent Systems and Multistep Reactions Consider chronopotentiometry in a solution containing two substances, O1 and O2 , reduced at different potentials (17, 21–23). The reaction O1 + n1 e → R1 occurs first; then, at more negative potentials, O2 + n2 e → R2 . Assuming semi-infinite linear diffusion, the following response-function equations can be written: [ ∗ ] i (s) 1∕2 C1 n1 FAD1 − C 1 (0, s) = 1 (9.5.1) s s1∕2 [ ∗ ] i (s) 1∕2 C2 n2 FAD2 − C 2 (0, s) = 2 (9.5.2) s s1∕2 where C1∗ and C2∗ are the bulk concentrations, and i1 (s) and i2 (s) are the transforms of the individual currents [i1 (t) and i2 (t)] involved in the reductions of O1 and O2 , respectively. The total applied current, i(t), is i1 (t) + i2 (t); therefore, i(s) = i1 (s) + i2 (s), and from (9.5.1) and (9.5.2), [ ∗ ] [ ∗ ] i(s) 1∕2 C1 1∕2 C2 n1 D1 − C 1 (0, s) + n2 D2 − C 2 (0, s) = (9.5.3) s s FAs1∕2 For a time after application of the current, only O1 is reduced. The potential is not sufficiently negative for O2 reduction to occur; hence, C 2 (0, s) = C2∗ ∕s and i2 (s) = 0. Consequently, (9.5.3) becomes identical to (9.4.3) and the behavior is unaffected by the presence of O2 . Eventually, the flux of O1 at the electrode surface becomes unable to handle all of the current, and the potential makes a transition at t = 𝜏 1 to the region where O2 begins to be reduced. For t > 𝜏 1 , C 1 (0, s) = 0, and (9.5.3) becomes [ ∗ ] 1∕2 n1 D1 C1∗ i(s) 1∕2 C2 + n2 D2 − C 2 (0, s) = (9.5.4) s s FAs1∕2 A second transition occurs at t = 𝜏 1 + 𝜏 2 , when the concentration of O2 drops to zero at the electrode surface, i.e., when C 2 (0, s) = 0. For a constant current, i(s) = i∕s, and (9.5.4) becomes 1∕2

n1 D1 C1∗

1∕2

n2 D2 C2∗

i = s s FAs3∕2 Inversion and substitution of t = 𝜏 1 + 𝜏 2 provides ( ) FA𝜋 1∕2 1∕2 1∕2 (n1 D1 C1∗ + n2 D2 C2∗ ) = i(𝜏1 + 𝜏2 )1∕2 2 +

1∕2

(9.5.5)

(9.5.6)

1∕2

For the special case where n1 D1 C1∗ = n2 D2 C2∗ , 𝜏 2 = 3𝜏 1 . Thus, two substances at equal concentration with equal diffusion coefficients and equal values of n show unequal transition times in chronopotentiometry. This is a contrast with behavior in voltammetry, where there are two waves of equal height. The long second transition in chronopotentiometry results from the continued diffusion of O1 to the electrode after 𝜏 1 , so that only a fraction of the applied current is available for reduction of O2 (Figure 9.5.1). Similar reasoning shows that for a stepwise process (17, 23): O + n1 e → R1

(9.5.7)

R1 + n2 e → R2

(9.5.8)

9.6 The Galvanostatic Double Pulse Method

100

7.7 × 10–6 M Cd2+

t/s

1.54 × 10–5 M Pd2+ 50

0 –0.2

–0.3

–0.4

–0.5

–0.6

–0.7

–0.8

E/V vs. Ag/AgCl

Figure 9.5.1 Consecutive reduction of Pb(II) and Cd(II) at a mercury pool electrode. An E − t curve plotted in this manner resembles a voltammogram. [Reilley, Everett, and Johns (22). © 1955, American Chemical Society.]

τ2 τ2 Potential

Figure 9.5.2 E − t curves for stepwise reductions at an Hg electrode. (a) 1 M LiCl saturated with O2 at 25 ∘ C. O2 + 2H2 O + 2e → H2 O2 + 2OH– ; H2 O2 + 2e → 2OH– ; 𝜏 2 /𝜏 1 ≈ 3. (b) 10−3 M uranyl nitrate in 0.1 M KCl + 0.01 M HCl. U(VI) + e → U(V); U(V) + 2e → U(III); 𝜏 2 /𝜏 1 ≈ 8. [Berzins and Delahay (17). © 1953, American Chemical Society.]

τ1

2s

(a)

the transition time ratio is given by ( )2 𝜏2 2n n2 = 2+ 𝜏1 n1 n1

τ1

Time

2s

(b)

(9.5.9)

Thus, for n2 = n1 , 𝜏 2 = 3𝜏 1 (Figure 9.5.2). By use of the response-function principle one can show (Problem 9.4) that, for a current i(t) = 𝛽t 1/2 , equal transition times result when n2 = n1 .

9.6 The Galvanostatic Double Pulse Method The single-pulse galvanostatic method (Section 9.3.3) cannot be applied to electron-transfer reactions with large i0 because the current is primarily nonfaradaic during the initial moments following the application of the current step. Faster kinetics can be examined using the galvanostatic double pulse (GDP) method (24–28), in which two current pulses are applied to the electrode (Figure 9.6.1). A large initial current, i1 , passes for time t 1 (typically 0.5–1 μs), principally to charge the double layer to a potential that becomes the point of interrogation by a following, smaller current, i2 . The basic idea is to use the first pulse to drive the system to an overpotential that exactly supports the second current. Then 𝜂 will not change when i2 is applied, and there is no significant effect from charging the double layer during the second phase.

401

9 Controlled-Current Techniques

Figure 9.6.1 Excitation waveform for the galvanostatic double pulse method.

i1

i

i2 t1

0

i1

t1

t

i2

i1

2 3

η (1 mV/div)

1

η (1 mV/div)

402

t1 i2

1 2 3 1 μF

5Ω

20 Ω t (0.5 μs/div) (a)

t (0.5 μs/div) (b)

Figure 9.6.2 (a) GDP traces of 𝜂 vs. t for reduction of 0.25 mM Hg2+ in 1 M HClO4 at an HMDE. Values of i2 /i1 2 were (1) 7.8, (2) 5.3, (3) 3.2; Curve 2 shows the desired response for use of (9.6.3). (b) Voltage–time traces for GDP applied to an equivalent circuit. i1 was (1) 7.6, (2) 5.5, (3) 3.3 mA; and i2 = 1 mA. [Kogoma, Nakayama, and Aoyagui (26). © 1972, Elsevier.]

When the ratio of pulse heights, i1 /i2 , is adjusted properly (by trial and error), the E − t curve following the cessation of the first pulse is horizontal (Figure 9.6.2). Under these conditions, the overpotential for a quasireversible one-step, one-electron process is given by (25) [ ] ) ( 4Ni )2 4Ni0 1∕2 ( RT i2 9𝜋 0 −𝜂 = 1+ t + 1− t +··· (9.6.1) F i0 32 3𝜋 1∕2 1 3𝜋 1∕2 1 where ⎛ ⎞ 1 ⎜ 1 1 ⎟ + FA ⎜ C ∗ D1∕2 C ∗ D1∕2 ⎟ ⎝ O O R R ⎠ At small values of t 1 , (9.6.1) can be linearized to ( ) RT 1 4N 1∕2 −𝜂 ≈ i2 + t1 F i0 3𝜋 1∕2 N=

(9.6.2)

(9.6.3)

Thus, one can carry out a series of experiments with different pulse widths t 1 , and plot the value 1∕2 of 𝜂 at the onset of i2 vs. t1 to obtain i0 from the intercept. For the calculation of i0 , the GDP method does not require knowledge of diffusion coefficients or C d . Values of k 0 in the range of 1 cm/s have been determined using this technique (27, 29).

9.7 Charge Step (Coulostatic) Methods

9.7 Charge Step (Coulostatic) Methods Suppose one has a working electrode at equilibrium in a solution containing species O and R, which engage in the electrode reaction O + e ⇌ R. In the charge-step (or coulostatic) method, a very short-duration current pulse is applied, delivering an increment of charge, Δq, and causing the potential to depart from the equilibrium value, Eeq . After the pulse, one measures the potential vs. time at open circuit.6 Ideally, the current pulse is sufficiently brief (perhaps 0.1–1 μs) that the electrode reaction does not proceed appreciably; thus, Δq serves only to charge the double layer. The method of charge injection or the shape of the injecting pulse is unimportant. An experimental system is shown schematically in Figure 9.7.1. When the electronic switch is in position A, the injection capacitor, C inj , is charged by the voltage source, V inj : (9.7.1)

Δq = Cinj Vinj

For example, if V inj = − 10 V and C inj = 10−9 F, then Δq = − 0.01 μC. When the electronic switch goes to position B, that charge is delivered to the electrochemical cell. If the double-layer capacitance, C d at the working electrode, is much larger than C inj , essentially all of the charge flows into the cell. The time required for charge injection depends on the total solution resistance between the working and counter electrodes, Rs , with the time constant for injection being essentially C inj Rs (Problem 9.6). Upon injection, the potential of the working electrode shifts from Eeq to E(t = 0), and an overpotential, 𝜂(t = 0), is established E(t = 0) − Eeq = 𝜂(t = 0) =

Δq Cd

(9.7.2)

where the sign on Δq determines the sign on 𝜂(t = 0). Because the electrode reaction is no longer at equilibrium with the electrode, a net reaction occurs. Either O is converted to R, removing electrons from the electrode, or R is converted to O, adding electrons. By the faradaic reaction, Trigger A

B Electronic switch

Cell Triggering/recording system (E vs. t)

Vinj

Cinj

Charge injection system

Figure 9.7.1 System for charge-step or coulostatic method. 6 This discussion is abridged. Section 8.7 of the second edition provides a fuller treatment.

403

404

9 Controlled-Current Techniques

the extra charge, Δq, stored temporarily on C d , discharges, and the open circuit potential moves back toward Eeq as 𝜂(t) decreases to zero. The transient 𝜂 vs. t is recorded.7 During the transient, the total external current is zero; therefore, the faradaic current, if , and the charging current, ic , are equal and opposite. ( ) d𝜂 if = −ic = Cd (9.7.3) dt or 𝜂(t) = 𝜂(t = 0) +

t

1 i dt Cd ∫0 f

(9.7.4)

One can solve (9.7.4) with an appropriate expression for if to derive an expression for E (or 𝜂) vs. t (30–33). If no faradaic reaction is possible at E(t = 0), C d remains charged, and the potential will not relax toward the starting point. 9.7.1

Small Excursions

If the potential excursion is small [𝜂(t = 0) ≪ RT/nF], one can use a linearized i − 𝜂 relation to describe if . If, in addition, the electrode kinetics are sluggish, the charge-transfer resistance, Rct can greatly exceed the mass-transfer resistance, so that (3.4.12)–(3.4.13) provide the linkage between faradaic current and overpotential, −𝜂 =

RT i = Rct if nFi0 f

(9.7.5)

and 𝜂(t) = 𝜂(t = 0) −

t

1 𝜂(t)dt Rct Cd ∫0

(9.7.6)

In Problem 9.7, the reader is invited to show that 𝜂(t) = 𝜂(t = 0)e−t∕Rct Cd

(9.7.7)

Thus, the potential is expected to relax exponentially toward Eeq with a time constant Rct C d , governed by the kinetics of the charge-transfer reaction. A plot of ln|𝜂| vs. t should be linear, with an intercept ln|𝜂(t = 0)| [which can be used to determine C d by (9.7.2)] and a slope −1/Rct C d , which yields the charge-transfer resistance and the exchange current. When Rct does not dominate the mass-transfer impedance, the equations describing the relaxation of 𝜂 become more complicated.8 Detailed discussions of the analysis of coulostatic data and relaxation curves have appeared (34, 35). Coulostatic experiments have some advantages. Since the measurement is made at open circuit with no net external current flow, ohmic drop does not appear, so work can be done in resistive media. Moreover, because the relaxation occurs by discharge of the double-layer capacitance, the usual competition between faradaic and charging current is replaced by an equality of ic and if , and C d no longer interferes in the measurement. However, high values of cell resistance increase the time required to deliver charge to the cell. Also, the large voltage V inj 7 We have assumed that the working electrode is initially at a true equilibrium established by bulk concentrations of O and R. In practice, the initial potential may have been imposed by a potentiostat, disconnected immediately before the charge injection. The potentiostat establishes “effective bulk concentrations” of O and R at the electrode surface, exactly as in differential pulse voltammetry (Section 8.4.2). These concentrations are in local equilibrium with the imposed potential, which is an “effective equilibrium potential” for the purposes of the coulostatic experiment. 8 Second edition, Section 8.7.2.

9.7 Charge Step (Coulostatic) Methods

appears across the cell at the instant of charge injection and complicates the measurement process, which must operate with high sensitivity, given the small changes in ΔE. The coulostatic method is useful for the determination of i0 values up to about 0.1 A/cm2 or k 0 values up to 0.4 cm/s. Experimental aspects and applications have been discussed in the literature (35, 36). 9.7.2

Large Excursions

If the potential change in a coulostatic experiment is large enough to reach the diffusion-limited region, then if is given by the Cottrell equation, (6.1.12). For the reduction of species O, and with C d taken as independent of potential, one obtains from (9.7.4) 1∕2

ΔE = E(t) − E(t = 0) =

∗ t 1∕2 2nFADO CO

𝜋 1∕2 Cd

(9.7.8)

The sign of ΔE is positive because the electrode relaxes from a more negative initial potential toward more positive values. A plot of ΔE vs. t 1/2 is linear with a zero intercept and a slope 1∕2 ∗ 2nFADO CO ∕𝜋 1∕2 Cd , which is proportional to the solution concentration. This method has been suggested for the determination of small concentrations (37, 38). For the reader, the “coulostatic idea” is perhaps more important than the coulostatic method. Here is the essence: An electrode not at equilibrium can continue to engage in electrolysis, even at open circuit, with the double-layer capacitance serving as the reservoir for charge. This concept is encountered in electrochemical situations apart from the coulostatic method. One instance is in Section 9.7.3; others are in Problems 9.9 and 9.10 and in Section 16.7.3(b), the latter dealing with resistance compensation by current interruption. 9.7.3

Coulostatic Perturbation by Temperature Jump

The potential of an electrode can be perturbed in a manner analogous to the coulostatic method by an abrupt change in any variable that shifts the electrode away from equilibrium. Perhaps most straightforward is to change the electrode temperature in a T-jump (39–43). This strategy is conveniently effected by using a thin (e.g., 1–25 μm) metal film on a dielectric (e.g., glass) and irradiating the film from the backside (through the dielectric) with a pulsed laser (41, 42). The film is sufficiently thick that no light passes through, and all absorbed light is converted to heat. Under these conditions, photoemission of electrons into the solution (39, 40) (Problems 9.9 and 9.10 and Section 20.4.1) does not occur. A schematic diagram of apparatus is shown in Figure 9.7.2. By using fast laser pulses and thin metal films, measurements in the nanosecond domain are possible. The T-jump perturbs the equilibrium at the electrode/solution interface and causes a change in the potential of the working electrode, which then relaxes back toward the original equilibrium point. Kinetic information about the electrode reaction is available from the E − t response. Although several different phenomena are involved in the potential shift with temperature (e.g., the temperature dependence of the double-layer capacitance and the Soret potential arising from the temperature gradient between the electrode and the bulk electrolyte), the response can be treated theoretically (40), and the rate constant for interfacial electron transfer can be obtained from experimental results. This approach has been used, for example, to measure the effect of alkyl chain length in a mixed organized film of alkyl thiols, some with ferrocene terminations (Sections 3.5.2 and 17.6.2), on the rate of electron transfer from the ferrocene moiety (44). Rate constants of 107 –108 s−1 were found. The coulostatic T-jump method has also been used to examine dynamics relating to double-layer formation at single-crystal electrode surfaces (45–47).

405

406

9 Controlled-Current Techniques

Cell

Amp. Digital scope

Diffuser Pot. ND Laser Mirror

Figure 9.7.2 Schematic diagram of apparatus for the T-jump method. The laser pulse passes through a neutral density (ND) filter and irradiates the thin film electrode at the bottom of the cell. The dark rectangles are a counter electrode and a QRE for measurement of the potential change. The potentiostat (Pot.) adjusts the electrode potential before irradiation, but is disconnected immediately before the laser pulse. The change in potential is measured with a fast amplifier (Amp.). [Smalley et al. (42). © 1993, Elsevier.]

9.8 References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

H. J. S. Sand, Philos. Mag., 1, 45 (1901). R. W. Murray and C. N. Reilley, J. Electroanal. Chem., 3, 64, 182 (1962). A. Molina, Curr. Top. Electrochem. 3, 201 (1994). Z. Karaoglanoff, Z. Elektrochem., 12, 5 (1906). P. Delahay and T. Berzins, J. Am. Chem. Soc., 75, 2486 (1953). L. B. Anderson and D. J. Macero, Anal. Chem., 37, 322 (1965). Y. Okinaka, S. Toshima, and H. Okaniwa, Talanta, 11, 203 (1964). W. T. de Vries, J. Electroanal. Chem., 17, 31 (1968). J. J. Lingane, J. Electroanal. Chem., 1, 379 (1960). A. J. Bard, Anal. Chem., 35, 340 (1963). P. E. Sturrock, G. Privett, and A. R. Tarpley, J. Electroanal. Chem., 14, 303 (1967). R. S. Rodgers and L. Meites, J. Electroanal. Chem., 16, 1 (1968). M. L. Olmstead and R. S. Nicholson, J. Phys. Chem., 72, 1650 (1968). A. J. Bard, Anal. Chem., 33, 11 (1961). H. A. Laitinen and I. M. Kolthoff, J. Am. Chem. Soc., 61, 3344 (1939). H. B. Herman and A. J. Bard, Anal. Chem., 35, 1121 (1963). T. Berzins and P. Delahay, J. Am. Chem. Soc., 75, 4205 (1953). A. C. Testa and W. H. Reinmuth, Anal. Chem., 33, 1324 (1961). E. Solon and A. J. Bard. J. Am. Chem. Soc., 86, 1926–1928 (1964). F. H. Beyerlein and R. S. Nicholson, Anal. Chem., 40, 286 (1968). P. Delahay and G. Mamantov, Anal. Chem., 27, 478 (1955). C. N. Reilley, G. W. Everett, and R. H. Johns, Anal. Chem., 27, 483 (1955). H. B. Herman and A. J. Bard, Anal. Chem., 36, 971 (1964). H. Gerischer and M. Krause, Z. Physik. Chem. N.F., 10, 264 (1957); 14, 184 (1958). H. Matsuda, S. Oka, and P. Delahay, J. Am. Chem. Soc., 81, 5077 (1959). M. Kogoma, T. Nakayama, and S. Aoyagui, J. Electroanal. Chem., 34, 123 (1972). T. Rohko, M. Kogoma, and S. Aoyagui, J. Electroanal. Chem., 38, 45 (1972). M. Kogoma, Y. Kanzaki, and S. Aoyagui, Chem. Instr., 7, 193 (1976).

9.9 Problems

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

N. Koizumi and S. Aoyagui, J. Electroanal. Chem., 55, 452 (1974). P. Delahay, J. Phys. Chem., 66, 2204 (1962); Anal. Chem., 34, 1161 (1962). P. Delahay and A. Aramata, J. Phys. Chem., 66, 2208 (1962). W. H. Reinmuth and C. E. Wilson, Anal. Chem., 34, 1159 (1962). W. H. Reinmuth, Anal. Chem., 34, 1272 (1962). J. M. Kudirka, P. H. Daum, and C. G. Enke, Anal. Chem., 44, 309 (1972). H. P. van Leeuwen, Electrochim. Acta, 23, 207 (1978). H. P. van Leeuwen, J. Electroanal. Chem., 12, 159 (1982). P. Delahay, Anal. Chem., 34, 1267 (1962). P. Delahay and Y. Ide, Anal. Chem., 34, 1580 (1962). V. A. Benderskii, S. D. Babenko, and A. G. Krivenko, J. Electroanal. Chem., 86, 223 (1978). V. A. Benderskii, I. O. Efimov, and A. G. Krivenko, J. Electroanal. Chem., 315, 29 (1991). J. F. Smalley, C. V. Krishnan, M. Goldman, S. W. Feldberg, and I. Ruzic, J. Electroanal. Chem., 248, 255 (1988). J. F. Smalley, L. Geng, S. W. Feldberg, L. C. Rogers, and J. Leddy, J. Electroanal. Chem., 356, 181 (1993). S. W. Feldberg, M. D. Newton, and J. F. Smalley, Electroanal. Chem., 22, 101 (2003). J. F. Smalley, S. W. Feldberg, C. E. D. Chidsey, M. R. Linford, M. D. Newton, and Y. P. Liu, J. Phys. Chem., 99, 13141 (1995). V. Climent, B. A. Coles, R. G. Compton, and J. M. Feliu, J. Electroanal. Chem., 561, 157 (2004). N. Garcia-Aráez, V. Climent, and J. M. Feliu, J. Am. Chem. Soc., 130, 3824 (2008). P. Sebastián, A. P. Sandoval, V. Climent, and J. M. Feliu, Electrochem. Commun., 55, 39 (2015). G. C. Barker, D. McKeown, M. J. Williams, G. Bottura, and V. Concialini, Faraday Discuss. Chem. Soc., 56, 41 (1973). G. C. Barker, Ber. Bunsenges. Phys. Chem., 75, 728 (1971).

9.9 Problems 9.1

Starting with (9.2.8) and (9.2.9), derive the dimensionless equations for the concen∗ tration profiles in Figure 9.2.1. Take the dimensionless concentrations as CO (x, t)∕CO ∗ , the dimensionless time as t/𝜏, and the dimensionless distance as and CR (x, t)∕𝜉CO 1/2 x/2(DO 𝜏) for the O profile and as x/2(DR 𝜏)1/2 for the R profile.

9.2

For current reversal chronopotentiometry involving the forward reduction of species O under conditions of semi-infinite linear diffusion, the reverse transition time, 𝜏 r , can be made equal to forward electrolysis time, t f , by a proper choice of currents during the forward (reduction) reaction, if , and the reverse (oxidation) reaction, ir . Find the ratio if /ir that will yield 𝜏 r = t f .

9.3 An analyst determines a mixture of Pb2+ and Cd2+ at a mercury pool cathode by chronopotentiometry. In the cell used in the determination, a 1.00 mM solution of Pb2+ reduced at i = 273 μA yielded 𝜏 = 25.9 s and E𝜏/4 = − 0.38 V vs. SCE. A 0.69 mM solution of Cd2+ reduced with i = 136 μA gave 𝜏 = 42.0 s and E𝜏/4 = − 0.56 V vs. SCE. An unknown mixture of Pb2+ and Cd2+ reduced at i = 56.5 μA produced a double wave, with 𝜏 1 = 7.08 s and 𝜏 2 = 7.00 s. Calculate the concentrations of Pb2+ and Cd2+ in the mixture. Neglect double-layer and other background effects.

407

408

9 Controlled-Current Techniques

9.4

(a) Derive the Sand equation for programmed-current chronopotentiometry using i(t) = 𝛽t 1/2 . (b) Show that the stepwise reduction of a substance with n1 = n2 gives 𝜏 1 = 𝜏 2 with this form of programmed current.

9.5

Examine the results in Figure 9.4.1. Estimate the transition times and work up the data to yield information about the electrode reaction.

9.6

Consider the circuit in Figure 9.9.1, which is characteristic of that used for the injection of charge in a coulostatic experiment. C inj is initially charged completely with a 10-V battery. At equilibrium, after the switch is closed, how much charge will reside on C d and on C inj ? About how long will it take to charge C d ? Figure 9.9.1 Dummy circuit for modeling coulostatic injection. Rs = 100 Ω Cinj = 1 nF Cd = 1 μF

9.7

Solve (9.7.6) by the Laplace transform method to obtain (9.7.7).

9.8

Consider 1 mM Cd2+ in 0.1 M HCl, examined coulostatically at an HMDE (A = 0.05 cm2 ). ′ For Cd2+ /Cd(Hg), E0 = − 0.61 V vs. SCE. The electrode is initially at rest at −0.4 V vs. SCE, then a sufficient charge is applied to shift its potential instantaneously to −1.0 V vs. SCE. Assume the differential and integral double-layer capacitances to be 10 μF/cm2 . How much charge is required for the potential excursion? How long would it take for the potential to fall back to −0.9 V after the charge injection? Assume that D = 10−5 cm2 /s.

9.9 Barker et al. (48) performed experiments in which 15-ns laser pulses illuminated a mercury pool working electrode. Each light pulse caused ejection of electrons from the electrode. These electrons apparently travel about 5 nm before becoming solvated and available for reaction. When electrons are emitted into a solution of N2 O in aqueous 1 M KCl, the following reaction occurs: eaq + N2 O + H2 O → OH ∙ +N2 + OH− The hydroxyl radicals are reduced at the electrode at potentials more negative than −1.0 V vs. SCE. The response of the illuminated working electrode to a flash is followed coulostatically. Curves like those shown in Figure 9.9.2 can be obtained. Explain their shapes. ΔE is measured with respect to the initial potential.

9.9 Problems

Figure 9.9.2 Potential–time response following a light pulse on a mercury electrode. ΔE (+)

With N2O

Einit = –1.3V vs. SCE

0

9.10

Without N2O

100 200 Time after flash/ns

300

Barker’s technique (Problem 9.9) can also be used to create hydrogen atoms and to study their electrochemistry. The reaction producing them in acid media is ∙ +H O H3 O+ + eaq → Haq 2

Investigators studying the hydrogen evolution reaction have often suggested that H∙ is an intermediate and that H2 is produced by reducing it further in a fast heterogeneous process [Mechanism (a)]: ∙ + e + H O+ → H + H O Haq 3 2 2

or through an adsorbed intermediate [Mechanism (b)] k ∙ →H Haq ads

Hads + e + H3 O+ → H2 + H2 O Whether H∙ is free or adsorbed has been debated. Barker addressed the question by comparing, in effect, the rate of H∙ electroreduction to the rate of its homogeneous reaction with ethanol (leading to electroinactive products). He found (49) that the fraction of H∙ undergoing electroreduction was independent of potential from −0.9 to −1.3 V vs. SCE. What do his observations say about the mechanistic alternatives above?

409

411

10 Methods Involving Forced Convection—Hydrodynamic Methods We now address electrochemical techniques in which the electrode and solution move with respect to each other. The electrode itself may be in motion (e.g., rotating disks, streaming mercury electrodes, vibrating electrodes), or the solution may be forced past a stationary electrode (e.g., tubular or packed-bed electrodes in fluid streams, channel electrodes, bubbling electrodes). Measurement approaches involving convective mass transport are sometimes called hydrodynamic methods. Two advantages of these methods are that a steady state is attained quickly and that measurements can be made easily at high precision. Moreover, double-layer charging does not enter measurements made at steady state. While it might first appear that the valuable time variable is lost in steady-state convective methods, this is not so, because time enters the experiment as the rotation rate of the electrode or the solution velocity with respect to the electrode. Dual-electrode techniques can be employed to provide the same kind of information that reversal methods do in stationary electrode techniques. Hydrodynamic methods are also of interest in the continuous monitoring of flowing liquids and in large-scale reactors, such as those employed for electrosynthesis (Section 12.5). The construction of hydrodynamic electrodes providing known, reproducible mass-transfer conditions is more difficult than for stationary electrodes. Theoretical treatments are also more difficult because they require one to solve a hydrodynamic problem (e.g., determining flow velocity profiles vs. rotation rate) before the electrochemical issues can be tackled. Rarely can closed-form or exact solutions be obtained. The range of electrode configurations and flow patterns is limited only by the imagination and resources of the experimenter; however, the rotating disk electrode (RDE) is by far the most convenient and widely used system. It is amenable to rigorous theoretical treatment and is readily constructed with a variety of electrode materials. Much of what follows deals with the RDE and its variations.

10.1 Theory of Convective Systems The simplest treatments of convective systems are based on the diffusion-layer concept (Section 1.3.2), in which one assumes that convection maintains the concentrations of all species uniformly equal to the bulk values beyond a certain distance from the electrode, 𝛿. In the layer between the electrode and 𝛿, convection is assumed not to occur, so that mass transfer takes place only by diffusion. In effect, the convection problem is converted to a diffusional one, in which an adjustable parameter, 𝛿, is introduced. This is a useful approach for many purposes; however, it does not show how currents are related to solution viscosity, Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

412

10 Methods Involving Forced Convection—Hydrodynamic Methods

electrode dimensions, or flow rates, nor can it be employed for dual-electrode techniques or for predicting relative mass-transfer rates of different substances. A more rigorous approach begins with the convective-diffusion equation and the velocity profiles in the solution. They are solved analytically or, more frequently, numerically. In most cases, only the steady-state solution is needed. 10.1.1

The Convective-Diffusion Equation

The general equation for the flux, Jj , of species j, is (4.1.10): Jj = −Dj 𝛁 Cj −

zj F

D C 𝛁𝜙 + Cj v (10.1.1) RT j j On the right-hand side, the first term represents diffusion, the second, migration, and the last, convection. For solutions containing an excess of supporting electrolyte, the ionic migration term can be neglected (Section 4.3.2). We assume this to be the case for most of this chapter, although Section 10.2.6 covers some exceptions. The velocity vector, v, represents the motion of the solution and is given in rectilinear coordinates by (10.1.2)

v(x, y, z) = ivx + jvy + kvz

where i, j, and k are unit vectors, and vx , vy , and vz are the component solution velocities in the x, y, and z directions at point (x, y, z). Similarly, in rectilinear coordinates, the concentration gradient is expressed as 𝛁Cj = grad Cj = i

𝜕Cj 𝜕x

+j

𝜕Cj 𝜕y

+k

𝜕Cj 𝜕z

(10.1.3)

The variation of C j with time is given by 𝜕Cj

= − 𝛁⋅Jj = div Jj (10.1.4) 𝜕t By combining (10.1.1) and (10.1.4), assuming that migration is absent and that Dj is not a function of x, y, and z, we obtain the general convective-diffusion equation, 𝜕Cj

= Dj ∇𝟐 Cj − v ⋅ 𝛁Cj (10.1.5) 𝜕t The forms for the Laplacian operator, ∇2 , are given in Table 4.4.2. For one-dimensional diffusion and convection along dimension y, (10.1.5) is 𝜕Cj 𝜕t

= Dj

𝜕 2 Cj 𝜕y2

− vy

𝜕Cj 𝜕y

(10.1.6)

In the absence of convection (i.e., v = 0 or vy = 0), (10.1.5) and (10.1.6) are reduced to the corresponding diffusion equations. Before a convective-diffusion equation like (10.1.5) can be solved for the concentration profile and, subsequently, for the current, one must obtain expressions for the velocity profile, v(x, y, z) in terms of spatial position, rotation rate, and other variables. 10.1.2

Determination of the Velocity Profile

While it lies beyond the scope of this chapter to treat hydrodynamics in any depth, a brief discussion of key concepts and terms is offered to provide the reader with a sense for the approach and the results that follow.

10.1 Theory of Convective Systems

For an incompressible fluid—a fluid whose density is constant in time and space—the velocity profile is obtained by two differential equations and appropriate conditions. The continuity equation, 𝛁 ⋅v = div v = 0

(10.1.7)

is a statement of incompressibility, whereas the Navier–Stokes equation, dv = − 𝛁P + 𝜂s ∇2 v + f (10.1.8) dt represents Newton’s first law for a fluid. The left side of (10.1.8) is ma per unit volume (where ds is the density), and the right side is the sum of forces per unit volume (where P is pressure; 𝜂 s is the viscosity; and f is the gravitational force per unit volume). The term 𝜂 s ∇2 v describes frictional forces internal to the fluid, and, through f, the effect of natural convection arising from density gradients is included. Equation 10.1.8 is usually written as ds

dv 1 f = − 𝛁P + 𝜈∇2 v + dt ds ds

(10.1.9)

where 𝜈 = 𝜂 s /ds is called the kinematic viscosity. For many solvents of electrochemical interest 𝜈 is on the order of 0.01 cm2 /s. As examples, the values for 0.1 M aqueous KCl, 0.1 M TEAP in DMF, and 0.1 M TEAP in MeCN are 0.008844, 0.008971, and 0.004536 cm2 /s, respectively (1). Any solution of the hydrodynamic equations requires physical specification of the system, writing the equations in the appropriate coordinate system (linear, cylindrical, etc.), defining boundary conditions, and, usually, solving the problem numerically. In electrochemical problems, only the steady-state velocity profile is normally of interest; therefore (10.1.9) is solved for dv/dt = 0. Two different types of fluid flow are usually considered in hydrodynamic problems (Figure 10.1.1). Smooth, steady flow occurs as if layers (laminae) of the fluid have steady and characteristic velocities. This kind of flow is said to be laminar. For example, the flow of water through a smooth pipe is typically laminar, with the flow velocity being zero at the walls (because of friction between the fluid and the wall) and having some maximum value in the middle of the pipe. The velocity profile under these conditions is typically parabolic. In contrast, turbulent flow involves unsteady and chaotic motion, in which only on the average is there a net flow in a particular direction. Turbulence might result, for example, from a barrier in a pipe, obstructing the stream. Theoretical descriptions of turbulent systems are far more difficult than for laminar cases. Hydrodynamic problems are often characterized by dimensionless groups of variables. A commonly encountered example is the Reynolds number, Re, which relates the fluid flow velocity to the physical size of the system and the kinematic viscosity of the fluid. For example, the Reynolds number for a spinning disk is 𝜔r2 /𝜈, where r is the total radius of the disk

(a)

(b)

Figure 10.1.1 Types of fluid flow. (a) Laminar flow and (b) turbulent flow. Arrows represent instantaneous local fluid velocities.

413

414

10 Methods Involving Forced Convection—Hydrodynamic Methods

(including any mantle) and 𝜔 is the angular velocity (2𝜋 times the rotation rate). High values of Re imply high flow (or high rotation rate). For Re below a certain critical value, Recr , the flow remains laminar, but for Re > Recr , it becomes turbulent. For a spinning disk, Recr ≈ 2 × 105 . General treatments of hydrodynamic problems, especially as they relate to problems in electrochemistry, are available (2–7).

10.2 Rotating Disk Electrode The RDE is one of the few convective electrode systems for which the hydrodynamic equations and the convective-diffusion equation have been solved rigorously for the steady state. This electrode consists of a disk of the electrode material imbedded in a rod of an insulating material, usually Teflon, epoxy resin, or another plastic (Figure 10.2.1). Commercial sources exist, and most electrodes are now purchased, rather than being constructed in the laboratory. Although the literature suggests that the shape of the insulating mantle is critical and that exact alignment of the disk is important (8), these factors are usually not troublesome in practice, except perhaps at high rotation rates, where turbulence and vortex formation may occur. It is more important that there is no leakage of the solution between the electrode material and the insulator. Commercial apparatus is normally used for rotation of the electrode and for electrical connection to it. Usually the shaft is attached directly to the motor by a chuck and is rotated at a selectable frequency, f (revolutions per second). The more useful descriptor of rotation rate is the angular velocity, 𝜔 (s−1 ), where 𝜔 = 2𝜋f . Electrical connection to the RDE is by a brush contact, the quality of which typically determines the noise level in measurements. Details of the construction and application of RDEs are given in reviews (8–12). 10.2.1

The Velocity Profile at a Rotating Disk

A spinning disk drags the fluid at its surface and, because of centrifugal force, flings the solution outward. The fluid at the disk surface is replenished by a flow normal to the surface. Because of the symmetry of the system, it is convenient to write the hydrodynamic equations in terms of cylindrical coordinates r, y, and 𝜙 (Figure 10.2.2). For cylindrical coordinates, v = 𝛍1 vr + 𝛍2 vy + 𝛍3 v𝜙

(10.2.1)

𝛁 = 𝛍1 (𝜕∕𝜕r) + 𝛍2 (𝜕∕𝜕y) + (𝛍3 ∕r)(𝜕∕𝜕𝜙)

(10.2.2)

Brush contact Shaft (brass)

Insulator (Teflon)

Disk (Pt)

Face view

Figure 10.2.1 Rotating disk electrode.

10.2 Rotating Disk Electrode

Figure 10.2.2 Cylindrical polar coordinates for the rotating disk. ϕ y=0 r +y r=0

where 𝛍1 , 𝛍2 , and 𝛍3 are unit vectors in the directions of positive changes of r, y, and 𝜙 at a given point. In contrast to the usual cartesian vectors, i, j, and k, the vectors 𝛍1 , 𝛍2 , and 𝛍3 have directions that depend on the position of the point; thus, the divergence and the Laplacian take on more complex forms. In particular, [ ] 1 𝜕 𝜕 𝜕 2 2 𝛁 ⋅v = 2 (v r ) + (vy r ) + v (10.2.3) 𝜕y 𝜕𝜙 𝜙 r 𝜕r r [ ( ( ) ( )] ) 𝜕 𝜕 1 𝜕 𝜕 𝜕 1 𝜕 2 r + ∇ = r + (10.2.4) r 𝜕r 𝜕r 𝜕y 𝜕y 𝜕𝜙 r 𝜕𝜙 Gravitational effects are assumed to be absent (f = 0), and the mantle surrounding the disk is assumed to be wide enough to eliminate special flow effects at the edge of the electrode. At the disk surface (y = 0), vr = 0, vy = 0, and v𝜙 = 𝜔r. These conditions imply that the solution is dragged along at the surface of the disk at the angular velocity, 𝜔. Far from the disk (y → ∞), there is no flow in the r and 𝜙 directions, but the solution flows toward the disk at a limiting velocity, U 0 , determined by the solution of the problem. Thus, lim vr = 0, lim v𝜙 = 0, and y→∞

y→∞

lim vy = −U0 .

y→∞

The velocity profile of a fluid near a rotating disk of infinite radius was obtained by von Karman (13) and Cochran (14) by solving the hydrodynamic equations under steady-state conditions (2). The treatment yielded results as infinite series based on the dimensionless variable 𝛾 = (𝜔/𝜈)1/2 y. For small values of y (i.e., for 𝛾 ≪ 1), ( ) 1 1 vr = r𝜔F(𝛾) = r𝜔 a𝛾 − 𝛾 2 − b𝛾 3 + · · · (10.2.5) 2 3 ( ) 1 v𝜙 = r𝜔G(𝛾) = r𝜔 1 + b𝛾 + a𝛾 3 + · · · (10.2.6) 3 ( ) 1 1 vy = (𝜔𝜈)1∕2 H(𝛾) = (𝜔𝜈)1∕2 −a𝛾 2 + 𝛾 3 + b𝛾 4 + · · · (10.2.7) 3 6 in which a = 0.51023 and b = − 0.6159. Equations for velocities at greater distances from the electrode (𝛾 ≫ 1) are given by Levich (2). The important velocities are vr and vy (Figure 10.2.3), which, near the disk surface (𝛾 → 0), are vy = (𝜔𝜈)1∕2 (−a𝛾 2 ) = −0.51𝜔3∕2 𝜈 −1∕2 y2

(10.2.8)

vr = r𝜔(a𝛾) = 0.51𝜔3∕2 𝜈 −1∕2 ry

(10.2.9)

A vector representation is shown in Figure 10.2.4. The limiting velocity in the y direction is U0 = lim vy = −0.88447(𝜔𝜈)1∕2 y→∞

(10.2.10)

415

416

10 Methods Involving Forced Convection—Hydrodynamic Methods

Figure 10.2.3 (a) Normal fluid velocity, vy vs. y. (b) Radial fluid velocity vr vs. y at two different radii, r and r′ .

–vy

–U0 = 0.88447(𝜔ν)1/2

y (a) vr

r′

r′ > r

r y (b)

r=0

r=0 y=0 vr y

vy

Figure 10.2.4 (a) Vector representation of fluid velocities near a rotating disk. These pictures are illustrative. The treatment in the text applies to a spinning disk of infinite radius, so that there is no discontinuity at the edge. (b) Schematic resultant streamlines (or flows). The circular picture at bottom represents the flow pattern over the face of the disk. The streamlines would not stop at the circumference, but would continue beyond.

r U0

(a)

(b)

At 𝛾 = (𝜔/𝜈)1/2 y = 3.6, vy = 0.8U0 . The corresponding distance, yh = 3.6(𝜈/𝜔)1/2 , is called the hydrodynamic (or sometimes the momentum or Prandtl) boundary layer thickness and quantifies the thickness of the liquid layer in which there is significant rotational motion imparted by the moving disk. Beyond yh , the liquid has little to no rotational component of motion. For water (𝜈 ≈ 0.01 cm2 /s) at 𝜔 = 102 and 104 s−1 , yh is 0.036 and 3.6 × 10−3 cm, respectively.

10.2.2

Solution of the Convective-Diffusion Equation

Once the velocity profile has been determined, the convective-diffusion equation, (10.1.5), can be solved with appropriate boundary conditions. We focus now on the steady-state limiting current. When 𝜔 is fixed and a steady velocity profile has been attained, a potential step to the limiting-current region [where C O (y = 0) ≈ 0] will cause a current transient similar to that of a Cottrell experiment (Section 6.1.1); however, the transient in an unstirred solution, (6.1.12), decays toward zero, while that at the RDE decays to a steady-state.

10.2 Rotating Disk Electrode

In cylindrical coordinates, the convective-diffusion equation becomes [ ( )] 2 𝜕CO 𝜕 2 CO 𝜕 2 CO 1 𝜕CO 1 𝜕 CO = DO + + + 2 𝜕t r 𝜕r 𝜕y2 𝜕r2 r 𝜕𝜙2 [ ( ) ( ) v ( )] 𝜕CO 𝜕CO 𝜕CO 𝜙 − vy + vr + 𝜕y 𝜕r r 𝜕𝜙 But we can make important simplifications based on the following considerations:

(10.2.11)

1) For reasons of symmetry, C O is not a function of 𝜙; therefore, 𝜕C O /𝜕𝜙 = 𝜕 2 C O /𝜕𝜙2 = 0 and the terms depending on these derivatives drop out. ∗ everywhere, and there is no net diffusion of O in any 2) Far from the electrode, CO = CO direction. 3) The velocity toward the electrode, vy , does not depend on r, as one can see in (10.2.7); therefore, convective delivery of O toward the electrode is uniform for all y over the face of the electrode.1 4) At the mass-transfer limit, C O (y = 0) = 0 over the entire electrode surface; hence, (𝜕C O /𝜕r)y = 0 = 0 for 0 ≤ r ≤ r1 , where r1 is the disk radius. 5) Condition 4 restricts diffusion near the electrode to the direction normal to the disk surface, except near the edge. Since both convection and diffusion provide uniform mass transfer in the y direction, 𝜕C O /𝜕r = 𝜕 2 C O /𝜕r2 = 0 for all y over the face of the disk (0 ≤ r ≤ r1 ). 6) At steady state, concentrations no longer depend on time; therefore, 𝜕C O /𝜕t = 0. Thus, for the mass-transfer limit at steady state, (10.2.11) becomes ( ) 𝜕CO 𝜕 2 CO vy = DO 𝜕y 𝜕y2 By substitution of vy from (10.2.8) and rearrangement, one obtains 𝜕 2 CO 𝜕y2

=−

y2 𝜕CO B 𝜕y

(10.2.12)

(10.2.13)

where B = DO 𝜔–3/2 𝜈 1/2 /0.51. Equation 10.2.13 can be solved directly by integration. To simplify, let X = 𝜕C O /𝜕y, so that 𝜕X/𝜕y = 𝜕 2 C O /𝜕y2 . At y = 0, let X = X 0 = (𝜕C O /𝜕y)y = 0 . Then (10.2.13) becomes y2 𝜕X =− X 𝜕y B X

(10.2.14) y

dX 1 =− y2 dy ∫X X ∫0 B 0 ( 3) −y X = exp X0 3B ( ) ( 3) 𝜕CO 𝜕CO −y = exp 𝜕y 𝜕y y=0 3B Integrating from the surface of the electrode to the bulk, we write ( ) ( 3) ∗ CO ∞ 𝜕CO −y dC O = exp dy ∫0 𝜕y y=0 ∫0 3B

(10.2.15) (10.2.16) (10.2.17)

(10.2.18)

1 Equation 10.2.7 depends on having an insulating mantle wide enough to eliminate any edge effects in convective transport to the disk electrode itself.

417

418

10 Methods Involving Forced Convection—Hydrodynamic Methods

where the limits on the left are set for limiting-current conditions (i.e., for C O (y = 0) = 0). The integral on the right side is (3B)1/3 Γ(4/3) or 0.8934(3B)1/3 . Thus, ( )1∕3 ( ) 𝜕CO 3DO 𝜔−3∕2 𝜈 1∕2 ∗ CO = 0.8934 (10.2.19) 𝜕y y=0 0.51 The current is proportional to the slope of C O at the electrode surface: ( ) 𝜕CO i = nFADO 𝜕y y=0

(10.2.20)

where, under the chosen current conditions, i = il,c . By combining (10.2.19) and (10.2.20), we obtain the Levich equation, 2∕3

∗ il,c = 0.62nFADO 𝜔1∕2 𝜈 −1∕6 CO

(10.2.21)

∗ , which is the RDE analog of the current One can define the Levich constant as il,c ∕𝜔1∕2 CO function in voltammetry or the transition time constant in chronopotentiometry. When we treated this problem using the steady-state diffusion-layer model (Section 1.3.2), we obtained ( ) DO ∗ = nFA ∗ il,c = nFAmO CO CO (10.2.22) 𝛿O

From the Levich equation, we now see that for the RDE2 mO =

DO 𝛿O

2∕3

= 0.62DO 𝜔1∕2 𝜈 −1∕6

(10.2.23)

1∕3

(10.2.24)

𝛿O = 1.61DO 𝜔−1∕2 𝜈 1∕6

It is often convenient to treat RDE problems using diffusion-layer model. If necessary, mO can then be substituted according to (10.2.23) to give final equations including all variables. While the Levich equation, (10.2.21), suffices for most purposes, improved forms based on more terms in the velocity expression are available (15). 10.2.3

Concentration Profile

The concentration profile for the limiting-current condition can be obtained from (10.2.18) by integrating between 0 and distance y, where the concentration is C O (y): ( ) ( 3) CO (y) y 𝜕CO −y dC O = CO (y) = exp dy (10.2.25) ∫0 𝜕y y=0 ∫0 3B From (10.2.19) we have ( ) ∗ CO 𝜕CO = (3B)−1∕3 𝜕y y=0 0.8934

(10.2.26)

2 From the expression for yh and (10.2.24), we obtain yh /𝛿 O ≈ 2(𝜈/D)1/3 . For H2 O, 𝜈 = 0.01 cm2 /s, DO ≈ 10−5 cm2 /s, so that 𝛿 O ≈ 0.05 yh . The dimensionless ratio 𝜈/D occurs frequently in hydrodynamic problems and is called the Schmidt number, Sc.

10.2 Rotating Disk Electrode

1.2

1.0

0.8 CO(y) 0.6 C*O 0.4 y = 𝛿O

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

y/𝛿O

Figure 10.2.5 Concentration profile of species O in dimensionless coordinates.

This can be put in a more convenient form by letting u3 = y3 /3B, so that dy = (3B)1/3 du. Then (10.2.25) becomes ( ) ∗ Y CO CO (y) = exp(−u3 )du (10.2.27) 0.8934 ∫0 where Y = y/(3B)1/3 . The profile for C O when i = il,c is shown in Figure 10.2.5. 10.2.4

General i–E Curves at the RDE

For non-limiting-current conditions, only a change in integration limits in (10.2.18) is required. At y = 0, C O = C O (y = 0), so (𝜕C O /𝜕y)y = 0 is given by an analogue to (10.2.19), ( ∗ CO

− CO (y = 0) =

𝜕CO 𝜕y

(

) 0.8934

3DO 𝜔−3∕2 𝜈 1∕2

y=0

)1∕3

0.51

(10.2.28)

In exactly the manner leading to (10.2.21), one can show that 2∕3

∗ − C (y = 0)] i = 0.62nFADO 𝜔1∕2 𝜈 −1∕6 [CO O

By substitution from (10.2.21), one then obtains [ ∗ ] CO − CO (y = 0) i = il,c ∗ CO

(10.2.29)

(10.2.30)

Alternatively, (10.2.29) can be written in terms of 𝛿 O , as defined in (10.2.24), to yield i=

∗ − C (y = 0)] nFADO [CO O

𝛿O

∗ − C (y = 0)] = nFAmO [CO O

which is the same as derived from the steady-state model in Section 1.3.2.

(10.2.31)

419

420

10 Methods Involving Forced Convection—Hydrodynamic Methods

The i − E curves for the reversible reaction O + ne ⇌ R can be derived from (10.2.31) and the equivalent expression for the reduced form: [ ∗ ] CR − CR (y = 0) (10.2.32) i = il,a CR∗ where 2∕3

il,a = −0.62nFADR 𝜔1∕2 𝜈 −1∕6 CR∗

(10.2.33)

Starting with the Nernstian relationship linking potential and surface concentrations, one can make substitutions based on (10.2.30) and (10.2.32) to obtain the familiar voltammetric wave equation: E = E1∕2 +

RT (il,c − i) ln nF (i − il,a )

(10.2.34)

where E1∕2 10.2.5

RT =E + ln nF 0′

(

DR

)2∕3

DO

(10.2.35)

The Koutecký–Levich Method

The shape of the wave for a reversible reaction is independent of 𝜔, because i varies as 𝜔1/2 at all potentials. Thus, a plot of i vs. 𝜔1/2 would be a straight line passing through the origin. A deviation from this behavior often indicates a kinetic limitation in the electrode process. For example, consider a totally irreversible one-step, one-electron reaction, for which the disk current is i = FAk f (E)CO (y = 0)

(10.2.36) 0′

where k f (E) = k 0 exp[−𝛼f (E − E )]. From (10.2.30), ( ) i ∗ i = FAk f (E)CO 1 − il,c

(10.2.37)

By defining ∗ iK = FAk f (E)CO

(10.2.38)

one obtains, with rearrangement, the Koutecký–Levich (KL) equation, 1 1 1 1 1 = + = + i iK il,c iK 0.62nFAD2∕3 𝜔1∕2 𝜈 −1∕6 C ∗ O O

(10.2.39)

The parameter iK expresses the current in the absence of mass-transfer effects, i.e., the current that would flow under the kinetic limitation if mass transfer were efficient enough to keep the concentration at the electrode surface equal to the bulk value, regardless of the electrode reaction. In a case like this, i/𝜔1/2 C is a constant only when iK [or k f (E)] is very large. Otherwise, i vs. 𝜔1/2 will be curved, tending toward a limit at iK as 𝜔1/2 → ∞ (Figure 10.2.6). According to (10.2.39), 1/i vs. 1/𝜔1/2 (a Koutecký–Levich plot) should be linear and provide 1/iK as the intercept (Figure 10.2.7). By determining iK in this way for different values of E, one can evaluate of k 0 and 𝛼. Figure 10.2.8 illustrates a typical application of this procedure for the reduction of O2 to HO− at a gold electrode in alkaline solution. 2

10.2 Rotating Disk Electrode

Figure 10.2.6 i vs. 𝜔1/2 at constant E D for an electrode reaction with slow kinetics at an RDE.

Levich line (il ∝ ω1/2)

i iK

i independent of ω1/2

ω1/2

Figure 10.2.7 Koutecký–Levich plots at potential E 1 , where the rate of electron transfer is sufficiently slow to act as a limiting factor, and at E 2 , where electron transfer is rapid, for example, in the limiting-current region. The slope of both 2∕3 lines is (0.62nFAC∗O DO 𝜈 −1∕6 )−1 .

1/i

E = E1 E = E2

1/iK

ω–1/2

The KL method originated with the RDE to deal with heterogeneous electron transfer kinetics, just as we have discussed; however, it has proven broadly useful with other forms of steady-state mass transfer or kinetic limitation. Indeed, we first encountered it in the context of SSV in Section 5.4.6, and we will encounter it again with respect to modified electrodes (Section 17.5). Accordingly, it is worthwhile to think more generally about (10.2.39). Whenever mass transport and any form of electrode-based process occur in series at steady-state, both must operate at the same rate. If the electrode-based process has a rate proportional to C O (y = 0), one can write [ ] ∗ − C (y = 0) = k ′ C (y = 0) mO CO (10.2.40) O O where k ′ is a constant of proportionality. Problem 10.10 invites the reader to show that (10.2.40) leads directly to (10.2.39) when iK is defined generally as iK = nFAk ′ CO (y = 0)

(10.2.41)

For an irreversible one-step, one electron electrode reaction, this approach gives the overall outcome presented above, but for a different rate-limiting process (e.g., diffusion of species O through a polymer film on the RDE), k ′ and iK would take different forms (e.g., involving the film thickness and the diffusion coefficient for O inside the film). The KL equation applies in all such cases, with extrapolation to 𝜔−1/2 → 0 providing k ′ . Many different types of rate limitation are discussed in Section 17.5.2.

421

10 Methods Involving Forced Convection—Hydrodynamic Methods

Corrected disk current (mA)

0.8 il

0.6 0.4 0.2 0 0.9

0.7 0.5 0.3 Disk potential (V vs. H/β – Pd)

0.1

(a)

4.0 0.75 V 0.70 V

3.0 0.60 V

1/i1 (mA)–1

422

2.0

1.0

0

0

0.01

0.02

0.03

0.04

ω–1/2 (rpm)–1/2 (b)

Figure 10.2.8 (a) i vs. E at 2500 rpm and (b) Koutecký–Levich plots for the reduction of O2 to HO− at a gold 2 electrode in O2 -saturated (∼1 mM) 0.1 M NaOH at an RDE (A = 0.196 cm2 ). The potential was swept at 1 V/min. T = 26 ∘ C. (i1 represents the corrected current attributable to O2 reduction.) [Adapted from Zurilla, Sen, and Yeager (16). © 1978, The Electrochemical Society, Inc.]

10.2 Rotating Disk Electrode

Quasireversible electron transfer kinetics are not among the rate-limiting cases just discussed, because their net rate depends on both C O (y = 0) and C R (y = 0). However, a treatment resembling the KL approach can be derived. The i − 𝜂 equation for a one-step, one-electron reaction, (3.4.10), can be written [ ] [ ] CO (y = 0) −𝛼 CR (y = 0) 1−𝛼 i = b − b (10.2.42) ∗ i0 CO CR∗ where b = exp(F𝜂/RT). With substitution from (10.2.30) and (10.2.32), one obtains ) ( 1 b𝛼 1 b−𝛼 b1−𝛼 = + − i 1 − b i0 il,c il,a

(10.2.43)

which can be re-expressed as ⎡ 1 b𝛼 ⎢ 1 1 = + i 1 − b ⎢ i0 0.62FA𝜈 −1∕6 𝜔1∕2 ⎣

⎛ ⎞⎤ −𝛼 b1−𝛼 ⎟⎥ ⎜ b + 2∕3 ∗ ⎟⎥ ⎜ D2∕3 C ∗ ⎝ O O DR CR ⎠⎦

(10.2.44)

Thus, 1/i vs. 1/𝜔1/2 at a given value of 𝜂 is predicted to be linear, and the intercept of the plot allows the determination of a kinetic expression. By determining intercepts for multiple values of 𝜂, one can sort out individual parameters (e.g., i0 and 𝛼). Alternative forms of (10.2.39) and (10.2.44) are sometimes given in the literature and are provided here for convenience: • If the more general kinetic relation for the one-step, one-electron process, (3.2.8), is used in the derivation, then the equation for 1/i at the disk becomes −2∕3 −2∕3 ⎡ kf + DR kb ⎤ D 1 1 ⎢1 + O ⎥ = ∗ − k C∗ ) ⎢ i FA(kf CO 0.62𝜈 −1∕6 𝜔1∕2 ⎥ b R ⎣ ⎦

(10.2.45)

• If the reverse (e.g., anodic) reaction can be ignored, then (10.2.45) yields i=

∗ FAk f CO 2∕3

1 + kf ∕(0.62𝜈 −1∕6 DO 𝜔1∕2 )

=

∗ FAk f CO

1 + kf 𝛿O ∕DO

(10.2.46)

where 𝛿 O is as defined in (10.2.24). This equation, easily derived from (10.2.40), is useful for defining conditions for kinetic or mass-transfer control at the RDE. When k f 𝛿 O /DO ≪ 1, the current is completely under kinetic (or activation) control. When k f 𝛿 O /DO ≫ 1, the mass-transfer-controlled equation results. If the RDE is to be used for kinetic measurements, k f 𝛿 O /DO should be small, say less than 0.1; i.e., k f ≤ 0.1DO /𝛿 O . Applications of RDE techniques to electrochemical kinetic problems have been reviewed (8–11, 15, 17). 10.2.6

Current Distribution at the RDE

In the preceding derivations, we assumed that the resistance of the solution was small, allowing the expectation that the current density remains uniform across the disk and independent of the radial distance. Although this is frequently the case in real systems, the actual current distribution depends on the solution resistance, as well as the mass- and charge-transfer parameters of the electrode reaction. This topic has been treated (18) and discussed (19).

423

424

10 Methods Involving Forced Convection—Hydrodynamic Methods

The primary current distribution describes the spatial pattern of current density at an RDE when the system is entirely controlled by solution resistance. No limitations are imposed by mass transfer of electroreactants or electrode kinetics. Thus, activation and concentration overpotentials are neglected, and the electrode is taken as an equipotential surface. The actual current distribution at an RDE would approach the primary current distribution in cases where the resistivity of the solution is relatively high, and the faradaic process is operating far below the mass-transfer limit and with low activation overpotential. For a disk electrode of radius r1 embedded in a large insulating plane and with a counter electrode at infinity, the potential distribution under such conditions is shown in Figure 10.2.9.3 The current in solution moves perpendicularly to the equipotential surfaces. The current density is not uniform across the disk surface, but is larger at the edge (r = r1 ) than at the center (r = 0). This distribution arises because the ionic flux at the edge comes from the side, as well as from the direction normal to the disk. The total current flowing to the disk is given by Ohm’s law (5, 18), (10.2.47)

i = ΔE∕Rs r

r1 0.8 0.7 0.6 0.5 0.4

y

0.3 0.2

ϕ = 0.1 ϕ0

Figure 10.2.9 Primary current distribution at an RDE. Solid arcs represent surfaces of fixed values of 𝜙/𝜙0 , where 𝜙0 is the potential at the electrode surface. The potential 𝜙 is that of the disk measured against an infinitesimal (unperturbing) reference electrode. Dotted lines are lines of current. The current density is higher near the edge of the disk than at the center. [Newman (18). © 1966, The Electrochemical Society, Inc.] 3 Figure 10.2.9 was encountered previously as Figure 5.2.3, because Newman’s mathematics (18) can be used to address two distinct electrochemical problems. In the prior case, the contours in solution connected points of equal concentration, and the flux in solution was for a diffusing electroreactant. Here, the contours connect points of equal electric potential, and the flux is for ionic conduction in solution.

10.2 Rotating Disk Electrode

where ΔE is the potential difference in solution between the disk and counter electrodes. The overall solution resistance, Rs , is given for this situation by Rs = 1∕4𝜅r1

(10.2.48)

with 𝜅 being the specific conductivity of the bulk solution. When the effects of electrode kinetics and mass-transfer are included, the actual current distribution (sometimes called the secondary current distribution) is more nearly uniform than the primary one. Albery and Hitchman (19) showed that the current distribution can be considered in terms of the dimensionless parameter, 𝜌, given by 𝜌=

Rs

(10.2.49)

RE

where RE is an electrode resistance encompassing both charge transfer and concentration polarization (Section 3.4.6). The secondary current distribution as a function of 𝜌 is shown in Figure 10.2.10. As 𝜌 → ∞ (i.e., high solution resistance and small RE ), the current distribution approaches the primary one. Conversely, for small values of 𝜌 (highly conductive solutions and large RE ) a fairly uniform current distribution is obtained. To avoid a nonuniform distribution, the conditions must be such that 𝜌 < 0.1 (19). By taking RE + Rs =

dE di

(10.2.50)

1.8

10 8 9 Curve 1 2 3 4 5 6 7 8 9 10

1.6

1.4

1.2

𝜌 0.079 0.204 0.393 0.899 1.571 2.594 3.927 5.918 15.708 ∞

7

6 5 4 3 2 1

j/javg 1

1.0 2

3 4

0.8 7

5 6 8

0.6 9 10 0.4 0.0

0.2

0.4

0.6

0.8

1.0

r/r1

Figure 10.2.10 Secondary current distribution at an RDE. Ratio of local current density, j, to average current density, javg , vs. ratio of radius, r, to disk radius, r1 . Parameter 𝜌 defined in (10.2.49). [Albery and Hitchman (19). © 1971, Oxford University Press.]

425

426

10 Methods Involving Forced Convection—Hydrodynamic Methods

(where di/dE is the slope of the i − E curve at a given value of E) and combining with (10.2.48) and (10.2.49), one obtains the condition for a uniform distribution (19) as di < 0.36r1 𝜅 (10.2.51) dE For investigators focused on routine use of the RDE, the principal message here is to maintain a low solution resistance by using adequate supporting electrolyte, so that the current distribution at the disk is controlled by the electrode process, rather than the solution resistance. At the limiting current, di/dE approaches zero, so that a uniform current distribution is always obtained. 10.2.7

Practical Considerations for Application of the RDE

The equations derived for the RDE do not apply at very small or very large values of 𝜔. If 𝜔 is small, the hydrodynamic boundary layer becomes large, and when its thickness, yh = 3.6(𝜈/𝜔)1/2 , approaches the disk radius, r1 , the approximations break down. The lower limit for 𝜔 is obtained from the condition that r1 > 3.6(𝜈/𝜔)1/2 , which provides 𝜔 > 10𝜈∕r12 . For 𝜈 = 0.01 cm2 /s and r1 = 0.1, for example, 𝜔 should be larger than 10 s−1 . Still another problem occurs when i − E curves are recorded at low values of 𝜔. The curve shape derived in Section 10.2.4 was based on a steady-state concentration at the electrode surface (i.e., 𝜕C O /𝜕t = 0); hence, the electrode potential must be scanned slowly enough to allow the steady-state to be maintained. If the scan rate is too large for a given 𝜔, the i − E curves will not have the predicted wave shape, but will instead show a peak, as in LSV at a stationary electrode. The time response at the RDE is covered further in Section 10.4.1. The upper limit for 𝜔 is governed by the onset of turbulent flow. At real electrodes, the transition to turbulent flow occurs at much lower rotation rates than one would predict on the basis of critical Reynolds number, because of defects in polishing, small bends or eccentricities in the RDE shaft, or cell walls that are too close to the electrode. At very high rotation rates, one can encounter excessive splashing or vortex formation. In practice, the maximum rotation rate is frequently near 10,000 rpm or 𝜔 ≈ 1000 s−1 . In sum, the practical ranges for 𝜔 and f are normally 10 < 𝜔 < 1000 s−1 or 100 < f < 10,000 rpm. The theory developed for the RDE also assumes that the disk is precisely centered on the axis of rotation. If it is off-axis, because of its construction, its alignment in the chuck, or a bent shaft, the observed currents will be larger than those for the centered disk. In this situation, the disk sweeps a wider area of solution and gains an additional mass-transfer contribution from radial diffusion, as in the case of the rotating ring electrode (Section 10.3.1). The theory for eccentric alignment is discussed in the second edition.4 In standard RDE theory, one assumes a negligible contribution to the current from radial diffusion at the edges of the disk. This holds if the disk radius is sufficiently large that linear diffusion applies [Section 6.1.3(a)].

10.3 Rotating Ring and Ring-Disk Electrodes Reversal techniques based on potential steps or potential sweeps are not available with the RDE, because the product of the electrode reaction is continuously swept away from the surface of the disk. Reversal of a potential sweep under steady-state conditions will just retrace the i − E curve of the forward scan. However, information equivalent to that available from reversal techniques at a stationary electrode can be obtained by adding an independent ring electrode 4 Second edition, Section 9.3.6.

10.3 Rotating Ring and Ring-Disk Electrodes

Electrode material (e.g., Pt) Shaft (e.g., brass) r1

r2

r3

Insulator (e.g., Teflon)

Figure 10.3.1 Ring-disk electrode.

surrounding the disk (Figure 10.3.1), to create a rotating ring-disk electrode (RRDE). By measuring the current at the ring electrode, one can obtain knowledge about what is occurring at the disk electrode. For example, if species O is being reduced at the disk according to O + ne → R, the product, R, will be swept over to the ring by the radial flow, where it can be oxidized back to O (or be collected) if the ring is held at a potential on the positive side of the O/R wave. The ring can also be used alone as an electrode (the rotating ring electrode), e.g., when the disk of an RRDE is inactive. Mass transfer to a ring electrode is larger than to a disk at a given A and 𝜔, because a flow of fresh solution occurs radially from the area inside the ring, as well as normally from the bulk solution. The theoretical treatment of the ring electrode is more complicated than for the RDE because the radial mass-transfer term must be included in the convective-diffusion equation. While the mathematics sometimes become difficult, the results are still easy to understand and apply. The treatment will only be outlined here; details are available elsewhere (8, 19). 10.3.1

Rotating Ring Electrode

Consider a ring electrode with inner radius r2 , outer radius r3 , and area Ar = 𝜋(r32 − r22 ).5 When this electrode is rotated at angular velocity 𝜔, the solution velocity profile is that discussed in Section 10.2.1. The steady-state convective-diffusion equation that must be solved is ( ) ( ) ( ) 𝜕CO 𝜕CO 𝜕 2 CO + vy = DO (10.3.1) vr 𝜕r 𝜕y 𝜕y2 which is obtained by simplifying (10.2.11) in three ways: 1) For the steady-state, time dependences disappear, so 𝜕C O /𝜕t = 0. 2) As with the RDE, symmetry requires that concentrations be independent of 𝜙, so the derivatives with respect to 𝜙 vanish. 3) Mass transfer by diffusion in the radial direction, represented by the terms DO [𝜕 2 C O /𝜕r2 + (1/r)𝜕C O /𝜕r], is small compared to convection in the radial direction, [vr (𝜕C O /𝜕r)]; hence, the diffusive terms are neglected. The boundary conditions for the limiting ring current are: ∗ lim CO = CO

(10.3.2)

y→∞

CO (y = 0) = 0

(r2 ≤ r < r3 )

(10.3.3)

(𝜕CO ∕𝜕y)y=0 = 0

(r < r2 )

(10.3.4)

5 The electrode is assumed to be constructed with a wide outside mantle, as in the case of the RDE, and also with an interior insulator filling the space from r = 0 to r = r2 . The overall assembly has a flush surface from r = 0 to the outside radius of the mantle.

427

428

10 Methods Involving Forced Convection—Hydrodynamic Methods

From (10.2.8) and (10.2.9) the values of vr and vy are introduced to give ( ) ( ) ( ) 𝜕CO 𝜕CO 𝜕 2 CO ′ ′ 2 (B ry) −B y = DO 𝜕r 𝜕y 𝜕y2 ) ( ) ( ) ( ) ( 𝜕CO 𝜕CO DO 1 𝜕 2 CO r −y = 𝜕r 𝜕y B′ y 𝜕y2 where B′ = 0.51𝜔3/2 𝜈 −1/2 . The current at the ring electrode is6 r3 ( 𝜕C ) O iR = nFDO 2𝜋 r dr ∫r 𝜕y y=0

(10.3.5)

(10.3.6)

(10.3.7)

2

The solution to these equations yields the limiting ring current (20): 2∕3

∗ iR,l,c = 0.62nF𝜋(r33 − r23 )2∕3 DO 𝜔1∕2 𝜈 −1∕6 CO

(10.3.8)

A general result, covering both the limiting region and the rising portion of a wave, is [ ] CO (y = 0) iR = iR,l,c 1 − (10.3.9) ∗ CO In terms of the current at a disk of radius r1 observed under identical conditions, one finds from (10.2.30) and (10.2.21) that iR = iD

(r33 − r23 )2∕3

or iR iD

( =

𝛽 2∕3

(10.3.10)

r12

=

r33 r13

−

r23

)2∕3

r13

(10.3.11)

∗ and 𝜔, a ring electrode will produce a larger current than a disk of the same area. For given CO

10.3.2

The Rotating Ring-Disk Electrode

Experiments at an RRDE involve the control of two potentials (ED and ER , for the disk and ring, respectively) and the measurement of two currents (iD and iR ); therefore, the representation of the results is more complex than for experiments with a single working electrode. Measurements with an RRDE are normally carried out using a bipotentiostat (Section 16.4.4), which allows separate adjustment of ED and ER (Figure 10.3.2). At an operating RRDE, the i − E characteristics of the disk are unaffected by the presence of the ring and are as described in Section 10.2.7 Different types of experiments are possible at the RRDE, but most fall into two categories: • Collection experiments, where a disk-generated species is observed at the ring. • Shielding experiments, where a reaction at the disk perturbs the flow to the ring of an electroactive species present in the bulk. 6 The area of an infinitesimal ring section at a radius r and width 𝛿r is 𝜋(r + 𝛿r)2 – 𝜋r2 ≈ 2𝜋r𝛿r. The current through this section, (iR )δr , is ( 𝜕C ) (iR )δr (iR )δr O = = DO nFA nF2𝜋r𝛿r 𝜕y y=0 The total ring current is the summation of all (iR )δr , giving (10.3.7) as 𝛿r → 0. 7 If the disk current is found to change upon variation of ER or iR , one should suspect a defective RRDE or an undesirable coupling of the ring and disk through uncompensated

10.3 Rotating Ring and Ring-Disk Electrodes

Figure 10.3.2 Block diagram of a bipotentiostat controlling an RRDE.

Counter

Bipotentiostat Ref Ring Disk

(a) Collection Experiments

Consider an experiment in which the disk is held at a potential, ED , where the reaction O + ne → R produces a cathodic current, iD , while the ring is maintained at a sufficiently positive potential, ER , that any R reaching the ring is re-oxidized at the mass-transfer-limited rate. Species R is absent from the bulk, so all that arrives and reacts at the ring has come from the disk. We are interested in the magnitude of iR ; that is, we want to know what fraction of the R is collected at the ring. The approach is again to solve the steady-state ring convective-diffusion equation, (10.3.6), but this time for species R: ) ( ) ( ) ( ) ( 𝜕CR 𝜕CR DR 1 𝜕 2 CR r −y = (10.3.12) 𝜕r 𝜕y B′ y 𝜕y2 The boundary conditions are more complex: 1) At the disk (0 ≤ r < r1 ), the flux of R is related to that of O by the usual conservation equation: ( ) ( ) 𝜕CR 𝜕CO DR = −DO (10.3.13) 𝜕y y=0 𝜕y y=0 From the results in Section 10.2.2, ( ) 𝜕CR −iD −i = = 2 D 𝜕y y=0 nFADR 𝜋r1 nFDR 2) In the insulating gap (r1 ≤ r < r2 ), there is no current; hence, ( ) 𝜕CR =0 𝜕y y=0

(10.3.14)

(10.3.15)

3) At the ring (r2 ≤ r < r3 ) under limiting current conditions, CR (y = 0) = 0

(10.3.16)

Since R is not present in the bulk, lim CR = 0. However, species O is present in the bulk at y→∞

∗ . As in (10.3.7), the ring current is given by concentration CO r3 ( 𝜕C ) R iR = nFDR 2𝜋 r dr ∫r 𝜕y y=0

(10.3.17)

2

This problem can be solved in terms of dimensionless variables using the Laplace transform method and results have been reported (21, 22). The ring current is related to the disk current by the collection efficiency, N, N=

−iR iD

(10.3.18)

429

430

10 Methods Involving Forced Convection—Hydrodynamic Methods ∗ , D , and D . For a given RRDE, N which depends only on r1 , r2 , and r3 , independent of 𝜔, CO O R can be calculated from

N = 1 − F(𝛼∕𝛽) + 𝛽 2∕3 [1 − F(𝛼)] − (1 + 𝛼 + 𝛽)2∕3 {1 − F[(𝛼∕𝛽)(1 + 𝛼 + 𝛽)]} where 𝛼 = (r2 /r1 is given by (10.3.11), and (√ ) { } ( 1∕3 ) 3 (1 + 𝜃 1∕3 )3 3 2𝜃 −1 1 F(𝜃) = ln + arctan + 1∕2 4𝜋 1+𝜃 2𝜋 4 3

(10.3.19)

)3 − 1, 𝛽

(10.3.20)

The function F(𝜃) and values of N for different ratios r2 /r1 and r3 /r2 have been tabulated (21). One can also find N experimentally for a given electrode by measuring −iR /iD for a system where R is stable. Once N is determined, it is a known constant for that RRDE. For example, an RRDE with r1 = 0.187 cm, r2 = 0.200 cm, and r3 = 0.332 cm, has N = 0.555. Thus, 55.5% of a stable product generated at the disk is collected at the ring. The collection efficiency becomes larger as the gap thickness, (r2 − r1 ), decreases or as the ring size (r3 − r2 ) increases. Concentration profiles of R in the vicinity of the RRDE surface are illustrated in Figure 10.3.3. In a typical collection experiment, iD and iR are plotted as functions of ED (at a constant ER ), as in Figure 10.3.4a. Stability of the product is assured if N is independent of both iD and 𝜔. If R decomposes at a rate sufficiently high that some is lost in the passage from disk to ring, ∗ . Information about the the collection efficiency will be smaller and will depend on 𝜔, iD , or CO rate and mechanism of decay of R is available from RRDE collection experiments (Chapter 13). The kinetic reversibility of the electrode reaction can be examined by plotting the ring voltammogram (iR vs. ER ) at a constant value of ED , and comparing the E1/2 with that of the disk voltammogram (Figure 10.3.4b). (b) Shielding Experiments

When the disk is at open circuit, iR for the reduction of O to R is given by (10.3.8)–(10.3.11). The mass-transfer-limited ring current with iD = 0, denoted i0R,l is given by (10.3.11), rewritten as i0R,l = 𝛽 2∕3 iD,l Disc

r1

(10.3.21) Gap

r2

Ring

r3

6 5 4 3

2

Concentration contours

1

Figure 10.3.3 Concentration profiles of species R at an RRDE. The curves are contours connecting points of equal concentration, with increasing concentration from curve 1 to curve 6. For the disk (0 ≤ r < r1 ), 𝜕C R /𝜕r = 0; in the gap (r1 ≤ r < r2 ), (𝜕C R /𝜕y)y = 0 = 0; and at the ring surface (r2 ≤ r < r3 ), C R (y = 0) = 0. [Albery and Hitchman (19). © 1971, Oxford University Press.]

10.3 Rotating Ring and Ring-Disk Electrodes

i

iD

1

O + ne → R

iD,l,c E1

E2 iR,l,a = –NiD,l,c

2 iR

ED

R → O + ne

(a) iR

3 NiD,l,c

4 E1

iR,l,c = i0R,l,c – NiD,l,c

O + ne → R i0R,l,c = 𝛽2/3iD,l,c

ER

iR,l,a = –NiD,l,c

(b)

Figure 10.3.4 (a) Voltammograms in a collection experiment. (1) iD vs. E D and (2) iR vs. E D with E R = E 1 . (b) Ring voltammograms showing both shielding and collection. (3) iR vs. E R , iD = 0 (E D = E 1 ) and (4) iR vs. E R , iD = iD, l, c (E D = E 2 ).

where iD, l is the limiting current that would be achieved at the disk electrode if it were active. If the disk is made active, so that O is also reduced there, giving current, iD , the flux of O to the ring must be decreased. We learned above that when the ring is set in the mass-transfer-controlled region for collection of a species produced at the disk, the collected fraction is N. In this shielding experiment, the disk creates a deficit of the species being reduced at the ring, and the ring collects a fraction N of that deficit. Thus, the limiting ring current iR, l becomes iR,l = i0R,l − NiD

(10.3.22)

This equation holds for any value of iD from iD = 0 to iD = iD, l . For the special case iD = iD, l , one obtains by substitution from (10.3.21), iR,l = i0R,l (1 − N𝛽 −2∕3 )

(10.3.23)

Thus, when the disk current is at its limiting value, the ring current is decreased by the factor 1 – N𝛽 –2/3 . This factor, always less than unity, is called the shielding factor. These relations are easier to grasp when the complete i − E curves are considered (Figure 10.3.4b). One sees that the effect of switching from iD = 0 to iD = iD, l is to shift the entire ring voltammogram (iR vs. ER ), which is assumed to be reversible, downward by the amount NiD, l . Other dual electrode systems (23), including the scanning electrochemical microscope (SECM; Chapter 18) and microelectrode arrays (Section 5.6.3), can operate at steady state and show similar shielding and collection effects. However, these systems differ from the RRDE in that convective effects are absent, so that the time for interelectrode transit is governed by diffusion across the space between the electrodes.

431

432

10 Methods Involving Forced Convection—Hydrodynamic Methods

10.4 Transient Currents The observation of current transients following a potential step at the disk or ring can sometimes be useful for understanding an electrochemical system. For example, the adsorption of a component, A, on a disk electrode can be studied by observing the transient shielding of the ring current for electrolysis of A when the disk potential is stepped to a value where A becomes adsorbed. 10.4.1

Transients at the RDE

Treatment of the transient response at the RDE requires a solution of the convective-diffusion equation, (10.2.13), but with retention of 𝜕C O /𝜕t: ( ) ( ) 𝜕CO 𝜕 2 CO 𝜕CO ′ 2 = DO −B y (10.4.1) 𝜕t 𝜕y 𝜕y2 where B′ = 0.51𝜔3/2 𝜈 −1/2 . Solutions have been obtained by approximation methods and by digital simulation. For a potential step to the limiting current region, the instantaneous value of il , denoted il (t), is given approximately by (24) ( ) ∞ ∑ il (t) −m2 𝜋 2 DO t R(t) = =1+2 exp (10.4.2) 2 il (ss) 𝛿O m=1 where il (ss) is the value of il as t → ∞, and 𝛿 O is given in (10.2.24). An implicit approximate equation for R(t) obtained by the “method of moments” has also been proposed (25): { )]} ( ( ) √ [ ( ) DO t 1 − R(t)3 2R(t) + 1 1 1.8049 2 1 𝜋 = ln + 3 − arctan √ 6 1.6116 2 6 [1 − R(t)]3 𝛿2 3 O

(10.4.3) 2.2 2.0 1.8 i/iss 1.6 1.4 1.2 1.0 0.0

0.5

1.0

1.5

2.0

2.5

ωt(D/ν)1/3(0.51)2/3

Figure 10.4.1 Disk-current transient for potential step at the disk: curve from simulation; squares from (10.4.2); circles from (10.4.3). [Prater and Bard (26). © 1970, The Electrochemical Society, Inc.]

10.4 Transient Currents

Both expressions are in good agreement with simulation (26), as shown for a typical disk transient in Figure 10.4.1. At short times, when the diffusion layer thickness is much thinner than 𝛿 O , the potential step transient follows the Cottrell equation, (6.1.12), but it departs toward the steady-state current as the diffusion layer becomes thick enough to be affected by convection. The time required for the current to attain its steady-state value can be obtained from the curve in Figure 10.4.1. The current is within 1% of il (ss) at a time, 𝜏, when 𝜔𝜏(D∕𝜈)1∕3 (0.51)2∕3 > 1.3

(10.4.4)

With the approximation (D/𝜈)1/3 ≈ 0.1, one finds that 𝜏 > 20/𝜔. Thus, for 𝜔 = 100 s−1 (∼1000 rpm), 𝜏 ≈ 0.2 s. 10.4.2

Transients at the RRDE

Consider an experiment in which the ring of an RRDE is maintained at a potential where species R can be oxidized to O, and the disk is at open circuit or at a potential where no R is produced. If R is then generated at the disk by a potential step or by a constant-current step, a time will be required for R to cross the gap from the outside of the disk to the inside edge of the ring (the transit time). An additional time will be required before the disk current attains its steady-state value. The rigorous solution for the ring current transient, iR (t), involves solving the time-dependent form of (10.3.6): ( ) ( ) ( ) 𝜕CR 𝜕 2 CR 𝜕CR 𝜕CR ′ y2 ′ ry = DR + B − B (10.4.5) 𝜕t 𝜕y 𝜕r 𝜕y2 This difficult problem is discussed by Albery and Hitchman (27), who present several mathematical approaches and approximate results. One may also obtain results by simulation (26). Figure 10.4.2 shows ring-current transients for several different electrodes. As the gap increases and the ring becomes wider, the onset of the ring current becomes delayed, and the rise toward steady-state becomes slower.

1.0 a i/iss b

0.5

c d e 0.0 0.0

0.5 𝜔t(D/𝜈)

1/3

(0.51)

1.0 2/3

Figure 10.4.2 Simulated ring transients for electrodes of different geometries (r2 /r1 , r3 /r1 ). (a) 1.02, 1.04; (b) 1.05, 1.07; (c) 1.07, 1.48; (d) 1.09, 1.52; (e) 1.13, 1.92. Points show transit times from (10.4.8). [Prater and Bard (26). © 1970, The Electrochemical Society, Inc.]

433

10 Methods Involving Forced Convection—Hydrodynamic Methods

An approximation for the transit time, t ′ , has been presented (28). The radial velocity near the electrode surface is given by (10.2.9), which can be written dr vr = = 0.51𝜔3∕2 𝜈 −1∕2 ry (10.4.6) dt A molecule of R generated at the edge of the disk (r = r1 ) must diffuse normal to the disk to reach the ring, since vr is zero at y = 0. It is then swept in a radial direction and then moves by diffusion and convection in the y direction to reach the inner edge of the ring. This path can be described by some average trajectory and some time-dependent distance, y, from the electrode surface. Integration of (10.4.6) yields ( ) t′ r2 3∕2 −1∕2 ln = 0.51𝜔 𝜈 y dt (10.4.7) ∫0 r1 If one makes the approximation y ≈ (Dt)1/2 and carries out the integration of (10.4.7), the result is [ ( )]2∕3 r ′ 1∕3 𝜔t = 3.58(𝜈∕D) log 2 (10.4.8) r1 For (D/𝜈)1/3 ≈ 0.1, 𝜔 = 100 s−1 , and r2 /r1 = 1.07 (representing a rather narrow gap), the estimated transit time from (10.4.8) is 30 ms. In Figure 10.4.2, a value of t ′ calculated from (10.4.8) is shown as an open circle on each curve where that curve begins to rise. The transit time is a measure of first arrival of the electroreactant at the ring, so it marks the onset of the ring transient. In Figure 10.4.2, these times correspond to only about 2% attainment of the steady value. Ring-current transients can be useful for detecting adsorption of a disk-generated intermediate at the disk, because adsorption will delay the appearance of that species at the ring (29). Transients can also be used in a qualitative way to study electrode processes, as illustrated in a “shielding transient” experiment (30) involving a Cu-ring/Pt-disk electrode immersed in an air-saturated solution containing 2 × 10−5 M Cu(II). The events are summarized in Figure 10.4.3. With ED initially at 1.00 V vs. SCE, no reaction occurs at the disk. At the same 600

12 Ring

8 iRIμA

400 iDIμA

434

200

Disk potential switched from 1.0 to 0.0 V

4

Disk residual current

Disk

10

30

20 Time/s

Figure 10.4.3 Time dependence of the oxygen reduction current at the Pt disk and the Cu(II) reduction current at the Cu ring of an RRDE. Air-saturated 0.2 M H2 SO4 and 2 × 10−5 M Cu(II). Rotation speed, 2500 rpm. Disk potential at 0.00 V vs. SCE for t > 0; ring potential at −0.25 V vs. SCE for all t. Disk area, 0.458 cm2 ; 𝛽 2/3 = 0.36; N = 0.183. [Bruckenstein and Miller (30). © 1977, American Chemical Society.]

10.5 Modulation of the RDE

time, ER is held at −0.25 V vs. SCE, and there is a cathodic ring current of about 11 μA from Cu(II) + 2e → Cu. Oxygen is not reduced at the Cu ring at this potential, even though the reversible potential for the reaction O2 + 4H+ + 4e → 2H2 O is much more positive, because the kinetics are very slow. The Pt disk is stepped to 0.0 V, and a cathodic disk current of about 700 μA flows, representing the simultaneous reduction of O2 and Cu(II). The plating of copper on the platinum substrate occurs at potentials more positive than those where copper would deposit on bulk copper and is called underpotential deposition (Section 15.6.3). The reduction of Cu(II) at the disk shields the ring, so that a drop in iR is observed. As copper deposits on the disk, the reduction of oxygen becomes kinetically hindered, and the disk current falls. After about a monolayer of copper has deposited on the disk, further underpotential deposition is no longer possible, Cu(II) reduction at the disk ceases, and the ring current returns to its unshielded value. This simple experiment demonstrates the “poisoning” of the oxygen reduction process by copper. It also shows that underpotential deposition of a monolayer of copper from solutions containing as little as 1 ppm Cu can drastically affect the behavior of a platinum electrode. Since Cu(II) is a common impurity in distilled water and mineral acids, this is not a casual risk in electrochemical research. Adsorption of small amounts of other impurities can also affect the behavior of a solid electrode. Electrochemical work often requires great efforts to establish and to preserve solution purity.

10.5 Modulation of the RDE So far in this chapter, we have assumed that the rotation rate of the electrode remains constant at 𝜔 when measurements of currents are carried out. However, one can also usefully measure currents when 𝜔 is changing with time.8 The simplest case involves a monotonic variation of 𝜔 as a function of time (e.g., 𝜔 ∝ t 2 ) and an automatic plotting of iD vs. 𝜔1/2 . These “automated Levich plots” can save valuable time vs. point-by-point measurements, especially when the electrode surface is changing with time and a rapid examination is needed (e.g., during an electrodeposition, or with impurity or product adsorption). This technique and related methods have been reviewed (31). Another useful technique features the sinusoidal variation of 𝜔1/2 (called hydrodynamic modulation) (30, 32). The square-root of rotation rate of an RDE is varied about a fixed center speed, 𝜔0 , at a frequency 𝜎, so that the instantaneous value of 𝜔1/2 is given by 1∕2

𝜔1∕2 = 𝜔0

+ Δ𝜔1∕2 sin(𝜎t)

(10.5.1)

where Δ𝜔 is usually only about 1% of 𝜔0 . Modulation of the rotation rate modulates the disk current for any process in which the electroreactant is convectively transported. The modulation amplitude of the current is Δi = (Δ𝜔∕𝜔0 )1∕2 i𝜔

0

(10.5.2)

The varying component of the disk current can be readily separated from the overall current by lock-in detection. The modulated current contains essentially no contribution from double-layer charging, oxidation or reduction of the electrode, or electrode reactions of adsorbed species. Moreover, it is relatively insensitive to the anodic and cathodic background currents. Figure 10.5.1 shows results from a study in which 0.2 μM Tl(I) was reduced at an amalgamated gold RDE (which can be used when a mercury-like surface is desired) (33). Even though the 8 This discussion is abridged in this edition. More is available in the second edition, Section 9.6.

435

436

10 Methods Involving Forced Convection—Hydrodynamic Methods

i (A, A2) or Δi (B, B2)

A A, A2: 200 nA

A2

B2

B, B2: 2 nA

B

0.00

–0.25

–0.50

–0.75

–1.00

E(V vs. SCE)

Figure 10.5.1 Voltammetric scans of Tl(I) at an amalgamated gold RDE. All A traces are steady-state voltammograms; all B traces involve hydrodynamic modulation. (A, B) 0.01 M HClO4 ; (A2, B2) 2.0 × 10−7 M Tl+ in 0.01 M HClO4 . Current sensitivities indicated by markers; zero current for all curves is the dashed line. For all 1∕2

curves, 𝜔0 = 60 rpm1∕2 . For B curves, Δ𝜔1/2 = 6 rpm1/2 , 𝜎/2𝜋 = 3 Hz, averaging time constant = 3 s, and scan rate = 2 mV/s. [Miller and Bruckenstein (33). © 1974, American Chemical Society.]

faradaic current for the Tl(I) reduction cannot be distinguished from the residual current on the iD − E scan, a clear reduction wave is found by measuring Δi. An RDE can also be modulated thermally by irradiating the back of the disk electrode with a laser (34). This method is the RDE analog of the temperature-jump experiment discussed in Section 9.7.3.

10.6 Electrohydrodynamic Phenomena Electrohydrodynamics deal with fluid motion induced by electric fields, or vice versa. The phenomena involved differ fundamentally from those of convective flow induced by a mechanical pressure gradient, such as we have been considering in this chapter. In the absence of gravitational effects (natural convection), the steady-state fluid velocity vector, v, is obtained in the latter cases from the equation 𝜂s ∇2 v = 𝛁P

(10.6.1)

where 𝜂 s is the fluid viscosity. In electrohydrodynamic systems, the force inducing fluid flow arises from the interaction of an electric field with excess charge density in the fluid, 𝜂s ∇2 v = −𝜌E E

(10.6.2)

where E is the electric field vector (V/cm) and 𝜌E is the charge per unit volume. Let us consider the application of an electric field along a glass capillary filled with an electrolyte solution, as shown schematically in Figure 10.6.1. At most pH values, the walls of the capillary are charged because of protonation/deprotonation equilibria involving surface Si–OH groups. At pH > 3, the surface charge is negative, and it must be countered by a diffuse layer of positive charge in solution, just as at a charged electrode surface (Section 2.2.2 and Chapter 14).

10.6 Electrohydrodynamic Phenomena

Fixed charges on glass wall

Anode

Cathode Fluid velocity profile

U

Ex

Figure 10.6.1 Representation of electroosmotic flow of a fluid (e.g., water) in a glass capillary. Only ions near the glass walls are shown. The velocity profile here is flatter (so-called “plug flow”) than the parabolic profile seen with an external pressure drop along a tube.

When an electric field is applied along the capillary axis, the excess charge on the solution side of the double layer (in the form of electrolyte cations) begins to move along the field, and the net viscous drag on the solvent induces convection of the solution in the same direction. The fluid is literally pumped toward the cathode by this process, which is called electroosmotic flow. In most cases, the radius of the capillary is large compared to the thickness of the diffuse layer; hence, curvature can be neglected, and the flow can be considered as taking place along a plane. Then, (10.6.2) can be written (7) in terms of the field, Ex , along the axis of the capillary, while potential, 𝜙, and velocity, vx , are functions of distance, y, above the surface. 𝜂S

𝜕 2 vx 𝜕y2

= −𝜌E Ex = 𝜀𝜀0

𝜕2𝜙 E 𝜕y2 x

(10.6.3)

In (10.6.3), the charge density is obtained from the Poisson equation, (14.3.5), 𝜌E = −𝜀𝜀0

𝜕2𝜙 𝜕y2

(10.6.4)

A single integration of (10.6.3), with 𝜕vx ∕𝜕y = 0 and 𝜕𝜙/𝜕y = 0 far from the walls (y → ∞), yields 𝜂s (𝜕vx ∕𝜕y) = 𝜀𝜀0 (𝜕𝜙∕𝜕y)Ex

(10.6.5)

A second integration from a position near the wall, where vx = 0 and 𝜙 = 𝜁 , to the middle of the channel, where vx = U and 𝜙 = 0, gives the Helmholtz–Smoluchowski equation: U = −𝜀𝜀0 𝜁 Ex ∕𝜂s

(10.6.6)

The parameter 𝜁 is called the zeta potential, which is the potential at a position in the diffuse layer called the shear plane (35), and U is the electroosmotic solution velocity past a planar charged surface. For 𝜁 = 0.1 V and Ex = 100 V/cm in an aqueous solution, U ∼ 0.1 cm/s. A complete treatment of electroosmotic flow in capillaries, without the assumption that the diffuse double layer thickness is small relative to the capillary radius, may be found in reference (7). Electroosmosis is one of several electrokinetic effects associated with the relative motion of a charged solid and a solution. A second example is the streaming potential that arises between two electrodes placed as in Figure 10.6.1 when a solution streams down the tube. This is

437

10 Methods Involving Forced Convection—Hydrodynamic Methods

essentially the inverse of the electroosmotic effect. There is also electrophoresis, where charged particles in a solution move in an electric field. This effect is widely exploited for separations of proteins, DNA fragments, many other substances. Electrokinetic effects have a long scientific history (35, 36). Electroosmosis and electrophoresis also underly the phenomenon of ionic current rectification (ICR) in conical nanopores. ICR refers to the non-ohmic current–voltage characteristic frequently observed prepared by pulling the end of glass capillaries into a conical or tapered geometry (37). The behavior arises from high and low ionic conductance states at the small orifice of the nanopore that depend on the sign of the applied voltage (38, 39). The dual requirements for observing ICR are that the nanopore have an asymmetric geometry and that the interior surface be charged. Both are satisfied by conical nanopores in glass. Consider a conical nanopore formed at the tip of a tapered glass pipette and having a uniform negative surface charge, as depicted in Figure 10.6.2. Identical KCl solutions are placed inside and outside the nanopore. When a negative potential is applied to the Ag/AgCl electrode inside the nanopore vs. the Ag/AgCl electrode in the bulk, a K+ flux is directed from the external solution to the pore interior, while Cl− moves in the opposite direction. As the surface of the pore is cation-selective, Cl− ions are impeded by the glass surface because of electrostatic repulsion, and their transport rate through the orifice from the pore interior to the bulk solution is reduced. A consequence of this anion rejection is an accumulation of K+ and Cl− inside the nanopore near the orifice, resulting in local conductivity greater than in the bulk solution. Conversely, when a positive potential is applied inside the pore interior relative to the external solution, the transport of Cl− from the external solution to the internal solution is impeded by the negative surface charge, and Cl− becomes depleted within the pore interior, thus, decreasing the nanopore conductivity and the observed ionic current. Typically, ICR is observed in conical nanopores that have an orifice radius between 10 and 500 nm. Applications have been discussed in the literature (40–42). In the kinds of electrochemical experiments considered in this book, electrokinetic effects are usually unimportant because electric fields are generally small. However, experiments in which very large fields are intentionally applied can produce convection (43, 44). Related experiments involve the effects of applied magnetic fields on electrochemical processes (43, 45). Hydrodynamic instabilities can produce convective patterns in thin-layer cells with resistive solutions. For example, a form of convection in thin-layer cells (sometimes called the Felici instability) can produce hexagonal patterns during ECL experiments (Section 20.5) in organic solvents with very low concentrations of supporting electrolyte (46, 47). V0

0.0 i(nA)

438

–0.2 –0.4 –0.6 –0.4 –0.2 0.0 0.2 0.4 Voltage(V) (a)

(b)

(c)

Figure 10.6.2 (a) i − V curve recorded for a glass nanopipette with a ∼20 nm radius orifice in 0.01 M KCl. The voltage corresponds to the potential of an Ag/AgCl electrode inside the nanopipette vs. an identical Ag/AgCl electrode in the external solution. Scan rate, 20 mV/s. (b, c) Schematic representations of the ion distributions at negative (V < 0) and positive (V > 0) applied voltages. [Part (a) from Wei, Bard, and Feldberg (37). © 1997, American Chemical Society.]

10.7 References

10.7 References 1 M. Tsushima, K. Tokuda, and T. Ohsaka, Anal. Chem., 66, 4551 (1994). 2 V. G. Levich, “Physicochemical Hydrodynamics,” Prentice-Hall, Englewood Cliffs, NJ, 1962. 3 R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena,” 2nd ed., Wiley,

New York, 2002. 4 J. N. Agar, Disc. Faraday Soc., 1, 26 (1947). 5 J. Newman, J. Electroanal. Chem., 6, 187 (1973). 6 J. S. Newman and K. E. Thomas-Alyea, “Electrochemical Systems,” 3rd ed., Wiley, Hoboken,

NJ, 2004. 7 R. F. Probstein, “Physicochemical Hydrodynamics—An Introduction,” 2nd ed., Wiley,

New York, 1994. 8 A. C. Riddiford, Adv. Electrochem. Electrochem. Eng., 4, 47 (1966). 9 R. N. Adams, “Electrochemistry at Solid Electrodes,” Marcel Dekker, New York, 1969, 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

pp. 67–114. C. Deslouis and B. Tribollet, Adv. Electrochem. Sci. Eng., 2, 205 (1992). W. J. Albery, C. C. Jones, and A. R. Mount, Compr. Chem. Kinet., 29, 129 (1989). V. Yu. Filinovskii and Yu. V. Pleskov, Comprehensive Treatise Electrochem., 9, 293 (1984). T. von Kármán, Z. Angew. Math. Mech., 1, 233 (1921). W. G. Cochran, Math. Proc. Cambridge Philos. Soc., 30 (3), (1934). J. S. Newman, J. Phys. Chem., 70, 1327 (1966). R. W. Zurilla, R. K. Sen, and E. Yeager, J. Electrochem. Soc., 125, 1103–1109 (1978). V. Yu. Filinovskii and Yu. V. Pleskov, Prog. Surf. Membr. Sci., 10, 27 (1976). J. Newman, J. Electrochem. Soc., 113, 501, 1235 (1966). W. J. Albery and M. L. Hitchman, “Ring-Disc Electrodes,” Clarendon, Oxford, 1971, Chap. 4. V. G. Levich, op. cit., p. 107. W. J. Albery and M. Hitchman, op. cit., Chap. 3. W. J. Albery and S. Bruckenstein, Trans. Faraday Soc., 62, 1920 (1966). E. O. Barnes, G. E. M. Lewis, S. E. C. Dale, F. Marken, and R. G. Compton, Analyst, 137, 1068 (2012). Yu. G. Siver, Russ. J. Phys. Chem., 33, 533 (1959). S. Bruckenstein and S. Prager, Anal. Chem., 39, 1161 (1967). K. B. Prater and A. J. Bard, J. Electrochem. Soc., 117, 207 (1970). W. J. Albery and M. Hitchman, op. cit., Chap. 10. S. Bruckenstein and G. A. Feldman, J. Electroanal. Chem., 9, 395 (1965). S. Bruckenstein and D. T. Napp, J. Am. Chem. Soc., 90, 6303 (1968). S. Bruckenstein and B. Miller, Acc. Chem. Res., 10, 54 (1977). S. Bruckenstein and B. Miller, J. Electrochem. Soc., 117, 1032 (1970). J. V. Macpherson, Electroanalysis, 12, 1001 (2000). B. Miller and S. Bruckenstein, Anal. Chem., 46, 2026 (1974). J. L. Valdes and B. Miller, J. Phys. Chem., 92, 525 (1988). A. W. Adamson and A. P. Gast, “Physical Chemistry of Surfaces,” 6th ed., Wiley, New York, 1997. D. A. MacInnes, “The Principles of Electrochemistry,” Dover, New York, 1961, Chap. 23. C. Wei, A. J. Bard, and S. W. Feldberg, Anal. Chem., 69, 4627 (1997). D. Woermann, Phys. Chem. Chem. Phys., 5, 1853 (2003). H. S. White and A. Bund, Langmuir, 24, 2212 (2008). W.-J. Lan, M. A. Edwards, L. Luo, R. T. Perera, X. Wu, C. R. Martin, and H. S. White, Acc. Chem. Res., 49, 2605 (2016). N. Laohakunakorn and U. F. Keyser, Nanotechnology, 26, 275202 (2015). R. A. Lucas, C.-Y. Lin, and Z. S. Siwy, J. Phys. Chem. B,

439

10 Methods Involving Forced Convection—Hydrodynamic Methods

43 J.-P. Chopart, A. Olivier, E. Merienne, J. Amblard, and O. Aaboubi, Electrochem. Solid-State 44 45 46 47 48

Lett., 1, 139 (1998). D. A. Saville, Annu. Rev. Fluid Mech., 29, 27 (1997). J. Lee, S. R. Ragsdale, X. Gao, and H. S. White, J. Electroanal. Chem., 422, 169 (1997). H. Schaper and E. Schnedler, J. Phys. Chem., 86, 4380 (1982). M. Orlik, J. Rosenmund, K. Doblhofer, and G. Ertl, J. Phys. Chem., 102, 1397 (1998). S. Bruckenstein and P. R. Gifford. Anal. Chem., 51(2), 250–255 (1979).

10.8 Problems 10.1 Consider an RDE with r1 = 0.20 cm immersed in an aqueous solution of a substance ∗ = 10−2 M, D = 5 × 10−6 cm2 /s). The disk is rotated at 100 rpm. Species A is A (CA A reduced in a 1e reaction. 𝜈 = 0.01 cm2 /s. Calculate: (a) vr and vy at a distance 10−3 cm normal to the disk surface at the edge of the disk. (b) vr and vy at the electrode surface. (c) U 0 , il,c , mA , 𝛿 A , and the Levich constant. 10.2 What radii, r2 and r3 , can a rotating ring electrode have to produce the same limiting current as an RDE with r1 = 0.20 cm? (Many combinations of r2 and r3 are suitable.) What is the area of the ring electrode? 10.3 From the data in Figure 10.2.8, calculate the diffusion coefficient for O2 in 0.1 M NaOH and k f for the reduction of oxygen at 0.75 V. Take 𝜈 = 0.01 cm2 /s. The electrode reaction is not a one-step, one-electron process, so the interpretation given to k f depends on the proposed mechanism (Section 15.3.1). 10.4 Figure 10.8.1 shows i − E curves at an RRDE for 5 mM CuCl2 in 0.5 M KCl. For the electrode employed, N = 0.53. iR (iD = 0)

2 1000

800

2000

1600 iD

1 600

1200

400

800

200

400

0

0 0.2

0.0

–0.2

–0.4

–0.6

ED or ER (V vs. Ag/AgCl)

Figure 10.8.1 Voltammograms at an RRDE in 5 mM CuCl2 (iD = 0). 𝜔 = 201 s−1 ; disk area = 0.0962 cm2 ; 𝜈 = 0.011 cm2 /s.

iR(μA)

iD(μA)

440

10.8 Problems

(a) Analyze the data to determine D, 𝛽, and any available information about the first reduction step, Cu(II) + e → Cu(I). (b) If the ring voltammogram were obtained with ED = − 0.10 V, what value of iR,l,c would be expected for ER = − 0.10 V? (c) If ER is held at +0.4 V and ED = − 0.10 V, what value of iR is expected? (d) What process occurs at the second wave? Account for the wave shape. (e) Assume the ring is held at +0.4 V. Sketch the expected plot of iR vs. ED as ED is scanned from +0.4 to −0.6 V. 10.5 Ring voltammograms for an RRDE in 5 mM K3 Fe(CN)6 and 0.1 M KCl are shown in Figure 10.8.2. Calculate N for this electrode and D for Fe(CN)3− . What is the predicted 6 1/2 slope of iD vs. 𝜔 ? What are the predicted values of the limiting disk current, iD,l,c , and the limiting ring current, iR,l,c , with iD = 0 and iD = iD,l,c at 5000 rpm? Assume 𝜈 = 0.01 cm2 /s. 1 1200 2

iR(μA)

800

400

0

iD = 0 iD = 302 μA

–400

0.4

0.2

0.0

–0.2

–0.4

ER (V vs. SCE)

Figure 10.8.2 Ring voltammograms at an RRDE with (1) iD = 0 and (2) iD at iD, l, c = 302 μA. The RRDE (r2 = 0.188 cm, r3 = 0.325 cm) was rotated at 48.6 rps in 5.0 mM K3 Fe(CN)6 and 0.10 M KCl.

10.6 Experiments are performed at an RRDE with r1 = 0.20 cm, r2 = 0.22 cm, r3 = 0.32 cm. A disk voltammogram (iD vs. ED ) is to be recorded for a rotation rate of 2000 rpm. What maximum potential sweep rate should be employed to prevent non-steady-state effects from occurring? What is the transit time with this electrode? 10.7 The diffusion coefficient of an electroactive species can be obtained from a limitingcurrent measurement at an RDE and a transient measurement (e.g., a potential step measurement) at the same electrode under identical conditions, but at 𝜔 = 0. It is not necessary to know A, n, or C * . Explain how this is accomplished and discuss the possible errors in this procedure. 10.8 Bruckenstein and Gifford (48) proposed that ring shielding measurements at the RRDE can be employed for the analysis of micromolar solutions by using the equation ΔiR,l = 0.62nF𝜋r12 D2∕3 𝜈 −1∕6 𝜔1∕2 NC ∗ where ΔiR,l represents the change in the limiting

441

442

10 Methods Involving Forced Convection—Hydrodynamic Methods

(a) Derive this equation. (b) Figure 10.8.3 shows iR, l vs. ED for the reduction of Bi(III) to Bi(0) in 0.1 M HNO3 . Mass-transfer-limited reduction occurs at −0.25 V. From data during the forward scan (ED varied from +1.0 to −0.2 V), calculate N for this RRDE. (c) Explain the large ring current transient observed during the reverse scan (ED from −0.2 to +1.0 V).

iR (cathodic)

0

0.1 μA

1.0

0.8

0.6

0.4 0.2 ED(V vs. SCE)

0.0

–0.2

Figure 10.8.3 iR,l,c vs. E D for reduction of Bi(III) to Bi(0) for a solution containing 4.86 × 10−7 M Bi(III) and 0.1 M HNO3 . The ring electrode was held at −0.25 V, and the disk potential was scanned from +1.0 V at 200 mV/s. For this electrode, the slope of iD, l, c vs. C∗Bi(III) is 0.934 μA/μM. [Bruckenstein and Gifford (48). © 1979, American Chemical Society.]

10.9 Species O, reduced to R in a one-step, one-electron process, is studied by steady-state voltammetry at an RDE and at UMEs of various sizes. (a) Calculate values of the mass-transfer coefficient, mO , for the RDE at rotation rates of 500, 1000, 5000, and 10,000 rpm, with the latter figure approximating the highest practical rotation rate. Assume DO = 1 × 10−5 cm2 /s and 𝜈 = 0.01 cm2 /s. Show your results on a plot of log mO vs. rotation rate. (b) Calculate values of mO for UME disks with r0 = 25, 10, 1, 0.1, and 0.05 μm. Place these values on your log plot as horizontal lines. (c) Suppose you are interested in evaluating the standard heterogeneous rate constant, k 0 , for the O/R process. Section 5.4.3 shows that Λ0 = k 0 /mO must be less than 10 to observe any kinetic limitation in steady-state voltammetry. What is the largest value of k 0 that could be addressed using the RDE? Using a UME with a size in the range given in (b)? 10.10 Show how (10.2.40) and (10.2.41) lead directly to (10.2.39) without substitution from any other relationship.

443

11 Electrochemical Impedance Spectroscopy and ac Voltammetry In Chapters 5–10, we mainly discussed ways of studying electrode reactions through large perturbations. By imposing potential sweeps, potential steps, or current steps, we typically drive a working electrode to a condition far from equilibrium, and we observe the response, usually a transient signal. A contrasting approach is to perturb an electrochemical system with an alternating potential, Eac , and to observe the resulting alternating current, iac . That current has an amplitude and phase angle containing electrochemical information. If the perturbation is purely sinusoidal at frequency f and has an amplitude smaller than a few millivolts, the relationship between iac and Eac can be expressed in terms of an electrical impedance, Z. That is, one can find a unique Z such that a current identical to iac will result when the voltage across the impedance is varied exactly as Eac . In electrochemistry, one can normally express Z as a series combination of a specific resistance, R, and a specific capacitance, C. Thus, the cell behaves at frequency f indistinguishably from a physical resistor and a physical capacitor with these values in series. A real electrochemical system is, however, much more complex than a series combination of a single resistance and a single capacitance. While the values R and C accurately describe the cell for frequency f , they typically will not describe it for a different frequency, f ′ . An equivalent resistance–capacitance combination will still exist at f ′ , but with other values, R′ and C ′ , making up a different impedance, Z′ . By varying the frequency of perturbation, one probes the system on different time scales; hence, the impedance becomes a function of frequency, Z(𝜔), where 𝜔 = 2𝜋f . This impedance spectrum embodies everything that affects the flow of current in the electrochemical system, including • • • •

Electrical resistances in the current path. Interfacial capacitances. Mass transfer of reactants. Kinetics of all relevant heterogeneous and homogeneous reactions.

One can analyze Z(𝜔) to identify controlling processes and to extract parameters quantifying them. Electrochemical impedance spectroscopy (EIS) is the name given to this area (1–11). Automated systems are used to apply sinusoidal potential variations covering a frequency range, to measure the resulting currents, and to deliver the results as Z(𝜔). The practice of EIS has grown extensively because it is convenient and informative for a wide range of electrochemical problems. Most of this chapter relates to EIS and its interpretation. However, there are other methods based on the idea of electrochemical impedance. Most important among them is ac voltammetry, in which Eac is a sinusoidal component imposed on

Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

444

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

a potential sweep, and iac is detected and measured (12–16). The resulting ac voltammogram is the amplitude of iac vs. the sweep potential, in effect, expressing the variation of Z with sweep potential. Important advantages attend impedance-based techniques, including • The experimental ability to achieve high precision, because the response may be indefinitely steady and can be averaged over a period. • The freedom to operate over a wide range of frequencies (or time scales)—10−4 to 106 Hz (or 104 to 10−6 s). • The ability to treat responses theoretically by linearized (or otherwise simplified) current–potential characteristics. The interpretation of impedance is different from our experience in earlier chapters, where we predicted electrochemical responses directly from chemical and physical fundamentals. Here, interpretation is a two-phase process involving an intermediate representation of the chemical system as a network of electrical components. The more difficult task is to understand the components of that network—the resistors and capacitors—in terms of chemical and physical processes. In Sections 11.1–11.3, we will focus on a few important concepts that turn out to be widely applicable.

11.1 A Simple Measurement of Cell Impedance There is a manual experiment that nicely demonstrates the electrical equivalence of a resistor–capacitor combination and an electrochemical cell at a single frequency, f . It is worthwhile to think briefly about this experiment, just to bring home the concept of electrochemical impedance. The measurement resembles the operation of a Wheatstone bridge (Figure 11.1.1a). In the four-resistor network shown there, Rx is an unknown resistance that one desires to measure. A dc voltage is applied across the top and bottom vertices, and one looks for current passing between the lateral vertices, A and B. The resistors on the left and right sides of the network form voltage dividers that establish voltages at A and B according to the proportions of the two pairs of resistances. Under most conditions, A and B are not at the same voltage,

Edc

R1

Null idc

A

Eac

R1

B

R1

R1 Eadj

Null idc

A

B

Null iac

Radj Rx

Radj

Cadj

Cell = RB

(a)

CB

(b)

Figure 11.1.1 (a) A Wheatstone bridge for determining an unknown resistance, Rx . (b) An impedance bridge for determining the series equivalent of the cell, RB and C B .

11.1 A Simple Measurement of Cell Impedance

because the resistances do not have the same proportions on both sides; consequently, a current passes between A and B and is registered at the null detector (a galvanometer). Since the upper resistances on the left and on the right are identical, A and B can have the same voltage only when Radj equals Rx exactly. The unknown, Rx , is measured simply by changing Radj until i = 0 between A and B. A measurement like this, based on the principle of null balance, is capable of excellent precision and has a long history of scientific application. The same principle can be applied to measure the impedance of an electrochemical cell (11), as illustrated in Figure 11.1.1b. There are five differences vs. Figure 11.1.1a: 1) The cell is the unknown, occupying the place held earlier by Rx . 2) A purely sinusoidal ac voltage of small amplitude (perhaps 5 mV) is applied across the top and bottom vertices, instead of the dc voltage previously used. 3) The adjustment arm of the bridge contains a variable capacitance, C adj , in addition to the variable resistance, Radj . 4) There are two null detectors, one for ac current and another for dc current. 5) There is a variable voltage in the dc null detection path. While this impedance bridge is more complex than the Wheatstone bridge, the operational idea is the same. We seek a complete null-current condition between vertices A and B, and it is achieved when Radj and C adj are adjusted to the values that, in series combination, behave exactly like the cell at the frequency employed.1 The cell behaves toward the ac signal exactly like a resistor–capacitor combination RB and C B , and the bridge becomes balanced when Radj and C adj are adjusted to these values. The very fact that balance is achieved provides palpable confirmation that the cell behaves exactly as an identifiable resistance–capacitance combination at the frequency of measurement. One can change the frequency of the ac signal and balance the bridge at a different pair of RB and C B values. In this way, the impedance spectrum of the cell, Z(𝜔), becomes defined. The two-electrode cell usually would contain a solution with both forms of a redox couple, so that an equilibrium potential, Eeq , is defined at the working electrode. For example, one might use 1 mM Eu2+ and 1 mM Eu3+ in 1 M NaClO4 . A gold disk might be employed as the working electrode, and it might be paired with a nonpolarizable reference such as Ag/AgCl, which would also act as the counter electrode. In this experiment, the mean potential of the working electrode is Eeq , as determined by the bulk ratio of oxidized and reduced forms of the couple. Measurements can be made at other potentials by preparing solutions with different concentration ratios. The approach just described is called the faradaic impedance method, which was important in the history of electrochemistry for providing much precise information about heterogeneous kinetics and double-layer structure (11). In fact, it is the root of contemporary EIS. While automated EIS systems with much greater capabilities have now fully supplanted this method, it still furnishes the simplest illustration that a real cell does behave under defined conditions like an identifiable equivalent circuit made up of RB and C B in series. To interpret an impedance spectrum and to translate it into useful quantitative parameters, one must have a theory linking Z(𝜔) to fundamental electrochemical processes. We now proceed in that direction, beginning with a refresher on the behavior of ac circuits. 1 The cell is a voltage source and its voltage must also be nulled for proper operation of the bridge. This is the purpose of the variable voltage source and the dc null detector. This issue need not distract us now.

445

446

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

11.2 Brief Review of ac Circuits A purely sinusoidal voltage can be expressed as e = E sin 𝜔t

(11.2.1)

where 𝜔 is the angular frequency, 2𝜋f , with f being the conventional frequency in Hz. It is ̇ as pictured convenient to think of this voltage as a rotating vector (or phasor) quantity, E, in Figure 11.2.1. Its length is the amplitude, E, and its angular frequency of rotation is 𝜔. The observed voltage at any time, e, is the component of the phasor projected on some particular axis (normally that at 90∘ ). One often wishes to consider the relationship between two sinusoidal signals of the same frequency, such as the current, i, and the voltage, e. Each is represented as a separate phasor, ̇ both rotating at 𝜔 (Figure 11.2.2). They generally are not in-phase; thus, their phasors İ or E, ̇ is taken as a reference signal, and 𝜙 are separated by a phase angle, 𝜙. One of them, usually E, is measured with respect to it. In the figure, the current lags the voltage. It can be expressed generally as i = I sin(𝜔t + 𝜙)

(11.2.2)

where 𝜙 is a signed quantity, negative in this case. The relationship between two phasors at the same frequency remains constant as they rotate; hence, the phase angle is constant. Consequently, we can usually drop the references to rotation in the phasor diagrams and study the relationships between phasors simply by plotting them as vectors having a common origin and separated by the appropriate angles. Rotation at ω π/2

E •

E π

e

0

ωt→

π

2π

π/2

0

3π/2

–E

–π/2

Figure 11.2.1 Phasor diagram for an alternating voltage, e = E sin 𝜔t. ω i

π/2

E π

π–ϕ

e

•

ϕ •

I

0

e or i

π

ωt→ 2π

0 –ϕ

–π/2

Figure 11.2.2 Phasor diagram showing the relationship between current and voltage signals at frequency 𝜔.

11.2 Brief Review of ac Circuits

Let us apply these concepts to the analysis of some simple circuits. Consider first a pure resistance, R, across which a sinusoidal voltage, e = E sin 𝜔t, is applied. Since Ohm’s law always holds, the current is (E/R) sin 𝜔t or, in phasor notation, Ė İ = R ̇ Ė = IR

(11.2.3) (11.2.4)

The phase angle is zero, and the vector diagram is that of Figure 11.2.3. Suppose we now substitute a pure capacitance, C, for the resistor. The fundamental relation of interest is then q = Ce, or i = C(de/dt); thus, i = 𝜔CE cos 𝜔t ( ) 𝜋 E sin 𝜔t + i= XC 2

(11.2.5) (11.2.6)

where X C is the capacitive reactance, 1/𝜔C. The phase angle is 𝜋/2, and the current leads the voltage, as shown in Figure 11.2.4. Since the vector diagram has now expanded to a plane, it is convenient to represent phasors in terms of complex notation (Section A.5). Components along the ordinate are assigned as imaginary and √ are multiplied by j = −1. Components along the abscissa are real. Introducing complex notation here is a bookkeeping measure to help keep the vector components straight. We handle them mathematically as “real” or “imaginary,” but both types are real in the sense of being measurable by phase angle. In circuit analysis, it turns out to be advantageous to plot the current i e

•

•

e or i

I

E

ωt→

π

2π

0

Figure 11.2.3 Relationship between the voltage across a resistor and the current through the resistor. i e

•

I

•

•

E = –jXCI

e or i

π

ωt→ 2π

0

Figure 11.2.4 Relationship between an alternating voltage across a capacitor and the alternating current through the capacitor.

447

448

11 Electrochemical Impedance Spectroscopy and ac Voltammetry •

•

•

ER = RI

I

R ϕ

ϕ •

•

•

•

E = ZI

EC = –jXC I

Z

–j XC

(a)

(b)

Figure 11.2.5 (a) Phasor diagram showing the relationship between the current and the voltages in a series RC ̇ while Ė and Ė are voltage components across the network. The voltage across the whole network is E, R C resistance and the capacitance. (b) An impedance vector diagram derived from the phasor diagram in (a).

phasor along the abscissa, as shown in Figure 11.2.5, even though the current’s phase angle is measured experimentally with respect to the voltage. If that is done, it is clear that Ė = −jX C İ

(11.2.7)

This relation must hold regardless of where İ is plotted with respect to the abscissa, since only the relative positioning of Ė and İ is significant. A comparison of (11.2.4) and (11.2.7) shows that X C must carry dimensions of resistance, but, unlike R, its magnitude falls with increasing frequency. ̇ is applied across Now consider a resistance, R, and a capacitance, C, in series. A voltage, E, them, and it must always equal the sum of the individual voltage drops; thus, Ė = Ė R + Ė C ̇ − jX ) Ė = I(R C Ė = İ Z

(11.2.8) (11.2.9) (11.2.10)

In this way, we find that the voltage is linked to the current through a vector Z = R − jX C called the impedance. Figure 11.2.5 shows the relationships between these various quantities. In general, the impedance can be represented as2 Z = ZRe + jZIm

(11.2.11)

where ZRe and ZIm are the real and imaginary parts. For the example here, ZRe = R and ZIm = − X C = − 1/𝜔C. The magnitude of Z, written |Z| or Z, is given by |Z|2 = R2 + XC2 = (ZRe )2 + (ZIm )2

(11.2.12)

and the phase angle, 𝜙, is tan 𝜙 = −ZIm ∕ZRe = XC ∕R = 1∕𝜔RC

(11.2.13)

The impedance is a generalized resistance, and (11.2.10) is a generalized version of Ohm’s law. It embodies both (11.2.4) and (11.2.7) as special cases. The phase angle expresses the balance between the capacitive and resistive components in the series circuit. For a pure resistance, 𝜙 = 0; for a pure capacitance, 𝜙 = 𝜋/2; and for series combinations, intermediate phase angles are observed. 2 In many treatments (and in the earlier editions of this book), the definition of impedance is taken as Z = ZRe − jZIm , but in this edition, we convert to the more common practice by using the definition in (11.2.11). The literature also features varied symbols for the real and imaginary components of impedance. Other authors may use Z′ or Zr instead of ZRe and Z′′ or Zj instead of ZIm .

11.2 Brief Review of ac Circuits

The variation of impedance with frequency is often of interest and can be displayed in different ways. In a Bode plot, both log|Z| and 𝜙 are plotted against log 𝜔. An alternative representation, a Nyquist plot (or an impedance-plane plot), displays −ZIm vs. ZRe for different values of 𝜔.3 Plots for the series RC circuit are shown in Figures 11.2.6 and 11.2.7. Similar plots for a parallel RC circuit are shown in Figures 11.2.8 and 11.2.9. More complex circuits can be analyzed by combining impedances according to rules analogous to those applicable to resistors. For impedances in series, the overall impedance is the sum of the individual values (expressed as complex vectors). For impedances in parallel, the reciprocal of the overall impedance is the sum of the reciprocals of the individual vectors. Figure 11.2.10 shows a simple application. Sometimes, it is advantageous to analyze ac circuits in terms of the admittance, Y, which is the inverse impedance, 1/Z, and, therefore, represents a kind of conductance. The generalized ̇ These concepts are useful in the form of Ohm’s law, (11.2.10), can then be rewritten as İ = EY. 80

8

60

6

ϕ/deg

log |Z|/Ω

–3

4

40

2

20

–1

1 3 log ω/s–1 (a)

5

7

–3

–1

1 3 log ω/s–1 (b)

5

7

Figure 11.2.6 Bode plots of (a) impedance and (b) phase angle (i vs. e) for a series RC circuit with R = 100 Ω and C = 1 μF.

0.01 s–1

100 80

ω increasing

60 –ZIm/MΩ 40 20 ω→∞

0.1 s–1

0 0

100 ZRe/Ω

200

Figure 11.2.7 Nyquist plot for a series RC circuit with R = 100 Ω and C = 1 μF. Numbers by points are corresponding values of 𝜔. 3 In electrical engineering, a Nyquist plot is ZIm vs. ZRe , but in EIS, the vertical axis is usually negated. This is because an electrochemical ZIm is almost always capacitive and, therefore, negative. By using the indicated convention, Nyquist plots in EIS lie in the first quadrant.

449

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

3 90

log |Z|/Ω

450

ϕ/deg

1

–3

–1

1

3

5

7

70 50 30

–1

10

–2

–3

log ω/s–1

–1

1 3 log ω/s–1

5

7

Figure 11.2.8 Bode plots for a parallel RC circuit with R = 100 Ω and C = 1 μF.

50 104 40

30 –ZIm

∞

20

10

105

ω 0

103

ω→∞

0 0

50

100

ZRe

Figure 11.2.9 Nyquist plot for a parallel RC circuit with R = 100 Ω and C = 1 μF. Numbers by points are corresponding values of 𝜔.

Z1

Z2 Z3

=

Zp

Z1 Zp =

Z2 Z3 Z2 + Z3

=

ZT Z2 Z3 ZT = Z1 + Z2 + Z3

Figure 11.2.10 Calculation of a total impedance from component impedances.

analysis of parallel circuits, because the overall admittance of parallel elements is the sum of the individual admittances.

11.3 Equivalent Circuits of a Cell In Section 11.1, we saw that a whole cell responds as a series combination of RB and C B to an ac signal at a given frequency, 𝜔. This is the simplest equivalent circuit of the cell. Unfortunately, it is too comprehensive. It uses just two quantities to encompass everything affecting the ac current in the cell, so it rarely supports a focus on any specific aspect of cell performance. We need to dissect RB and C B into physically and chemically identifiable elements; hence, we

11.3 Equivalent Circuits of a Cell

will begin to define more elaborate circuits. All will be equivalent circuits in that they behave as the cell (and as RB and C B ), but sometimes just in a range of frequencies. Different electrochemical situations require different elaborated forms of RB and C B . This is true because the factors controlling the ac response vary broadly in real systems. Sometimes mass-transfer controls, sometimes electrode kinetics. Sometimes both. At other times, the behavior of an electroactive layer on the electrode surface is important, or perhaps the kinetics of a process in solution linked to the electrode reaction. In the presentation below, the term functional equivalent circuit is used to identify a specific derivative of the general equivalent circuit RB and C B that supports the interpretation of EIS data in a particular situation. It is literally an electrical expression of the controlling processes occurring in the cell for a defined range of frequencies, and it features electrical components that are identified quantitatively with those processes. Functional equivalent circuits take many forms in EIS, but we will focus on just a few, suitable for the more common patterns of behavior. 11.3.1

The Randles Equivalent Circuit

For the paradigm that we have usually considered (O + ne ⇌ R, not necessarily reversible, but with both O and R being unabsorbed, dissolved, and stable), the Randles equivalent circuit (Figure 11.3.1a) is a common starting point (17–20). It is an elaboration of RB and C B , so it is a general equivalent circuit, meant to be describe the system for all frequencies. Parallel elements are introduced because the total current through the working interface is the sum of distinct contributions, if and ic , from the faradaic process and double-layer charging, respectively. The double-layer capacitance is nearly ideal;4 hence, it is represented in the equivalent circuit by the element C d . The faradaic process cannot be represented by simple linear elements like, R and C, whose values are independent of frequency. Instead, it is represented as a general impedance, Zf . All current must pass through the uncompensated solution resistance; therefore, Ru is a series element in the Randles circuit.5, 6 The faradaic impedance, Zf , characterizes the electrode reaction itself, including the effects of kinetics and mass transfer. It is usually further dissected, and Figure 11.3.1b shows two ic

Figure 11.3.1 (a) Randles equivalent circuit of an electrochemical cell. (b) Subdivision of Z f into Rs and C s or into Rct and Z W .

Ru if + ic

Cd Zf if (a)

Zf

Rs Cs =

Rct =

ZW

(b) 4 A “pure,” or “ideal,” capacitance or resistance provides a linear response to voltage and has a value independent of frequency. In electrochemical systems, Ru and C d are usually treated as ideal elements. Although C d often varies with potential, it behaves essentially linearly toward ac signals varying over just a few mV. At low electrolyte concentrations, both Ru and C d can begin to depend on frequency (i.e., they dsiplay frequency dispersion) (21, 22). 5 In the bridge measurements in Section 11.1, the entire cell resistance is uncompensated. 6 In faradaic impedance measurements (Section 11.1), processes at the counter electrode also contribute. Since they are not usually of interest, the impedance at that interface is intentionally reduced to insignificance by employing a counter electrode of large area relative to the working electrode.

451

452

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

common equivalences. The simpler approach is to take the faradaic impedance as a series resistance, Rs , and a series capacitance, C s .7 An alternative is to separate a kinetic resistance, Rct , the charge-transfer resistance [Section 3.4.3(b)], from another general impedance, ZW , the Warburg impedance, embodying the effects of mass transfer. We will shortly see the basis for this latter dissection. 11.3.2

Interpretation of the Faradaic Impedance

In contrast to Ru and C d , the components of the faradaic impedance are not “pure,” because they change with frequency, 𝜔. Indeed, a chief objective of an EIS experiment focused on electrode kinetics is to discover those functions. Theory is then applied to transform them into chemical information. In this section, we will see how that is done. (a) Electrical Behavior of Rs and Cs in Series

Suppose a sinusoidal current is forced through Zf , expressed as Rs and C s in series (Figure 11.3.1b). The total voltage drop is q E = iRs + (11.3.1) Cs hence, di i dE = Rs + dt dt Cs

(11.3.2)

If the current is i = I sin 𝜔t

(11.3.3)

I dE = Rs I𝜔 cos 𝜔t + sin 𝜔t dt Cs

(11.3.4)

then

This equation is the link we will use to identify Rs and C s in electrochemical terms. It describes the response of the electrode reaction to a sinusoidal current, (11.3.3), in terms of the electrical equivalent. We will next find that same response in terms of the chemical process; then, by comparison, we will learn how Rs and C s depend on concentrations, kinetic parameters, and diffusivities. (b) Properties of the Chemical System

For the standard system, O + ne ⇌ R, with both O and R soluble and stable, we can write

hence,

E = E[i, CO (0, t), CR (0, t)]

(11.3.5)

[ ] [ ] ( ) dC O (0, t) dC R (0, t) dE 𝜕E 𝜕E 𝜕E di = + + dt 𝜕i dt 𝜕CO (0, t) dt 𝜕CR (0, t) dt

(11.3.6)

7 In some treatments, Rs and C s are called the polarization resistance and the pseudocapacity, respectively. However, those names or others like them are applied to other variables in electrochemistry, so we avoid them here. Pseudocapacitance, in particular, is a term frequently used for something entirely different in the battery community (23). Also, elsewhere in this book, Rs symbolizes the solution resistance, Ru + Rc . In this chapter, Rs always refers to the series resistance in Zf .

11.3 Equivalent Circuits of a Cell

or dC (0, t) dC (0, t) dE di = Rct + 𝛽O O + 𝛽R R dt dt dt dt where

( Rct = [

𝜕E 𝜕i

(11.3.7)

) (11.3.8)

CO (0, t),CR (0, t)

] 𝜕E 𝜕CO (0, t) i,C (0, t) [ ] R 𝜕E 𝛽R = 𝜕CR (0, t) i,C (0, t)

𝛽O =

(11.3.9) (11.3.10)

O

Obtaining an expression for dE/dt depends on our ability to evaluate the six factors on the right of (11.3.7). The three parameters Rct , 𝛽 O , and 𝛽 R depend on the kinetic properties of ∗ and C ∗ , can be the electrode reaction. The remaining three factors, the derivatives of i, CO R evaluated generally for a current given by (11.3.3). One of them is trivial: di = I𝜔 cos 𝜔t dt

(11.3.11)

The others are evaluated by considering mass transfer.8 ∗ and Assuming semi-infinite linear diffusion with initial conditions CO (0, t) = CO CR (0, t) = CR∗ , we can use (9.2.5) and (9.2.6) from Section 9.2.1 to obtain:9 C O (0, s) = C R (0, s) =

∗ CO

s CR∗ s

+

−

i(s)

(11.3.12)

1∕2

nFADO s1∕2 i(s)

(11.3.13)

1∕2

nFADR s1∕2

Inversion by convolution gives ∗ + CO (0, t) = CO

CR (0, t) = CR∗ −

1 1∕2 nFADO 𝜋 1∕2

1 1∕2 nFADR 𝜋 1∕2

t

∫0 t

∫0

i(t − u) du u1∕2

(11.3.14)

i(t − u) du u1∕2

(11.3.15)

From (11.3.3), we can substitute for i(t − u); hence, the problem becomes one of evaluating the integral common to both of these relations. 8 The equivalent impedance was analyzed just above in terms of current as it is usually defined for circuit analysis. That is, a positive change in E causes a positive change in i. The electrochemical current convention followed elsewhere in this book denotes cathodic currents as positive; hence, a negative change in E causes a positive change in i. If we adhere to this convention now, confusion will reign when we try to make comparisons between the electrical equivalents and the chemical systems. We must have a common basis for the current. Since the interpretations of impedance measurements are closely linked to electronic circuit analysis, it is advantageous to adopt the electronic convention. For this chapter, we take an anodic current as positive. This expedient will turn out not to cause much trouble, because we rarely follow the instantaneous sign of the current in ac experiments. Instead, we measure the amplitude of iac and its phase angle with respect to Eac . The phase angle would depend on our choice of current convention, but it is advantageous even here to take the electronic custom, because the electronic devices used to measure phase angle are based on it. 9 With recognition that the sign of the current is opposite in this chapter.

453

454

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

We begin with the trigonometric identity: sin 𝜔(t − u) = sin 𝜔t cos 𝜔u − cos 𝜔t sin 𝜔u

(11.3.16)

which implies that t

t t I sin 𝜔(t − u) cos 𝜔u sin 𝜔u du = I sin 𝜔t du − I cos 𝜔t du (11.3.17) ∫0 ∫0 u1∕2 ∫0 u1∕2 u1∕2 ∗ and C ∗ , and after a few Before the current is turned on, the surface concentrations are CO R cycles we can expect them to reach a steady condition in which they cycle repeatedly through the same pattern. We can be sure of this point because no net electrolysis takes place in any full cycle of current flow. Our interest is not in the transition from the initial condition to steady state, but in the steady state itself. The two integrals on the right side of (11.3.17) embody the transition period. Because u1/2 appears in their denominators, the integrands are appreciable only at short times. After a few cycles, each integral must reach a constant value characteristic of the steady state. We can obtain it by letting the integration limits go to infinity: ∞ ∞ I sin 𝜔(t − u) cos 𝜔u sin 𝜔u du = I sin 𝜔t du − I cos 𝜔t du 1∕2 1∕2 ∫Steady ∫ ∫ u u u1∕2 0 0

(11.3.18)

state

It is easy to show that both integrals on the right side of (11.3.18) are equal to (𝜋/2𝜔)1/2 ; hence, we have by substitution into (11.3.14) and (11.3.15) I ∗ + CO (0, t) = CO (sin 𝜔t − cos 𝜔t) (11.3.19) nFA(2DO 𝜔)1∕2 I CR (0, t) = CR∗ − (sin 𝜔t − cos 𝜔t) (11.3.20) nFA(2DR 𝜔)1∕2 Now we can evaluate the derivatives of the surface concentrations as required above:10 ( )1∕2 dC O (0, t) I 𝜔 = (sin 𝜔t + cos 𝜔t) (11.3.21) dt nFA 2DO ( )1∕2 dC R (0, t) I 𝜔 =− (sin 𝜔t + cos 𝜔t) (11.3.22) dt nFA 2DR By substitution of (11.3.11), (11.3.21), and (11.3.22) into (11.3.7), we obtain ( ) dE 𝜎 = Rct + I𝜔 cos 𝜔t + I𝜎𝜔1∕2 sin 𝜔t dt 𝜔1∕2

(11.3.23)

where 𝜎=

⎛ 𝛽 𝛽R ⎞ 1 O √ ⎜ 1∕2 − 1∕2 ⎟ nFA 2 ⎜⎝ DO DR ⎟⎠

(11.3.24)

(c) Identification of Rs and Cs

To identify Rs and C s , we just need to compare the two response equations, (11.3.4) and (11.3.23). They both describe the behavior of the electrode reaction toward the current, 10 In general, we ought to consider the current as i = idc + I sin 𝜔t, where idc is steady or varies only slowly with time. However, we are interested now in derivatives of surface concentrations, and they will be dominated by the higherfrequency ac signal. Relations (11.3.21) and (11.3.22) will still apply to a good approximation. This is a mathematical manifestation of the way in which the ac part of the experiment can usually be uncoupled from the dc part.

11.3 Equivalent Circuits of a Cell

(11.3.3)—the first in electrical terms and the second in chemical terms. By inspection, we see that Rs = Rct + 𝜎∕𝜔1∕2

(11.3.25)

1 𝜎𝜔1∕2

(11.3.26)

Cs =

The complete evaluation of Rs and C s depends on finding relations for Rct , 𝛽 O , and 𝛽 R . We will see below that Rct is primarily determined by the heterogeneous charge-transfer kinetics, and we have already observed above that the terms 𝜎/𝜔1/2 and 1/𝜎𝜔1/2 come from mass-transfer effects. These points underlie the division of the faradaic impedance into the charge-transfer resistance, Rct , and the Warburg impedance, ZW , as shown in Figure 11.3.1b. Equations 11.3.25 and 11.3.26 demonstrate that ZW can be regarded as a frequency-dependent resistance, RW = 𝜎/𝜔1/2 , in series with a frequency-dependent capacitance, C W = C s = 1/𝜎𝜔1/2 . Thus, the total faradaic impedance, Zf , can be written Zf = Rct + RW − j∕(𝜔CW ) = Rct + [𝜎𝜔−1∕2 − j(𝜎𝜔−1∕2 )]

(11.3.27)

with the term in brackets representing the Warburg impedance. 11.3.3

Behavior and Uses of the Faradaic Impedance

(a) Obtaining Kinetic Parameters

Conceptually, impedance measurements are made with the working electrode’s mean potential at equilibrium. Since the amplitude of the sinusoidal perturbation is small, the resulting departures from equilibrium are always small, and we can use the linearized i − 𝜂 characteristic to describe the electrical response. For a one-step, one-electron process, the linearized relationship is (3.4.31), which can be rewritten in terms of the electronic current convention as ] [ RT CO (0, t) CR (0, t) i 𝜂= − + (11.3.28) ∗ F i0 CO CR∗ Hence, RT Fi0

(11.3.29)

RT FC ∗O

(11.3.30)

Rct = 𝛽O =

𝛽R = −

RT FC ∗R

and now we see that 1 RT Rs − = Rct = 𝜔Cs Fi0

(11.3.31)

(11.3.32)

so that the exchange current, and, therefore, k 0 , can be evaluated when Rs and C s are known. Impedance measurements allow a precise definition of these electrical equivalents; thus, they can yield kinetic data of high quality. Equation (11.3.32) shows that one can, in principle, evaluate i0 from data taken at a single frequency. However, doing so is unwise, because one has no experimental assurance that the

455

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

Rs Resistance or reactance/Ω

456

Figure 11.3.2 Dependence of Rs and 1/𝜔C s on 𝜔−1/2 .

1/ωCs

Rct

Slope = σ

ω–1/2/s1/2

equivalent circuit faithfully represents the performance of the system. The best way to check for agreement is to examine the frequency dependence of the impedance. For example, (11.3.25) and (11.3.26) predict that Rs and 1/𝜔C s should both be linear with 𝜔−1/2 and should have a common slope, 𝜎, quantitatively predictable from the constants of the experiment: 𝜎=

⎛ ⎞ 1 RT ⎜ 1 ⎟ + √ 1∕2 ∗ 1∕2 F 2 A 2 ⎜⎝ DO CO DR CR∗ ⎟⎠

(11.3.33)

Figure 11.3.2 shows these relationships. The plot of Rs should have an intercept, Rct , from which i0 can be evaluated. Extrapolation to the intercept is equivalent to estimating the system’s performance at infinite frequency. The Warburg impedance drops out at high frequencies, because the time scale becomes so short that diffusion cannot manifest itself as a factor influencing the current. Since the surface concentrations never change greatly from the mean values [see (11.3.19) and (11.3.20)], charge-transfer kinetics alone dictate the current. If the linear behavior typified in Figure 11.3.2 is not observed, the electrode process is not as simple as we assume here, and a more complex situation must be considered. The availability of this kind of check for internal consistency is an important asset of impedance methods. Section 11.4 presents much more detail. (b) Reversible Electrode Reactions

When charge-transfer kinetics are very facile, i0 → ∞; hence, Rct → 0. Thus, Rs → 𝜎/𝜔1/2 . The corresponding impedance plot is shown in Figure 11.3.3a. Since the resistance and the capacitive reactance are exactly equal, the magnitude of the faradaic impedance is ( )1∕2 2 Zf = 𝜎 (11.3.34) 𝜔 which is the magnitude of the Warburg impedance alone. Since this is a mass-transfer impedance applicable to any electrode reaction, it is a minimum impedance. If kinetics are observable, another factor, Rct , contributes, and Zf must be greater, as Figure 11.3.4a depicts. Thus, the amplitude of the sinusoidal current flowing in response to a

11.3 Equivalent Circuits of a Cell

Rs = σ/ω1/2

•

I

1/ωCs = σ/ω1/2

ϕ = 45°

ϕ = 45° •

Eac

Zf = ZW

(a)

(b)

Figure 11.3.3 (a) Vector diagram showing the components of the faradaic impedance for a reversible system. (b) Phase relationship between the ac current and the ac component of potential. σ/ω1/2

Rct

•

I

ϕ < 45°

ϕ < 45°

σ/ω1/2

•

Eac |Zf| > |ZW|

(a)

(b)

Figure 11.3.4 (a) Vector diagram showing the effect of Rct on the impedance. (b) Phase relationship between İ and Ė for a system with significant R . ac

ct

given excitation signal, Ė ac , is maximal for a reversible system and decreases correspondingly for more sluggish kinetics. If the heterogeneous redox process is very immobile, Rct and Zf are so large that there is only a very small ac component to the current, and the limit of detection sets the lower bound on rate constants that can be measured. Section 11.4.2(d) presents more detail about practical ranges. (c) Effect of Mean Potential ∗ ∕C ∗ , which one can It is also important to understand the effect of the concentration ratio, CO R change experimentally to vary the mean potential for a series of impedance measurements. ∗ ∕C ∗ imply that one of the concentrations is small; hence, 𝜎 Both large and small values of CO R and Zf must be large. In either case, the ac response is small, because the supply of one reagent is insufficient to permit a high reaction rate for the cyclic, reversible electrode process that causes the ac current. Only when both electroreactants are present at comparable concentrations can we expect appreciable ac current. By this simple reasoning, we can expect Zf to have a minimum ′ near E0 , and the fact is easily shown mathematically (Section 11.5.1). ′ The important point now is that impedance measurements are most easily made near E0 ′ and gradually become more difficult as one moves away from E0 either positively or negatively. This effect presages the shape of the ac voltammetric response, which will be derived in Section 11.5.1. It also is an important consideration in selecting the dc potential used in EIS (Section 11.4.1).

457

458

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

(d) Phase Angle

The phase angle, 𝜙, between the current phasor, İ ac , and the potential, Ė ac is also of interest. Since İ ac lies along Rs , while Ė ac lies along Zf (Figure 11.3.3), 𝜙 is 𝜙 = tan−1

𝜎∕𝜔1∕2 1 = tan−1 𝜔Rs Cs Rct + 𝜎∕𝜔1∕2

(11.3.35)

For the reversible case, Rct = 0; hence, 𝜙 = 𝜋/4 or 45∘ . A quasireversible system shows Rct > 0; hence, 𝜙 < 𝜋/4. However, 𝜙 must always be greater than zero, unless Rct → ∞; but then the reaction would be so sluggish that little ac response would be seen. This sensitivity of 𝜙 to kinetics suggests that Rct might be extracted from the phase angle. It can be, and it often is, using ac voltammetry (Section 11.5.2). (e) Multistep Electrode Reactions

Even though this section has been developed with the assumption that the electrode reaction is a one-step, one-electron process, many of the conclusions also apply for chemically reversible multistep mechanisms. The nernstian limit is still described by (11.3.34) and Figure 11.3.3, but with 𝜎 given by 𝜎=

⎛ ⎞ 1 1 ⎟ + √ ⎜ 1∕2 1∕2 ∗ n2 F 2 A 2 ⎜⎝ DO CO DR CR∗ ⎟⎠ RT

(11.3.36)

When charge-transfer kinetics manifest themselves in a chemically reversible n-electron system, they follow the pattern discussed in relation to Figure 11.3.4. For a quasireversible multistep mechanism, Rct is Rct =

RT nFi0

(11.3.37)

See Section 3.7.5(b) for more about the interpretation of i0 in such a system.

11.4 Electrochemical Impedance Spectroscopy EIS has become the most important impedance-based technique because it offers a powerful combination of broad applicability, diagnostic range, and precision (1–11). One can use it to probe the behavior of practically any electrochemical system over wide ranges of timescale and potential, then one can display results in comprehensive ways, offering diagnostic value. Once a system is understood in behavioral terms, it is often possible to focus EIS with respect to time scale and potential to obtain precise measurements of parameters. As an example of the comprehensive presentation of results, Figure 11.4.1 provides a set of impedance spectra recorded as the potential of a zinc working electrode in KOH solution was changed stepwise over a large range. The spectra are in the form of Nyquist plots (or impedance-plane plots) of −ZIm vs. ZRe , which is a common format. We will learn below how to read these plots. 11.4.1

Conditions of Measurement

EIS instrumentation (Section 11.8) typically is built around a three-electrode potentiostat; therefore, the investigator has general control of the working-electrode potential. It is common

11.4 Electrochemical Impedance Spectroscopy

500

–ZIm/Ω

400 300 200 100 0 400 300 ZR / 200 e Ω

100 0

–1800

–1200 –1400 l C –1600 g . Ag/A E/V vs

–1000

Figure 11.4.1 Impedance spectra (Nyquist plots) obtained for a zinc electrode in 0.01 M KOH. The electrode potential was changed incrementally along the scale shown at right (negative to positive), and the impedance spectrum was recorded while the potential was held constant for 200 ms. The frequency range was 5 Hz to 25 kHz. For each curve, the high-frequency limit is at the front of the diagram, along the potential axis, and the low-frequency limit is at the far end of the curve, toward the left wall of the diagram. In the potential range examined, the zinc electrode begins to oxidize, largely to Zn(OH)2− , but also with formation of a film of 4 ZnO/Zn(OH)2 . The impedance spectra show the influence of the film on the electrode kinetics. [Reprinted with permission from Ko and Park (24). © 2012, American Chemical Society.]

to examine an electrode at an arbitrary Edc , and even to change Edc stepwise over a range, as was done for Figure 11.4.1. However, it is also common to carry out EIS on a system at open circuit. In Section 11.1, we discussed an example in which the impedance of a whole two-electrode cell was measured. Many EIS studies are carried out similarly on entire electrochemical devices, such as batteries, essentially at open circuit. A potentiostatic EIS system can handle this kind of measurement by having the reference- and counter-electrode leads connected jointly to one terminal of the device and the working-electrode lead to the other terminal. The open-circuit potential of the device is measured, and the dc potential, Edc , is set to that value. The impedance spectrum is then recorded. In this situation, the value of Ru is the entire solution resistance between the working and counter electrodes (plus any resistance in the electrodes or connections internal to the system). If the focus is only on the working electrode, the three-electrode capability is used, and Edc is chosen by the investigator. An impedance spectrum can be obtained for any potential, even if a “dc current” (i.e., a slowly varying current) also flows. The choice of Edc for an EIS spectrum often falls in one of three categories: 1) If the working electrode is poised (i.e., both members of a redox couple of interest are present), it becomes natural to work at the equilibrium potential. Potentiostatic experiments done in this way are essentially equivalent to the faradaic impedance measurements described in Section 11.1. 2) If the working electrode has a potential range in which it resembles an ideally polarizable electrode (IPE; Section 1.6.1), the working electrode might be set in this range (ideally at the open-circuit potential). Faradaic kinetics make little to no contribution. The value of Edc is chosen in this mode when the investigator wishes to concentrate on cell resistance or interfacial capacitance.

459

460

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

3) If there is an electrode process (e.g., O + ne ⇌ R) that is reversible on a long time scale compared to the ac time scale, then one can use Edc to control the mean surface concentrations, C O (0, t)m and C R (0, t)m . They adhere to the nernstian relation: CO (0, t)m CR (0, t)m

= 𝜃m = exp

[

′ nF (Edc − E0 ) RT

] (11.4.1)

∗ , but species R is absent, one can combine this If species O is present in the bulk at CO relationship with (7.11.1) to obtain ( ) 𝜉𝜃m ∗ CO (0, t)m = CO (11.4.2a) 1 + 𝜉𝜃m

( CR (0, t)m =

∗ CO

𝜉 1 + 𝜉𝜃m

) (11.4.2b)

where 𝜉 = (DO /DR )1/2 . If Edc is steady with time, a diffusion layer becomes established in which these surface concentrations extend further into the solution than any distance that will be affected by the ac signal. Consequently, C O (0, t)m and C R (0, t)m act as “effective bulk concentrations” with respect to that signal, much as we saw in Section 8.4.2 for DPV. This approach is functionally equivalent to carrying out faradaic impedance measurements using poised systems with different ratios of O and R, as described in Section 11.1. If the experimental goal is to explore the impedance characteristics of the O/R electrode ′ reaction, it is essential to set Edc near E0 for that process. As shown in Section 11.3.3(c), the electrode reaction can generate an informative ac signal only when both redox forms are comparably present at the electrode surface. When a three-electrode system is used, the impedance measurement carries no information about the counter electrode or about compensated solution resistance. The value of Ru is exactly as we have previously understood for three-electrode cells (Section 1.5.4). 11.4.2

A System with Simple Faradaic Kinetics

Once again, let us focus on the one-step, one-electron reaction, kf

−→ O+e← − R

(11.4.3)

kb

where O and R are dissolved, stable, and unadsorbed. These conditions are the same as in Section 11.3.2; therefore, the Randles equivalent circuits of Figure 11.3.1 apply, as do all results of Section 11.3.2 relating to Zf , Rs , and C s . We will first work with the general equivalent circuit in Figure 11.4.2a. The real part of Z can be derived according to the methods of Section 11.2 as ZRe = Ru +

Rs

A2 + B2 where A = (C d /C s ) + 1 and B = 𝜔Rs C d . Likewise, −ZIm =

B2 ∕𝜔Cd + A∕𝜔Cs A2 + B2

(11.4.4)

(11.4.5)

11.4 Electrochemical Impedance Spectroscopy

Cd

Cd

Ru

Ru Rs

Cs

Rct ZW

(a) All ω

(b) All ω

low ω XCd >> Rct + ZW

high ω ZW Rct + ZW Ru

Ru

(b) –ZIm/Ω

low ω ZW → RL + CL

Rct

Rct ZW

(c)

Diffusion control

Charge saturation

ω

Sl

op

e

=

1

Kinetic control

∞←

468

Ru

Ru + Rct

Ru + Rct + RL

ZRe/Ω

Figure 11.4.9 Expected Nyquist plot for an electroactive layer on an electrode surface together with functional equivalent circuits for different segments: (a) Randles equivalent circuit expressed with the Warburg impedance (applicable at all 𝜔). (b) Functional equivalent circuit for high 𝜔 (kinetic control). (c) Functional equivalent circuit for mid-𝜔 (diffusion control). (d) Functional equivalent circuit for low 𝜔 (charge saturation). Labeled arrows show how the Randles equivalent circuit evolves to the functional forms (b), (c), and (d) as 𝜔 changes. [Adapted from Hunter, Tyler, Smyrl, and White (25). Reprinted with permission of the publisher, The Electrochemical Society, Inc.]

11.4 Electrochemical Impedance Spectroscopy

The theoretical treatment of the Warburg impedance (25, 29) identifies dC −Fc+ CL = nFAl dE ( ) dC −Fc+ −1 l RL = 3nFADE dE

(11.4.22) (11.4.23)

where dC −Fc+ ∕dE describes the incremental change in concentration caused by an incremental change in potential. In EIS, Eac remains small, usually less than 5 mV; thus, the relationship between dE and dC −Fc+ can be taken as linear, so that dC −Fc+ ∕dE is constant. Thus, all factors in (11.4.22) and (11.4.23) are constants of the system. The origin of (11.4.22) is easy to understand. The differential charge passed upon a differential change in potential, dE, is dQ = nFAl ⋅ dC. Dividing by the potential change and recognizing that dQ/dE = C L , one arrives at (11.4.22). Experimentally, RL is available from the intercept of the Nyquist plot in the charge saturation −1 vs. 𝜔 for data in the charge saturation region. One can obtain C L as the slope of a plot of −ZIm region. The product of these parameters eliminates dC −Fc+ ∕dE to give RL CL =

l2 3DE

(11.4.24)

which can provide DE or l if the other is known. Figure 11.4.10 shows experimental results for a PVFc layer on a Pt electrode (25), in which all three zones can be clearly distinguished. The Nyquist plot in the charge saturation region 300

225

–ZIm/Ω Charge saturation

Diffusion control

150

0.5 Hz

Sl

op e

=

1

Kinetic control

75

5000 Hz 500 Hz

0

0

75

5 Hz

150 ZR/Ω

225

300

Figure 11.4.10 Nyquist plot (0.01–5000 Hz) for a PVFc layer on a Pt electrode in MeCN with 0.1 M TBAP. E dc = 0.4 V vs. SSCE. Surface coverage of pendant ferrocene moieties was 1.06 × 10−7 mol/cm2 . [From Hunter, Tyler, Smyrl, and White (25). Reprinted with permission of the publisher, The Electrochemical Society, Inc.]

469

470

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

departs sharply upward from the diffusion-controlled unit-slope line; however, it is not vertical. The authors ascribed the observed behavior to variability in the thickness of the PVFc layer across the surface of the Pt electrode. 11.4.5

Other Applications

The approaches discussed above can be applied to the EIS of more complicated electrochemical systems, such as those with coupled homogeneous reactions or adsorbed intermediates. Ideally, experimental Nyquist plots are eventually interpreted by comparison with accurate theoretical models based on the relevant processes and the physical structure of the system. However, in the diagnostic phase of investigation, neither the processes nor the structure is fully understood. It may aid early interpretation to represent the system by an equivalent circuit involving different components. Insight can come from this direction; but one must also be on guard. Equivalent circuits are not unique, and one cannot always assign components in an equivalent circuit to specific physical or chemical processes (31). As in other areas of electrochemistry, interpretation is aided by varied experience. EIS is very often used for the characterization of complex structures in which electrode surface roughness and heterogeneity are significant factors. The method has been applied to a broad range of electrochemical systems, including those involved in power sources, corrosion, electrodeposition, polymer films, and semiconductor electrodes. Representative studies can be found in specialized references (1, 3–11) or in symposium proceedings.

11.5 ac Voltammetry A variation on the faradaic impedance method is ac voltammetry (12–16). A three-electrode cell is used in the conventional manner, and the potential program imposed on the working electrode is a dc value, Edc , which is stepped or scanned slowly with time, plus a sinusoidal component, Eac , of perhaps 5-mV amplitude. The measured responses are the magnitude of iac at the frequency of Eac and its phase angle with respect to Eac .15 A typical experimental arrangement is shown schematically in Figure 11.5.1. The role of Edc is to set the mean surface concentrations of O and R, C O (0, t)m and C R (0, t)m (Section 11.4.1). In general, Edc differs from the equilibrium value, Eeq ; hence, C O (0, t)m and ∗ and C ∗ , and a diffusion layer exists. Because E is effectively steady C R (0, t)m differ from CO dc R on the ac time scale, this layer soon becomes much thicker than the zone affected by the rapid perturbations from Eac . Thus, C O (0, t)m and C R (0, t)m become effective bulk concentrations for the ac part of the experiment. One usually starts with a solution containing only one redox form, for example Eu3+ , and obtains voltammograms of ac current amplitude and phase angle vs. Edc . In effect, these plots represent the faradaic impedance at continuously changing ratios of C O (0, t) and C R (0, t). The chief strength of ac voltammetry is the access it gives to precise quantitative information about electrode processes. 11.5.1

Reversible Systems

Let us consider the ac response at a working electrode immersed in a solution containing initially only species O, which participates in the nernstian process O + ne ⇌ R. The dc potential 15 Alternatively, one could measure the current components in phase with Eac and 90∘ out of phase with Eac . They provide equivalent information.

11.5 ac Voltammetry

Potentiostat Cell

i/V conversion

Phase-sensitive detection ac signal at ω, 2ω, ...

Waveform generation

Low-pass filter E

• ac amplitude + ϕ • ac component at ϕdet • ac voltammogram

dc signal

frequency ω

• dc voltammogram

t

Figure 11.5.1 Schematic diagram of apparatus for ac voltammetry. The applied signal is a potential sweep with a superimposed ac signal at frequency 𝜔. Filtration and phase-sensitive detection at the output allows separation of the current signal into dc (or slowly changing) and ac (rapidly changing) components. Tuned detection allows separation of ac currents at the fundamental frequency, 𝜔, and at harmonics, 2𝜔, 3𝜔, …. Any of these signals can be reported, as can the ac component at a chosen detection angle, 𝜙det vs. E ac . ′

starts at a value considerably more positive than E0 and is scanned slowly in a negative direction. The dc and ac time scales are such that Edc remains effectively constant for many ac cycles. Since the charge-transfer resistance is negligible, (11.3.36) applies where 𝜎=

⎡ ⎤ 1 1 ⎥ + √ ⎢ 1∕2 1∕2 ⎢ 2 2 n F A 2 ⎣ DO CO (0, t)m DR CR (0, t)m ⎥⎦ RT

(11.5.1)

The mean surface concentrations C O (0, t)m and C R (0, t)m are given by (11.4.2a and 11.4.2b); hence, the faradaic impedance is obtained by substitution into (11.5.1) and then into (11.3.34): ( ) RT 1 Zf = + 2 + 𝜉𝜃m (11.5.2) 1∕2 ∗ 𝜉𝜃m n2 F 2 A𝜔1∕2 DO CO Let us write 𝜉𝜃 m as 𝜉𝜃m = ea

(11.5.3)

where a=

nF (E − E1∕2 ) RT dc ′

(11.5.4) 1∕2

1∕2

and E1∕2 = E0 + (RT∕nF) ln(DR ∕DO ). The parenthesized term in (11.5.2) is e−a + 2 + ea , which is also 4 cosh2 (a/2). Thus, we have ( ) 4RT a (11.5.5) Zf = cosh2 1∕2 ∗ 2 1∕2 2 2 n F A𝜔 DO CO

471

472

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

In Section 11.3.3(d), we saw that the faradaic current for a reversible system leads Ė ac by exactly 45∘ . If Ė ac = ΔE sin 𝜔t, then ) ( ΔE 𝜋 (11.5.6) iac = sin 𝜔t + Zf 4 and the amplitude of this current is 1∕2

∗ ΔE n2 F 2 A𝜔1∕2 DO CO ΔE I= = Zf 4RT cosh2 (a∕2)

(11.5.7)

Figure 11.5.2a is a display of an ac voltammogram defined by this equation. The bell shape arises from the factor cosh−2 (a/2), and it reflects the potential dependence of the impedance, ′ Zf . The maximum occurs at a/2 = 0, which is at Edc = E1/2 , near E0 . As one moves away from that potential, either positively or negatively, the impedance rises sharply, and the current falls off. The physical basis for this behavior is outlined in Section 11.3.3(c). In effect, the current is controlled by the smaller of C O (0, t)m or C R (0, t)m . The peak current at Edc = E1/2 comes easily from (11.5.7), since cosh(0) = 1: 1∕2

Ip =

∗ ΔE n2 F 2 A𝜔1∕2 DO CO

(11.5.8)

4RT

From (11.5.7) and (11.5.8), one can show (Problem 11.7) that the shape of the voltammogram is given by Edc

( )1∕2 ( ) ⎡ Ip − I 1∕2 ⎤ 2RT ⎢ Ip ⎥ = E1∕2 + ln − ⎥ ⎢ I nF I ⎦ ⎣

(11.5.9)

An experimental example is presented in Figure 11.5.3a. 24.0

Figure 11.5.2 Calculated ac voltammograms for one-step, one-electron systems. (a) k0 → ∞; (b) k0 = 1; (c) k0 = 0.1; (d) k0 = 0.01 cm/s. Other parameters: 𝜔 = 2500 s−1 , 𝛼 = 0.500, D = 9 × ∗ = 1.00 × 10−3 M, 10−6 cm2 /s, A = 0.035 cm2 , CO T = 298 K, ΔE = 5.00 mV. [Reprinted from Smith (16), by courtesy of Marcel Dekker, Inc.]

(a)

(b)

16.0 (c) I(ωt)/μA

8.0

(d) 160

80

0 –80 (E – E1/2)/mV

–160

11.5 ac Voltammetry

I

I

25 μA

0.8

0.7 0.6 E/V vs. Ag/AgCl (a)

0.5

–0.650

–0.550 E/V vs. SCE (b)

Figure 11.5.3 Voltammograms of ac current amplitude vs. potential. (a) For 5 × 10−4 M Fe(P2 dtc)3 at a Pt disk in acetone/0.1 M TEAP. ΔE = 5 mV, 𝜔/2𝜋 = 200 Hz. P2 dtc− = piperidyldithiocarbamate. [Reprinted with permission from Bond, O’Halloran, Ruzic, and Smith (32). © 1976, American Chemical Society.] (b) For 3 × 10−3 M Cd2+ at a DME (ac polarogram) in 1.0 M Na2 SO4 . ΔE = 5 mV, 𝜔/2𝜋 = 320 Hz. [Reprinted with permission from Smith (33). © 1963, American Chemical Society.]

According to (11.5.8), I p for a reversible ac voltammogram is directly proportional to n2 , ∗ . There is also an indicated proportionality to ΔE; however, it is limited, because and CO the linearized i − E characteristic underlying the derivation of Zf becomes compromised if ΔE is too large. For linearity within a few percent, ΔE must be less than about 10/n mV. For smaller ΔE, the width of the peak at half height is 90.4/n mV at 25 ∘ C. At larger ΔE, the peak broadens. All results of this section also hold for the DME. Indeed, ac voltammetry was invented as a polarographic method, and is still called ac polarography when applied at the DME or SMDE. Because the growth and fall of drops at the DME is slow compared to the ac part of the experiment, the only significant effect on theory is the need to express the time-dependent area, A, by substitution of (8.1.2) into (11.5.7). The current oscillates as successive drops grow and fall, with the largest current at the end of each drop’s life (Figure 11.5.3b). The envelope of the ac polarogram can be treated by all relations derived above, with A defined at t max . 𝜔1/2 ,

11.5.2

Quasireversible and Irreversible Systems

When heterogeneous kinetics become sluggish enough to be visible, one requires a more elaborate theory of the ac voltammetric response. Even for a one-step, one-electron process, the general case, in which k 0 can adopt any value, is complex (13–16). Here we examine only an important special case, which provides good insight into the effects of interest. That case is where the dc response from a one-step, one-electron system is effectively nernstian, while the ac response is not. This situation is common in real systems, because the time

473

474

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

scales of the two aspects can differ greatly. That is, k 0 can be sufficiently large that C O (0, t)m and C R (0, t)m are kept in the nernstian ratio dictated in Edc , even though it is not large enough to assure negligible charge-transfer resistance toward the much faster ac perturbation. The faradaic impedance in this situation involves both Rct and 𝜎, and the magnitude can be written from (11.3.27): [( )2 ( )2 ]1∕2 𝜎 𝜎 Zf = Rct + + (11.5.10) 𝜔1∕2 𝜔1∕2 The assumption of dc reversibility allows us to use mean surface concentrations given by (11.4.2a and 11.4.2b); thus, we can develop 𝜎 exactly as for the reversible system (Section 11.5.1) to obtain ( ) 4RT a (11.5.11) 𝜎=√ cosh2 1∕2 2 ∗ 2F 2 ADO CO where we have recognized n = 1 and a as defined in (11.5.4). In (11.3.29), Rct , is expressed in terms of the exchange current, i0 . Normally we speak of i0 as an equilibrium property defined by bulk concentrations of O and R according to (3.4.6). Since the mean surface concentrations act like bulk values for the ac process, we can recognize an effective exchange current for the ac perturbation as (i0 )eff = FAk 0 [CO (0, t)m ](1−𝛼) [CR (0, t)m ]𝛼

(11.5.12)

By determining the mean surface concentrations, Edc controls (i0 )eff and, therefore, Rct . A more explicit expression of this dependence is obtained by substitution from (11.4.2a,b) and (11.5.4): ( 𝛽a ) e ∗ 𝜉𝛼 (i0 )eff = FAk 0 CO (11.5.13) 1 + ea where 𝛽 = 1 − 𝛼. Since Rct = RT/F(i0 )eff , we have ( ) RT 1 + ea Rct = e𝛽a F 2 Ak 0 C ∗ 𝜉 𝛼

(11.5.14)

O

Now that Rct and 𝜎 have been written out, let us examine the limiting behavior for high and low frequencies, which can be discerned from (11.5.10). At very low frequencies, Rct is small compared to 𝜎/𝜔1/2 ; hence, the system looks reversible. Everything we found in Section 11.5.1 about the reversible ac response should apply to a quasireversible system at the low-frequency limit. As the frequency is elevated, the heterogeneous kinetics become strained. As 𝜎/𝜔1/2 becomes smaller, Rct grows in relative importance. At the high-frequency limit, it exceeds 𝜎/𝜔1/2 , and Zf approaches Rct . The amplitude of the alternating current then tends toward ∗ ΔE𝜉 𝛼 ( 𝛽a ) F 2 Ak 0 CO ΔE e I= = (11.5.15) Rct RT 1 + ea The ac voltammogram remains bell-shaped; however, unless 𝛼 = 0.5, it is not symmetric, but skewed. These equations, together with those describing the reversible, low-frequency limit, give a good picture of the behavior of the system as 𝜔 becomes elevated. The peak current is at first linear with 𝜔1/2 , manifesting control by the diffusion-based Warburg impedance. The lack of a frequency dependence in (11.5.15) comes about because the current is totally controlled at high

11.5 ac Voltammetry

𝜔 by heterogeneous kinetics. Thus, I is proportional to k 0 at high 𝜔, and it is totally insensitive ∗ hold at all frequencies. to k 0 at low 𝜔. The proportionalities of I with ΔE and CO Kinetic control of I at high frequencies implies that the current must be smaller than for a truly reversible system, as illustrated in Figure 11.5.2. The k 0 values for all curves shown there are sufficiently great that the assumption of dc reversibility holds. One can see that any totally irreversible process (on the ac time scale) will be almost invisible in ac voltammetry. This fact is useful for analytical work (Section 11.7).16 The phase angle of İ ac with respect to Ė ac is of great interest as a source of kinetic information. This point, suggested in Section 11.3.3(d), is rooted in (11.3.35), which we can rewrite as cot 𝜙 = 1 +

Rct 𝜔1∕2

𝜎 Substitution from (11.5.11) and (11.5.14) and rearrangement gives cot 𝜙 = 1 +

𝛽 (2DO D𝛼R 𝜔)1∕2 [

k0

1 e𝛽a (1 + e−a )

(11.5.16)

] (11.5.17)

For a reversible system, k 0 → ∞ and the second term goes to zero; therefore, cot 𝜙 = 1, and 𝜙 is always 𝜋/4, or 45∘ . Quasireversibility is seen when k 0 is small enough for the second term to be significant. The bracketed factor shows that cot 𝜙 depends on the dc potential. It has a maximum at Edc (max cot 𝜙) = E1∕2 +

RT 𝛼 ln F 𝛽

(11.5.18)

The value of Edc (max cot 𝜙) is independent of nearly all experimental variables, including ΔE, ∗ , A, and 𝜔. The difference E (max cot 𝜙) − E CO dc 1/2 provides 𝛼 directly. Figure 11.5.4 shows data for the one-electron reduction of Ti(IV) to Ti(III) in oxalic acid solution (36), where one can see that Edc (max cot 𝜙) is independent of 𝜔, as predicted. From (11.5.17), we see that a plot of cot 𝜙 vs. 𝜔1/2 yields k 0 , once 𝛼 has been obtained from the position of Edc (max cot 𝜙) and the diffusion coefficients have become available from other measurements. Although the plot may be made for any value of Edc , the usual practice is to make it for Edc = E1/2 , where a = 0. Then, (11.5.17) becomes 1∕2

[cot 𝜙]E

1∕2

⎛ D𝛽 D𝛼 ⎞ = 1 + ⎜ O R⎟ ⎜ 2 ⎟ ⎝ ⎠

𝜔1∕2 k0

(11.5.19)

𝛽

If one can take DO = DR = D, then DO D𝛼R = D, and the slope becomes independent of 𝛼. Figure 11.5.5 is an example in which the data from Figure 11.5.4 at Edc = E1/2 = − 0.290 V vs. SCE have been plotted vs. 𝜔1/2 . Quantitative information about heterogeneous charge-transfer kinetics obtained from ac voltammetry often comes from the behavior of cot 𝜙 with potential and frequency, rather than from the heights, shapes, or positions of the peaks. One reason is that many experimental 16 The totally irreversible case does yield an ac current, contrary to the impression one might gain from this line of argument. The current arises from the simple modulation of the dc wave (34, 35). Since the shape of that wave is independent of k 0 (Section 7.4.1), the ac peak height is also independent of k 0 . The peak lies near the half-wave ′ potential of the dc wave; hence, it is shifted substantially from E0 by an amount related to the size of k 0 .

475

476

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

Figure 11.5.4 cot 𝜙 vs. E dc for 3.36 mM TiCl4 in 0.200 M H2 C2 O4 at Hg. ΔE = 5.00 mV, T = 25 ∘ C. Points are experimental; curves are calculated for k0 = 4.6 × 10−2 cm/s, 𝛼 = 0.35, and D = 6.60 × 10−6 cm2 /s, by (11.5.17). [Reprinted with permission from Smith (36). © 1963, American Chemical Society.]

4.60 4.20 1020 Hz

3.80 3.40

626 Hz

cot ϕ 3.00

315 Hz

2.60

153 Hz

2.20 79.1 Hz

1.80

39.6 Hz 19.6 Hz

1.40 10.3 Hz

1.00

80

40

0 –40 (Edc – E1/2)/mV

–80

4.20 3.80 3.40 3.00 cot ϕ 2.60 2.20 1.80 1.40 1.00

0

8

16

24

32

40

48 ω1/2

56

64

72

80

88

96

Figure 11.5.5 cot 𝜙 vs. 𝜔1/2 at E dc = E 1/2 = –0.290 V vs. SCE for the system and conditions of Figure 11.5.4. [Reprinted with permission from Smith (36). © 1963, American Chemical Society.] ∗ and A. variables do not have to be controlled closely or even be known, among them being CO Freedom from knowing A can be a particular advantage. However, the more important reason for evaluating kinetics through cot 𝜙 is that (11.5.17)–(11.5.19) hold for any quasireversible or irreversible system. We have derived them for the situation in which dc reversibility applies; however, they are valid regardless of that condition (15, 16). This fact frees the experimenter from having to achieve special limiting conditions.

11.6 Nonlinear Responses

11.5.3

Cyclic ac Voltammetry

One can perform cyclic ac voltammetry (CACV) simply by adding the reversal scan in Edc (12, 32, 37, 38). The resulting voltammograms have characteristics very similar to those of difference-current voltammograms from cyclic square-wave voltammetry (CSWV; Section 8.5.4) carried out at the same frequency and sweep rate. Both CACV and difference-current CSWV generate their responses from small-amplitude cyclic variations of potential, so they ′ typically yield peaks only near the E0 values for couples with reversible or quasireversible electrode kinetics. In a reversible case, the peaks for both the forward and reverse scans are expected at E1/2 . The relative heights of the forward and reverse peaks in CACV are diagnostic for coupled homogeneous reactions, just as for conventional CV (Chapter 13). In principle, cyclic ac voltammetry retains much of the diagnostic utility of conventional cyclic measurements, but with a response function that facilitates quantitative evaluations. However, the technique has not seen extensive use, perhaps because CSWV has now been extensively developed and offers a greater variety of useful voltammetric output (difference-current voltammograms, forward-sample voltammograms, and reverse-sample voltammograms, all for both the forward sweep and the reverse sweep; Section 8.5.4). The second edition includes a significant discussion of CACV, to which the interested reader is referred.17

11.6 Nonlinear Responses In the theory for EIS and ac voltammetry, we have consistently relied upon the linearized i − 𝜂 relationship, applicable for small overpotentials. The assumption of linearity was made when we invoked (11.3.28), from which we developed all expressions of impedance. Since EIS and ac voltammetry normally involve ac excitation signals of low amplitude, this approach has been valid for our purposes to this point. In a linear system, excitation by a purely sinusoidal Eac at frequency 𝜔 provides a current also of frequency 𝜔—and only of frequency 𝜔—just as we have considered. However, the i − 𝜂 function for an electrode reaction is not actually linear. It is curved over moderate ranges of overpotential, and the effects of curvature can be observed and exploited. Any nonlinear i − E relation gives a distorted response to a purely sinusoidal excitation at frequency 𝜔; therefore, iac is not purely sinusoidal. However, it is periodic, so it can be represented as a superposition (a Fourier synthesis; Section A.6) of signals at the fundamental frequency, 𝜔, and its harmonics 2𝜔, 3𝜔, …. A nonlinear system also produces a dc component upon excitation, even if the latter is purely sinusoidal (Section A.6). This effect is faradaic rectification. If a system is excited by a superposition of two pure sinusoids at frequencies 𝜔1 and 𝜔2 , the resulting current will, of course, have components at 𝜔1 , 2𝜔1 , 3𝜔1 , … and 𝜔2 , 2𝜔2 , 3𝜔2 , … . However, there are also mixing effects, yielding intermodulation responses at the combination frequencies 𝜔1 + 𝜔2 and 𝜔1 − 𝜔2 .18 By tuned detection, any of these signals can be selectively measured and put to use (if they rise above the noise). An advantage common to all techniques based on nonlinear response is comparative freedom from nonfaradaic interference. The double-layer capacitance is generally much more linear than the faradaic impedance; hence, charging currents are largely restricted to the fundamental frequency. 17 Second edition, Section 10.5.4. 18 The sum and difference frequencies are commonly called sidebands when 𝜔1 ≫ 𝜔2 or beat frequencies when 𝜔1 ≈ 𝜔2 . Sidebands are near 𝜔1 , flanking it above and below. Beat frequencies are generally much lower and much higher than 𝜔1 .

477

478

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

11.6.1

Second Harmonic ac Voltammetry

The most common methodology based on nonlinear response is second harmonic ac voltammetry, in which the cell is excited exactly as in ac voltammetry (Figure 11.5.1), but the detection system measures only the current contribution at the second harmonic, 2𝜔.19 The resulting voltammogram is I 2𝜔 vs. Edc . An exact treatment of higher-harmonic voltammetry is straightforward, but lengthy (13, 15, 16). We will follow an intuitive approach that reveals the distinctive features, and for simplicity, we consider only a reversible system in which R is initially absent. The mean surface concentrations, C O (0, t)m and C R (0, t)m , are set by the value of Edc and are given by (11.4.2a,b). In Figure 11.6.1, C R (0, t)m is depicted graphically. The ac response is determined by the way in which Ė ac causes small perturbations in the surface concentrations about the mean values. The ac response at the fundamental frequency is controlled essentially by the linear elements of variation, which are the slopes 𝜕C O (0, t)m /𝜕E and 𝜕C R (0, t)m /𝜕E. The higher harmonics reflect curvature; hence, they are sensitive to the second and higher derivatives. This point alone allows us to predict the general shape of the second harmonic response. Consider potentials E1 , E3 , and E5 in Figure 11.6.1. They have the common feature that the curvature in C O (0, t)m and C R (0, t)m is zero; thus, the second derivatives vs. potential are zero, and there is no second harmonic current. Of course, E1 and E5 lie at extreme values where there is also no fundamental response; but E3 lies at the inflection point, E = E1/2 , where the fundamental response is maximal. The potentials E2 and E4 are at points of maximum curvature; hence, they should be the potentials of peak second-harmonic current. If we detect only the amplitude, I 2𝜔 , then we can expect a double-peaked voltammogram. Since the curvature at E2 is opposite to that at E4 , the second harmonic component must undergo a 180∘ phase shift when Edc passes through the null point at E1/2 . Phase-selective detection of I 2𝜔 at a given phase angle would, therefore, produce a sign inversion at E1/2 . Figure 11.6.2 provides an example (39). A nernstian reaction detected at any phase angle shows positive and negative lobes that are symmetrical around the point of intersection on the potential axis. The dc potential corresponding to this intersection is E1/2 at all phase angles (40).

CO* (DO/DR)½ CO(0,t)m

E1

E2 E3 E4 E/V

E5

Figure 11.6.1 Dependence of the mean surface concentration of species R on the electrode potential. 19 This nomenclature differs from that used in electrical engineering, where the signal at 𝜔 is the fundamental and that at 2𝜔 is the first harmonic. We adhere to the usual electrochemical usage.

11.6 Nonlinear Responses

4 3

80 mV

2 1 40 mV

Iʹ2ω 0 –1 –2 –3 –4

–200

–100

0 (Edc – E½)/mV

100

200

Figure 11.6.2 Second-harmonic phase-selective ac voltammograms at an HMDE for 0.30 mM Ru(NH3 )3+ in 6 0.1 M KCl + 1.0 M HNO3 . 𝜔/2𝜋 = 35.00 Hz. Data points are experimental. Curves are theoretical for large′ . Values of ΔE are amplitude excitation. The ordinate is the in-phase (0∘ ) second-harmonic amplitude, I2𝜔 shown by the curves. Curve shape for small-amplitude excitation (ΔE ≤ 10 mV) is similar. [From Engblom, Myland, Oldham, and Taylor (39), with permission.]

For the reversible case with ΔE ≤ 10 mV, the second-harmonic amplitude tends toward (15, 16) I2𝜔 =

n3 F 3 AC ∗O (2𝜔DO )1∕2 ΔE2 sinh(a∕2) 16R2 T 2 cosh3 (a∕2)

(11.6.1)

∗ , 𝜔1/2 , and where a = (nF/RT)(Edc − E1/2 ). This equation embodies the proportionalities of CO 1∕2

DO that we have come to expect of diffusion-controlled processes. However, I 2𝜔 is proportional to ΔE2 , reflecting the greater importance of nonlinear effects for perturbations of larger magnitude. The two peak potentials are located at Edc = E1/2 ± 34/n mV at 25 ∘ C. Second-harmonic techniques can be useful for analytical purposes and for the quantitative evaluation of heterogeneous kinetic parameters (13, 15, 16, 41). The near absence of capacitive background is helpful. 11.6.2

Large Amplitude ac Voltammetry

As we have seen, most theory and practice of ac voltammetry is based on ac signals with small amplitude, so that linearity can be assumed between potential, current, and surface concentrations. While this practice is greatly simplifying, it is not essential. Theoretical approaches capable of treating much larger ac perturbations (ΔE = 10 − 100 mV) have been developed (42, 43), and experimental methodology has evolved in parallel (12, 39, 44, 45). This domain is known as large-amplitude ac voltammetry. We have already learned that higher harmonics appear in the current because of nonlinearity in the i − E characteristic. Moreover, we have seen that i2𝜔 is expected (at low ΔE) to be proportional to ΔE2 , where the square dependence reflects the influence of greater nonlinearity with

479

80

8

40

4

I/µA 0

i/µA 0

–40 –80

60

15

I/µA 0

I/µA 0

–4 –60 0

10

20 t/s

30

40

–800

–400 0 E/mV vs. Ag/AgCl

(a) 4

2

I/µA 0

I/µA 0

I/µA 0

0

10

20 t/s

30

40

–4

0

10

(e) 0.15

I/µA 0

I/µA 0

0

10

20 t/s

(i)

20 t/s

30

40

(f)

0.3

–0.3

10

30

40

–0.15

0

10

20 t/s

20 t/s

30

40

0

10

(c)

10

–10

–15 0

(b)

–2

30

40

30

40

0.6

I/µA 0

–0.6 0

10

20 t/s

(g)

30

20 t/s

(d)

30

40

0

10

20 t/s

(h)

40

(j)

Figure 11.6.3 Large-amplitude cyclic ac voltammetry of 0.5 mM Ru(NH3 )3+ at glassy carbon in 0.5 M KCl. (a) Total current; (b) dc current (similar to conventional CV, 6 but distorted by faradaic rectification); (c–j) fundamental through 8th harmonic cyclic ac voltammograms. f = 9.54 Hz; ΔE = 80 mV; v = 50 mv/s. Scan starts at 200 mV vs. Ag/AgCl and first moves negatively. In all frames but (b), the graphs record instantaneous ac currents on a time base corresponding to the CV in (b). Individual cycles are unresolved on the timescale used here. The first response in each frame relates to the negative-going E dc scan, and the second, to the positive-going scan. [Adapted with permission from Zhang, Guo, Bond, and Marken (44). © 2004, American Chemical Society.]

11.7 Chemical Analysis by ac Voltammetry

increasing ΔE. We can intuitively understand that very much larger values of ΔE would explore the i − E characteristic much more extensively, perhaps even to the mass-transfer-controlled limits in both directions from Edc . Therefore, such experiments should yield currents containing harmonics of greater amplitude and should bring more harmonics into significance. These expectations are borne out in the experimental results of Figures 11.6.2 and 11.6.3, showing large-amplitude ac voltammetric results for aqueous solutions of Ru(NH3 )3+ . For 6 Figure 11.6.3, the experiment was cyclic ac voltammetry, starting at 200 mV vs. Ag/AgCl with an initial negative-going scan. Signals for the fundamental through the eighth harmonic are observed with good signal to noise. Because this methodology can test a large portion (even essentially all) of the i − E characteristic and is able to provide results of high precision, a case has been made that it can manifest kinetic detail more faithfully than competing approaches (42, 43, 46). In the analysis of results, sophisticated statistical methods are used to fit the contributions of multiple harmonics to kinetic models. This approach was used to examine the kinetics of Fc+ /Fc at boron-doped diamond electrodes in the viscous solvent 1-butyl-3-methylimidazolium hexafluorophosphate (47). The kinetics were found to depart from the Butler–Volmer model and to proceed in a dual mode. The investigators proposed that sp2 carbon at edges supports much more facile electron-transfer kinetics than the sp3 diamond surfaces. Given the focus on multiple harmonics, the ideal experimental strategy for large-amplitude ac voltammetry is to employ a system based on Fourier transformation of time-domain data (Section 11.8.2). Consequently, the method is sometimes abbreviated in the literature as FTAC.

11.7 Chemical Analysis by ac Voltammetry Fundamental and second-harmonic ac voltammetry offer good analytical sensitivities, because both methods have ready means for discrimination against capacitive currents (13, 15, 16). Detection limits are comparable with those of DPV and SWV, sometimes reaching the order of 10−7 M. To use the fundamental mode for analysis, one measures the current component in-phase with the excitation signal, Ė ac . Figure 11.7.1 illustrates the idea for a reversible system. Since the charging current is ideally 90∘ out of phase with Ė ac (being of purely capacitive origin), it has no component in-phase with Ė ac . Thus, we expect the in-phase current to be purely faradaic. In contrast, the current at 90∘ (the quadrature current) should contain an equal faradaic component plus the full nonfaradaic contribution. By taking the in-phase current as the analytical signal, we discriminate against the capacitive interference. Figure 11.7.1 Phasor diagram showing ideal relationships between the faradaic (İ f ) and capacitive (İ c ) components to the total current (I)̇ for a reversible system. İ f has a component along Ė ac , but İ c does not.

· I(total) · Ic

· If

· If(90°) · I(90°)

· · I(0°) = If(0°)

· Eac

481

482

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

A limitation is imposed by the uncompensated resistance, Ru . Since the charging current passes through Ru and C d in series, this current does not lead Ė ac by exactly 90∘ , but instead by some smaller angle. Thus, the current in phase with Ė ac does contain a nonfaradaic element, which becomes significant to the measurement as the analyte concentration drops. Second-harmonic ac voltammetry gains its freedom from nonfaradaic interference from the relative linearity of the double-layer capacitance as a circuit element. There is only a small second-harmonic capacitive current, although it, too, can become important at low analyte concentrations. The shapes of the voltammograms generated in ac measurements are convenient for analysis. Detection of the fundamental current produces a peak whose height is readily measured and is linear with concentration. Phase-selective second-harmonic voltammetry gives the second-derivative waveform of Figure 11.6.2. The peak-to-peak amplitude is linear with concentration and can be read with high precision. It is also relatively unaffected by the background signal (41). Analytical measurements are usually carried out at excitation frequencies ranging from 10 Hz to 1 kHz, but it is often better to select a frequency in the upper part of this range (48), because of the much smaller ac responses from systems with slow kinetics. For example, one can work directly with aerated solutions, significantly saving analysis time, because the reduction of oxygen in most aqueous solutions is irreversible and does not interfere with determinations made by ac voltammetry. Also, one can often improve selectivity toward certain analytes by careful choice of the medium. Transition metals lend themselves to this strategy, because their electrode kinetics are often strongly affected by coordination; thus, one can enhance their ac responses or mask them through electrolyte composition. Since many supporting electrolytes show irreversible reductions, there is considerable freedom to manipulate composition without introducing serious interferences.

11.8 Instrumentation for Electrochemical Impedance Methods Impedance measurements can be made either in the frequency domain, with a frequency response analyzer (FRA), or in the time domain, using a spectrum analyzer based on Fourier transformation. Both strategies are employed in commercial instrumentation, which must incorporate the necessary specialized hardware and software. 11.8.1

Frequency-Domain Instruments

The use of an FRA to measure the impedance of an electrochemical cell (8, 9, 49) is schematically illustrated in Figure 11.8.1. The FRA generates a signal, Eac = ΔE sin 𝜔t, which is fed to the potentiostat, where it is added to Edc and applied to the cell. In instrument design, care must be taken to minimize phase and amplitude errors introduced by the potentiostat, particularly at higher frequencies. A transduced voltage proportional to the current, i(t), is delivered back to the analyzer, where it is mixed with the input signal, and integrated over several signal periods to yield voltages that are proportional to the real and imaginary parts of the impedance (or, equivalently, to the magnitude and phase angle of the impedance) at the applied 𝜔. To obtain an impedance spectrum, the frequency is altered stepwise over a given range, with values of ZRe and ZIm being recorded vs. 𝜔. Commercial FRAs are available with a frequency range of 10 μHz to 30 MHz; however, EIS is generally limited to frequencies below 1 MHz by bandwidth limitations in the electrochemical cell and the potentiostat (Sections 16.7 and 16.9).

11.8 Instrumentation for Electrochemical Impedance Methods

Fequency response analyzer Generator

Analyzer

cos ωt

π/2

Multiply

Integrate

ZIm

Multiply

Integrate

ZRe

sin ωt

0

i/E converter output (iac) Eac Edc

Potentiostat Ctr

Ref Wk

Cell

Figure 11.8.1 System for measuring the impedance of an electrochemical cell based on an FRA.

11.8.2

Time-Domain Instruments

A different strategy is to subject the electrochemical system to an Eac that is the resultant of many frequencies, such as a pulse or a white noise signal (12, 45, 50–54), and then to sort out responses at individual frequencies by Fourier analysis ex post facto. As usual, Eac is added to Edc at the potentiostat, and the resulting time-dependent current from the cell is recorded. Both the stimulus and the response are then converted via Fourier transformation (FT) to spectral representations of amplitude and phase angle vs. frequency, from which the values of ZRe and ZIm can be computed as a functions of 𝜔. The FT allows one to interpret experiments in which several different excitation signals are simultaneously applied to a chemical system. Although the responses to those signals are also superimposed, an FT algorithm provides a means for resolving them (Section A.6). This capacity for simultaneous measurement is sometimes called the multiplex advantage. The operational concept is outlined in Figure 11.8.2, which shows that the excitation signal, Eac , is a noise waveform, rather than a pure sinusoid. Of course, it stimulates a current flow showing related “noisy” variations. During a brief period, lasting at least one cycle of the lowest frequency to be studied, the outputs of the potentiostat’s follower and i/E converter (Section 16.4.3) are digitized simultaneously and stored. FT of these two transients gives the distribution of pure sinusoids that make up the signals (Section A.6). One, therefore, knows the amplitude of excitation and the corresponding amplitude and phase angle for the current at each frequency in the Fourier distributions. Accordingly, one has the faradaic impedance vs. 𝜔 for the potential Edc . Since the period of data acquisition and analysis is short, it is feasible to repeat the whole procedure, so that ensemble-averaged results are obtained or changes with Edc or time can be followed.

483

484

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

Potentiostat Multi-ω Eac + Edc

PC

F

S/H + ADC

Clock

Cell

i/E

Figure 11.8.2 Physical apparatus and computational steps in a time-domain system for ac impedance measurements. Devices: F = voltage follower; i/E = current-to-voltage converter; PC = potential control amplifier; S/H = sample/hold amplifier; ADC = analog-to-digital converter. Steps in the large dashed box are computational. FFT = fast Fourier transform algorithm (Section A.6).

S/H + ADC

Computational Stored potential waveform FFT

Stored current waveform FFT

E(t) in frequency domain

i(t) in frequency domain

I(ω1), ϕ(ω1), E(ω1) → Z(ω1) I(ω2), ϕ(ω2), E(ω2) → Z(ω2) • • •

I(ωn), ϕ(ωn), E(ωn) → Z(ωn)

In practice, it is desirable to use a special kind of noise for excitation. Especially favorable (51, 53) is an odd-harmonic, phase-varying pseudorandom white noise, as displayed in Figure 11.8.3. This noise is the superposition of signals at several frequencies (15 in the example), all being odd harmonics of the lowest frequency. The choice of odd harmonics ensures that second-harmonic components will not appear in the currents measured for the 15 fundamental frequencies. The amplitudes of the excitation frequencies are equal (“white” noise) so that each carries equal weight. Their phase angles are randomized, so that the total excitation signal does not show large swings in amplitude. One can readily generate this kind of Eac by inverting the process used for signal analysis (Figure 11.8.3). One starts with the amplitude and phase-angle arrays, which have been tailored according to specifications. These are transformed into the complex plane; then the fast inverse FT is invoked, so that one obtains a digital representation of the time-domain noise signal. Feeding these numbers sequentially to a digital-to-analog converter (DAC) at the desired rate yields an analog signal, which is filtered and sent to the potentiostat’s input. Repeated passage through the D/A conversion and filtering steps yields a repetitive excitation waveform, which is applied until a single measurement pass is completed. A new waveform with different randomized phase angles is generated for the next pass, and so on. The ability of the FT to dissect a complex waveform into its components can also be used to obtain multiple harmonics (44, 52, 53). One would excite with a pure sinusoid and examine the transformed current waveform, which will provide the dc current plus the amplitudes and phase angles at the fundamental frequency and the harmonics. By making small, progressive changes in Edc and repeating the time-domain measurement cycle after each change, one can trace out all of the corresponding voltammograms from data obtained on a single potential scan (as for Figure 11.6.3). In addition to its use as a component of the time-domain measurement process, an FT algorithm can be valuable for signal-conditioning operations, including smoothing, convolution, and correlation (53, 55).

Phase angle

11.10 References

(f)

Amplitude

Frequency (a) Low-pass filter

(e) Frequency

(b) Digital-to-analog conversion

Magnitude

Inverse Fourier transform

Magnitude

Polar-coordinate to complex-plane transform

Frequency (c)

Time (d)

Figure 11.8.3 Procedure for generating a complex excitation waveform. (a, b) Randomized phase angles and uniform amplitudes for the chosen frequencies. (c) Complex-plane representation of (a) and (b). (d) Timedomain representation. (e) DAC output. (f ) Filtered E ac for addition to E dc at the potentiostat. Only a small part of the waveform period is shown in (e) and (f ). [From Creason, Hayes, and Smith (51), with permission.]

11.9 Analysis of Data in the Laplace Plane It is often difficult in electrochemistry to obtain an explicit mathematical linkage between current, potential, and time. Either the system itself is intrinsically too complex, or the experimental conditions are less than ideal. Simpler relationships may exist in the Laplace domain; thus, it can be fruitful to transform the data and to carry out the analysis in transform space (56–59). Methodology and results in this area were presented in the second edition.20

11.10 References 1 C. Gabrielli, Electrochim. Acta, 331, 135324 (2020). 2 E. Barsoukov and J. R. Macdonald, “Impedance Spectroscopy,” 3rd ed., Wiley, Hoboken, NJ,

2018. 3 M. E. Orazem and B. Tribollet, “Electrochemical Impedance Spectroscopy,” 2nd ed., Wiley,

Hoboken, NJ, 2017. 4 A. Lasia, “Electrochemical Impedance Spectroscopy and Its Applications,” Springer, New

York, 2014. 5 V. F. Lvovich, “Impedance Spectroscopy: Applications to Electrochemical and Dielectric Phe-

nomena,” Wiley, Hoboken, NJ, 2012. 6 U. Retter and H. Lohse, in “Electroanalytical Methods,” F. Scholz, Ed., 2nd ed., Springer,

Berlin, 2010. 20 Second edition, Section 10.9.

485

486

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

7 A. Lasia, Mod. Asp. Electrochem., 32, 143 (1999). 8 C. Gabrielli, in “Physical Electrochemistry,” I. Rubinstein, Ed., Marcel Dekker, New York,

1995, Chap. 6. 9 F. Mansfeld and W. J. Lorenz, in “Techniques for Characterization of Electrodes and

10 11 12 13 14 15 16 17 18 19 20 21 22 23

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Electrochemical Processes,” R. Varma and J. R. Selman, Eds., Wiley, New York, 1991, Chap. 12. D. D. Macdonald, in “Techniques for Characterization of Electrodes and Electrochemical Processes,” R. Varma and J. R. Selman, Eds., Wiley, New York, 1991, Chap. 11. M. Sluyters-Rehbach and J. H. Sluyters, Electroanal. Chem., 4, 1 (1970). K. B. Oldham, J. C. Myland, and A. M. Bond, “Electrochemical Science and Technology,” Wiley, Chichester, 2012, Chap. 15. A. M. Bond, “Modern Polarographic Methods in Analytical Chemistry,” Marcel Dekker, New York, 1980. D. D. Macdonald, “Transient Techniques in Electrochemistry,” Plenum, New York, 1977. D. E. Smith, Crit. Rev. Anal. Chem., 2, 247 (1971). D. E. Smith, Electroanal. Chem., 1, 1 (1966). J. E. B. Randles, Disc. Faraday Soc., 1, 11 (1947). D. C. Grahame, J. Electrochem. Soc., 99, C370 (1952). L. Pospisil and R. de Levie, J. Electroanal. Chem., 22, 227 (1969). H. Moreira and R. de Levie, J. Electroanal. Chem., 29, 353 (1971); 35, 103 (1972). D. C. Grahame, J. Am. Chem. Soc., 68, 301 (1946). G. C. Barker, J. Electroanal. Chem., 12, 495 (1966). G. Crabtree, G. Rubloff, and E. Takeuchi, “Next Generation Electrical Energy Storage,” Office of Basic Energy Sciences, Washington, DC, 2017. https://science .osti.gov/-/media/bes/pdf/brochures/2017/BRN-NGEES_rpt-low-res.pdf?la=en& hash=0DEB8D525053595FEEEB75BB52F02D19E43F149F. Y. Ko and S.-M. Park, J. Phys. Chem. C, 116, 7260 (2012). T. B. Hunter, P. S. Tyler, W. H. Smyrl, and H. S. White, J. Electrochem. Soc., 134, 2198 (1987). J. H. Sluyters and J. J. C. Oomen, Rec. Trav. Chim. Pays-Bas, 79, 1101 (1960). H. Kojima and A. J. Bard, J. Electroanal. Chem., 63, 117 (1975). H. Kojima and A. J. Bard, J. Am. Chem. Soc., 97, 6317 (1975). C. Ho, I. D. Raistrick, and R. A. Huggins, J. Electrochem. Soc., 127, 343 (1980). C. Gabrielli and H. Perrot, Mod. Asp. Electrochem., 44, 151 (2009). R. de Levie, Ann. Biomed. Eng., 20, 337 (1992). A. M. Bond, R. J. O’Halloran, I. Ruzic, and D. E. Smith, Anal. Chem., 48, 872 (1976). D. E. Smith, Anal. Chem., 35, 1811 (1963). B. Timmer, M. Sluyters-Rehbach, and J. H. Sluyters, J. Electroanal. Chem., 14, 169, 181 (1967). D. E. Smith, and T. G. McCord, Anal. Chem., 40, 474 (1968). D. E. Smith, Anal. Chem., 35, 610 (1963). A. M. Bond, J. Electroanal. Chem., 50, 285 (1974). A. M. Bond, R. J. O’Halloran, I. Ruzic, and D. E. Smith, Anal. Chem., 50, 216 (1978). S. O. Engblom, J. C. Myland, K. B. Oldham, and A. L. Taylor, Electroanalysis, 13, 626 (2001). C. P. Andrieux, P. Hapiot, J. Pinson, and J.-M. Savéant, J. Am. Chem. Soc., 115, 7783 (1993). H. Blutstein, A. M. Bond, and A. Norris, Anal. Chem., 46, 1754 (1974). S. O. Engblom, J. C. Myland, and K. B. Oldham, J. Electroanal. Chem., 480, 120 (2000).

11.11 Problems

43 D. J. Gavaghan and A. M. Bond, J. Electroanal. Chem., 480, 133 (2000). 44 J. Zhang, S.-X. Guo, A. M. Bond, and F. Marken, Anal. Chem., 76, 3619 (2004). 45 A. M. Bond, D. Elton, S.-X. Guo, G. F. Kennedy, E. Mashkina, A. N. Simonov, and J. Zhang,

Electrochem. Commun., 57, 78 (2015). 46 S. Y. Tan, P. R. Unwin, J. V. Macpherson, J. Zhang, and A. M. Bond, Anal. Chem., 89, 2830

(2017). 47 J. Li, C. L. Bentley, S.-Y. Tan, V. S. S. Mosali, M. A. Rahman, S. J. Cobb, S.-X. Guo, J. V.

Macpherson, P. R. Unwin, A. M. Bond, and J. Zhang, J. Phys. Chem. C, 123, 17397 (2019). 48 A. M. Bond, Anal. Chem., 45, 2026 (1973). 49 C. Gabrielli, “Identification of Electrochemical Processes by Frequency Response Analy-

50 51 52 53 54 55 56 57 58 59

sis,” Technical Report 04/83, Solartron Analytical, 1998. https://www.ameteksi.com/library/ application-notes/-/media/76e935901dd74ceea01ccaed28ffd9a2.ashx. H. Kojima and S. Fujiwara, Bull. Chem. Soc. Jpn., 44, 2158 (1971). S. C. Creason, J. W. Hayes, and D. E. Smith, J. Electroanal. Chem., 47, 9 (1973). D. E. Glover and D. E. Smith, Anal. Chem., 45, 1869 (1973). D. E. Smith, Anal. Chem., 48, 221A, 517A (1976). J. Házì, D. M. Elton, W. A. Czerwinski, J. Schiewe, V. A. Vincente-Beckett, and A. M. Bond, J. Electroanal. Chem., 437, 1 (1997). J. W. Hayes, D. E. Glover, D. E. Smith, and M. W. Overton, Anal. Chem., 45, 277 (1973). M. D. Wijnen, Rec. Trav. Chim., 79, 1203 (1960). E. Levart and E. P. D. A. D’Orsay, J. Electroanal. Chem., 19, 335 (1968). A. A. Pilla, J. Electrochem. Soc., 117, 467 (1970). A. A. Pilla, in “Computers in Chemistry and Instrumentation: Electrochemistry,” Vol. 2, J. S. Mattson, H. B. Mark, Jr., and H. C. MacDonald, Eds., Marcel Dekker, New York, 1972.

11.11 Problems 11.1 Derive formulas for converting a parallel resistance–capacitance network (Rp and C p in parallel) to a series equivalent (Rs and C s in series). 11.2 The faradaic impedance is sometimes represented as a resistance and a capacitance in parallel, rather than in series. Find expressions for the components of the parallel representation in terms of Rct , 𝛽 O , 𝛽 R , and 𝜔. [Hint: Use known expressions for series elements together with equations for series-to-parallel circuit conversion (Problem 11.1).] 11.3 The faradaic impedance method is employed to study the reaction O + e ⇌ R by imposing a small sinusoidal signal (5-mV amplitude) and measuring the equivalent series resistance, RB , and capacitance, C B , of the cell. The following data are obtained for ∗ = C ∗ = 1.00 mM, T = 25 ∘ C, and A = 1 cm2 : CO R (𝝎/2𝝅)/s−1

RB (𝛀)

C B (𝛍F)

49

146.1

290.8

100

121.6

158.6

400

63.3

41.4

900

30.2

25.6

487

488

11 Electrochemical Impedance Spectroscopy and ac Voltammetry

In a separate experiment under exactly the same conditions, but in the absence of the electroactive species, the cell resistance, Ru , is found to be 10 Ω, and the double-layer capacitance of the working electrode, C d , is determined as 20.0 μF. (a) From these data calculate, at each frequency, Rs and C s plus the phase angle, 𝜙, between the components of the faradaic impedance. (b) Calculate i0 and k 0 for the reaction and estimate D (assuming DO = DR ). 11.4 Derive (11.4.13) and (11.4.16) from (11.4.6) and (11.4.7). 11.5 Derive (11.4.11) from (11.4.9) and (11.4.10). 11.6 Devise and justify an equivalent circuit for a system in which O and R are bound to the surface of the electrode by chemical modification. Follow the steps in Section 11.3 to evaluate the expected frequency dependence of the faradaic impedance for a nernstian electrode reaction. What phase angle is expected? 11.7 Derive (11.5.9), describing the shape of a reversible ac voltammetric peak, from (11.5.7). 11.8 From the data in Figures 11.5.4 and 11.5.5, evaluate 𝛼 and k 0 for the reduction of Ti(IV) to Ti(III) at Hg in oxalic acid solution. From other experiments, we know that n = 1 and DO = 6.6 × 10−6 cm2 /s. Assume that DO = DR . 11.9 The reduction of nitrobenzene to its radical anion in DMF is reported to occur at Hg with k 0 = 2.2 ± 0.3 cm/s (28). The value of DO is given as 1.02 × 10−5 cm2 /s at 22 ± 2 ∘ C, where k 0 was evaluated. The transfer coefficient, 𝛼, is 0.70. Calculate the phase angles expected for 𝜔/2𝜋 = 10, 100, 1000, and 10,000 s−1 . Draw the corresponding plot of cot 𝜙 vs. 𝜔1/2 for E = E1/2 . Describe a means for obtaining cot 𝜙 from the in-phase and quadrature phase-selective voltammograms, and comment on the frequency range where it might be experimentally feasible to obtain cot 𝜙 values sufficiently precise to allow a determination of k 0 for the system at hand. 11.10 Plot the amplitude and phase arrays (as in Figure 11.8.3a,b) for generating a complex waveform having components at 100, 200, 300, … Hz, all with phase angles equal to 𝜋/2. Let these arrays have 128 elements, with the 0th element representing the dc level and the 127th element representing 𝜔/2𝜋 = 1270 Hz. What disadvantages would the waveform resulting from your arrays have with respect to that generated in Figure 11.8.3? Would your waveform have any advantages?

489

12 Bulk Electrolysis The methods described in the preceding chapters generally feature a small ratio of electrode area, A, to solution volume, V . Such “small A/V conditions” allow experiments to be carried out over fairly long time periods without appreciable changes of concentration in the bulk solution. However, there are other circumstances in which one desires to alter the composition of the bulk by electrolysis. For example, if the goal is to carry out electrosynthesis or coulometric analysis, then one must perform a bulk (or exhaustive) electrolysis featuring the best possible mass transfer and “large A/V conditions.” Although bulk electrolysis generally involves sizable electrodes, large currents, and experimental time scales of minutes or even hours, one can still rely on basic principles governing electrode reactions presented in Chapters 1–4. General treatments and reviews of bulk electrolysis are available (1–5). Bulk transformations are required in many practical electrochemical devices and processes, especially batteries, fuel cells, and chemical manufacturing built on electrosynthesis. This chapter introduces the fundamental concepts of bulk electrolysis; however, the focus, in keeping with the nature of this book, is on laboratory methods, rather than technology. Electrochemical engineering is the field more generally dedicated to managing and optimizing performance in technological systems, sometimes at very large scale (6–8). Methods of bulk electrolysis are often differentiated by the controlled variable. In controlled-potential techniques, the potential of the working electrode is held constant with respect to a reference electrode. In controlled-current techniques, the current passing through the cell is held constant or perhaps is programmed to change over time or in response to some signal from the cell. Major sections below are dedicated to each of these categories. Bulk electrolysis methods may also be classified according to the purpose or mode of operation: • Preparative bulk electrolysis or electrosynthesis (9–14) describes the electrolytic production of substances or materials. The scale covers an enormous range. On the smaller end, an investigator might seek to prepare a few micrograms to a few grams of a product. This can be done in laboratory-size cells using techniques we will discuss (Sections 12.1–12.3 and 12.5). On the larger end, the quantities are measured in tons. Essentially, all global production of metallic aluminum and chlorine is electrosynthetic, consuming a quite significant fraction of all electric power (8). • Coulometry (Sections 12.2.3 and 12.3.2) describes electroanalytical measurements based on the quantity of electricity required for exhaustive electrolysis of a targeted substance. • Electrogravimetry (Section 12.2.4) describes electroanalysis based on weighing a deposit on the working electrode. • Electroseparation (Section 12.2.5) describes bulk electrolysis intended for selective removal of constituents from a solution. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

490

12 Bulk Electrolysis

• Flow electrolysis (Section 12.5) describes the exhaustive electrolysis of a solution as it flows through a cell. • Thin-layer electrolysis (Section 12.6) describes systems in which a large A/V ratio is attained by trapping a tiny volume of solution in a thin (20–100 μm) layer against a working electrode. The currents and time scales of this approach resemble those of LSV and CV. • Stripping analysis (Section 12.7) begins with an electrolysis designed to preconcentrate a material in a small volume or on the surface of an electrode, before a voltammetric analysis is carried out, usually by LSV, DPV, or SWV. All topics in this list are addressed in this chapter, but not all are confined to dedicated sections. Electrosynthesis, in particular, relates to several sections because it can be carried out at controlled potential or controlled current, using either a batch cell or a flow cell. The methodological principles of electrosynthesis are the same as for bulk electrolysis toward any other purpose in any of these modes. Also included in this chapter are microfluidic flow cells, such as electrochemical detector cells for liquid chromatography, capillary electrophoresis, and other techniques based on liquid flow (Section 12.5.3). Although such systems do not always operate in a bulk electrolysis mode, they relate to other flow cells described here.

12.1 General Considerations In this section, we develop two fundamental concepts of bulk electrolysis, then we turn to a set of important practical aspects, all broadly applicable regardless of experimental mode or purpose. 12.1.1

Completeness of an Electrode Process

An important principle is that the potential of the working electrode ultimately determines the degree of conversion achieved in a bulk electrolysis. This idea is easiest to understand for a controlled-potential electrolysis of a nernstian system; however, it is relevant to bulk conversions generally, even if the conditions do not involve controlled potential or nernstian exchange at the working electrode. Let us proceed by considering the reversible process O + ne ⇌ R

(12.1.1)

where both O and R are soluble and stable, and R is initially absent. The working electrode is held at a potential, E, in the mass-transfer-controlled region for reduction, and the system is continuously and efficiently stirred, so that uniform concentrations apply everywhere, except in a negligibly thin mass-transfer layer near the working electrode. As O is converted to R, the reduction current falls off. Eventually, the electrolysis reaches an effective conclusion as the bulk concentrations of O and R converge on the equilibrium values defined by the applied potential through the nernstian balance, lim C ∗ RT t→∞ O E=E + ln nF lim CR∗ 0′

(12.1.2)

t→∞

As the system arrives at the equilibrium defined by E, there is no longer an impetus to convert O into R at the electrode.

12.1 General Considerations ∗ (0) is the initial concentration of O, and x is the fraction converted when equilibrium is If CO reached, then ∗ = (1 − x)C ∗ (0) lim CO O

(12.1.3a)

lim CR∗ = xC ∗O (0)

(12.1.3b)

t→∞ t→∞

hence, ′

E = E0 +

RT (1 − x) ln nF x

(12.1.4)

and the completeness of reduction is x=

1 0′ 1 + 10n(E−E )∕0.059

(25∘ C, E in V)

For 99% reduction of O to R, the potential of the working electrode must be ) ( ′ ′ (0.059)(2) 0.059 0.01 ≈ E0 − E = E0 + log n 0.99 n ′ 0 ∘ or 118/n mV more negative than E at 25 C.

(12.1.5)

(12.1.6)

If species R does not return to the solution, perhaps because it forms an amalgam with a mercury electrode or is electrodeposited as a metal, then (12.1.5) is not applicable. However, an analogous relationship can be readily developed using the applicable Nernst equation (Problem 12.4).1 12.1.2

Current Efficiency

When faradaic reactions can occur simultaneously at an electrode, the fraction of the total current, itotal , going to the jth process is called the instantaneous current efficiency: Instantaneous current efficiency for jth process = ij ∕itotal

(12.1.7)

A current efficiency of unity (or 100%) implies that a single process is occurring. Over an elapsed time, one can define the overall current efficiency in terms of the fraction of the total charge, Qtotal , given to the jth process: Overall current efficiency for jth process = Qj ∕Qtotal

(12.1.8)

It is generally desirable for bulk electrolysis to be carried out with high current efficiency. Achieving it requires that the working electrode potential and other conditions be chosen so that side reactions are minimized (e.g., reduction or oxidation of solvent, supporting electrolyte, electrode material, or impurities). 12.1.3

Experimental Concerns

Because of the long time scales and large currents usually required for bulk electrolysis, one encounters different experimental challenges than in experiments based on small A/V conditions. Here, we review matters that broadly apply to bulk electrolysis regardless of experimental mode or purpose. Experimental aspects particular to a given mode (controlled potential, controlled current, or flow electrolysis) are discussed in the corresponding sections. 1 Amalgamation is treated in the second edition, Section 11.2.1(b) and the first edition, Section 10.2.1(b).

491

492

12 Bulk Electrolysis

A good start is to grasp the huge quantity of charge needed to electrolyze a mole of any substance. The Faraday constant, F, is the total charge on a mole of electrons, 96,485 C. To convert a mole of analyte or to prepare a mole of product requiring n electrons per molecule, one must deliver nF coulombs at the working electrode. If this were to be done in an hour, the average current would be 26.8n amperes, a large figure, exceeding the delivery capability of even the most powerful electrochemical instruments and presenting large control problems. While a mole might be electrolyzed or produced in a laboratory-size cell, the job would have to be done over quite a few hours or in several batches. Most laboratory-scale bulk electrolysis is aimed at smaller quantities, but still involves the delivery and control of sizable currents, often on the ampere scale. The size of F is what makes batteries practical and electrosynthesis difficult. Cell design is important in bulk electrolysis, especially with regard to • • • • •

The area of the working electrode, which should be as large as possible. Mass transport, which should be as effective as possible. Symmetry in the placement of the working and counter electrodes. Isolation of the counter electrode. Resistance of the electrolyte, which should be as small as practical.

Typical cells for batch electrolysis are shown in Figure 12.1.1. Cell design for flow electrolysis will be discussed in Section 12.5. Counter electrode Ag QRE Counter electrode chamber (fritted tube)

Reference electrode

Pt counter electrode

Au voltammetric working electrode

Pt foil coulometric working electrode

Gas inlet

Rotatable sidearm

Teflon cap RVC working electrode 75-mL glass or water-jacketed cell Stir bar

Septum Liquid level Bridge Frits

(a)

(b)

Figure 12.1.1 Cells for bulk electrolysis. (a) Commercial cell designed for electrosynthesis or coulometry in aqueous or nonaqueous media under an inert atmosphere. The counter electrode chamber allows symmetric positioning of counter and working electrodes. RVC = reticulated vitreous carbon. [Courtesy of Bioanalytical Systems, Inc.] (b) Three-compartment cell with ground glass joints for coulometric and voltammetric studies after deaeration on a vacuum line. Rotatable side arm is for solid sample addition. Septum is for additions by injection. Arm for attachment to the vacuum line is not shown. [Reprinted with permission from Smith and Bard (15). © 1975, American Chemical Society.]

12.1 General Considerations

(a) Working Electrodes

Solid electrodes are often wire gauzes or cylinders made of metal foil or reticulated vitreous carbon (RVC),2 although packed beds of powders, slurries, or fluidized beds are sometimes used. The aim is to have as large a working electrode area as possible, so as to carry out the electrolysis as quickly as possible. (b) Mass Transport

Also important to the speed of electrolysis is the effectiveness of mass transport, not only in the bulk solution but also inside a porous electrode material. In batch cells like those in Figure 12.1.1, convection is usually driven by a stirring bar. It is helpful to stir as vigorously as possible without creating a vortex. Ultrasonic stirring has also been employed effectively (Section 12.2.1). When porous electrodes are used, the entire surface area is typically not effective for the intended reaction, either because mass transfer is poor to interior locations or because solution resistance limits interior current densities [Section 12.1.3(c)]. A better overall rate of electrolysis may be achieved with a porous electrode having less total area, but with better mass transport to interior surfaces. In flow cells, the flow rate of the electrolyte through the working electrode determines the mass transport characteristics of the system, as discussed in Section 12.5.1. (c) Geometry of Electrode Placement

The large currents in bulk electrolysis typically produce significant differences in potential within the electrolyte phase, even over distances on the millimeter scale. Spatial variations in electric potential within the solution cause the interfacial potential difference to vary over the surface of the working electrode (especially inside pores), which, in turn, causes variations in the local rate of the electrode reaction and, therefore, the current density. In general, current densities on the working and counter electrodes are highest at points where these electrodes are closest to each other. Such effects are often more severe for working electrodes of large physical size. In consequence, the effective area of the electrode can be far smaller than the microscopic area or even the geometric area. Figure 12.1.2 illustrates the expected behavior for the two cells depicted in Figure 12.1.1. In neither case, is the back side of the working electrode active,3 even though it is exposed to the electrolyte. In the cell of Figure 12.1.1a, one can anticipate a fairly uniform current density on the front side of the working electrode, but it is likely that current density declines with distance into the porous material, so that interior zones further from the counter electrode become progressively less active. It is always valuable to maximize the effective area of the working electrode by symmetrically situating the working and counter electrodes and by making the electrolyte phase as conductive as possible. However, a highly symmetric placement is not always practical, as in the case of Figure 12.1.1b, because of other operational demands on the cell, such as arrangements for stirring, deaeration in vacuo, or inclusion of voltammetric electrodes. The placement of the reference electrode’s tip is also important for controlled-potential experiments. In general, it should be adjacent to the position of highest current density on the working electrode and as close to the working electrode as practicality allows. The long-term stability of the reference electrode potential is also important. 2 RVC is a form of glassy carbon having a porous structure with large surface area and the ability to support good fluid flow. 3 The front side faces toward the counter electrode; the back side faces away.

493

494

12 Bulk Electrolysis

Working electrode Counter electrode

Cell body

Counter electrode

Reference electrode

Working electrode

Cylindrical frit separator (a)

Reference electrode

Frit separators Cell body (b)

Figure 12.1.2 Representation of current distribution in the two cells shown in Figure 12.1.1. View is from the top in each case. (a) The cylindrical symmetry in the cell of Figure 12.1.1a provides equivalent paths of least resistance from points on the counter electrode to points on the working electrode (dashed lines). (b) In the cell of Figure 12.1.1b, the radius of curvature of the working electrode is smaller than the distance between the working and counter electrodes, so the edges of the working electrode are closer to the counter electrode than other points on the working electrode. Current paths to the edges, therefore, involve less solution resistance, and the current density will be greater at the edges. Spacing of dashed lines is an indicator of current density.

(d) Isolation of the Counter Electrode

In exhaustive electrolysis, the products at the counter electrode, often unknown and potentially reactive, are generated in amounts commensurate with the desired product at the working electrode. If soluble, a product from the counter electrode may react with the desired product at the working electrode or may itself be electroactive there. The default solution is to compartmentalize the two electrodes, just as one can see in both cells of Figure 12.1.1. An electrical junction must always be made between the compartments using an ionically conducting separator—typically a fritted glass disk, an ion-exchange membrane (e.g., Nafion; Figure 17.4.1), a microporous membrane, a porous ceramic, or a fiber mat (16). Usually, the separator contributes appreciably to cell resistance and often accounts for the largest portion of ohmic drop between the working and counter electrodes. The choice of separator is always important in bulk electrolysis but becomes critical in the design of electrochemical power sources, such as batteries and fuel cells. An “undivided cell” (without a separator) can be used when the product at the counter electrode presents no threat as an interference, e.g., when solid products or innocuous gaseous products are formed. Examples might involve the use of a silver anode in halide media (e.g., Ag + Cl− → AgCl + e) or hydrazine as an “anodic depolarizer” (Section 12.3.1) at platinum anodes (N2 H4 → N2 + 2H+ + 2e). Electrosynthetic systems are sometimes built to create useful products at both the working and counter electrodes, in which case the system is said to be employing paired reactions. An important classic example is the chloralkali cell, which electrolyzes brine, producing Cl2 at the anode and NaOH plus H2 at the cathode (8). (e) Cell Resistance

High cell resistances greatly complicate experiments involving high currents because large values of i2 R mean wasted power, a need for high voltage capability at the potentiostat or other power source, and undesirable heat evolution (Section 16.7.1). Moreover, it may become impractical to place the reference electrode tip close enough to the working electrode to avoid

12.2 Controlled-Potential Methods

unmanageable ohmic drops [Section 12.2.2(b)]. The minimization of cell resistance merits particular attention when nonaqueous solvents are employed (e.g., MeCN, DMF, THF, DCE, CH2 Cl2 , and NH3 ) because the solubilities of electrolytes and the mobilities of ions are often lower than in water. (f) Effect of Long Time Scale

The time scale of bulk electrolysis (∼10–60 min) is normally orders of magnitude longer than that of voltammetry; consequently, perturbing homogeneous chemical reactions following electron transfer, which might not affect voltammetry, may be important in a bulk transformation. For example, consider the reaction sequence: O+e⇌R

(12.1.9)

R→A

(slow, t1∕2 ∼ 2 − 5 min)

(12.1.10)

A+e→B

(A reduced at less negative E than O)

(12.1.11)

This sequence occurs, for example, in the reduction of o-iodonitrobenzene (O = IPhNO2 ) in liquid ammonia (with 0.1 M KI as supporting electrolyte). A voltammetric experiment (e.g., cyclic voltammetry at scan rates of 50–500 mV/s) shows a 1e reaction with the formation of the radical anion, IPhNO2 −∙, which is stable on this time scale. However, controlled-potential coulometric reduction (Section 12.2.3) shows n approaching 2 for reductions requiring 1-hour durations. In this time, the radical anion loses I− to form a radical, ∙PhNO2 , which is reduced at these potentials to −∶PhNO2 . This then protonates to form nitrobenzene. Effects of this kind are often relevant to electrosynthesis.

12.2 Controlled-Potential Methods Bulk electrolysis at controlled potential, if practical, is preferred because control of potential provides the best possible selectivity and defines the completeness of electrolysis. Except in special cases (Section 12.3.3), it also provides the highest achievable current efficiency and the shortest time to completion. If one can realize all these benefits, there is no reason to consider another strategy. 12.2.1

Current–Time Behavior

The simple mass-transfer model described in Section 1.3.2 also applies to a working electrode used for bulk electrolysis; however, the bulk concentration is a function of time, decreasing during the electrolysis. Thus, i − E curves occasionally recorded at the working electrode during electrolysis (assumed to be taken at such a rapid rate that no appreciable change of bulk concentration occurs during the scan itself ) would show a continuously decreasing limiting current, il (Figure 12.2.1). Suppose the electrolysis is O + ne → R, occurring at an electrode of area A held at a potential, Ec , in the mass-transfer-limited region. The current at any time is given by (1.3.10): ∗ (t) il (t) = nFAmO CO

(12.2.1)

Assuming 100% current efficiency, this relationship also reflects the total rate of electrolytic consumption of O, dN O (t)/dt (mol/s): [ ] dN O (t) il (t) = −nF (12.2.2) dt

495

496

12 Bulk Electrolysis

t1 = 0

i(0)

t2 i

t3 t4 t5

E

Background current

EC

Figure 12.2.1 Current–potential curves at progressively increasing times during a controlled-potential bulk electrolysis at E = E c .

where N O (t) is the total number of moles of O. If the solution is homogeneous (neglecting the small volume of the diffusion layer, 𝛿 O A, near the electrode), then ∗ (t) = CO

NO (t)

V where V is the total solution volume. Thus, [ ∗ ] dC O (t) il (t) = −nFV dt By equating (12.2.1) and (12.2.4), we obtain ( ) dC ∗O (t) mO A ∗ (t) = −pC ∗ (t) =− CO O dt V

(12.2.3)

(12.2.4)

(12.2.5)

This describes a first-order decay, where p = mO A/V is the rate constant. The solution is ∗ (t) = C ∗ (0) exp(−pt) CO O

(12.2.6)

∗ (0) is the initial concentration of O. The i − t behavior is given by substitution into where CO (12.2.1),

i(t) = i(0) exp(−pt)

(12.2.7)

where i(0) is the initial current (17, 18). Thus, a controlled-potential bulk electrolysis is like a first-order reaction, with the concentration and the current decaying exponentially with time during the electrolysis (Figure 12.2.2) and eventually attaining the background (residual) current level. One can use (12.2.6) or (12.2.7) to determine the duration of the electrolysis for a given conversion: [ ∗ ] [ ] CO (t) i(t) 2.303 2.303 log (12.2.8) t=− ∗ (0) = − p log i(0) p CO ∗ (t)∕C ∗ (0) = 10−2 and t = 4.6/p. For 99.9% completion, t = 6.9/p. With For 99% completion, CO O effective stirring, mO ≈ 10−2 cm/s, so that for A(cm2 ) ≈ V (cm3 ), p ≈ 10−2 s−1 , and a 99.9%

12.2 Controlled-Potential Methods 1.0

0

i(t) i(0)

In

Q(t) Q0

i(t) i(0)

0.5

0.0

1.0

–2

0

1

2

pt (a)

3

4

–4

0.5

0

1

2

pt (b)

3

4

0.0

0

1

2

pt (c)

3

4

Figure 12.2.2 (a) Current–time curve during controlled-potential electrolysis (in dimensionless form). (b) log[i(t)/i(0)] vs. pt. (c) Charge–time curve (in dimensionless form).

electrolysis would require ∼690 s or ∼12 min. Typically, bulk electrolyses are slower than this, requiring 30–60 min, although cell designs with very large A/V and very effective stirring (e.g., using ultrasonics), giving p as large as 10−1 s−1 , have been described (19). For effective rates of electrolysis, A should be as large as possible. In many practical devices (e.g., preparative cells or fuel cells), porous electrodes and flow systems are employed (Section 12.5). The total quantity of electricity consumed in the electrolysis, Q(t), is given by the area under the i − t curve (Figure 12.2.2c): t

Q(t) =

∫0

i(t) dt

(12.2.9)

Electrolysis at controlled potential is the most efficient method of carrying out a direct bulk electrolysis because the current is always maintained at the maximum achievable value consistent with 100% current efficiency, given the cell design and mass-transport conditions. The ∗ (0); hence, electrolysis of a 0.1 M solution or a 10−6 M rate constant, p, is independent of CO solution should require the same amount of time in the same cell under the same conditions. 12.2.2

Practical Aspects

Bulk electrolysis at controlled potential can be experimentally demanding, requiring the largest currents and sometimes presenting serious problems with potential control (Section 16.7.1). A suitable potentiostat must have substantial output-power capability (e.g., 100 W, with compliances4 of ±1 A at ±100 V or ±5 A at ±20 V). In large systems, potentiostatic control of bulk electrolysis may simply be unachievable, but it is usually practical in laboratory-size cells for modest electrosynthesis or other purposes like those discussed below. (a) Compliance-Limited Electrolysis

Even the most powerful commercial potentiostats may not be able to deliver the demanded current or voltage during part of the experiment (especially at the beginning, when the current is generally at its highest). Usually, a compliance limitation does not compromise a bulk electrolysis, except that electrolysis will proceed at less than the maximum rate. In such a situation, the potential always remains less extreme than desired (more positive for a reduction and more negative for an oxidation) and will not activate a new electrode process. 4 Voltage compliance and current compliance are terms used to describe the electronic delivery limits of a potentiostat [Section 16.7.1(a)].

497

498

12 Bulk Electrolysis

(b) When Controlled Potential is Impractical

Even though electrolysis at controlled potential is usually the optimal choice for bulk conversion, one does not always have this option. The issue is the effectiveness of potential control, which comes down to an issue of ohmic drop. As we learned in Section 1.5.4, E = Eappl + iRu

(12.2.10)

where E is the actual potential of the working electrode and Eappl is the controlled voltage between the working and reference electrodes. The uncompensated ohmic drop, iRu , is, therefore, a potential-control error. The question is whether this error is small enough to be manageable. In a bulk electrolysis, the best possible positioning of the reference electrode may still leave an uncompensated resistance, Ru , amounting to 10% of the total resistance between the working and counter electrodes. In a favorable case, Ru might be as low as 1 Ω. If the current during an electrosynthesis is 100 mA (a relatively modest value), the control error would be 100 mV, a fairly large value, but still manageable for a bulk electrolysis (i.e., one can accept the error and still keep the actual potential of the working electrode in the mass-transfer-controlled region for the desired process). However, the current might be larger or the uncompensated resistance might be greater, even by orders of magnitude. Suppose, for example, the uncompensated resistance is 10 Ω and the current is 1 A (both realistic figures). In that case, the potential control error, iRu , would be 10 V—a huge number on the scale of potentials. The variations in potential over the face of the electrode or in response to fluctuations in mass transfer could be a volt or more. In any system where iRu exceeds perhaps 250 mV, operation at controlled potential becomes invalid in a practical sense. Such situations are common in industrial cells, where currents can exceed kiloamperes, and they can easily be encountered in laboratory-size electrosynthetic cells when currents are sustained above 1 A. With resistive media (e.g., with nonaqueous solvents), controlled-potential electrolysis may be impractical even for much more modest sustained currents. The better option may then be to use controlled current (Section 12.3.3) or to employ a flow cell with more dilute solutions (Section 12.5.1). 12.2.3

Coulometry

In controlled-potential coulometry, the total number of coulombs consumed in an electrolysis is used to determine the amount of substance electrolyzed. The method is absolute; no standardization is required; however, the electrode reaction must • Be of known stoichiometry. • Be a single reaction or at least have no side reactions of different stoichiometry. • Occur with virtually 100% current efficiency. The experimental system is generally as described for bulk electrolysis in Section 12.1.3. In addition, there must be a means for measurement of the total charge passed. Normally, this task is accomplished ex post facto in a suitable electrochemical workstation by numeric integration of the i − t curve; however, one can employ a free-standing electronic device (called a coulometer) based on an operational-amplifier integrator circuit (Section 16.2.4) or other design. By either approach, one obtains a Q − t curve during electrolysis (Figure 12.2.2c). For an uncomplicated system, the shape of this curve is available from (12.2.7) and (12.2.9): Q(t) =

i(0) (1 − e−pt ) = Q0 (1 − e−pt ) p

(12.2.11)

12.2 Controlled-Potential Methods

where Q0 is the value of Q at the completion of the electrolysis (t → ∞) and is given by Q0 = nFN O (0) = nFVC∗O (0)

(12.2.12)

Here, N O (0) represents the total number of moles of O initially present. Equation (12.2.12) is just a statement of Faraday’s law and is the basis for any coulometric method of analysis. The current is monitored during the electrolysis, so that the background current and the completion of electrolysis can be ascertained. The shape of the i − t curve can be diagnostic for experimental problems or for the mechanism of the electrode reaction. For example, if the final current following electrolysis is constant, but appreciably higher than the pre-electrolysis background current of the supporting electrolyte solution alone, a reaction of the electrolysis product may be regenerating starting material or some other electroactive substance (Section 13.1.1). This symptom can also indicate leakage of a product from the counter-electrode compartment. If the current at the start of the electrolysis remains constant for some time before showing the usual exponential decay (Figure 12.2.2a), the current or voltage compliance of the potentiostat is probably insufficient to maintain the working ∗ , cell electrode at the chosen potential, given the electrolysis conditions (electrode area, CO resistance, stirring rate). Coulometric determinations are broadly applicable (1, 5, 20, 21). One can readily work in all kinds of media, and there is no need for an isolable product, so electrode reactions may produce soluble products or gases. Typical applications are listed in Table 12.2.1. Controlled-potential coulometry is also useful for studying the mechanisms of electrode reactions and for determining an n-value without prior knowledge of electrode area or diffusion coefficient.5 As previously noted [Section 12.1.3(f )], coulometric results, including Table 12.2.1 Typical Controlled-Potential Coulometric Determinations Substance

Electrode Electrolyte(a)

Potential(b) Overall Reaction

Li

Hg

0.1 M TBAP (MeCN)

−2.16

Li(I) → Li(Hg)

Cr

Pt

1 M H2 SO4

+0.50

Cr(VI) → Cr(III)

Fe

Pt

1 M H2 SO4

+0.20

Fe(III) → Fe(II)

Zn

Hg

2 M NH3 +

−1.45

Zn(II) → Zn(Hg)

1 M (NH4 )3 citrate

Te2−

Hg

1 M NaOH

−0.60

Te2− → Te

Br−

Ag on Pt

0.2 M KNO3 (MeOH)

0.0

Ag + Br− → AgBr

I−

Pt

1 M H2 SO4

+0.70

2I− → I2

U

Hg

0.5 M H2 SO4

−0.325

U(VI) → U(IV)

Pu

Pt

1 M H2 SO4

+0.70

Pu(III) → Pu(IV)

Ascorbic acid

Pt

0.2 M phthalate buffer, pH 6 +1.09

Oxdn. n = 2

Aromatic hydrocarbons (e.g., DPA, rubrene)

Hg or Pt

0.1 M TBAP (DMF)

Redn. Ar → Ar−∙

Aromatic nitro compounds

Hg

0.5 M LiCl (DMSO)

Redn. ArNO2 → ArNO2−∙

(a) With water as solvent unless indicated otherwise. (b) V vs. SCE. 5 In voltammetric methods, if n is to be determined from the limiting current, D and A usually must be known. An exception is highlighted in Problem 6.11. To determine n from potential measurements, knowledge about the reversibility of the reaction is required.

499

500

12 Bulk Electrolysis

determinations of n, can be affected by chemical processes following the electron transfer that are too slow to show up in voltammetry. One must be on guard for such effects. 12.2.4

Electrogravimetry

The determination of a metal by selective deposition on an electrode, followed by weighing, is among the oldest of electroanalytical methods, with the earliest examples dating from about 1800. In a controlled-potential process, the potential of the electrode is adjusted to a value where the desired plating reaction occurs, while no interfering reaction leading to the deposition of another insoluble substance takes place. Metals determined by electrogravimetry and their deposition potentials are given in Table 12.2.2. Electrogravimetry is not restricted to the determination of metals but can be applied to any electrode process that adds mass to an electrode securely enough to survive the removal, drying, and weighing of the electrode (1, 3, 22). The sensitivity is limited by the ability to determine the small difference in weight between the electrode itself and the electrode plus deposit. The physical characteristics of a deposit depend on the chemical form of the analyte in the solution, the presence of adsorbable surface-active agents in the solution, and other factors (23–26). Metal depositions from solutions of complex ions frequently are smoother than those obtained from solutions containing only the aquo form. For example, brighter deposits are ] than from a nitrate obtained from solutions of Ag+ in a CN− medium [containing Ag(CN)− 2 medium. The addition of a surfactant (a “brightener”), such as gelatin, often leads to improved deposits. Hydrogen evolution during deposition can lead to a rougher deposit, and deposits at very large current densities tend to be less adherent and rougher than those obtained at lower ones.

Table 12.2.2 Deposition Potentials (V vs. SCE) for Metals at a Platinum Electrode Supporting Electrolyte 1.2 M NH3 + 0.2 M NH4 Cl

0.4 M KCN + 0.2 M KOH

EDTA + NH4 OAc(a)

Metal

0.2 M H2 SO4

0.4 M NaTart + 0.1 M NaHTart

Au

+0.70

(+0.50)(c)

—

−1.00

+0.40

Hg

+0.40

(+0.25)(c)

−0.05

−0.80

+0.30

Ag

+0.40

(+0.30)(c)

−0.05

−0.80

+0.30

Cu

−0.05

−0.30

−0.45

−1.55

−0.60

Bi

−0.08

−0.35

—

(−1.70)(c)

−0.60

Sb

−0.33

−0.75

—

−1.25

−0.70

Sn(b)

—

—

—

—

—

Pb

—

−0.50

—

—

−0.65

Cd

−0.80

−0.90

−0.90

−1.20

−0.65

Zn

—

−1.10

−1.40

−1.50

—

Ni

—

—

−0.90

—

—

Co

—

—

−0.85

—

—

(a) 5 g NH4 OAc + 200 mL H2 O (pH ≈ 5); [EDTA]:[metal] = 3:1. (b) Tin can be deposited from solutions of Sn(II) in HCl or HBr media. (c) Metal deposits obtained are not suitable for electrogravimetric analysis. Adapted from Tanaka (3).

12.3 Controlled-Current Methods

x1 = 0.999

1.0 x

1 0.5

2 0.36 V

x2 = 0.001

0.0

EC

E

Figure 12.2.3 Conditions for separation of metals M1 and M2 at an Hg electrode (n1 = n2 = 1) at the part-per-thousand level.

Electrogravimetry is also performed extensively with the quartz crystal microbalance (QCM) (27, 28), but this mode does not involve bulk electrolysis. It is better considered with other applications of the QCM (Section 21.2). 12.2.5

Electroseparations

Electrochemical separations historically have been focused on metals selectively deposited into a mercury electrode. The considerations of Section 12.1.1 concerning the completeness of elec0′ is the formal potential for the n -electron trolysis as a function of potential are applicable. If Ea1 1 reduction of metal M1 to the amalgam, and if the volume of the mercury electrode equals the 0′ − volume of the electrolyte, then complete (i.e., ≥99.9%) deposition of M1 requires E ≤ Ea1 ′ 0 + 0.18∕n V. Therefore, the separation 0.18∕n1 V at 25 ∘ C. For ≤0.1% deposition of M2 , E ≥ Ea2 2 −1 ) V (Figure 12.2.3). If |E 0′ − E 0′ | between the formal potentials must be at least 0.18(n−1 + n 1 2 a2 a1 is smaller than this, a separation at the 99.9% level cannot be accomplished. In that case, changing the supporting electrolyte to one that complexes one or both of the metals will often give an improved separation. This approach is applicable to most transition metals (29).

12.3 Controlled-Current Methods 12.3.1

Characteristics of Controlled-Current Electrolysis

The course of a bulk electrolysis under controlled-current conditions can be worked out from i − E curves like those in Figure 12.3.1, describing the reduction of species O. As long as the applied current, iapp , is less than the limiting current, il (t), the electrode reaction proceeds ∗ (t) and i (t) decrease linearly with time. with 100% current efficiency. During this phase, CO l Eventually, il (t) falls to the point where it equals iapp , at which ∗ (t) = CO

iapp nFAmO

(12.3.1)

At any subsequent time, iapp > il (t); hence, the potential must adopt a more negative value, so that an additional electrode reaction can furnish the current, iapp − il (t), that can no longer be supported by direct reduction of O. The current efficiency drops below 100%. In this phase, the potential remains on the mass-transfer-controlled plateau of the O → R reduction; therefore, the electrolysis of O proceeds as if it were being carried out under controlled-potential

501

12 Bulk Electrolysis

i t1 = 0 t2

iapp

t3

E3

E1

E4

E2

E5 E6

t4

Background current

t5 t6

E

Figure 12.3.1 Current–potential curves at the working electrode for times increasing from t1 to t6 during bulk electrolysis with a constant applied current, iapp . The electrode potential shifts from E 1 to E 6 during the course of the electrolysis, with the largest shift occurring (between curves 3 and 4) as il falls below iapp . ↑ (–)

E4

E5

E6

E E1 0

E3

E2

t3

t→ (a)

Current efficiency for O → R

502

100% Current to second process

50%

0

0

t3

t→ (b)

Figure 12.3.2 (a) Potential and (b) current efficiency for the electrolysis illustrated in Figure 12.3.1.

conditions. The current supported by this reaction decays exponentially, as in (12.2.7) (Figure 12.3.2b). If iapp is larger than the initial limiting current, the rate of electrolysis of O will be the same as if the reduction were carried out under controlled-potential conditions, but with much lower current efficiency.6 The selectivity of a constant-current electrolysis is intrinsically poorer than the corresponding controlled-potential method because a new electrode reaction always occurs after the potential shift. One might avoid the additional process by using iapp 1),

Cathodic→

12.4 Electrometric End-Point Detection

0.0

0 ←Anodic

E/V vs. NHE

i

0.2

0.4

0.6

0.8 f (a)

1.0

1.2

1.4

0.0

0.2

0.4

0.6

0.8 f (b)

1.0

1.2

1.4

Figure 12.4.2 Electrometric titration curves. (a) Potentiometric curve at one polarizable electrode based on the measurement of the open-circuit (zero-current) potential. Points correspond to the open circles in Figure 12.4.1. (b) Amperometric curve for an indicator cell with one polarizable electrode held at 0.95 V vs. NHE. Points correspond to the filled circles in Figure 12.4.1. There is a slight slope change at the equivalence point. More important is the change in sign of the current at f = 1.

the electrode is poised, first by Fe3+ /Fe2+ and then by Ce4+ /Ce3+ . During both phases, the potential is well-defined and becomes more slightly more positive with increasing f . At the beginning (f = 0) and at the equivalence point (f = 1), the potential is unpoised and is able to make a rapid transition. These features are evident in the i − E curves and in the titration curve. Although the few points represented do not fully represent the shape of the titration curve, one can readily understand the effects of the poised and unpoised phases in defining that shape. Potentiometry with one polarizable electrode can also be carried out with an impressed nonzero current. Many pH/voltmeters used for potentiometric end-point detection can support this option, although it is infrequently employed. The impressed current would be small on the scale of Figure 12.4.1 and can be either anodic or cathodic at the indicator electrode. The measured potentials would be at the intersections of the i − E curves with the current level applicable at the indicator electrode. If the impressed current is anodic, the intersection points would be more positive than the open circles in Figure 12.4.1. If the impressed current is cathodic, they would be more negative. In either case, the titration curve would still have basically the same shape near the equivalence point. An impressed current can be advantageous if one or both couples involved in the titration have sluggish electrode kinetics. Another variant is two-electrode potentiometry, where two identical polarizable indicator electrodes are employed. A constant current is always impressed. The set of i − E curves applies separately to each indicator electrode; however, one is a cathode and the other is an anode, each having the same current magnitude. The titration curve can be worked out by finding the potential difference, ΔE, between the intersection points of each i − E curve with the small, equal cathodic and anodic currents. The resulting plot of ΔE vs. f shows a peak at the equivalence point. 12.4.3

Amperometric Methods

One-electrode amperometry involves maintaining the potential of a polarizable indicator electrode at a constant value with respect to a nonpolarizable reference. The current is then

509

510

12 Bulk Electrolysis

measured as a function of f . In Figure 12.4.1, the filled points at E = 0.95 V vs. NHE show the amperometric values for a polarizable indicator electrode set at that potential. The resulting titration curve appears in Figure 12.4.2b. Because the currents are typically linear with the concentrations of species involved in the titrations, these titration curves are normally made up of straight sections. Usually, there is a slope change at the equivalence point because the mass-transfer coefficients of the analyte couple and titrant couple differ. Depending on the choice of potential, the titration curve may pass through the zero-current axis at the equivalence point, as in the example of Figure 12.4.2b. Two-electrode amperometry involves the use of two polarizable indicator electrodes with a small constant potential difference, ΔE, impressed between them. Since they are in the same current loop, the anodic current in one will be equal in magnitude to the cathodic current in the other. The two electrodes are always ΔE apart, flanking the zero-current potential, and situated so that their currents are equal and opposite. The magnitude of the current is roughly proportional to the derivative of the i − E curve at the zero-current potential. Problem 12.6 invites the reader to work out the concept and to understand why this method is sometimes called dead-stop end-point detection. It is a well-used approach, lending itself to automation and being standard for the coulometric Karl Fischer titration.

12.5 Flow Electrolysis An alternative method of bulk electrolysis involves flowing the solution to be electrolyzed continuously past a working electrode surface (45–47). The flow may arise from mechanical or electrohydrodynamic pumping, or even by gravity feed. Flow electrolytic methods can result in high efficiencies and rapid conversions. They are especially convenient where large amounts of solution are to be treated. Flow methods are used in industrial situations (e.g., removal of metals such as copper from waste streams) and have been broadly applied to electrosynthesis, separations, and analysis (45). Flow electrolysis cells (Figure 12.5.1) are designed to show high conversions with a minimum length of electrode and maximum flow velocities. Normally, they contain a working electrode of large surface area, composed, for example, of screens of finely meshed metal, beds of conductive material (e.g., graphite or glassy carbon grains, metal shot, or powder), or a conductive foam such as RVC. If a divided cell is not necessary, as in metal deposition, the counter electrode can be interleaved with the working electrode and insulated from it with simple separators. Divided cells require more complex structures (including separators such as porous glass, ceramics, or ion-exchange membranes). If the cell is intended to operate at controlled potential, careful placement of the counter and reference electrodes is also required to minimize iR drops. However, flow cells are often designed to operate at controlled current, and, in that case, no reference electrode is required. Many practical flow cells are designed to be used as modules in multi-stage systems (called stacked cells), in which the output of one cell feeds the input of the next, with all cells in a stack operating identically (45). Cell stacking allows the size of an electrochemical reactor to be readily tailored to achieve high conversion efficiency for the electrochemical system of interest. The cell in Figure 12.5.1b is designed to facilitate stacking. 12.5.1

Mathematical Treatment

Consider a flow-through porous electrode of length L (cm) and cross-sectional area A (cm2 ) immersed in a stream of volumetric flow rate v (cm3 /s) (Figure 12.5.2). The linear flow velocity

Outlet Glass wool Working electrode contact

Glass frit Sat’d KCI solution Counter electrode contact

Glassy carbon granules

Working electrode

Working electrode contact

Elastomer RVC working gasket electrode

Ag/AgCl reference electrode Solution outlet

Ag wire counter electrode Porous carbon tube

Solution outlet

Working electrode contact

Counter electrode contact

Solution inlet

Solution inlet

Electrolyte Membrane chamber separator

Elastomer gasket

Inlet (a)

(b)

Figure 12.5.1 Flow electrolytic cells. (a) Cell for coulometry at controlled potential utilizing a working electrode of glassy carbon granules. [Reprinted with permission from Fujinaga and Kihara (48). © CRC Press, Inc., Boca Raton, FL.] (b) A two-compartment flow cell for electrosynthesis featuring a three-dimensional working electrode of reticulated vitreous carbon (RVC). This cell involves separate flow streams for the two compartments. The blocks defining the flow compartments are machined from an insulating polymer. The elastomer gaskets on both sides provide seals to prevent leakage, but each has an opening in the shape of the flow channel for liquid access and electrical contact. The counter electrode is mounted in the plate at right. When this cell is operated under controlled current, a reference electrode is not required. A similar cell is discussed in detail elsewhere (49). [Pletcher, Green, and Brown (45)/American Chemical Society/Licensed under CC BY 4.0.]

512

12 Bulk Electrolysis

dx

A/cm2

CO(x)

υ/cm3 s–1

CO(in)

CO(out)

CR(in) = 0

CR(out)

x=0

x

L

Figure 12.5.2 Schematic diagram of the working electrode of a flow electrolysis cell.

of the stream, U (cm/s), is given by v U= (12.5.1) A The reaction being carried out at the electrode, O + ne → R, is assumed to occur with 100% current efficiency. The inlet concentration of O is C O (in), while C R (in) is assumed to be zero. At the outlet, the concentrations are C O (out) and C R (out). If R is the fraction of O converted (R = 0, no conversion; R = 1, 100% conversion), then CO (out) = CO (in)(1 − R)

(12.5.2a)

CR (out) = CO (in)R

(12.5.2b)

To obtain an expression for R, let us consider operation of the electrode for a time period, Δt, during which solution volumes of vΔt enter and exit the electrode (46, 50–54). The numbers of moles of O entering and exiting are CO (in)vΔt and CO (out)vΔt. Since the rate of conversion of O to R in the electrode is i/nF (mol/s), the number of moles converted during Δt is iΔt/nF. Consequently, CO (out)vΔt = CO (in)vΔt −

iΔt nF

(12.5.3)

or CO (out) = CO (in) − R=1−

CO (out) CO (in)

=

i nFv

(12.5.4)

i nFvCO (in)

(12.5.5)

We also desire to express the current in terms of certain characteristics of the electrode that first require definition. The total internal area, which encompasses the sum of the areas of all pores, is a (cm2 ), and the total electrode volume is LA (cm3 ). Porous electrodes are frequently characterized by their specific area, s, given by s(cm−1 ) = a(cm2 )∕LA(cm3 )

(12.5.6)

Figure 12.5.3 and its caption define relationships between these and other electrode characteristics. The concentration of O decreases continuously with distance from the front face of the electrode (x = 0), and the local current density, j(x), varies with x, where j(x) relates to internal area. Since a slab of thickness dx has an internal area of sA ⋅ dx, the current in the slab is j(x)sA ⋅ dx

12.5 Flow Electrolysis

0.008 cm

0.008 cm

1 cm

1 cm

0.016 cm

υ =1 cm3/s 1 cm

Figure 12.5.3 Ideal porous electrode illustrating calculation of the specific area, s, and porosity, 𝜀. Consider the electrode as a cube 1 cm × 1 cm × 1 cm, containing straight pores, each 0.008 cm in diameter, spaced with centers 0.016 cm apart. The total number of pores, N, on a 1-cm2 face is ∼3900; the internal area of each pore is 2𝜋rL = 𝜋(0.008 cm)(1 cm) = 0.025 cm2 ; the total internal electrode area is a = (3900)(0.025 cm2 ) = 98 cm2 ; the total electrode volume is 1 cm3 ; the specific area is s = 98 cm2 /1 cm3 = 98 cm−1 ; the facial area of each pore is 𝜋r2 = 5.0 × 10–5 cm2 ; the total open area on faces is ap = (3900)(5 × 10−5 cm2 ) = 0.2 cm2 ; and the porosity is 𝜀 = 0.2 cm2 /1 cm2 = 0.2. If the volumetric flow rate is v = 1 cm3 /s, the linear flow velocity is U = (1 cm3 /s)/(1 cm2 ) = 1 cm/s, and the interstitial velocity is W = (1 cm/s)/0.2 = 5 cm/s.

and the corresponding rate of conversion is j(x)sA ⋅ dx/nF (mol/s). From (12.5.4), we can express the differential of concentration at x as −dC O (x) =

j(x)sA ⋅ dx nFv

(12.5.7)

This is a steady-state system; therefore, we can write for mass-transfer-limited conditions (Section 1.3.2) that j(x) = nFmO CO (x)

(12.5.8)

Combining (12.5.7) and (12.5.8) yields −

dC O (x) dx

=

mO CO (x)sA

(12.5.9)

v

and integration from the input to location x produces ∫C

CO (x)

dC O (x)

(in)

CO (x)

O

=

−mO sA v (

CO (x) = CO (in) exp

x

∫0

−mO sA

(12.5.10)

dx )

(12.5.11)

x v ( ) −mO sA j(x) = nFmO CO (in) exp x v

(12.5.12)

The total current in the electrode is then L

i=

∫0

(

L

j(x)sA dx = nFmO CO (in)sA

∫0

[ ( )] −mO sAL i = nFC O (in)v 1 − exp v

exp

−mO sAx v

) dx

(12.5.13)

(12.5.14)

513

514

12 Bulk Electrolysis

which can be combined with (12.5.5) to yield (50) ( ) −mO sAL R = 1 − exp v

(12.5.15)

The mass-transfer coefficient, mO , is a function of flow velocity, U, and is sometimes given as mO = bU 𝛼

(12.5.16)

where b is a proportionality factor and 𝛼 is a constant frequently having values between 0.33 and 0.5 for laminar flow and increasing up to nearly 1 for turbulent flow. With (12.5.16) and (12.5.1), equations 12.5.14 and 12.5.15 take the forms i = nFAUCO (in)[1 − exp(−bU 𝛼−1 sL)]

(12.5.17)

R = 1 − exp(−bU 𝛼−1 sL)

(12.5.18)

Thus, the conversion efficiency, R, increases with decreasing flow velocity and increasing specific area and length of electrode. From (12.5.11), one can see that the concentration of O varies exponentially with distance along the electrode. At the output, ( ) −mO sAL CO (out) = CO (in) exp (12.5.19) v The local current density, j(x), is highest at the front face of the electrode and decreases exponentially with x. These equations can also be cast in a form comparable with those for batch bulk electrolysis. If the total front-surface open area of the pores is ap , then the porosity is defined (Figure 12.5.3) as 𝜀=

ap

(12.5.20) A The linear flow velocity in the liquid stream, U, increases upon entering the electrode to an interstitial velocity, W , given by U v v W= = = (12.5.21) 𝜀 A𝜀 ap A volume element of solution moves down a pore at this velocity, and if it entered the electrode at time t = 0, then at time t, it will be a distance x, given by Ut x = Wt = (12.5.22) 𝜀 This development allows the equations to be formulated in terms of time, so that substitution of (12.5.22) into (12.5.11) yields ( ) −mO s CO (t) = CO (in) exp t (12.5.23) 𝜀 which is of the same form as that for a batch electrolysis, (12.2.6), with mO s/𝜀 = p (compared to mO A/V = p). Thus, the cell factor, p, increases with mass-transfer rate and specific area but decreases with porosity. From (12.5.15), the length of porous electrode required for a given conversion R can be obtained as v ln (1 − R) (12.5.24) L=− mO sA

12.5 Flow Electrolysis

The time for an element of solution to transit the electrode, 𝜏, sometimes called the residence time, is derived from (12.5.22) and (12.5.24) as 𝜏=

L𝜀 = p−1 ln(1 − R) U

(12.5.25)

An alternative, simplified approach to the efficiency of electrolysis in a porous electrode is ′ based on the time, t , required for O in the center of a pore of radius r to diffuse to the wall (55): t′ ≈

r2 2DO

(12.5.26)

From (12.5.21) and (12.5.22), one finds that the time required to move through the electrode down a pore of length L is 𝜏=

Lap

(12.5.27)

v ′

which is an alternate estimate of the residence time. If 𝜏 is greater than or equal to t , a high conversion (R ≈ 1) will be attained. Thus, we find that the flow velocity required for high conversion must satisfy the expression: v≤

2ap LDO r2

(12.5.28)

For example, consider a porous silver electrode with A = 0.2 cm2 , 𝜀 = 0.5, L = 50 μm, r = 2.5 μm, DO = 5 × 10−6 cm2 /s, and ap = 𝜀A = 0.1 cm2 . To obtain R → 1, flow velocity should not exceed 0.1 cm3 /s, so that the residence time is at least ∼5 ms. The simple treatment given here can be employed to find general conditions for an efficient flow electrolysis involving limiting-current conditions. However, we have neglected (a) resistive drops in the electrode and in the solution in the pores, (b) kinetic limitations on the electron-transfer reaction, and (c) the possibility of a current efficiency less than unity. Treatments taking these other effects into account are available (56–58) but usually require numerical solution. Flow cells operating at R = 1 are convenient for the continuous analysis of liquid streams, since the measured current is directly proportional to the concentration of the substance undergoing electrolysis; that is, from (12.5.5), CO (in) = i∕nFv. This case is a continuous coulometric analysis; hence, the method is absolute and requires neither calibration nor knowledge of mass-transfer parameters or electrode area (59). 12.5.2

Dual-Electrode Flow Cells

Flow cells that incorporate two working electrodes in the flow channel (Figure 12.5.4) have also been described (48, 55). These can be considered as the flow-coulometric equivalent of the rotating ring-disk electrode, where convective flow carries material from the first working electrode to the second. This strategy has been used in the coulometric analysis of plutonium (48), where the two working electrodes were large beds of glassy carbon particles, with the first electrode used to adjust the oxidation state of the plutonium to a single known level, Pu(IV), and the second used for a coulometric analysis based on Pu(IV) + e → Pu(III). Dual-electrode flow cells can also be used to analyze for the products of the first electrode (the generator electrode), by electrolysis of those products at the second (the collector electrode).

515

516

12 Bulk Electrolysis

Figure 12.5.4 Schematic representation of a dual-electrode flow cell. The concept follows that in reference (55), which shows apparatus for gravity-based solution flow from an upper reservoir.

Solution flow Counter electrode 1 Reference electrode 1 Working electrode 1 (generator) Gap Working electrode 2 (collector)

Reference electrode 2

Counter electrode 2

The most obvious case involves reversal electrolysis, such as O + ne → R at the generator and R → O + ne at the collector. Thin, but efficient, working electrodes separated by a small gap, g, are desired. A system involving two porous Ag-disk working electrodes (50-μm average pore diameter), separated by a 200-μm layer of porous Teflon, has been described (55). Each working electrode was provided with its own reference and counter electrodes; hence, this was a six-electrode cell, requiring two separate potential control circuits. The characteristics of the working electrodes were as given in the example following (12.5.28). In an experiment involving the reduction of Fe(III) to Fe(II) at one working electrode, the flow rate where the current began to deviate from that predicted by (12.5.5) with R = 1 was in reasonable agreement with the estimated maximum flow velocity for good conversion (∼0.1 cm3 /s). At this flow rate, the transit time across the gap is about 40 ms. High collection efficiencies, i.e., |igenerator /icollector |, were found even for flow rates where R < 1. Such a system can be useful for studying homogeneous reactions coupled to the electron-transfer reaction. Not only is collection efficiency informative about the stability of the product at the generator, but also the ultimate products of the electrode process are available in the effluent and can be readily analyzed. Application of this cell to a study of the isomerization of the radical anion of diethylmaleate in DMF was reported (60). Dual-electrode flow cells also find use in microfluidic systems [Sections 12.5.3(b,c)]. 12.5.3

Microfluidic Flow Cells

An important growth area is the use of flow cells for electrochemical detection, characterization, or electrosynthesis in microfluidic systems. In this book, the word microfluidic describes a system in which an electrolyte is handled on a microliter scale or smaller, so that an electrochemical process can be carried out. In general, a fluid can be manipulated by flow or as individual droplets, but this section dwells entirely on microfluidic flow cells, which largely originated during the highly successful development of liquid chromatography with electrochemical detection (LCEC) (61–64). Microfluidic detectors for capillary electrophoresis (CE) and flow injection (FI) soon followed and are also now widely employed (61–63). Subsequent research has focused on the creation of smaller systems, often with chemically tailored surfaces to produce selective sensors. For a fuller view of goals, strategies, and progress, the interested

12.5 Flow Electrolysis

reader is referred to published reviews (65, 66). Here, we will focus on the principles underlying electrochemical flow cells in microfluidic systems. Most of the applications mentioned in this section involve detection of an analyte; therefore, noise is a concern. It is usually important to drive the flow using a mechanism that does not increase the noise level from the cell. Since this is not an electrochemical matter, we will not discuss it in detail, but the reader should be aware of the issue. (a) Liquid Chromatography with Electrochemical Detection

Because LCEC motivated the development of microfluidic flow cells and still is the largest area of application, it is appropriate to introduce the general principles in that context. LCEC cells may be coulometric, where all of the material flowing into the cell is electrolyzed, but more frequently they are amperometric or voltammetric. Many different cell geometries and flow arrangements have been employed. General requirements (67) include • • • • • •

Well-defined hydrodynamics. Low dead volume. High mass-transfer rate. High signal/noise. Robust design. Reproducible working- and reference-electrode behavior.

An important factor is the nature of solution flow with respect to the electrode, which can follow any of several patterns (Figure 12.5.5). For all of them, the hydrodynamic equations governing the cell current can be solved using the approaches discussed in Chapter 10 (67, 68). Table 12.5.1 summarizes the resulting equations for mass-transfer-limited currents. Inlet

Outlet to reference electrode

Inlet Counter electrode

b Outlet to reference electrode

Working electrode (a)

Working electrode (b)

Outlet to counter electrode

Electrode

Electrode r

Electrode

w Flo

r

Flow a

l Flow Electrode

b (c)

(d)

(e)

Flow (f)

Figure 12.5.5 Common cell configurations: (a) Thin-layer cell. (b) Wall-jet cell. Various electrode geometries: (c) tubular electrode; (d) planar electrode with parallel flow; (e) planar electrode with perpendicular flow; and (f ) wall-jet electrode. [Adapted from Gunasingham and Fleet (67), by courtesy of Marcel Dekker, Inc.]

517

518

12 Bulk Electrolysis

Table 12.5.1 Limiting Currents for Different Cell Geometries Electrode Geometry

Limiting Current Equation

Tubular

i = 1.61nFC(DA∕r)2∕3 v1∕3

Planar, parallel flow in channel

i = 1.47nFC(DA∕b)2∕3 v1∕3

Planar, perpendicular flow

i = 0.903nFCD2/3 𝜈 −1/6 A3/4 U 1/2

Wall jet

i = 0.898nFCD2∕3 𝜈 −5∕12 a−1∕2 A3∕8 v3∕4

a= diameter of jet, A= electrode area, b= channel height, C= concentration, D= diffusion coefficient, 𝜈= kinematic viscosity, r= radius of tubular electrode, v = average volume flow rate (cm3 /s), U= flow velocity (cm/s). Adapted from Elbicki, Morgan, and Weber (68). See Figure 12.5.5 for illustration of types.

The thin-layer and wall-jet configurations predominate in commercial devices.10 Electrode materials are typically different forms of carbon (e.g., carbon paste or glassy carbon), or Pt, Au, or Hg, although other metals, such as Cu, Ni, and Pb, find application for particular analyses (e.g., amino acids, carbohydrates). Within the basic thin-layer concept, one has different choices for placement of the reference and counter electrodes (Figure 12.5.6). The concept in Figure 12.5.6a is the simplest, but it produces both a nonuniform current distribution across the working-electrode surface and a high uncompensated resistive drop to the reference electrode. The design in Figure 12.5.6b produces a uniform current density but still shows uncompensated resistive drop. In this arrangement, potentially interfering products can be formed at the counter electrode and may react at the working electrode to produce an unwanted current. Such interference will not occur if the (a)

(c)

(e)

C

W

W

C

W

R

(d)

(b)

C

C, R

W

C

R

W

R Ionically conducting membrane

R

C

Figure 12.5.6 Thin-layer electrochemical detector cells involving different placements of the working (W), counter (C), and reference (R) electrodes. [Lunte, Lunte, and Kissinger (63), by courtesy of Marcel Dekker, Inc.] 10 This section references concepts of thin-layer electrochemistry, which are yet to be presented. Effort has been made to make this section independently readable, but some users may find it worthwhile to examine Section 12.6 in parallel.

12.5 Flow Electrolysis

5-HT

5-HIAA

0.4 mL/min

EPI

NE

Figure 12.5.7 Liquid chromatographic separation of tryptophan and tyrosine metabolites using amperometric detection with a glassy carbon working electrode at 0.65 V vs. Ag/AgCl in a thin-layer cell. NE, norepinephrine; EPI, epinephrine; DOPAC, 3,4-hydroxyphenylacetic acid; DA, dopamine; 5-HIAA, 5-hydroxyindole-3-acetic acid; HVA, homovanillic acid; 5-HT, serotonin (5-hydroxytryptamine). Structures in Figure 1. [From Huang and Kissinger (69), with permission.]

DOPAC DA

flow velocity is sufficiently high to carry the detected electroactive species through the cell in a short time compared to the time needed for diffusion across the cell (perpendicular to the flow direction). At the expense of complexity in cell design and greater maintenance, a separator membrane can be added between the two parallel electrodes, as shown in Figure 12.5.6c. In principle, the reference electrode can be placed nearer the working electrode, as shown in Figure 12.5.6d,e, but this is difficult with conventional reference electrodes. The simplest electrochemical technique for use in flow cells is amperometry, where the working-electrode potential is fixed at a value where the analytes of interest are oxidized or reduced and are detected by the corresponding current as they pass through the cell (Figure 12.5.7). Sensitivity is determined by the current produced by the electroactive species compared to the background current. A detection limit at about the 0.1 pmol level can be achieved for oxidizable substances. Higher detection limits, at the 1 pmol level, are found with reducible substances because of higher background currents from oxygen reduction and other processes. Amplification of the response is possible when the cell configuration is that in Figure 12.5.6b and the redox couple is reversible (70). If the gap is small enough, the product at the working electrode can diffuse to the counter electrode and be converted back to the detected analyte, which can, in turn, return to the working electrode to be detected again. This process of redox cycling (Sections 12.6.3 and 19.6) can be repeated for each molecule of analyte while it remains between the electrodes. The effectiveness of cycling depends upon the flow rate (determining residence time) and the spacing between electrodes C and W (determining the diffusion time to cross the gap). Redox cycling is practical when the diffusion time is much smaller than the residence time. In that case, many more electrons are passed per molecule of analyte than in the absence of the cycling.

HVA

1 nA

4 min

519

520

12 Bulk Electrolysis

Greater selectivity and better information for qualitative identification can be obtained with the detector electrode operated in the voltammetric mode, where its potential is scanned over a given potential window during elution. However, the detection limits are much higher in the voltammetric mode because of the large background current, which arises partly from double-layer charging, but more significantly from slow faradaic processes associated with adaptation of the electrode surface to a changing potential (Section 8.4.4). The situation can be improved by use of square-wave or staircase voltammetry, but the best sensitivity is always associated with an electrode operating at a fixed potential in a mobile phase of unchanging composition. An alternative is to utilize cells with dual working electrodes maintained at different potentials (70), while monitoring the current at each simultaneously (Figure 12.5.8). If the electrodes are placed side-by-side, perpendicular to the solution flow (parallel arrangement), each is exposed to the same sample components, with one used to establish a background current level and the other to detect the species of interest. The electrodes can also be arranged along the direction of solution flow (70), as in Section 12.5.2 (series arrangement). In this case, the downstream working electrode monitors (collects) products from the upstream one. The effect can be used to improve selectivity when the analyte of concern has chromatographic interferences that do not generate products detectable at the second electrode. An important problem with flow cells is electrode fouling with continued use. Although an LC column is effective in removing some impurities that can foul the electrode surface, sometimes Outlet

Inlet

Locking collar Bushing Reference electrode O-ring Counter electrode Gasket Quick release mechanism

Working electrode Parallel Series

Figure 12.5.8 Cell with dual working electrodes and cross-flow design. [Reprinted from Lunte, Lunte, and Kissinger (63), by courtesy of Marcel Dekker, Inc.]

12.6 Thin-Layer Electrochemistry

the electrode reaction used for detection, such as the oxidation of phenols, will itself create insulating layers on the electrode surface. In such a case, it is necessary to clean the electrodes, perhaps by cycling the potential between anodic and cathodic limits (Sections 6.8.1 and 14.6) or using other types of potential programs to obtain reproducible behavior. Steps like these can oxidize or desorb surface impurities and return the electrode surface to a reproducible state. Automated operation in this mode is sometimes called pulsed amperometric detection (PAD) (61, 64, 71, 72). (b) Capillary Electrophoresis

Detector cells much like those we have just discussed are commonly used for CE (61–63, 73–75); however, cell design for CE is more demanding because much smaller volumes are involved and because one can encounter electrical interference from the high applied electric field and associated current flow that drive the electrophoretic separation. (c) Dual-Electrode Cells

The serial dual-electrode concept presented in Section 12.5.2 has already been presented in the context of LCEC above; however, it has also been employed in miniaturized microfluidic cells designed for other purposes (76–79). A parallel dual-electrode concept has been introduced for electrosynthesis (80), in which the flow cell consists of working and counter electrodes facing each other across a gap of only 25–500 μm. The intent is to facilitate cross reaction of electrolytic products from the two electrodes. The electrode processes are chosen to produce separate intermediates that can react in the flow channel to generate a final product, isolable from the exiting stream. This concept has been shown to be effective in a stacked-cell configuration, leading to high overall conversion efficiencies. Gram-scale production in good yield has been demonstrated.

12.6 Thin-Layer Electrochemistry An alternative approach to large A/V conditions is to confine a tiny solution volume (a few μL) in a thin layer (2–100 μm) at the electrode surface. The concept is illustrated in Figure 12.6.1a, and a practical cell is shown in Figure 12.6.1b. If the solution thickness, l, is much smaller than the diffusion layer thickness for a given experimental time, i.e., l ≪ (2Dt)1/2 , the solution will be homogenized by diffusion, and special bulk electrolysis equations will apply. At shorter times, mass-transfer by diffusion (and sometimes by migration) must be considered. After discovery of the effect (82–84), the theory and applications of thin-layer electrochemical cells were developed extensively and have been reviewed (81, 85, 86). Systems like that in Figure 12.6.1b are quite useful for certain purposes; however, the thin-layer concept has gained much broader importance in electrochemistry. Here, we develop the thin-layer concept in its original form, but we will enlarge it repeatedly, beginning in Sections 12.6.3 and 12.6.4 and continuing in Chapters 17–20. 12.6.1

Chronoamperometry and Coulometry

Most thin-layer cells adhere to the model in Figure 12.6.1a, in which there is an active electrode only on one boundary; however, systems in which both boundaries are active have also been realized. Much early theory of thin-layer systems was developed for “twin-working-electrode” cells in which both boundaries are active electrodes held at the same potential. The treatment

521

522

12 Bulk Electrolysis

Pressurized N2 Counter electrode

Stopcock Heavy Pt wire Reference electrode

Pinhole Pt rod

Working electrode

Reference electrode compartment

Glass wall Thin solution layer

Solution

0 Counter electrode

Working electrode

Precision-bore Pyrex tubing

l

x

Counter electrode Thin solution layer

Fine glass frit

(a)

Soft-glass bead

(b)

Figure 12.6.1 (a) Conceptual diagram of a single-electrode thin-layer cell. (b) A type of thin-layer cell used in routine work. The solution layer is in the small space between the Pt rod, which is the working electrode, and the inner surface of the precision-bore capillary. The layer thickness is typically 2.5 × 10−3 cm. The metal rod may be concentrically positioned within the capillary by machining three small flanges onto the surface of the rod near each end. Highly reproducible rinsing and filling of the thin layer cavity are accomplished by alternately applying and releasing nitrogen pressure with the stopcock. The heavy Pt wire connected to the rod is the working electrode connection. It also acts as a spring to re-seat the working electrode after solution is expelled from the thin-layer zone and is replaced by capillary action. [From Hubbard and Anson (81), by courtesy of Marcel Dekker, Inc.]

of thin-layer electrolysis in the prior editions of this book was based on that concept. In this edition, the theory is based on a single active boundary because this is the more common model in practice. Suppose the working-electrode potential is stepped from E1 , where no current flows, to E2 , where the reaction O + ne → R occurs, and the concentration of O at the electrode surface goes essentially zero. To obtain the i − t behavior and the concentration profile one must solve the linear diffusion equation, ( ) 𝜕CO (x, t) 𝜕 2 CO (x, t) = DO (12.6.1) 𝜕t 𝜕x2 with the boundary conditions ∗ (0 ≤ x ≤ l) CO (x, 0) = CO

(t > 0)

CO (0, t) = 0 [ JO (l, t) = −DO

(12.6.2)

𝜕CO (x, t) 𝜕x

(12.6.3) ] =0

(12.6.4)

x=l

With equations (12.6.1)–(12.6.3), this problem begins exactly as for the Cottrell experiment (Section 6.1.1). However, the semi-infinite boundary condition does not apply to the thin-layer cell and must be replaced by (12.6.4), which states that no flux can cross the inert boundary at

12.6 Thin-Layer Electrochemistry

1.0

(a) (b)

0.8 CO CO*

(c) 0.6 (d) 0.4 0.2

(e) (f)

0.0 0.0

0.2

0.4

x/l

0.6

0.8

1.0

Figure 12.6.2 Dimensionless concentration profiles of O during reduction in a thin-layer cell with a single active surface at x = 0. Curves correspond to times when (2Dt)1/2 /l is equal to (a) 0, (b) 0.06, (c) 0.18, (d) 0.57, (e) 1.26, and (f ) 1.79.

x = l. Solution of these equations by the Laplace transform method yields (81) 11 [ ] ∗ ∑ ∞ ( ) 4CO −(2m − 1)2 𝜋 2 DO t (2m − 1)𝜋x 1 exp CO (x, t) = sin 2 𝜋 m=1 2m − 1 2l 4l Dimensionless concentration profiles are shown in Figure 12.6.2. For an electrode of area A, [ ] 𝜕CO (x, t) i(t) = nFADO 𝜕x x=0 Using (12.6.5) to evaluate the derivative, one obtains [ ] ∗ ∑ ∞ 2nFADO CO −(2m − 1)2 𝜋 2 DO t i(t) = exp l 4l2 m=1

(12.6.5)

(12.6.6)

(12.6.7)

At later times, the current decay can be expressed using only the m = 1 term, since the (2m − 1)2 factor in the exponential causes the terms for m = 2, 3, … to be small for any appreciable value of 𝜋 2 DO t/l2 . Then ) ( ∗ 2nFADO CO −𝜋 2 DO t i(t) ≈ exp (12.6.8) l 4l2 or i(t) ≈ i(0) exp(−pt)

(12.6.9)

with p=

𝜋 2 DO 4l2

=

𝜋 2 DO A 4Vl

=

mO A V

(12.6.10)

where mO =

𝜋 2 DO 4l

11 The twin-electrode case is covered in Section 11.7.2, second edition, and Section 10.7.2, first edition.

(12.6.11)

523

524

12 Bulk Electrolysis

and 8nFAC∗O mO

(12.6.12) 𝜋2 The form of (12.6.9) is the same as (12.2.7), which would apply to the thin-layer cell if the concentration within the cell could be considered completely uniform throughout the electrolysis. The total charge passed by the electrolytic reaction is the integral of (12.6.7), which is { [ ]} ∞ ( )2 −(2m − 1)2 𝜋 2 DO t 8 ∑ 1 ∗ Q(t) = nFVCO 1 − 2 exp (12.6.13) 𝜋 m=1 2m − 1 4l2 i(0) =

In the long-time limit, this becomes Q(t → ∞) = nFVC∗O = nFN O

(12.6.14)

where N O is the number of moles of species O in the cell when the step is applied. Equation (12.6.14) is the same as the coulometry equation, (12.2.12), and it shows that determinations of N O or n are possible without knowledge of DO . The electrolysis rate constant for a thin-layer cell, p, can be quite large. For example, for DO = 5 × 10−6 cm2 /s and l = 10−3 cm, p = 49 s−1 , and the electrolysis would be 99% complete in t = 4.6/p (Section 12.2.1), or about 0.1 s. In actual coulometric experiments, the measured charge will be larger than that given by (12.6.14) because of additional contributions from double-layer charging, electrolysis of adsorbed species, and background reactions (Section 17.3.2) 12.6.2

Potential Sweep in a Nernstian System

Consider again the system of Section 12.6.1, but now for the situation where the working electrode potential is swept from an initial value, Ei , where no reaction occurs, toward negative values. Under conditions where the concentrations of O and R can be considered uniform in thin layer [i.e., C O (x, t) = C O (t) and C R (x, t) = C R (t) for 0 ≤ x ≤ l], the current is given, as in (12.2.4), by [ ] dC O (t) i = −nFV (12.6.15) dt For a nernstian reaction, ′ RT CO (t) E = E0 + ln (12.6.16) nF CR (t) Let us define C ∗ = CO (t) + CR (t) Combination of (12.6.16) and (12.2.17) yields C∗𝜃 CO (t) = 1+𝜃 𝜃 = enf (E−E

0′ )

(12.6.17)

(12.6.18a) (12.6.18b)

Differentiation of (12.6.18a) and substitution into (12.6.15), with the sweep rate, v, recognized as |dE/dt|, yields the following expression for the current (using the positive sign on the right-hand side): i=±

n2 F 2 vVC ∗ 𝜃 RT (1 + 𝜃)2

(12.6.19)

12.6 Thin-Layer Electrochemistry

1.0

Negative-going scan 0.5

i/µA

0.0

+0.1

0.0

–0.1

(E – E0ʹ)/V

–0.5 Positive-going scan

–1.0 ∗ = 1.0 mM, Figure 12.6.3 Cyclic voltammogram for a nernstian reaction with n = 1, V = 1.0 μL, v = 1 mV/s, CO T = 298 K. [From Hubbard and Anson (81), by courtesy of Marcel Dekker, Inc.

Although we have developed this problem for a negative-going (reductive) scan, the same equation applies to the positive-going (oxidative) case, but with the use of the negative sign on the right-hand side. Thus, a cyclic transition from O to R and back again yields a CV with mirror-image forward and reverse scans (Figure 12.6.3). ′ The peak current occurs at 𝜃 = 1, or E = E0 , and is given by ip = ±

n2 F 2 vVC ∗O 4RT

(+ for cathodic, − for anodic)

(12.6.20)

The peak current is directly proportional to v, but the total charge under the i − E curve is independent of v and is given by (12.6.14). In this treatment, we have assumed that the O/R couple is in equilibrium with the electrode potential, E, everywhere in the cell. This is an approximation that can hold only if the scan rate is low. The rigorous solution for this problem (81), accounting for nonuniform concentrations within the cell, shows that the approximate form, (12.6.19), holds when ( ) RT 𝜋 2 D 1−𝜀 (12.6.21) |v| ≤ log nF 3l2 1+𝜀 where 𝜀 is the relative error in the calculation of ip from (12.6.20). Thin-layer methods have been suggested for determination of kinetic parameters of electrode reactions (81, 87),12 but they have not been widely applied. A difficulty with conventional 12 A treatment of the totally irreversible, one-step, one-electron reaction is provided in Section 11.7.3, second edition, and Section 10.7.3, first edition.

525

526

12 Bulk Electrolysis

thin-layer cells is the high solution resistance in the thin layer, especially when nonaqueous solutions or low supporting electrolyte concentrations are employed. Since the reference and auxiliary electrodes are placed outside the thin layer, one can have seriously nonuniform current distributions and high uncompensated iR drops (producing for example, nonlinear potential sweeps) (88, 89). Although cell designs that minimize this problem have been devised (81), careful control of experimental conditions is required in any measurement of voltammetric response. 12.6.3

Dual-Electrode Thin-Layer Cells

Early in Section 12.6.1, a mention was made of thin-layer cells in which both boundaries are active electrodes. In the “twin-electrode” case, both boundaries are at the same potential. Conceptually more important is the “dual-electrode” cell, having independently controlled electrodes facing each other on opposite boundaries of the thin layer. This idea was originally realized (90) in thin-layer systems such as we have been discussing, capturing electrolyte of micron-scale thickness. When one electrode operates as an anode and the other as a cathode, and when the electrode processes are based on a single stable couple, these cells are able to support a steady state in which the reduction product at the cathode is re-oxidized at the anode and vice versa. Each electroactive molecule in the thin layer is thereby placed in a steady pattern of roundtrips between the electrodes. Over time, each molecule contributes many electrons to the current. Thus, these cells provided the first demonstration of redox cycling between neighboring electrodes, a phenomenon introduced in Section 5.6.3. The effect is especially important in scanning electrochemical microscopy (Section 18.1), where electrochemistry is confined to a nanometer-scale gap between a movable electrode and a conducting or insulating substrate. Redox cycling is treated in detail in Section 19.6 because certain experimental methods for single-molecule electrochemistry rest upon it. In Problem 19.5, the reader is invited to work out some of the relevant dynamics, which are applicable whether the system is of “classic” size (l in the μm range) or much smaller (l in the nm range). Redox cycling has also been exploited in microfluidic flow cells [Section 12.5.3(a)]. 12.6.4

Applications of the Thin-Layer Concept

Conventional thin-layer cells are standard tools for investigating adsorption (91, 92) and for determining n for an electrode process. They have also become widely employed for spectroelectrochemistry (Chapter 20). The theory and mathematical treatments underlying the thin-layer concept find application in many other important electrochemical problems. In Section 12.6.3, we have already noted their applicability to microfluidic cells, SECM, and single-molecule electrochemistry, but there is more: • The deposition of metals into thin films of mercury and their subsequent stripping (Section 12.7) is fundamentally a thin-layer problem. • The electrochemical oxidation or reduction of thin films (e.g., oxides, adsorbed layers, and precipitates) follows an analogous treatment (Section 17.2). • Thin-layer concepts often describe the behavior of synthetically modified electrodes featuring electroactive species bound to the surface (Chapter 17). In many problems involving surface films, mass transfer truly is negligible over wide time domains, and problems with uncompensated resistance are minimal; thus, relatively fast experiments can be performed.

12.7 Stripping Analysis

12.7 Stripping Analysis 12.7.1

Introduction

Stripping analysis is an analytical method represented pictorially in Figure 12.7.1. It begins with a preconcentration step, in which an analyte is collected from a sample solution, either onto the surface of an electrode or into a small volume (historically, a hanging mercury drop or a thin mercury film). After preconcentration, a rest period may be observed, then the deposited analyte is electrolytically re-dissolved (“stripped”) using a voltammetric technique, usually LSV, DPV, or SWV. If the preconcentration is reproducible, it is unnecessary for it to involve exhaustive electrolysis of the solution. By proper calibration, the measured voltammetric response (a peak current or an integrated charge) can be reliably employed to determine the analyte concentration in the sample solution. The major advantage vs. direct voltammetric analysis is the preconcentration (by factors of 100 to >1000), which amplifies the analyte response relative to background. Stripping analysis is especially useful for very dilute samples (down to 10−10 to 10−11 M). It is most frequently used to determine metal ions by cathodic deposition, followed by anodic stripping with a linear potential scan; thus, it is sometimes called anodic stripping voltammetry (ASV) or, less frequently, inverse voltammetry. The basic theoretical principles and typical applications will be described here. Extensive reviews are available (93–98).

Preconcentration

←Anodic Cathodic →

M+n + ne → M(Hg) (a) Stirrer off

i

E

Ed

Rest period

Stripping

(b)

M(Hg) → M+n + ne (c)

(10–100 s) ts td (1–15 min)

tr (30 s)

t

(–1.0 V vs. SCE)

–0.5 V 0

Ep

t

Figure 12.7.1 Anodic stripping at an HMDE. Values shown are typical of analysis for Cu2+ . (a) Preconcentration at E d ; stirred solution. (b) Rest period; stirrer off. (c) Anodic scan (v = 10 − 100 mV/s). [Adapted from Barendrecht (93), by courtesy of Marcel Dekker, Inc.]

527

528

12 Bulk Electrolysis

12.7.2

Principles and Theory

The mercury electrode used in most stripping analysis is either an HMDE [usually at an SMDE; Section 8.2.3(b)] or a mercury film electrode (MFE) deposited onto a rotating glassy carbon or wax-impregnated graphite disk. If an MFE is used, it is often created during the preconcentration period by co-deposition with the analyte. One usually adds mercuric ion (10−5 to 10−4 M) directly to the analyte solution to enable the formation of a suitable film, which is often less than 100 Å thick. Since the MFE has a much smaller volume than the HMDE, stripping analysis at an MFE shows higher sensitivity. There is evidence that mercury electrodes on Pt dissolve some platinum on prolonged contact, with possible deleterious effects; hence, a platinum substrate is usually avoided. Solid electrodes (e.g., Pt, Ag, C) may be used (without mercury) for elements that cannot be determined at mercury (e.g., Ag, Au, Hg). To avoid the use of mercury, alternatives are being sought (94). Especially promising is the bismuth film electrode (BiFE), which is co-deposited with the analyte in the same manner as an MFE (99). In stripping analysis for common metal ions, BiFE electrodes have shown performance comparable with an MFE. Preconcentration is carried out in a stirred solution at a potential Ed , which is several tenths ′ of a volt more negative than E0 for the least easily reduced metal ion to be determined. The behavior generally follows the principles for a controlled-potential electrolysis (Section 12.2.1). However, since the electrode area is so small, and t d is much smaller than the time needed for exhaustive electrolysis, the current for any electroactive species remains essentially constant (at its il ) during this step, and the number of moles of that species deposited is then il t d /nF. Because the electrolysis is not exhaustive, the deposition conditions (stirring rate, t d , temperature) must be the same for the sample and the standards. With an HMDE, one observes a rest period, during which the stirrer is turned off, the solution is allowed to become quiescent, and the concentration of the amalgam becomes more uniform. The stripping step is then carried out by LSV, DPV, or SWV, with the potential being scanned toward more positive values. When an MFE is used, the stirring during deposition is by rotation of the substrate disk. A rest period usually is not observed, and rotation continues during the stripping step. The behavior governing the i − E curve during the anodic scan depends on the type of elec∗ , at the start trode employed. For an HMDE of radius r0 , the concentration of reduced form, CM of the scan is uniform throughout the Hg drop. When the sweep rate, v, is sufficiently high that ∗ at the completion of the scan, then the concentration in the middle of the drop remains at CM the behavior is essentially that of semi-infinite diffusion. If LSV is used for stripping, the basic treatment of Section 7.2 applies (100). At sufficiently large scan rates, the effects of sphericity become negligible and linear diffusion applies, with ip proportional to v1/2 . Practical stripping measurements are usually carried out in this regime. Because the volume and thickness of an MFE are small, the stripping behavior with this electrode generally follows thin-layer behavior (Section 12.6). The theoretical treatment for stripping by LSV at the MFE has appeared (101, 102); a diagram of the model employed is shown in Figure 12.7.2. If the stripping reaction is assumed to be reversible, the Nernst equation holds at the outer surface. The solution of the diffusion equations with this condition plus the initial and boundary conditions shown in Figure 12.7.2 leads to an integral equation that must be solved numerically. At small v and l, thin-layer behavior predominates and ip ∝ v. For high v and large l, semi-infinite linear diffusion predominates and ip ∝ v1/2 . MFEs used in current

12.7 Stripping Analysis

Pt

Hg

* CM

DM

𝜕CM(x,t) 𝜕x

Solution CM+n(y,t)

CM(x,t) t=0

–DM

𝜕CM(x,t) 𝜕x

= DM+n x=0

𝜕CM+n(y,t) 𝜕y

=

i nFA

y=0

t=t

=0 x=l

t=0

* +n = 0 CM x=l

x=0 y=0

←x

y→

Figure 12.7.2 Initial conditions and boundary conditions for theoretical treatment of stripping at an MFE. Electrode reaction is M(Hg) → M+n + ne.

practice fall in the region where thin-layer behavior can be expected for the usual LSV sweep rates (≤500 mV/s). 12.7.3

Applications and Variations

Controlled-potential cathodic deposition followed by anodic stripping with a linear potential sweep has been applied to the determinations of a number of metals (e.g., Bi, Cd, Cu, In, Pb, and Zn), either alone or in mixtures (Figure 12.7.3). A significant increase in sensitivity can be (a)

(b)

Cd In 0.2 µA

← Anodic current

0.1 µA

Cu

Pb

In

Cd In 0.2 µA

(c)

Cu 0.0 –0.2

Pb

Cu

Cd

Cu

Pb –0.4

(d)

–0.6

–0.8 0.0 E/V vs. SCE

Cd Pb

–0.2

–0.4

In 0.2 µA –0.6

–0.8

Figure 12.7.3 Stripping curves for 2 × 10−7 M Cd2+ , In3+ , Pb2+ , and Cu2+ in 0.1 M KNO3 . v = 5 mV/s. (a) HMDE, td = 30 min. (b) Pyrolytic graphite, td = 5 min. (c) Unpolished glassy carbon, td = 5 min. (d) Polished glassy carbon, td = 5 min. For (b–d), 𝜔/2𝜋 = 2000 rpm and Hg2+ was added at 2 × 10−5 M. [From Florence (103), with permission.]

529

530

12 Bulk Electrolysis

obtained by using DPV or SWV for the stripping step. Other variants, such as stripping by a potential step or current step, have also been proposed (93–98). Interferences that sometimes occur with mercury electrodes involve • Reactions of the metals with the substrate material (e.g., Pt or Au). • Formation of a compound between a deposited metal and Hg (e.g., Ni–Hg) • Formation of an intermetallic compound between two metals deposited into the mercury at the same time (e.g., Cu–Cd or Cu–Ni). Such effects are much more serious with mercury films than with hanging drops because MFEs involve more concentrated amalgams and a high ratio of substrate area to film volume. Interferences like these can be overriding concerns in the choice between an MFE and an HMDE. On the other hand, MFEs offer much better sensitivity and better control of mass transfer during the deposition step. If an HMDE must be chosen (e.g., to reduce interferences), one can use differential-pulse stripping to obtain sensitivities comparable to those attained by LSV at an MFE. Since stripping at an MFE exhausts the thin film, the voltammetric peaks are narrow and often allow baseline resolution of multicomponent systems. Thin-layer properties and the sharpness of the peaks permit relatively fast stripping sweeps, which in turn shorten analysis times. In contrast, the falloff in current past the peak in a stripping voltammogram obtained at an HMDE is based on diffusive depletion, rather than exhaustion, and it continues for quite some time. Thus, the peaks are broader, and overlap of adjacent peaks is more serious. (Compare, for example, Figure 12.7.3a,d.) This problem is usually minimized for the HMDE by using slow sweep rates, at a cost of lengthened analysis time. Cathodic stripping voltammetry can be carried out for species (usually anions) that deposit in an anodic preconcentration. For example, the halides (X− ) can be determined at mercury by deposition as Hg2 X2 . Deposition on solid electrodes, such as Ag, is also possible (Problem 12.12). In such systems, surface problems (e.g., oxide films) and underpotential deposition effects13 often appear. On the other hand, the sensitivity for stripping from a solid electrode is high, since the deposit can be removed completely, even at high scan rates. Stripping of films has often been used to determine the thickness of coatings (e.g., Sn on Cu) and oxide layers (e.g., CuO on Cu).14 A variation of stripping analysis involves the electrolytic removal of species that have spontaneously adsorbed on the surface of an electrode in a non-electrolytic preconcentration. This technique, called adsorptive stripping voltammetry, can be applied to sulfur-containing species, organic compounds, and certain metal chelates that adsorb on Hg and Au (94–97, 105). Examples include cysteine (and proteins that contain this amino acid), dissolved titanium in the presence of the chelator solochrome violet RS, and the drug diazepam. The amounts found by this method are typically limited to monolayer levels. However, similar approaches can be employed with thicker polymer layers that can interact with solution species. Related experiments are described in Chapter 17.

13 Underpotential deposition refers to the phenomenon in which the first monolayer of a substance M (e.g., Cu) binds more strongly to a substrate (e.g., Pt) than subsequent layers of M bind to each other. An example is discussed in Section 10.5. The topic is covered further in Section 15.6.3. In stripping analysis, one can find that the last layer stripped is voltammetrically harder to remove than the earlier layers. 14 One of the earliest electroanalytical (coulometric) methods was the determination of the thickness of tin coatings on copper wires (104).

12.8 References

As the range of analytes has broadened due to the advent of adsorptive stripping or stripping based on complexation or binding in a polymer layer, much research has been dedicated to new types of electrode materials (94, 106) and to the incorporation of stripping into automatable systems, especially those involving flow-based analysis (107).

12.8 References 1 J. J. Lingane, “Electroanalytical Chemistry,” 2nd ed., Wiley-Interscience, New York, 1958,

Chaps. 13–21. 2 P. Delahay, “New Instrumental Methods in Electrochemistry,” Wiley-Interscience, New

York, 1954, Chaps. 11–14. 3 N. Tanaka, in “Treatise on Analytical Chemistry,” Part I, Vol. 4, I. M. Kolthoff and

P. J. Elving, Eds., Wiley-Interscience, New York, 1963, Chap. 48. 4 E. Leiva, Electrochim. Acta, 41, 2185 (1996). 5 D. G. Davis, in “Instrumental Analysis,” H. H. Bauer, G. D. Christian, and J. E. O’Reilly,

Eds., Allyn and Bacon, Boston, MA, 1978, pp. 93–110. 6 T. F. Fuller and J. N. Harb, “Electrochemical Engineering”, Wiley, Hoboken, NJ, 2018. 7 J. Newman and K. E. Thomas-Alyea, “Electrochemical Systems,” 3rd ed., Wiley, Hoboken,

NJ, 2004. 8 D. Pletcher and F. C. Walsh, “Industrial Electrochemistry,” 2nd ed.; Chapman and Hall,

London, 1990. 9 C. J. Burrows, Ed., “Electrifying Synthesis,” Acc. Chem. Res. (Special Issue), 2020. https://

pubs.acs.org/page/achre4/electrifying-synthesis. 10 D. Pletcher, Electrochem. Commun., 88, 1 (2018). 11 A. Wiebe, T. Gieshoff, S. Möhle, E. Rodrigo, M. Zirbes, and S. R. Waldvogel, Angew. Chem. 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Int. Ed., 57, 5594 (2018). M. Yan, Y. Kawamata, and P. S. Baran, Chem. Rev., 117, 13230 (2017). R. Francke and R. D. Little, Chem. Soc. Rev., 43, 2492 (2014). M. M. Baizer, Ed., “Organic Electrochemistry,” Marcel Dekker, New York, 1973. W. H. Smith and A. J. Bard, J. Am. Chem. Soc., 97, 5203 (1975). M. Paidar, V. Fateev, and K. Bouzek, Electrochim. Acta, 209, 737 (2016). J. J. Lingane, J. Am. Chem. Soc., 67, 1916 (1945). J. J. Lingane, Anal. Chim. Acta, 2, 584 (1948). A. J. Bard, Anal. Chem., 35, 1125 (1963). G. A. Rechnitz, “Controlled Potential Analysis,” Pergamon, New York, 1963. J. E. Harrar, Electroanal. Chem., 8, 1 (1975). H. J. S. Sand, “Electrochemistry and Electrochemical Analysis,” Blackie, London, 1940. J. A. Harrison and H. R. Thirsk, Electroanal. Chem., 5, 67–148 (1971). J. O. M. Bockris and A. Damjanovic, Mod. Asp. Electrochem., 3, 224 (1964). M. Fleischmann and H. R. Thirsk, Adv. Electrochem. Electrochem. Eng., 3, 123 (1963). M. Schlesinger and M. Paunovic, Eds., “Modern Electroplating,” 5th ed., Wiley, Hoboken, NJ, 2010. V. Tsionsky, L. Daikhin, M. Urbakh, and E. Gileadi, Electroanal. Chem., 22, 1 (2004). C. Gabrielli and H. Perrot, Mod. Asp. Electrochem., 44, 151 (2009). J. A. Maxwell and R. P. Graham, Chem. Rev., 46, 471 (1950). M. Pérez-Gallent, M. C. Figueiredo, I. Katsounaros, and M. T. M. Koper, Electrochim. Acta, 227, 77 (2017). J. J. Lingane, “Electroanalytical Chemistry,” op. cit., pp. 488–495.

531

532

12 Bulk Electrolysis

32 J. J. Lingane, C. H. Langford, and F. C. Anson, Anal. Chim. Acta, 16, 165 (1959). 33 A. J. Bard, in “Chemistry at the Nanometer Scale,” Proc. of the Welch Foundation 40th

Conf. Chem. Res., Robert A. Welch Foundation, Houston, TX, 1996, p. 235. 34 J. J. Lingane, “Electroanalytical Chemistry,” op. cit., Chap. 21. 35 H. L. Kies, J. Electroanal. Chem., 4, 257 (1962). 36 D. DeFord and J. W. Miller, in “Treatise on Analytical Chemistry,” Part I, Vol. 4, I. M.

Kolthoff and P. J. Elving, Eds., Wiley-Interscience, New York, 1963, Chap. 49. 37 P. A. Shaffer, Jr., A. Briglio, Jr., and J. A. Brockman, Jr., Anal. Chem., 20, 1008 (1948). 38 R. S. Braman, D. D. DeFord, T. N. Johnston, and L. J. Kuhns, Anal. Chem., 32, 1258 (1960). 39 J. J. Lingane, “Electroanalytical Chemistry,” op. cit., Chaps. 5–8, 12 and references to the

older literature contained therein. 40 C. N. Reilley and R. W. Murray (Chap. 43) and N. H. Furman (Chap. 45) in “Trea-

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

64 65 66 67

tise on Analytical Chemistry,” Part I, Vol. 4, I. M. Kolthoff and P. J. Elving, Eds., Wiley-Interscience, New York, 1963. J. T. Stock, “Amperometric Titrations,” Interscience, New York, 1965. W. D. Cooke, C. N. Reilley, and N. H. Furman, Anal. Chem., 23, 1662 (1951). I. M. Kolthoff and J. J. Lingane, “Polarography,” 2nd ed., Vol. 2, Interscience, New York, 1952, Chap. 47. L. Meites, “Polarographic Techniques,” 2nd ed., Wiley-Interscience, New York, 1965, Chap. 9. D. Pletcher, R. A. Green, and R. C. D. Brown, Chem. Rev., 118, 4573 (2018). R. E. Sioda and K. B. Keating, Electroanal. Chem., 12, 1 (1982). R. de Levie, Adv. Electrochem. Electrochem. Eng., 6, 329 (1967). T. Fujinaga and S. Kihara, CRC Crit. Rev. Anal. Chem., 6, 223 (1977). A. A. Folgueiras-Amador, A. E. Teuten, D. Pletcher, and R. C. D. Brown, React. Chem. Eng., 5, 712 (2020). R. E. Sioda, Electrochim. Acta, 13, 375 (1968); 15, 783 (1970); 17, 1939 (1972); 22, 439 (1977). R. E. Sioda, J. Electroanal. Chem., 34, 399, 411 (1972); 56, 149 (1974). R. E. Sioda and T. Kambara, J. Electroanal. Chem., 38, 51 (1972). I. G. Gurevich and V. S. Bagotsky, Electrochim. Acta, 9, 1151 (1964). I. G. Gurevich, V. S. Bagotsky, and Yu. R. Budeka, Electrokhim, 4, 321, 874, 1251 (1968). J. V. Kenkel and A. J. Bard, J. Electroanal. Chem., 54, 47 (1974). J. A. Trainham and J. Newman, J. Electrochem. Soc., 124, 1528 (1977). R. Alkire and R. Gould, J. Electrochem. Soc., 123, 1842 (1976). B. A. Ateya and L. G. Austin, J. Electrochem. Soc., 124, 83 (1977). E. L. Eckfeldt, Anal. Chem., 31, 1453 (1959). A. J. Bard, V. J. Puglisi, J. V. Kenkel, and A. Lomax, Faraday Discuss. Chem. Soc., 56, 353 (1973). J. Fedorowski and W. R. LaCourse, Anal. Chim. Acta, 861, 1 (2015). P. T. Kissinger, in “Encyclopedia of Chromatography,” 3rd ed., Vol. 1, J. Cazes, Ed., CRC Press, Boca Raton, FL, 2010, pp. 695–697. S. M. Lunte, C. E. Lunte, and P. T. Kissinger, in “Laboratory Techniques in Electroanalytical Chemistry,” 2nd ed., P. T. Kissinger and W. R. Heineman, Eds., Marcel Dekker, New York, 1996, Chap. 27. D. C. Johnson and W. R. LaCourse, Anal. Chem., 62, 589A (1990). D. G. Rackus, M. H. Shamsi, and A. R. Wheeler, Chem. Soc. Rev., 44, 5320 (2015). W. B. Zimmerman, Chem. Eng. Sci., 66, 1412 (2011). H. Gunasingham and B. Fleet, Electroanal. Chem., 16, 89 (1989).

12.8 References

68 69 70 71 72 73 74

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

J. M. Elbicki, D. M. Morgan, and S. G. Weber, Anal. Chem., 56, 978 (1984). T. Huang and P. T. Kissinger, Curr. Sep., 14, 114 (1996). D. A. Roston, R. E. Shoup, and P. T. Kissinger, Anal. Chem., 54, 1417A (1982). M. B. Jensen and D. C. Johnson, Anal. Chem., 69, 1776 (1997). W. R. LaCourse, Analysis, 21, 181 (1993). T. Sierra, A. G. Crevillen, and A. Escarpa, Electrophoresis, 40, 113 (2019). B. White, in “Capillary Electrophoresis and Microchip Capillary Electrophoresis,” K. Y. Chumbimuni-Torres, E. Carrilho, and C. D. García, Eds., Wiley, Hoboken, NJ, 2013, Chap. 9. J. Wang, Electroanalysis, 17, 1133 (2005). R. G. Compton, B. A. Coles, J. J. Gooding, A. C. Fisher, and T. I. Cox, J. Phys. Chem., 98, 2446 (1994). I. Dumitrescu, D. F. Yancey, and R. M. Crooks, Lab Chip, 12, 986 (2012). M. J. Anderson and R. M. Crooks, Anal. Chem., 86, 9962 (2014). Z. Kostiuchenko and S. G. Lemay, Anal. Chem., 92, 2847 (2020). Y. Mo, Z. Lu, G. Rughoobur, P. Patil, N. Gershenfeld, A. I. Akinwande, S. L. Buchwald, and K. F. Jensen, Science, 368, 1352 (2020). A. T. Hubbard and F. C. Anson, Electroanal. Chem., 4, 129 (1970). E. Schmidt and H. R. Gygax, Chimia, 16, 156 (1962). J. H. Sluyters, Rec. Trav. Chim., 82, 100 (1963). C. R. Christensen and F. C. Anson, Anal. Chem., 35, 205 (1963). A. T. Hubbard, CRC Crit. Rev. Anal. Chem., 2, 201 (1973). C. N. Reilley, Pure Appl. Chem., 18, 137 (1968). A. T. Hubbard, J. Electroanal. Chem., 22, 165 (1969). G. M. Tom and A. T. Hubbard, Anal. Chem., 43, 671 (1971). I. B. Goldberg and A. J. Bard, Anal. Chem., 38, 313 (1972). L. B. Anderson and C. N. Reilley, J. Electroanal. Chem., 10, 295 (1965). A. T. Hubbard, Chem. Rev., 88, 633 (1988). A. T. Hubbard, Acc. Chem. Res., 13, 177 (1980). E. Barendrecht, Electroanal. Chem., 2, 53–109 (1967). A. Economou and C. Kokkinos, RSC Detect. Sci. Ser., 6, 1 (2016). M. Lovri´c, in “Electroanalytical Methods,” 2nd ed., F. Scholz, Ed., Springer, Berlin, 2010, Chap. II.7. J. Wang, “Analytical Electrochemistry,” 3rd ed., Wiley, Hoboken, NJ, 2006, pp. 85–97. J. Wang, “Stripping Analysis: Principles, Instrumentation, and Applications,” VCH, Dearfield Beach, FL, 1985. F. Vydra, K. Štulík, and E. Juláková, “Electrochemical Stripping Analysis,” Halsted, New York, 1977. J. Wang, Electroanalysis, 17, 1341 (2005). W. H. Reinmuth, Anal. Chem., 33, 185 (1961). W. T. de Vries and E. Van Dalen, J. Electroanal. Chem., 8, 366 (1964). W. T. de Vries, J. Electroanal. Chem., 9, 448 (1965). T. M. Florence, J. Electroanal. Chem., 27, 273 (1970). G. G. Grower, Proc. Am. Soc. Test. Mater., 17, 129 (1917). J. Wang, Electroanal. Chem., 16, 1 (1989). A. Economou, Sensors, 18, 1032 (2018). A. Economou, Anal. Chim. Acta, 683, 38 (2010). H. A. Laitinen and N. H. Watkins, Anal. Chem., 47, 1352 (1975).

533

534

12 Bulk Electrolysis

12.9 Problems 12.1 When a 0.010-M solution of a metal ion, M2+ , is examined voltammetrically in a rapid scan at a large-area (10 cm2 ) rotating disk electrode, a limiting current of 193 mA is observed for reduction to metal M. Calculate the value of the mass transport coefficient, mM2+ , in cm/s. If 100 cm3 of the solution is electrolyzed at this electrode at controlled potential in the limiting current region, what time will be required for 99.9% of the M2+ to be plated out? How many coulombs will be required? 12.2 If the solution in Problem 12.1 is electrolyzed at a constant current of 80 mA under the same conditions: (a) What is the concentration of M2+ remaining in solution when the current efficiency drops below 100%? (b) How long does it take to reach this point? (c) How many coulombs have been passed? (d) How much longer will it take to decrease the M2+ concentration to 0.1% of its initial value? What is the overall current efficiency for removal of 99.9% of M2+ by this constant-current electrolysis? 12.3 Fifty milliliters of a ZnSO4 solution are transferred to an electrolytic cell with a mercury cathode; then, enough solid KNO3 is added to make the solution 0.1 M in KNO3 . The electrolysis of Zn2+ is carried to completion at −1.3 V vs. SCE with the passage of 241 C. Calculate the initial concentration of Zn2+ . 12.4 A solution contains 1.0 × 10−3 M X2+ and 3.0 × 10−3 M Y2+ , where X and Y are metals. A 200-cm3 volume is to be electrolyzed at a mercury pool electrode of area 50 cm2 and volume 100 cm3 . Under the stirring conditions and cell geometry, both X2+ and Y2+ ′ have mass-transfer coefficients, m, of 10−2 cm/s. The E0 values for reduction of X2+ and Y2+ to the metal amalgams are −0.45 and −0.70 V vs. SCE, respectively. (a) An i − E curve for the solution is taken under the above conditions. Make a neat, labeled, quantitatively correct sketch of the i − E curve that would be obtained, assuming no changes in concentrations in solution during the scan. (b) Derive an equation for completeness of electrolysis analogous to (12.1.5) for the case of deposition into a mercury pool. (c) If the electrolysis is to be performed at a controlled potential, at what potentials can X2+ be quantitatively deposited ( E10 ), so that species R1 diffusing away from the 2 Even though the medium is “aprotic,” there is often a proton source (such as a trace of water) capable of supplying H+ to a strongly basic reactant, such as the carbanion in (13.1.24).

543

544

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

electrode is capable of reducing O2 in solution. For the example mentioned above, the following homogeneous reaction is possible: XC6 H4 NO2 −∙ + ∙ C6 H4 NO2 ⇌ XC6 H4 NO2 + −∙∙ C6 H4 NO2

(13.1.25)

This step can be so important that the second electron transfer occurs entirely in bulk solution, rather than at the electrode surface (17) [Section 13.3.7(c)]. → − ← − (c) E C E Reaction

A distinctive reaction pattern is found when the product of a chemical reaction following the reduction of A at the electrode is oxidized at potentials where A is reduced (hence, the backward arrow on the second E) (18): A + e ⇌ A−

(13.1.26)

A− → B−

(13.1.27)

B− − e ⇌ B

(13.1.28)

Charges are explicitly shown only to emphasize the different directions of the two E-steps. As with EE and ECE reactions, there is also the possibility of a homogeneous electron transfer: A− + B ⇌ B− + A

(13.1.29)

The overall reaction is simply A → B. Thus, if (13.1.27) is fast, the conversion of A to B does not require a net transfer of charge from the electrode (i.e., n = 0). An example is the reduction of Cr(CN)3− in 2 M NaOH. When the stable complex Cr(CN)3− 6 6 − (A) is reduced (in the absence of CN ) to Cr(CN)4− (A− ), there is rapid loss of CN− to form 6 Cr(OH)n (H2 O)2−n (B− ), which is immediately oxidized to Cr(OH)n (H2 O)3−n (B). In general, 6−n 6−n → − ← − EC E reactions are based on structural changes triggered by the first E-step. An interesting extension is electron-transfer-catalyzed substitution (equivalent to the organic chemist’s SRN 1 mechanism) (1, 5, 7): RX + e ⇌ RX−

(13.1.30)

RX− → R + X−

(13.1.31)

R + Nu− → RNu−

(13.1.32)

RNu− − e ⇌ RNu

(13.1.33)

which may be accompanied by the solution-phase reaction, RX + RNu− → RX− + RNu

(13.1.34)

The overall reaction involves no net transfer of electrons and is a simple substitution, RX + Nu− → RNu + X−

(13.1.35)

(d) Square Schemes

Two electron-transfer reactions are sometimes coupled via two chemical reactions into a cyclic pattern called a “square scheme” (19): A + e ⇌ A− ↑↓

↑↓

B + e ⇌ B−

(13.1.36)

13.2 Impact of Coupled Reactions on Cyclic Voltammetry

This mechanism often occurs when there is a structural change on reduction, such as a cis-trans isomerization. An oxidative example is found in the electrochemistry of cis-W(CO)2 (DPE)2 [DPE = 1,2-bis(diphenylphosphino)ethane], where the cis-form (C) is oxidized to C+ , which isomerizes to the trans species, T+ . More complex reaction mechanisms arise by coupling square schemes into meshes, ladders, or fences (4). (e) Elaborations

Although we have now reviewed the most important basic reaction patterns, a great variety of others exist. Many can be treated as combinations or variants of the cases just discussed. A complex example is the reduction of nitrobenzene (PhNO2 ) to phenylhydroxylamine in liquid ammonia in the presence of proton donor (ROH). This case has been analyzed as an EECCEEC process (20): PhNO2 + e ⇌ PhNO2 −∙

(13.1.37)

PhNO2 −∙ + e ⇌ PhNO2− 2

(13.1.38)

O − − PhNO2− 2 + ROH ⇌ PhNOH + RO

(13.1.39)

O PhNOH− → PhNO (nitrosobenzene) + OH−

(13.1.40)

PhNO + e ⇌ PhNO−∙

(13.1.41)

PhNO−∙ + e ⇌ PhNO2−

(13.1.42)

H PhNO2− + 2ROH → PhNOH + 2RO−

(13.1.43)

Vast research has been dedicated to the elucidation of electrode reaction schemes by application of electrochemical methods. Many such studies have also included the identification of products and intermediates by spectroscopic techniques (Chapter 21) or chemical approaches.

13.2 Impact of Coupled Reactions on Cyclic Voltammetry Let us now turn to the ways in which coupled reactions influence responses from electrochemical experiments. The goal is practical—to learn how to use the responses for mechanistic diagnosis and for evaluation of kinetic or thermodynamic parameters. The path starts here with the theory, but one should understand that instincts and skill develop over time from both a sound grasp of theory and practical experience. It is easier to approach this topic by concentrating on CV and LSV, which we will do through the end of Section 13.3. In the laboratory, one almost always does the equivalent, turning first to CV for diagnosis of a new electrochemical system. Although all transient methods permit exploration of the i − E − t space and can often support equivalent diagnostic work, CV allows one to visualize the effects of E and t on the current in a single experiment (Figure 7.1.1). It also naturally illuminates linkages between multiple electroactive species in a system. 13.2.1

Diagnostic Criteria

In the electrochemical literature, the term diagnostic criteria is commonly used to encompass the most important observables, such as ipf , Epf , ΔEp , and |ipr /ipf |, together with their dependences on scan rate or the concentrations of principal species (21). In Section 7.3, we became familiar with ΔEp vs. v as a criterion for the reversibility, quasi-reversibility, or total

545

546

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

irreversibility of heterogenous electron-transfer kinetics. For coupled homogeneous reactions, the most important criteria are based on • The peak current for the forward reaction, ipf . • The characteristic potential of the forward reaction, Epf . • The reversal peak current ratio, |ipr /ipf |. Our concern is usually comparative: How does a coupled reaction alter the behavior of these characteristics from that seen for the unperturbed electrode reaction? The unperturbed behavior (e.g., the reversible electron transfer, O + ne ⇌ R) is the reference case, presumed to be already understood, either from theory or by experimental determination (e.g., in the absence of the species causing a following reaction). (a) Forward Peak Current

The details of a reaction scheme determine the extent to which coupled chemistry can affect ipf . For an EC case, for example, the flux of the electroreactant, O, is hardly changed by the C-step; therefore, any index of that flux, such as ipf in CV or LSV, is only slightly perturbed. In contrast, ipf for a catalytic system (EC′ ) is increased by the C′ step because it regenerates O. The extent of the increase depends on the duration (or characteristic time) of the observation (Section 13.2.2), which is controlled in CV or LSV by the scan rate. For short-duration experiments (high v), ipf may be near that for the unperturbed reaction because the regenerating reaction does not have time to regenerate O. For long-duration experiments (low v), the ipf will be larger than in the unperturbed case, perhaps markedly so. (b) Forward Peak Potential

The effect on Epf depends not only on the type of coupled reaction and the experimental duration, but also on the reversibility of electron transfer. Consider the Er Ci case (a nernstian electrode reaction followed by an irreversible chemical reaction): O + ne ⇌ R → X

(13.2.1)

During the experiment, the potential of the working electrode relates to the solution composition at the interface by the nernstian condition: ′ RT CO (0, t) E = E0 + ln (13.2.2) nF CR (0, t) The effect of the following reaction is to decrease C R (0, t) and to increase C O (0, t)/C R (0, t). The potential at any level of current will be more positive than in the absence of the perturbation, so the wave shifts toward positive potentials.3 For an Ei Ci case, where the electron transfer is totally irreversible, the following reaction causes no change in characteristic potential because the i − E characteristic contains no term involving C R (0, t). (c) Peak Current Ratio

Reversal results are usually very sensitive to following chemical reactions. For example, in the Ei Ci case, |ipr /ipf | < 1, if the following reaction creates a significant effect during the time of observation, as R is removed by reaction (as well as by diffusion) from the zone near the electrode. If the sweep is so fast that the following reaction cannot proceed appreciably during the voltammogram, |ipr /ipf | → 1, but for slower sweeps, |ipr /ipf | falls off toward lower v. For very slow sweeps, |ipr /ipf | = 0. 3 This result was shown for SSV using the simplified mass-transfer model in Section 1.4.2.

13.2 Impact of Coupled Reactions on Cyclic Voltammetry

13.2.2

Characteristic Times

As we have just seen, the effect of a perturbing reaction depends on the extent to which it can proceed during the observation; thus, it becomes valuable to define and to compare a characteristic time for reaction, 𝜏 rxn , and a characteristic time for observation, 𝜏 obs . Suppose, for example, that an electroproduct, R, engages in a chemical decay with rate constant k, • If the reaction is first order (e.g., R → X), then, 𝜏 rxn ≡ 1/k. One can easily show that 𝜏 rxn is the average lifetime of R [Problem 3.6(a)]. • If the reaction is a second-order process in which R reacts with species Z, then 𝜏rxn ≡ 1∕kC ∗Z . For the special case where CZ∗ is large compared to any concentration of R in the diffusion layer, the reaction is pseudo-first order, and 𝜏 rxn is again the average lifetime of R. For a given electrochemical method, 𝜏 obs characterizes the period for which time-dependent properties near the working electrode (such as concentrations in the diffusion layer) are interrogated. For CV or LSV, 𝜏 obs is usually defined as RT/nFv = 1/nfv, which is the time required to scan 25.7/n mV at 25 ∘ C (i.e., kT per electron). Thus, 𝜏 obs is not the duration of the entire experiment, or even of the forward scan, but is roughly the time required to scan the rising portion of a reversible wave. Events occurring in that time range or shorter control the shape of the wave. Those taking place on much shorter timescales will be fully manifested, while those occurring on much longer timescales will have hardly begun before the observation is over. Both 𝜏 rxn and 𝜏 obs are approximate indicators, not precise measures. Even so, they are valuable aids, mostly for order-of-magnitude comparisons. If, for example, the event of interest is a coupled reaction, it is now easy to see that the CV or LSV will be largely unperturbed if 𝜏 rxn ≫ 𝜏 obs and will reflect only the heterogeneous electron transfer kinetics. Conversely, if 𝜏 rxn ≪ 𝜏 obs , the perturbing reaction will be essentially complete during the observation and will fully manifest its effect on the CV. 13.2.3

An Example

Let us bring the ideas of this section together using Figure 13.2.1, which presents voltammograms for oxidation and re-reduction of Mo(CN)4− in a pH 10 aqueous solution to which incre8 ments of cysteine have been successively added (22). In the absence of cysteine (CV marked with open circles in Figure 13.2.1), Mo(CN)4− undergoes a 1e oxidation to the Mo(V) complex, 8 3− Mo(CN)4− 8 ⇌ Mo(CN)8 + e

(13.2.3)

At pH 10, this reaction is quasireversible on boron-doped diamond (BDD), with the peaks separating slightly as the scan rate is increased from 25 to 400 mV/s. In Figure 13.2.1, ΔEp ≈ 90 mV at v = 50 mV/s. For our present purpose, we need not concern ourselves with the quasireversibility. In the absence of cysteine, |ipr /ipf | is close to unity and ipf is very nearly linear with v1/2 , indicating diffusion control. The progressive addition of cysteine has two notable effects on the CV: 1) Measured against the charging-current background, the reverse peak becomes smaller in magnitude, almost disappearing at 500 μM cysteine. 2) ipf becomes much bigger. At 500 μM cysteine, the forward peak is threefold larger than at zero added cysteine. Effect 1 is a clear indicator that the product of (13.2.3), Mo(CN)3− , is removed from the dif8 fusion layer by a reaction promoted by cysteine. This is an EC case of some kind. The effect is

547

548

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

Figure 13.2.1 CV at v = 50 mV/s for 0.5 mM Mo(CN)4− in pH 10 borate buffer with 8 sequential additions of 50 μM cysteine. Dashed line shows CV for direct oxidation of 100 μM cysteine. Working electrode was boron-doped diamond (BDD). More positive potentials are to the right, and anodic currents are up. Three curves are marked with circles at the forward and reverse peaks: open circles, no cysteine; gray-filled circles, 250 μM cysteine; black-filled circles, 500 μM cysteine. [Adapted from Nekrassova et al. (22), with permission.]

40

30

20 Increasing cysteine

i/μA 10

0

Increasing cysteine

–10 0.00

0.25

0.50 E/V vs. SCE

0.75

1.00

progressively greater with increasing cysteine concentration, indicating that the characteristic time of reaction, 𝜏 rxn , shortens concomitantly. This behavior is consistent with a bimolecular reaction between Mo(CN)3− and cysteine, for which 𝜏rxn = 1∕kC ∗cysteine . Since the scan rate is 8 the same for all voltammograms, 𝜏 obs is constant. For the smallest concentrations of cysteine, 𝜏 rxn ≫ 𝜏 obs , and the effect of the following reaction does not strongly perturb the voltammetry, but at the highest cysteine concentrations, it must be true that 𝜏 rxn ≪ 𝜏 obs because the diffusion layer is almost cleared of Mo(CN)3− by the following reaction. 8 Effect 2 indicates that somehow Mo(CN)4− can deliver much more charge when cysteine is 8 present. We already know that ipf is diffusion-controlled at zero cysteine, so this result cannot represent any mass-transfer effect or change in the bulk concentration of Mo(CN)4− . Also, the 8 voltammetry remains at essentially the same mean potential, (Epf + Epr )/2; hence, the E-step seems to remain unaltered. Consequently, the C-step must generate a species that is oxidizable at the same potential as Mo(CN)4− . Note, however, that the increase of ipf shows no sign of 8 being bounded. If the system were an ECE case, there would be an upper bound for ipf because the n-value of the overall process would settle at n1 + n2 , where n1 applies to the first E-step, (13.2.3), and n2 , to the second. In sum, the results indicate that this is a catalytic system, Eq C′i , in which the C-step can be written as 4− Mo(CN)3− 8 + cysteine → Mo(CN)8 + Y

(13.2.4)

Since Mo(CN)4− is regenerated, it can react again at the electrode, elevating the apparent 8 n-value of the overall process. At high cysteine concentrations, the Mo complexes in the diffusion layer appear to undergo several catalytic cycles during 𝜏 obs . In Section 13.3, we will proceed through a survey of expected CV or LSV behavior for the principal mechanistic patterns defined in Section 13.1. In the process, diagnostic criteria such as those we have discussed here will be prominent. 13.2.4

Including Kinetics in Theory

For most electrochemical methods operating in diffusion-based systems, the theoretical methods developed in earlier chapters can be adapted to include the effects of coupled chemical reactions. One still must solve a set of partial differential equations expressing Fick’s second law

13.2 Impact of Coupled Reactions on Cyclic Voltammetry

under the appropriate initial and boundary conditions (Section 4.5); however, coupled kinetics add two types of complications: • The diffusion problem usually involves more participants than O and R and requires more differential equations. The concentrations of homogeneous reactants and intermediates (e.g., X, Y, or Z) also evolve in the diffusion layer and may need to be described. For any essential participant in the mechanism, there is an added PDE to be solved simultaneously with all others. • The PDE describing any participant engaged in a homogeneous reaction must also account for any reactive production or consumption of that species. Doing so requires one or more added terms. For example, consider the Er Ci reaction scheme in a system of linear diffusion: O + ne ⇌ R k

R −−→ Y

(at electrode)

(13.2.5)

(in solution)

(13.2.6)

For species R, the basic Fick’s law equation must be modified because the concentration of R at any location is altered by reaction, as well as by diffusion. The local rate of change caused by the homogeneous reaction is [ ] 𝜕CR (x, t) = −kC R (x, t) (13.2.7) 𝜕t chem.rxn. which adds to the local rate of change caused by diffusion. Thus, the diffusion-kinetic equation for species R becomes 𝜕CR (x, t)

𝜕 2 CR (x, t)

− kC R (x, t) 𝜕t 𝜕x2 Species O is not involved in reaction (13.2.6); hence, simple diffusion still applies, 𝜕CO (x, t)

= DR

(13.2.8)

𝜕 2 CO (x, t)

(13.2.9) 𝜕t 𝜕x2 Six initial and boundary conditions are needed to solve PDEs (13.2.8) and (13.2.9). Five are the initial conditions, the semi-infinite conditions, and the flux balance in the general formulation (Section 4.5.2). The sixth condition depends on the electrochemical method and the kinetics of the electron-transfer reaction, (13.2.5), just as we have seen many times [e.g., Section 7.2.1(a)]. In this case, the E-step is reversible; therefore, CO (0, t) 0′ = 𝜃(t) = enf [E(t)−E ] (13.2.10) CR (0, t) = DO

If CV is the chosen method, then E(t) = Ei − v[t − 2S𝜆 (t)(t − 𝜆)]

(13.2.11)

where 𝜆 is the time when the sweep is reversed and S𝜆 (t) is the unit step function. For any other mechanism, the diffusion–kinetic equations are deduced in the same manner from Fick’s second law and the appropriate homogeneous reaction rate equations. Table 13.2.1 provides the equations for several different reaction schemes, as well as appropriate boundary conditions for CV or LSV. Equations and conditions are not required for species that serve only as final products. For example, in (13.2.5)–(13.2.6), the concentration of species Y does not affect C O (x, t),

549

550

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

Table 13.2.1 Diffusion–Kinetic Equations and Conditions for Coupled Homogeneous Reactions Initial and Boundary Conditions (x,t) Case

Reactions

1. Cr Er

Y⇌O

Diffusion–Kinetic Equations

𝜕CY

kf

𝜕t 𝜕CO

kb

O + ne ⇌ R

= DR

𝜕t

Y⇌O

𝜕 2 CY

= DO

𝜕t 𝜕CR

kf

2. Cr Ei

= DY

𝜕x2 𝜕 2 CO

(x,0) and (∞, t)

− kf CY + kb CO

C O /C R = 𝜃(t) (Note a) CR = 0

𝜕x2

(as Case 1)

kb

C O /C Y = K C O + C Y = C*

+ kf CY − kb CO

𝜕x2 𝜕 2 CR

(as Case 1)

O + ne → R 𝜕CO

O + ne ⇌ R

3. Er Cr

𝜕t 𝜕CR

kf

R⇌Y

O + ne ⇌ R kf

R −−−→ Y

C O /C R = 𝜃(t) (Note a)

CY = 0

(Fluxes: Note b)

(as Case 3)

(as Case 3)

𝜕x2 𝜕 2 CY

(equation for C Y not required) 𝜕CO

= DO

𝜕t 𝜕CR

kf

= DR

𝜕t 𝜕CO

O + ne ⇌ R kf

= DR

𝜕t 𝜕CZ 𝜕t 𝜕CO

O1 + n1 e ⇌ R1

𝜕t 𝜕CR

kf

R1 −−−→ O2

𝜕t 𝜕CR 𝜕t

1

1

𝜕t 𝜕CO

O2 + n2 e ⇌ R2

= DZ

2

2

𝜕 2 CO

𝜕x2 𝜕 2 CR

= DO

𝜕t 𝜕CR

R + Z −−−→ O + Y

7. Er Ci Er

∗ CO = CO

= DY + kf CR − kb CY 𝜕t 𝜕x2 (as Case 3 with k b = 0)

2R −−−→ X 6. Er C′i

𝜕x2 𝜕 2 CR

[ ] DO 𝜕CO ∕𝜕x x=0 = k ′ CO ebt (Note c) (Fluxes: Note b)

CR = 0

O + ne ⇌ R

5. Er C2i

𝜕 2 CO

(Fluxes: Note b)

− kf CR + kb CY

= DR

𝜕t 𝜕CY

kb

4. Er Ci

= DO

(0, t > 0)

𝜕x2 𝜕 2 CO

𝜕x2 𝜕 2 CR

1

𝜕x2 𝜕 2 CR

1

= DO

𝜕x2 𝜕CR

2

− kf CR

CR = 0

(Fluxes: Note d)

∗ CO = CO

C O /C R = 𝜃(t) (Note a)

CZ = CZ∗

(Fluxes: Note e)

CO = C ∗

CO ∕CR = 𝜃1 (t) 1 1 (Note f )

2

CR = 0

1

+ kf CR

1

1

2

2

C O /C R = 𝜃(t) (Note a)

1

1

𝜕x2 𝜕 2 CO

1

∗ CO = CO

CR = 0

− kf CZ CR 𝜕x2 2 𝜕 CO

= DO

= DR

+ kf CZ CR − kf CZ CR

𝜕x2 𝜕 2 CZ

= DR

− kf CR2

(Fluxes: Note d)

CO = 0 2

CR = 0

𝜕x2

2

CO ∕CR = 𝜃2 (t) 2 2 (Note f ) (Fluxes: Note g)

′ − E0 )]

(a) For CV: 𝜃(t) = exp[nf (Ei exp{−nfv[t − 2S𝜆 (t)(t − 𝜆)]}, where Ei is the initial potential and S𝜆 (t) is the unit step function for the scan reversal time, 𝜆. (b) DO (𝜕C O /𝜕x)x = 0 = − DR (𝜕C R /𝜕x)x = 0 and DY (𝜕C Y /𝜕x)x = 0 = 0. ′

(c) For LSV: k ′ = k 0 exp[−𝛼f (Ei − E0 )] and b = 𝛼fv, where Ei is the initial potential. (d) DO (𝜕C O /𝜕x)x = 0 = − DR (𝜕C R /𝜕x)x = 0. ∗ , so (e) DO (𝜕C O /𝜕x)x = 0 = − DR (𝜕C R /𝜕x)x = 0 and DZ (𝜕C Z /𝜕x)x = 0 = 0. Often this problem is solved for CZ∗ ≫ CO ∗ that CZ → CZ everywhere. Then, the diffusion equation in Z is not needed, and the homogeneous kinetics are kf′

pseudo first-order, with R −−−→ O and kf′ = kf CZ∗ . ′ (f ) For CV: 𝜃j (t) = exp[nf (Ei − Ej0 )] exp{−nfv[t − 2S𝜆 (t)(t − 𝜆)]}, where Ei is the initial potential, S𝜆 (t) is the unit ′

step function for the scan reversal time, and Ej0 pertains to Oj + nj e ⇌ (g) DO (𝜕CO ∕𝜕x)x=0 = −DR (𝜕CR ∕𝜕x)x=0 and DO (𝜕CO ∕𝜕x)x=0 = −D 1

1

1

1

2

2

13.2 Impact of Coupled Reactions on Cyclic Voltammetry

C R (x, t), i, or E. However, if (13.2.6) were reversible, the concentration of species Y would appear in the analogue of (13.2.8), and a diffusion–kinetic equation for C Y (x, t) and initial and boundary conditions for Y would have to be supplied. This is Case 3 in Table 13.2.1. While systems based on semi-infinite linear diffusion encompass a great many cases of interest, others lie outside that domain. In foregoing chapters, we have addressed situations in which 1) 2) 3) 4)

Diffusion is geometrically simple, but nonlinear, as in SSV at a disk electrode (Chapter 5). The system is convective, as at an RDE or RRDE (Chapter 10). Migration is important, as in SSV with a low concentration of added electrolyte (Section 5.7). A complex geometry is involved, e.g., in SECM (Chapter 18), or with an array of electrodes (Section 5.6.3).

For any such case, one can still treat coupled kinetics essentially as described above; however, the mass-transfer part of the problem must differ from Fick’s second law for linear diffusion. It might just be Fick’s second law for some different geometry (Case 1 above). However, it might also require the convective term (Case 2) or the migration term (Case 3) from the modified Nernst–Planck equation. Or it might involve a more general mass-transfer expression, together with a complex set of boundary conditions defining an electrode array (Case 4). In every instance, the mass-transfer part of the problem is modified by kinetic terms exactly as above and the resulting set of PDEs is solved. Section 4.5 discusses relevant issues and options for solution. 13.2.5

Comparative Simulation

In current practice, solutions of diffusion–kinetic problems like those encountered in this chapter are nearly always obtained by digital simulation (8, 23–25).4 All simulators are, of course, numeric solvers for problems that are formulated mathematically just as we have discussed here. Versatile simulation packages are commercially available and are even integrated with some electrochemical workstations. The simulators allow easy definition of the reaction mechanism according to the user’s interest. Indeed, the practice with simulation goes well beyond just solving a problem involving mass transfer and kinetics. The standard method for evaluating parameters in an electrochemical kinetic model is now by an iterative approach that one might call comparative simulation. Repeated simulations of the system are carried out, varying the model parameters (homogeneous and heterogeneous rate constants, equilibrium constants, transfer coefficients, standard potentials, etc.) and comparing simulated voltammograms with the data set until the best fit is achieved (8, 9, 23). The fitting process can simultaneously encompass a complex voltammogram with multiple waves, or even a whole set of voltammograms embodying changes of experimental variables, such as scan rates or concentrations. In the most sophisticated approaches, all of this is done automatically by an algorithm guided by a statistically grounded measure of fit (e.g., the sum of squared deviations). In Section 13.3, we will encounter many examples where comparative simulation has been employed.5 There are limits to the complexity of a system that can be treated in this manner. We address that issue in Section 13.3.9. 4 The foundational work on electrochemical systems with coupled homogeneous reactions was mostly carried out analytically (often with numeric solutions). As an example of an analytical solution, the Er Ci case is treated for chronopotentiometry in Section 12.2.2 of the second edition and Section 11.2.2 of the first edition. 5 The application of this method is often signaled with just the single word, “simulation,” and the basis for comparison may be unclear. Sometimes, it is statistically well-based, but it may be based only on personal judgment.

551

552

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

13.3 Survey of Behavior This section covers theoretical expectations in CV or LSV for the full set of reaction schemes introduced in Section 13.1. The focus is on important limiting cases and on useful tools for mechanistic diagnosis and estimation of rate constants. 13.3.1

Following Reaction—Case Er Ci

The Er Ci mechanism is the simplest case of homogeneous chemistry coupled to an electrode reaction. We write it as kf

O + ne ⇌ R −−−→ Y

(13.3.1)

As a starting point for our survey, it is helpfully uncomplicated: • The homogeneous chemistry is triggered by the electrode reaction, so there is none to consider before the experiment begins. • There is only one homogeneous rate constant. • Because the homogeneous reaction is irreversible, there is no need to treat the diffusion or further reaction of species Y. The qualitative behavior is easy to predict and is represented in Figure 13.3.1.6 At a fast scan rate (Figure 13.3.1a), the observation time, 𝜏 obs , can be short enough that the following reaction does not appreciably proceed. The system would then show an unperturbed, reversible CV for the O/R couple. At progressively slower scan rates (Figure 13.3.1b–d), the C-step begins to proceed appreciably during the observational time, so that species R is lost by chemical reaction. With decreasing v, the reversal peak current, ipr , becomes smaller relative to ipf ; thus, |ipr /ipf | falls, eventually approaching zero at a very slow scan rate (Figure 13.3.1d). The decay of R causes the ratio of surface concentrations, C O (0, t)/C R (0, t), to become larger all along the wave, by comparison to the unperturbed situation. Therefore, the entire wave shifts progressively positively at lower scan rates, as the following reaction has more of an effect. This behavior is most easily seen in the positive shift of the forward peak potential, Epf , with decreasing v. (a) Dimensionless Kinetic Parameters and Zone Diagrams

In the theory of electrochemical responses, the ratio 𝜏 obs /𝜏 rxn is a principal dimensionless kinetic parameter, 𝜆. For different mechanisms and different experimental methods, 𝜆 takes varied functional forms because 𝜏 rxn depends on the nature of the chemistry, while 𝜏 obs depends on the method employed. For an Er Ci system observed by CV or LSV, 𝜏 rxn = 1/k f and 𝜏 obs = RT/nFv = 1/nfv; therefore, 𝜆=

k RT kf ⋅ = f nF v nfv

(13.3.2)

For any particular chemical system, k f is a constant and changes of 𝜆 manifest changes in timescale. For CV and LSV, 𝜆 is tunable by altering v. Alternatively, if one is comparing different Er Ci systems at a fixed scan rate, then differences in 𝜆 manifest differences in the rate constant, k f . An advantage of plotting the voltammetric response in terms of 𝜆 and dimensionless current is that a range of kinetic behaviors can be displayed together, as in Figure 13.3.1e. 6 For Figure 13.3.1, the electrode area is 1 cm2 . This is larger by an order of magnitude or more than would normally be used in CV. This area has been chosen for this figure and for others in this chapter so that the current scale is numerically the same as the current density.

13.3 Survey of Behavior

4 i/mA

10 V/s

2

1 V/s

(b)

0.5

i/mA

0

0.0

400 0.3 i/mA

1.0

(a)

0.2

200 0 –200 –400 (E – E1/2)/mV

400 0.10

(c)

0.1 V/s

200 0 –200 –400 (E – E1/2)/mV 0.01 V/s

(d)

i/mA 0.05

0.1

0.00

0.0

Dimensionless current

400 200 0 –200 –400 (E – E1/2)/mV (e)

400

λ = 500 10

200 0 –200 –400 (E – E1/2)/mV

0.1, 0.01

0.4 0.2 0.0 0.1 –0.2

0.01 180 120 60 0 –60 n(E – E1/2)/mV

Figure 13.3.1 Cyclic voltammograms for the Er Ci case at 25 ∘ C. A + e ⇌ B; B → C. (a–d) 1e system where 0 EA∕B = 0 V, CA∗ = 1 mM, CB∗ = 0, A = 1 cm2 , DA = DB = 10−5 cm2 /s, and kf = 10 s−1 . Scan rates, v, are (a) 10, (b) 1, (c) 0.1, and (d) 0.01 V/s. The vertical scale changes from frame to frame. (e) Dimensionless current in an n-electron case for several values of 𝜆 = kf /nfv. Dimensionless current is 𝜋 1/2 𝜒(𝜎t), defined in (7.2.18). [Part (e) reprinted with permission from Nicholson Shain (21). © 1964, American Chemical Society.]

One can map the full spectrum of electrochemical behavior in a zone diagram (1, 26, 27) like that in Figure 13.3.2. 7 Usually, a zone diagram is a plot of an important descriptor of the system—often a diagnostic parameter—vs. log 𝜆. The logarithmic scale is used because 𝜆 can normally vary over orders of magnitude. The central position on the horizontal axis is log 𝜆 = 0, where 𝜏 obs = 𝜏 rxn . The diagram in Figure 13.3.2 describes the behavior of Epf , referenced to the reversible E1/2 for the O/R couple. Also provided are labels describing behavior of |ipr /ipf |. Three zones are identified in Figure 13.3.2: • When 𝜆 ≪ 1 (log 𝜆 < 1), the reaction timescale is much longer than that of observation, so the homogenous decay of R does not proceed appreciably during the observation and the effective overall process is just the reversible O/R electron transfer, for which Epf = E1/2 − 28.5/n mV at 25 ∘ C and |ipr /ipf | = 1 [Section 7.2.2(b)]. Changes of timescale 7 Zone diagrams and the system of zone labels were originated by Savéant and Vianello (26, 27). The concept has proven useful over decades (1). Many examples are covered in this chapter. In two-letter zone labels, the first letter usually defines the dominant factor (D for diffusion, K for kinetics) and the second letter is a qualifier (E = extraordinary, G = general, I = intermediate, O = ordinary, P = pure). If the label does not begin with K or D, it is usually derived from other regular usage (e.g., IR = irreversible, QR = quasireversible). The labels used in this edition follow the practice in Savéant’s comprehensive book (1). Historically, varied labels were sometimes employed for a given zone. In an effort to simplify, the authors have standardized on Savéant’s evolved usage; however, figures in this chapter also show historic alternatives in parentheses.

553

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

–90 –60 n(Epf – E1/2)/mV at 25 °C

554

∣ipr /ipf∣ = 1

∣ipr /ipf∣ = 1 → 0

∣ipr /ipf∣ = 0

DO (or DP) Ordinary diffusion

KO (or KD) Kinetics and diffusion

KP Pure kinetics

–28.5 0 30 60 90 120 150 –4

–3

–2

–1

0 log λ

1

2

3

4

Figure 13.3.2 Zone diagram for the Er Ci reaction scheme in CV or LSV. Zone KO is shown for simplicity as 2 orders of magnitude wide but is often drawn slightly narrower. Historic alternative labels are shown for zones DO and KO.

in this region do not change the nature of behavior because the following reaction is never important. This is the zone of ordinary (or pure) diffusion and is labeled DO. • When 𝜆 ≫ 1 (log 𝜆 > 1), the following reaction is fully manifested during the period of observation, so that the effective overall reaction is an irreversible conversion, O + ne → X. The value of Epf becomes steadily more positive by 30 mV per decade of increase in 𝜆, as implied by (13.3.3) below. Because R is fully converted, |ipr /ipf | = 0. The shape and position of the wave are determined by the homogeneous process; hence, this is called the zone of pure kinetics and is labeled KP. • When 𝜆 is on the order of 1 (−1 > log 𝜆 > 1), the conversion of R to X is partial and small changes of timescale allow for greater or lesser degrees of completion. Thus, |ipr /ipf | makes the transition from unity at smaller 𝜆 to zero at larger 𝜆. In this region, the electrochemical response provides the best opportunity to evaluate parameters (k f , E1/2 , n) from experimental data. Also, Epf begins to show its positive shift with increasing 𝜆, but the effect is not large. This is the region of joint control by kinetics and diffusion and is labeled KO.8 Zone diagrams appear in many publications and are normally developed and presented on the basis of detailed theory for a particular experimental method. Only CV and LSV are of concern to us now, but zone diagrams are available for diverse other methods, such as chronoamperometry, square-wave voltammetry, chronocoulometry, chronopotentiometry, hydrodynamic voltammetry, and even bulk coulometry. (b) Quantitative Behavior

In the KP region, the shape of the curve is essentially that of a totally irreversible charge transfer, but without the dependence on 𝛼 (21). The ratio ip /v1/2 changes only slightly with scan rate (e.g., increasing by ∼5% for an increase in 𝜆 from 1 to 10). The peak potential is more positive than 8 In Savéant’s nomenclature, KO means “ordinary kinetics.” This label signifies that the homogenous kinetics modify the “ordinary” (unperturbed, reversible) situation. In contrast, the homogeneous kinetics become fully controlling in the KP region.

13.3 Survey of Behavior

1.0 –20 |ipr /ipf|

n(Epf – E1/2)

0.8 –10 0

0.6 0.4 0.2

+10 –1.5

–1.0

–0.5 log λ (a)

0.0

0.5

0.0

–2

–1 log kfτrev

0

(b)

Figure 13.3.3 (a) E pf vs. log 𝜆 and (b) |ipr /ipf | vs. log kf 𝜏 rev for the Er Ci case, where 𝜏 rev is the time between E 1/2 and the switching potential E 𝜆 . [Adapted with permission from Nicholson and Shain (21). © 1964, American Chemical Society.]

for the unperturbed electrode reaction and is given by (21) Epf = E1∕2 − 0.780

RT RT + ln 𝜆 nF 2nF

(13.3.3)

The wave shifts toward negative potentials (toward the position of the unperturbed wave) by about 30/n mV (at 25 ∘ C) per decade of increase in v (per unit decrease in log 𝜆; Figure 13.3.2). If E1/2 is known, (13.3.3) can be used to estimate k f from experimental data. In zone KO, information can be obtained from |ipr /ipf |, which is measured as described in Section 7.2.2. Figure 13.3.3b is a working curve of |ipr /ipf | vs. k f 𝜏 rev (21), where 𝜏 rev is the time between E1/2 and the switching potential, E𝜆 . By fitting observed values of |ipr /ipf | to this curve, k f can be estimated. An alternative approach—now much more widely used—is comparative simulation (Section 13.2.5), which can provide not only k f , but also other parameters, such as ′ DO , E1/2 , or E0 . Since reversal data yield kinetic information only over a small range of 𝜆, it is often useful to extend the scan to more extreme potentials to see if the products of the following reaction are electroactive. For example, Figure 13.3.4 shows the cyclic voltammograms for the same reaction and conditions as in Figure 13.3.1b, but with the scan extended to more positive potentials to show waves for the oxidation of species C and, on the second reversal, the reduction of its oxidation product, D. A great strength of CV is the ability to highlight linkages between voltammetric features, just as in this instance. (c) Other Er C Mechanisms

The Er C pattern has variations beyond the irreversible first-order C-step. For example, in Case Er C2 , the product R dimerizes (27–31), k2

2R −−−→ R2

(13.3.4)

For this second-order following reaction, 𝜏rxn = hence, the dimensionless kinetic ∗ ∕nfv. The behavior can be distinguished from Case E C by parameter becomes 𝜆2 = k2 CO r i ∗ . In the KP region, E is given by the dependence of the electrochemical response on CO pf (27, 28, 32) ) ( RT RT 2 Epf = E1∕2 − 0.902 + ln 𝜆2 (13.3.5) nF 3nF 3 so that Epf shifts 20/n mV (at 25 ∘ C) for a tenfold change in scan rate. ∗; 1∕k2 CO

555

556

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

1.0 0.8 0.6 0.4 i/mA 0.2 0.0 –0.2

Scan start

–0.4 –0.6 800

600

400

200 0 (E – E1/2)/mV

–200

–400

Figure 13.3.4 Cyclic voltammograms for the Er Ci case. A + e ⇌ B; B → C. Parameters as in Figure 13.3.1b, ′

0 v = 1 V/s, with the reversal scan extended to show the waves for the couple D + e ⇌ C, ED∕C = 0.5 V. DA = D B = D C = D D .

Other Er C schemes, e.g., where the product R can react with the starting material, O (31, 33), have also been discussed. 13.3.2

Effect of Electrode Kinetics in ECi Systems

When the charge-transfer kinetics are more sluggish, the observed behavior depends on k 0 and 𝛼 for the one-electron E-step in (13.3.6), as well as on the kinetic parameter, 𝜆 = k f /fv, for the C-step: k 0 ,𝛼

k

f −−−−−−− → O+e ← − R −−−→ X

(13.3.6)

The effect of heterogeneous kinetics can be important even with relatively facile chargetransfer reactions because, as shown in Section 13.3.1, the irreversible C-step causes the voltammetric wave to shift toward positive values, which, in turn, decreases the rate of the heterogeneous reduction. It is convenient to define a dimensionless parameter, Λ, relating k 0 to the observational timescale of the CV (34): 1∕2

𝜏 k0 k0 Λ= = obs (Dfv)1∕2 D1∕2

(13.3.7)

The full spectrum of behavior is described in terms of Λ and 𝜆 by the zone diagram in Figure 13.3.5 (34, 35). The different zones can be explained as follows: • For log 𝜆 < − 1, the homogenous reaction has no effect and the behavior is characteristic of the unperturbed electrode reaction. Depending on the value of Λ, the CV may be reversible (zone DO, log Λ > 1), quasireversible (zone QR, log Λ ∼ 0), or totally irreversible (zone IR, log Λ < − 1), as described in Section 7.3.

13.3 Survey of Behavior

k0 –

kf

4

(

∂Epc ∂ log v

)

= 29.6 mV

25° C

v 3 2

DO (or DP) –

(

∂Epc ∂ log v

)

KP

KO (or KD)

=0 25° C

log Λ

KI QR

KG

0

–1

–

IR

–4

–3

–2

–1

0 log λ

1

(

∂Epc ∂ log v 2

)

25° C

29.6 mV α

3

4

=

Figure 13.3.5 Zone diagram for the 1e Eq Ci case: 𝜆 = kf /fv; Λ = k0 /(Dfv)1/2 . Axis legend at upper left shows the directions and magnitudes of effects caused by increases in v, kf , and k0 by 1 order of magnitude. The direction of the vector for v reflects the fact that both 𝜆 and Λ depend on scan rate. Historic alternative labels are shown for zones DO and KO. [Adapted from Nadjo and Savéant (34), with permission.]

• Large Λ values always imply that the electron transfer is reversible on the experimental timescale; thus, the upper portion of Figure 13.3.5 corresponds to the Er Ci case discussed in Section 13.3.1. Zones DO, KO, and KP exist as presented in Figure 13.3.2.9 • At the bottom of the diagram, the rate-determining step is always the irreversible electron transfer, whether the C-step is fast or slow (Case Ei Ci ). If the homogeneous reaction is fast (lower right), it can make the system behave like an irreversible electron transfer even for values of Λ that would correspond to quasireversibility in the absence of the C-step. This is because the fast reaction removes species R from the vicinity of the electrode before it can engage in the electrode kinetics. • In zone KG (general kinetics, –0.7 < log Λ < 1.3, –1.2 < log 𝜆 < 0.8), the effects of electron-transfer kinetics and chemical irreversibility are jointly manifested; thus, one has the opportunity to evaluate k 0 , 𝛼, and k f simultaneously. Figure 13.3.6 shows Epc in this region as a function of Λ and 𝜆, and one can see there that 𝜕Epc /𝜕 log v changes significantly with Λ. Tables with values of the electrochemical parameters as functions of Λ and 𝜆 are given in reference (34), and they can be used for evaluation; however, comparative simulation (Section 13.2.5) provides a more comprehensive test of the mechanistic model. An example of Case Eq Ci is found with the Mo(IV)/Mo(III) system discussed earlier (Section 13.2.3). The prior focus was on the catalytic behavior in the presence of added cysteine; however, the behavior is also notable in the absence of cysteine (22). At a BDD electrode, there is a quasireversible wave for the Mo(CN)3− ∕Mo(CN)4− couple; however, the voltammetry does 8 8 9 The reader may notice that the boundaries between zones in the upper half of Figure 13.3.5 differ somewhat from those chosen for Figure 13.3.2. Since any transition between neighboring zones is gradual, one can place the boundary using slightly different criteria. A small difference in placement is usually inconsequential.

557

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

Figure 13.3.6 Variation of E pc at 25 ∘ C for the 1e Eq Ci case in

log Λ 4

zone KG. 𝜆 = kf /fv; Λ = k0 /(Dfv)1/2 . Parenthesized numbers indicate the limiting slope of each curve in mV per decade. [Data from Nadjo and Savéant (34).]

–0.7

(0.3)

3 –f(Epc – E0′)

558

–0.3

(9.2)

2

–0.1

(20) 1 (51)

0.5 0

1.3

(59) –1

0 log λ

1

not match expectations for a simple Eq process because |ipr /ipf | shows progressively diminished values at lower scan rates. The effect is modest, but detectable and quantifiable. The investigators proposed the following mechanism, k 0 ,𝛼

3− −−−−→ Mo(CN)4− 8 ←−−−− Mo(CN)8 + e kf

Mo(CN)3− −−−→ [Mo(CN)7 H2 O]2− + CN− 8 − H2 O

(13.3.8) (13.3.9)

Using comparative simulation (Section 13.2.5), they found that k 0 = 4 × 10−2 cm/s, 𝛼 = 0.5, and k f = 0.3 s−1 successfully modeled a set of CVs with scan rates varying over more than a decade. 13.3.3

Bidirectional Following Reaction

In a general EC scheme, one allows the C-step to be bidirectional, while permitting any level of reversibility for the 1e E-step (1, 21, 27): k 0 ,𝛼

kf

−−−−−−− → O+e ← − R⇄X kb

K = kf ∕kb

(13.3.10)

The mechanisms that we have already discussed, Er Ci , Eq Ci , and Ei Ci , are subcases, as are Er Cr , Eq Cr , Ei Cr . All are encountered in practice. The four parameters in (13.3.10) are needed to fully specify the dynamics of such a system. It is convenient to express them dimensionlessly as Λ [defined in (13.3.7], K, 𝛼, and 𝜆; however, 𝜆 is defined differently here because 𝜏 rxn must reflect the rate constants for both directions in the C-step. Using 𝜏 rxn ≡ 1/(k f + k b ), we obtain 𝜆=

(kf + kb )

(13.3.11) fv Since 𝛼 typically falls in a fairly narrow range near 0.5, the behavioral zone boundaries do not depend critically on it. Accordingly, the zones are defined in a three-dimensional space with Λ, K, and 𝜆 as the axes.

13.3 Survey of Behavior

−1

v

kf + kb

−0.5 K

0 KG

DE

0.5 log K 1

DO

KE

(or DP)

KO

1.5

KP

(or KD)

2 2.5 3 −2

−1

0

1 log λ

2

3

4

Figure 13.3.7 Zone diagram for the 1e Case Er C. 𝜆 = (kf + kb )/fv. Axis legend at upper right shows the directions of effects caused by increases in v, kf + kb , and K. Historic alternative labels are shown for zones DO and KO. [Adapted from Savéant (1), with permission.]

For any given system, prior knowledge may allow simplification of the zone space. For example, if one knows the electron-transfer step to be kinetically facile, Λ is not an important variable. The behavior then becomes defined by 𝜆 and K and can be summarized in the two-dimensional zone diagram of Figure 13.3.7. This diagram, though simplified from the entire EC space, displays a richer range of behavior than we have previously seen. Let us draw just a few observations: • Case Er Ci (Section 13.3.1) corresponds to the lower section of the diagram, where K is large, so that the C-step is effectively unidirectional. As 𝜆 increases, the behavior proceeds rightward through zones DO, KO, and KP, just as we saw in Figure 13.3.2. • On the left side of the diagram, where 𝜆 is small, the homogeneous kinetics are too slow for the C-step to be significant. Thus, zone DO corresponds, as in Sections 13.3.1 and 13.3.2, to the reversible CV of species O, unperturbed by the following reaction. • In the upper right, the kinetics of the following chemistry are fast, so the C-step is always at equilibrium. The system is reversible and diffusion-controlled but is not “ordinary” because, in addition to diffusion of the O/R couple, it depends on the coupled equilibrium involving X. Thus, the zone corresponds to extraordinary diffusion and has the label DE. This is the fully reversible sector in Case Er Cr , which we showed in Sections 1.4.1, 5.3.2(c), and 7.2.2(g) to offer valuable thermodynamic and stoichiometric information. • In Zones KO, KG, and KE, the homogeneous kinetics modify behavior and one has the opportunity to measure both k f and k b (or, alternatively, k f and K) from CV data. Rate constants and equilibrium constants are usually obtained for any of these cases by comparative simulation (Section 13.2.5). Figure 13.3.8 shows experimental CV curves for the I− ∕I ∕I− system in the room3 2 temperature ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide,

559

45

60

45

3I– → I3– + 2e

1000 mV/s

2I– → I2 + 2e

40 mM

30

30 15 15 i/μA

i/μA 0 4 mM

0 100 mV/s I2 + 2e → 2I–

–15

–15 –30

–30 I3– + 2e → 3I– –45 –0.50

–0.25

0.00

0.25 0.50 E /V vs. I3–/I– (a)

0.75

1.00

–45 –0.50

–0.25

0.00

0.25 0.50 E /V vs. I3–/I– (b)

0.75

1.00

Figure 13.3.8 CV for I− in [C2 mim]+ [NTf2 ]− at a Pt disk electrode (r0 = 0.8 mm). Solid curves are experimental; circles are from simulations using a single set of values ′

for E 0 , k0 , 𝛼, K, and kf . Because the solvent is ionic, no supporting electrolyte is required. Positive potentials are plotted to the right, and anodic currents are up. Scans begin at −0.2 V and first move positively. (a) 25 mM I− ; (b) v = 250 mV/s. [Adapted with permission from Bentley et al. (36). © 2014, American Chemical Society.]

13.3 Survey of Behavior

[C2 mim]+ [NTf2 ]− (36). Using comparative simulation, the investigators analyzed these results using the following EC mechanism: ′

I2 + 2e ⇄ 2I−

(E0 , k 0 , 𝛼)

(13.3.12)

I2 + I− ⇄ I− 3

(K, kf )

(13.3.13)

In this model, the only electrode reaction is (13.3.12); hence, it is assumed that all electroreduction proceeds through I2 . Tri-iodide, I− , is viewed as a complexed form of I2 , which is 3 reduced only after dissociation via the back reaction of (13.3.13). Thus, the reaction I− + 2e ⇌ 3 3I− is not an elementary process, but the net of (13.3.12) plus (13.3.13) written in reverse. It is ′ tempting to think that the standard potential EI0− ∕I− should be needed for the model; how3

′

ever, this parameter is unnecessary because it is fully determined by E0 for (13.3.12) and K for (13.3.13). In the CV curves, there are two sets of peaks relating to complexed and uncomplexed I2 . If all kinetics were so fast as to be invisible, only the reversible couple for the complexed form (I− ∕I− ) would be seen in the CV and the system would be in zone DE of Figure 13.3.7. If all 3 homogeneous kinetics were slow and the heterogenous kinetics were fast, one would see only the uncomplexed couple (I2 /I− ) and the system would be in zone DO. The actual observations in Figure 13.3.8 are between these limits. Since all forms are visible, the system must be in a kinetically limited zone bordering DE, viz. KG or KE. In Figure 13.3.7, the pattern CVs for these zones do resemble the actual result for the I2 /I− peaks.10 As shown in Figure 13.3.8, the investigators were able to model the complex voltammetric pattern over a significant range of scan rates and concentrations using one optimized set of values for the parameters.11 13.3.4

Catalytic Reaction—Case Er Ci ′

In the catalytic reaction scheme, a species Z, usually electroinactive, reacts with the immediate electroproduct to regenerate the electroreactant, e.g., O + ne ⇌ R kf

R + Z −−−→ O + Y

(13.3.14) (13.3.15)

The general case involves a second-order reaction and the need to account for the diffusion of ∗ ) is commonly assumed, so that C (x, t) ≈ C ∗ species Z. However, a large excess of Z (CZ∗ ≫ CO Z Z everywhere. Then, diffusion of Z need not be addressed, and (13.3.15) becomes pseudo-first order. Since 𝜏rxn = 1∕kf CZ∗ , 𝜆=

kf CZ∗

(13.3.16) nfv Expected voltammograms (1, 21, 26, 32) are shown in Figures 13.3.9 and 13.3.10. At sufficiently late times in the forward sweep, all curves tend to a limiting current, iplateau , independent of v and given by iplateau = nFAC∗O (Dk f CZ∗ )1∕2

(13.3.17)

10 The patterns in zones KG and KE in Figure 13.3.7 do not include peaks for the complexed form (i.e., the DE form), ′ which are displaced from E0 by a variable amount determined by K. 11 A weakness of the mechanism in (13.3.12)–(13.3.13), recognized by the investigators, is that (13.3.12) probably would occur in two steps, so that the system would be an EEC case. However, it proved impractical to maintain fitting precision with another set of electron-transfer parameters added into the model as unknowns.

561

562

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

3

1.2

2 i/mA

i/mA

1

0.8

0

0.4

–1

0.0

–2 400

200 0 –200 n(E – E1/2)/mV (a)

–400

400

1.0

200 0 –200 n(E – E1/2)/mV (b)

–400

200 0 –200 n(E – E1/2)/mV

–400

1.0

0.8

0.8

i/mA 0.6

i/mA 0.6

0.4

0.4

0.2 0.0 400

0.2 0.0 400

200 0 –200 n(E – E1/2)/mV (c)

–400

(d) ′

0 Figure 13.3.9 CV for the Er C′i system in (13.3.14)–(13.3.15), where EO∕R = 0 V, C∗O = 1 mM, CR∗ = 0, CZ∗ = 1 M, A = 1 cm2 , D = D = D = 10−5 cm2 /s, T = 25 ∘ C, and k = 10 s−1 with (a) v = 10, (b) v = 1, (c) v = 0.1, and O

(d) v = 0.01 V/s.

R

Z

f

Figure 13.3.10 LSV for the Er C′i system in (13.3.14)–(13.3.15) with the following values of 𝜆 = kf CZ∗ ∕nfv (1) 1.00 × 10−2 ; (2) 1.59 × 10−2 ; (3) 2.51 × 10−2 ; (4) 3.98 × 10−2 ; (5) 6.30 × 10−2 ; (6) 1.00 × 10−1 ; (7) 1.59 × 10−1 ; (8) 2.51 × 10−1 ; (9) 3.98 × 10−1 ; (10) 1.00; (11) ∞. [Reprinted with permission from Savéant and Vianello (26). © 1965, Pergamon Press PLC.]

5 1

4 2

3

3 π1/2χ(σt) λ1/2

4 5

2

6 7 8 1 9 10 11 0 –7.5

0

10 20 –nf(E – E0′)

30

40

13.3 Survey of Behavior

4.0 30 3.0 KO

KP

(or KI)

ipf

ipf ipf,u

ipf,u

2.0

n(Ep/2 – E1/2)/mV

DO

(or DP)

= 2.24λ1/2

1.0

0.0 0.0

0.5

1.0 λ1/2 (a)

1.5

2.0

20 DO

KO

KP

(or KI)

(or DP)

10

0 –2

–1

0 log λ (b)

1

2

Figure 13.3.11 Zone diagrams for Case Er C′i . (a) Ratio of ipf to the peak current for the unperturbed E-step, ipf, u , as a function of 𝜆1/2 . This form of presentation illustrates the arbitrary growth of ipf with respect to ipf, u as 𝜆 increases (e.g., at lower v or larger CZ∗ ). Since ipf, u ∝ v 1/2 , the curve implies that ipf is also proportional to v 1/2 in zone DO, but independent of v in zone KP. (b) Forward half-peak potential, E p/2 , vs. 𝜆. Historic alternative labels are shown for zones DO and KO. [Part (a) adapted with permission from Nicholson and Shain (21). © 1964, American Chemical Society.]

This condition is established when the diffusion layer reaches a thickness where the supply of O at the electrode surface, both by production through (13.3.15) and by diffusion, exactly matches the consumption of O by electrolysis. At that point, the surface concentration, C O (0, t), becomes independent of time or scan rate. The presence of this plateau (when it can be observed before another wave is reached) is a distinguishing feature of a catalytic mechanism. Another distinguishing mark is that the forward peak current, ipf , can grow arbitrarily larger than in the unperturbed case.12 Figure 13.3.11 presents two zone diagrams for Er C′i systems. A few points are notable: • When 𝜆 is small, 𝜏 rxn ≫ 𝜏 obs , and the C-step does not have time to manifest itself during the experiment. As we have seen in previous cases, the behavior is that of the unperturbed system. This is zone DO. • At large 𝜆, 𝜏 obs ≫ 𝜏 rxn , and the homogeneous kinetics dominate the response (zone KP). The i − E curve loses its peak-shaped appearance and becomes a wave described by i=

nFAC∗O (Dk f CZ∗ )1∕2 1 + exp[nf (E − E1∕2 )]

Substitution of (13.3.17) for the numerator in (13.3.18) gives, ( ) iplateau − i RT E = E1∕2 + ln nF i

(13.3.18)

(13.3.19)

Thus, analysis in zone KP is easy and leads immediately to both the reversible E1/2 and k f . 12 In this discussion, ipf is the forward peak current when a peak exists (Figure 13.3.9a,b), but is the plateau current when there is no peak (Figure 13.3.9c,d).

563

564

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

• For −1 < log 𝜆 < 1 (zone KO), ipf makes the shift from being proportional with v1/2 (at the boundary with zone DO) to being independent of v (at the boundary with zone KP) (Figure 13.3.11a). The forward half-peak potential, Ep/2 , depends on 𝜆, showing a maximum in ΔEp/2 /Δ log v of about 24/n mV at 25 ∘ C (Figure 13.3.11b). In contrast, Ep/2 is independent of 𝜆 in zones DO and KP. In CV, |ipr /ipf | (with ipr measured from the reversed extension of the forward curve) is always unity, even in zone KP, where the current on the reverse scan tends to retrace the forward scan (Figure 13.3.9c,d). A real example of the Er C′i mechanism is presented in Section 13.2.3. A more complicated variation of the EC′ scheme is the situation where reaction (13.3.15) is reversible, but product Y is unstable and undergoes a fast decay (Y → X). This overall scheme is Er C′r Ci ; however, the instability of Y drives (13.3.15) to the right, so that the observed behavior converges on Er C′i . In this kind of process, known as redox catalysis (1, 6, 37), the redox couple, O/R, mediates the reduction of Z, ultimately producing X. An example (38) involves chrysene (Ch; Figure 1), and its radical anion as the O/R couple, and bromobenzene, PhBr, as species Z, Ch + e ⇌ Ch−∙

(13.3.20)

Ch−∙ + PhBr ⇌ Ch + PhBr−∙

(13.3.21)

kd

PhBr ∙ −−−→ Ph ∙ +Br−

(13.3.22)

The product Ph∙ is eventually stabilized through other chemistry. One can exploit redox catalysis to measure k d , even if (13.3.22) is too fast to measure by direct ′ electrochemistry. This is done by varying v and CZ∗ in CV with a system where E0 for the mediator couple (Ch∕Ch ∙ in this case) is chosen to be positive of that of the Z/Y couple (PhBr∕PhBr ∙ here). Redox catalysis has been extensively employed to study bond breaking following electron transfer (1, 5, 6, 39) (Section 13.3.8). 13.3.5

Preceding Reaction—Case Cr Er

Like a bidirectional EC case (Section 13.3.3), the Cr Er system has behavior that depends on two first-order homogeneous rate constants, k f and k b (s−1 ): kf

Y⇄O

(13.3.23a)

K = kf ∕kb

(13.3.23b)

kb

O + ne ⇌ R

(13.3.24)

Once again it is convenient to use 𝜆 and K as dimensionless parameters, where, for CV and LSV, 𝜆 = (k f + k b )/nfv. The heterogeneous electron transfer, (13.2.4), is assumed to be nernstian, so no additional parameters are needed to describe it. Figure 13.3.12 is the corresponding zone diagram (1, 26, 32). As in all previous cases, zone DO, where 𝜆 is small, corresponds to ineffective homogeneous kinetics. The characteristic time of reaction 𝜏 rxn = 1/(k f + k b ) greatly exceeds 𝜏 obs ; hence,

13.3 Survey of Behavior

1.0 K 0.5

1

v

k

0.0 KG

DE

(or KI)

–0.5

(or DM)

6 DO

log K –1.0

KE

(or DP)

(or KI)

5

KO

–1.5

(or KI)

KP

–2.0 4

3

2 –2.5 A

–3.0 –2

B –1

C 0

D 1 log λ

2

3

4

Figure 13.3.12 Zone diagram for Case Cr Er . 𝜆 = (kf + kb )/nfv. Legend at upper right shows the directions of effects caused by increases in v, kf + kb , and K. Historic alternative labels are shown for several zones. Numbered CV patterns and lettered dots are discussed in the text. [Adapted from Savéant (1), with permission.]

species Y and O do not appreciably interconvert during the experiment. The CV is the unperturbed nernstian response for a solution of O at whatever bulk concentration exists when the scan begins. If the preceding process, (13.3.23a) has time to equilibrate before the CV is carried out, then the bulk concentration of O is ∗ ∗ = C K (13.3.25) CO K +1 where C * is the total concentration of O + Y. When K is large, the equilibrium lies far to the right and most of O + Y is in the electroactive form, O. Voltammetric currents are near the maximum that C * can support (CV pattern 1). Conversely, when K is small, only a minor fraction of O + Y is in the O form and the voltammetric peaks might be difficult even to observe (CV pattern 2). In zone DE, at the upper right of Figure 13.3.12, both 𝜆 and K are large. In this region, (13.3.23a) is very mobile and is always at equilibrium on the timescale of observation. As we have seen in Sections 1.4.1 and 5.3.2, a homogeneous equilibrium coupled to a nernstian electrode reaction yields a diffusion-controlled nernstian response but shifted along the potential axis from its unperturbed position by an extent that depends on K. This shift is a thermodynamic effect reflecting the free energy by which species O is stabilized in the equilibrium; hence, thermodynamic and stoichiometric information is available from the voltammetry in zone DE. The peak height in CV corresponds to the total bulk concentration, C * (CV pattern 6). In all other zones, the response is limited by the homogeneous kinetics and the shape of the voltammogram is significantly altered (1, 21, 26, 32).

565

566

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

3

1

2 2

1

3

i/μA

4

0 4

–1

3 2

–2

1 –3 400

200

0 E/mV

–200

–400 ′

0 Figure 13.3.13 CV for the Cr Er case given in (13.3.23) and (13.3.24). EO∕R = 0 V, C * = 1 mM, A = 1 cm2 , −5 2 −3 −2 −1 −1 D = D = D = 10 cm /s, K = 10 , k = 10 s , k = 10 s , T = 25 ∘ C, and (1) v = 10; (2) v = 1; (3) v = 0.1; Y

O

R

f

(4) v = 0.01 V/s.

b

Figure 13.3.13 illustrates these effects as v is changed over three orders of magnitude. For the identified system, log K = − 3; hence, all operating points lie along the bottom axis of Figure 13.3.12. Three different zones are probed: • At the highest scan rate, (curve 1, v = 10 V/s), the operating point is A, in the DO region. The behavior is that of an unperturbed reversible reaction with a small initial concentration of O determined by the low value of K. • At v = 1 V/s (curve 2), the operating point has moved to B, in zone KO. One still sees a peak, reflecting depletion in the diffusion layer, but it is muted by comparison to the diffusion-controlled case (curve 1). Also, the diffusive falloff in current past the forward peak (approximately as t −1/2 in a diffusion-controlled case) becomes flattened toward a plateau, the presence of which is a clear indicator of a kinetic limitation. • At the two lowest scan rates (curve 3, v = 0.1 V/s, and curve 4, v = 0.01 V/s), the operating points lie at C and D, respectively, both in zone KP. The current attains a steady-state value, indicated by the cathodic plateau, which is independent of v. The similarities between the voltammograms in Figure 13.3.13 and the corresponding pattern CVs in Figure 13.3.12 are apparent. In the pure kinetic zone (KP), the steady state is established when the diffusion layer thickens to the point where the rate of reduction of O at the electrode surface is exactly matched by the rate of arrival of O, which is supported by steady production in the C-step. The current is given by (26, 40) iplateau = nFAD1∕2 C ∗ K(kf + kb )1∕2

(13.3.26)

The half-peak (i.e., half-plateau) potential, Ep/2 , is (40) RT RT − ln 𝜆 nF 2nF thus, the shift of Ep/2 with v in zone KP is ′

Ep∕2 = E0 − 0.24

(13.3.27)

dEp∕2 ∕d log v = 2.303RT∕2nF

(13.3.28)

13.3 Survey of Behavior

At 25 ∘ C, a tenfold increase in v causes the reduction peak to shift positively by 30/n mV. As v increases, 𝜆 decreases and the system can move from zone KP into zone KO (Figure 13.3.12), where dEp/2 /d log v becomes smaller. With further increase in v, the zone of ordinary diffusion (DO) may be reached, where Epf and Ep/2 are both independent of v. In cyclic voltammetry, the reverse scan is not greatly affected by the coupled reaction (Figure 13.3.13), because the electroproduct, R, is stored normally in the diffusion layer and is available for diffusion-controlled reoxidation. A consequence is that |ipr /ipf | (with ipr measured from the extension of the cathodic curve as described in Section 7.2.2) tends to be appreciably greater than unity when the system is operating in zones where homogeneous kinetics influence the voltammetry (KO, KG, KE, or KP). This behavior is evident in curves 2–4 of Figure 13.3.13 and is also indicated in pattern CVs 3–5 of Figure 13.2.12. As the scan rate increases, and the operating point moves toward the boundary between zones KO and DO, |ipr /ipf | declines toward unity, then becomes unity if the system enters zone DO. Principal diagnostic indicators of a Cr Er mechanism are • Broadening of the forward response at slower scan rates and, perhaps, flattening toward a plateau. • |ipr /ipf | significantly above unity, but declining toward higher scan rates, perhaps reaching an asymptote at unity. • Declining ipf /v1/2 with increasing scan rate. The basis for this behavior will be discussed below using an example. Comparative simulation (Section 13.2.5) is the most effective means for using CV results to evaluate K and k f (or, alternatively, k f and k b ). An example is found in the electrochemistry of the cyclooctadiene (COD) complex of Rh(I), Rh(COD)+ (41). Because this species is subject to reversible ligand replacement by solvent, 2 the electrode process can be described by the Cr Er pattern. With acetone as the solvent, the postulated mechanism (41) is Rh(COD)(acetone)+ 2

kf′

+ COD ⇄ Rh(COD)+ + 2 acetone 2

Rh(COD)+ + e ⇌ Rh(COD)2 2

kb′

(13.3.29) (13.3.30)

where kf′ and kb′ are both pseudo-first-order rate constants (with both COD and acetone in large excess). The electroproduct, Rh(COD)2 , decays to an electroinactive form. Experimental and simulated results (41) for this system are compared in Figure 13.3.14a. The recorded CV shows a broadened forward peak, a sharper reversal peak, and |ipr /ipf | ≈ 1.4. These features are diagnostically consistent with a kinetically limited Cr Er system. A simulation based on (13.3.29)–(13.3.30) effectively models the experimental voltammogram. The same model and parameters also account well for the behavior of the current function ipf /v1/2 vs. scan rate (Figure 13.3.14b).13 Figure 13.3.14b illustrates a notable behavioral aspect of Case Cr Er , which is a significant decline in ipf /v1/2 vs. v. In the diffusion-controlled zones (DO and DE), ipf /v1/2 is constant with v. Most likely, the Rh(COD)+ system approaches or reaches zone DO at the highest scan rates 2 13 The model used by the investigators allowed for quasireversibility of (13.3.30) and decay of Rh(COD)2 ; however, the reported parameters [k 0 and 𝛼 for (13.3.30) and k c for the decay] suggest that neither effect is significant in most of the range of 𝜈 covered by Figure 13.3.14b, for which 2.5 ms ≤ 𝜏 obs ≤ 1 s. Quasireversibility might be a factor at the highest scan rates, and decay of the electroproduct might be significant at the lowest. For most of the range, the Cr Er mechanism suffices. The equilibrium constant Kassoc = [Rh(COD)+ ]∕[Rh(COD)(acetone)+ ][COD] is not the same as 2 2 ′ ′ K = kf ∕kb .

567

i

anodic cathodic

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

0

–0.5

–1.0

–1.5

E/V vs. Fc+/Fc (a) 15 ipf v –1/2/μA s1/2 V–1/2

568

10

5 –1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

log v/ V s–1 (b)

Figure 13.3.14 (a) CV of 1.4 mM Rh(COD)+ at a Pt disk in acetone + 0.1 M TBAPF6 without added COD. 2 v = 300 mV/s. Points are experimental. Solid curve is simulated for (13.3.29) and (13.3.30) with K assoc = 300 M−1 (measured separately), kb′ = 7.3 s−1 , and E 1/2 = − 1.31 V. Also included in the model were k0 = 0.1 cm/s and 𝛼 = 0.5 for (13.3.30) and kc = 0.16 for decay of the electroproduct. The forward wave is closely followed by another wave to the right of this diagram. The beginning of that wave accounts for the discrepancy at the most negative potentials in the forward scan. (b) Current function ipf /v 1/2 vs. log v for 1.1 mM Rh(COD)+ without 2 added COD. Filled circles are experimental; open circles are calculated from the parameters given above. [Reproduced with permission from Orsini and Geiger (41). © 1999, American Chemical Society.]

used in this study, and the asymptote in Figure 13.3.14b is the consequence. In a kinetically limited zone (KO, KG, KI, or KP), the supply of electroreactant exceeds that for diffusion control because the preceding reaction has time to contribute. Thus, the current at lower scan rates is greater than one would project on the basis of diffusion control (proportionality with v1/2 ), and ipf /v1/2 increases toward lower scan rates, as observed in this case. In Problem 13.9, the reader is invited to work out the zone corresponding to the CV in Figure 13.3.14a.

13.3 Survey of Behavior

13.3.6

Multistep Electron Transfers

We now turn now to systems with sequential heterogeneous electron transfers (1, 42, 43). For two-step (EE) mechanisms, the scheme is k10 ,𝛼1

−−−−−−−−− → A+e ← − B k20 ,𝛼2

−−−−−−−−− → B+e ← − C

′

E10 E20

(13.3.31)

′

(13.3.32)

A full treatment requires all six indicated parameters. If either step is reversible, the corresponding standard rate constant and transfer coefficient are not needed and the treatment is simplified. (a) Er Er Reactions

When both electron transfers are rapid, the CV is determined entirely by the relative placement ′ ′ ′ of the formal potentials, as expressed by ΔE0 = E20 − E10 (Figure 13.3.15) (42). ′ Figure 13.3.16 illustrates shape variations in the CV for several values of ΔE0 , and Figure 13.3.17 shows corresponding changes in ipf /v1/2 and ΔEp = Epf − Epr for the first ′

(cathodic) wave when the two waves are resolved (ΔE0 < −125 mV), and for the overlapping ′ and fully merged wave when ΔE0 > −125 mV. In Figure 13.3.17a, ipf /v1/2 is expressed as 𝜋 1/2 𝜒 pf (𝜎t)n3/2 , where 𝜒 pf (𝜎t) is the peak value of the dimensionless current defined in (7.2.18). One can readily show that 𝜋 1∕2 𝜒pf (𝜎t)n3∕2 = (ipf ∕v1∕2 )(Ff 1∕2 AC ∗O D1∕2 ), where (Ff 1∕2 AC ∗O D1∕2 ) is a constant for a given system. There are four ranges of behavior: ′

1) When ΔE0 ≥ 100 mV, the second E-step occurs much more easily than the first, and one observes a single wave with characteristics indistinguishable from a nernstian 2e transfer (ΔEp = 29 mV, 𝜋 1/2 𝜒 pf (𝜎t)n3/2 = 1.26).14 Since the simultaneous transfer of two electrons is very unlikely (Section 3.7.1), the overall reaction is understood to occur in two steps, with the second following immediately after the first. ′ ′ 2) For 100 mV > ΔE0 > − 80 mV, there is a single wave with ΔEp increasing as ΔE0 becomes more negative. Figure 13.3.15 Different Er Er cases based on ′ ′ ′ ΔE 0 = E20 − E10 .

ΔE0ʹ > 0 (a)

E20ʹ

E E10ʹ

–

ΔE0ʹ = 0 (b)

E

E10ʹ E20ʹ

–

ΔE0ʹ < 0 (c)

E10ʹ

14 For a reversible wave, 𝜋 1∕2 𝜒pf (𝜎t) = 0.4463 (Table 7.2.1). The value 1.26 is 23/2 × 0.4463.

E E20ʹ

–

569

cathodic

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

100 mV

50 mV

0

0 mV

E (–)

anodic

570

(a)

(b)

–35.6 mV

–75 mV

(d)

(c) –90 mV

(e)

–110 mV

–150 mV

(g)

(f) –200 mV

(h)

(i)

Figure 13.3.16 Changing shapes in CV for the Er Er reaction scheme in (13.3.31) and (13.3.32), starting with a ′

solution of A. Each curve is labeled by the corresponding value of ΔE 0 . All voltammograms are based on the axes and scan directions in (a). ′

3) For −80 mV > ΔE0 > − 125 mV, two unresolved waves are visible. ′ 4) For ΔE0 ≤ − 125 mV, the two waves become resolved and each wave takes on the characteristics of a 1e transfer [ΔEp = 58 mV, 𝜋 1/2 𝜒 pf (𝜎t)n3/2 = 0.4463]. For an Er Er reaction, ipf /v1/2 and ΔEp are independent of scan rate. If these diagnostics are ′

satisfied, the working curves in Figure 13.3.17 can be employed to estimate ΔE0 . ′ The value of ΔE0 (or ΔE0 ) is determined by chemical and structural factors (43).15 • When successive electron transfers involve a single molecular orbital (Figure 13.3.18a), and no large structural changes occur upon electron transfer, then one expects two well-spaced waves (ΔE0 ≪ − 125 mV). This behavior is typical in the reduction of aromatic hydrocarbons. Anthracene, for example, shows two 1e waves spaced ∼500 mV apart. • When the transfers occur to separated orbitals in a molecule (e.g., in two different groups) (Figure 13.3.18b), one can observe closer spacing between the waves, or even a single wave, depending upon the extent of interaction between the groups. A case of particular interest occurs when there is no interaction. For this situation, ΔE0 is not zero but instead is −35.6 mV (at 25 ∘ C), where the curve crosses the horizontal dashed line in ′

15 We ordinarily deal with formal potential, E0 , rather than standard potential, E0 , because electrochemical ′ responses relate more directly to E0 . Discussions about chemical and structural factors often take a more thermodynamic approach and focus on ΔE0 . We follow that practice here, but the distinction is rarely important. If ′ ′ all participants have the same activity coefficient, then E0 = E0 and ΔE0 = ΔE0 .

13.3 Survey of Behavior

1.3 1.26 (2e) π1/2χpf (σt)n3/2

1.1 0.9 0.7 0.5

0.446 (1e)

0.3 –200

–100

0 ΔE0ʹ/mV (a)

100

200

ΔEp /mV

160 100 80 58 mV 40

29 mV

0 –200

–100

0 ΔE0ʹ/mV (b)

100

200

′

Figure 13.3.17 (a) Dimensionless peak-current function vs. ΔE 0 for an Er Er system. The vertical scale is proportional to ipf /v 1/2 but is given in terms of the dimensionless current defined in (7.2.18), for which 𝜒 pf (𝜎t) ′

is the value at the forward peak. (b) ΔE p vs. ΔE 0 for an Er Er reaction system. The discontinuity near ′

ΔE 0 = −100 mV occurs when the two merged waves are resolved.

+e

(a)

+e

R–•

R

R2–

+e

(b)

A~A

+e

A–• ~ A

A–• ~ A–•

Figure 13.3.18 (a) Stepwise addition of electrons to the same molecular orbital in a molecule, R, usually yielding two separated waves. (b) Stepwise addition to two separate groups, A, on a molecule, A∼A, where the spacing between the waves depends upon the extent of interaction between the groups.

571

572

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

Figure 13.3.17b. The corresponding value of ΔEp is 58 mV; thus, one observes the characteristic splitting of a 1e transfer, even though a single wave involving two electron transfers is recorded. A statistical (entropic) factor makes the second electron transfer slightly more difficult than the first in terms of free energy (44, 45). We will shortly find in (13.3.35) that ΔE0 = − (2RT/F) ln 2. Accordingly for an Er Er system, ΔE0 > − (2RT/F) ln 2 indicates a positive interaction (the second E-step assisted by the first), while ΔE0 < − (2RT/F) ln 2 indicates a negative interaction (the second E-step impeded by the first). In the latter case, a greater peak splitting is observed, as one can see, for example, in the voltammetry of 𝛼,𝜔-9,9′ -dianthracenylalkanes (Figure 13.3.19). Positive interactions are generally rooted in a structural rearrangement, an important solvation change, or an ion-pairing effect resulting from the first electron transfer (43, 47). Whenever an EE reaction takes place, one must also consider the homogeneous disproportionation and comproportionation reactions, kf

−−−−−−− ⇀ 2B ↽ − A+C

(13.3.33a)

Kdisp = kf ∕kb = [A][C]∕[B]2

(13.3.33b)

kb

′

The equilibrium constant, K disp , is determined by ΔE0 : ′

′

′

(RT∕F) ln Kdisp = ΔE0 = E20 − E10

(13.3.34)

′

For two well-separated waves (ΔE0 < 0), K disp is small and the comproportionation of A and C dominates the disproportionation of B. At potentials of the second wave, C diffusing away 10

10

n=0

8

8

6

6

4 i/μA 2

4 i/μA 2

0

0

–2

–2

–4

–4

–6 –0.9

–1.3

10

–1.7 –2.1 E/V vs. SCE (a)

–2.5

6

4 i/μA 2

4 i/μA 2

0

0

–2

–2

–4

–4 –1.3

–1.7 –2.1 E/V vs. SCE (c)

–2.5

–6 –0.9

–1.7 –2.1 E/V vs. SCE (b)

–2.5

n=6

8

6

–6 –0.9

–1.3

10

n=4

8

–6 –0.9

n=2

–1.3

–1.7 –2.1 E/V vs. SCE (d)

–2.5

Figure 13.3.19 Cyclic voltammograms for the reduction of 𝛼,𝜔-9,9′ -dianthracenylalkanes (An - (CH2 )n - An, where An = 9-anthracenyl) in 1:1 benzene:MeCN + 0.1 M TBAP at a Pt electrode. As the alkane chain lengthens (n = 0, 2, 4, 6…), the voltammograms show a decreasing negative interaction. [Adapted from Itaya, Bard, and Szwarc (46), with permission.]

13.3 Survey of Behavior

from the electrode can reduce A diffusing toward it, so that the concentration profiles of A, B, and C differ from those that would exist if the solution-phase reaction did not occur. However, the voltammogram for an Er Er system remains independent of k f and k b because, at any given potential, the average oxidation state in any layer of solution near the electrode is unchanged by (13.3.33a) (48). If species A is removed by comproportionation, two B molecules are produced. In either case, two electrons can be delivered at potentials of the second wave, for no net change. While disproportionation/comproportionation does not affect CV in diffusion-controlled Er Er systems, it can affect responses in other cases of multistep electron transfer: • When either of the E-steps is quasireversible or irreversible [Sections 13.3.6(c–e)]. • When there is intervening chemistry, as in ECE reactions (Section 13.3.7). • When migration contributes significantly to mass transfer (Section 5.7; Figure 5.7.5). (b) (Er )n Reactions

The ideas of the preceding section generally apply for reactions involving more than two reversible electron transfers, i.e., Er Er Er … or (Er )n schemes. If the system involves n steps, the observed behavior can vary from a set of n resolved 1e waves to a single ne wave. For example, a solution of C60 shows up to six separated 1e waves, attributed to the addition of electrons to three degenerate orbitals in the molecule. In contrast, only a single wave is observed for solutions of many polymers, such as poly(vinylferrocene) (PVFc; Figure 17.4.1). The latter behavior is consistent with a lack of interaction among redox centers on a polymer chain (45). For molecules containing k noninteracting centers, the ΔE0 between the first and kth electron transfers is given by ( ) 2RT ln k (13.3.35) Ek0 − E10 = − F The single CV wave, manifesting k merged waves, has ΔEp characteristic of a 1e process and ipf reflecting k. Thus, the oxidation of PVFc containing (on the average) 74 ferrocene units per molecule produces a 74-electron wave whose shape is essentially that of a nernstian 1e reaction. The forward peak height, ipf , is scaled relative to that of the monomer (e.g., ferrocene) by ∗ ), where the subscripts refer to the monomer (m) and polymer (p) (45). k(Dp ∕Dm )1∕2 (Cp∗ ∕Cm This factor is linear with k, rather than k 3/2 , which would be expected for a (mythical) simultaneous k-electron event [Section 7.2.1(b)]. The difference arises because each redox unit in the polymeric system must become separately situated at the electrode for its electron-transfer event. In effect, the polymeric system behaves as a solution of independent monomers having concentration kC ∗p and diffusivity Dp . (c) Er Eq Reactions

If the first electron transfer is nernstian and the second somewhat slower, one must bring k20 ′ and 𝛼 2 into the treatment, in addition to ΔE0 . The appearance of the CV then varies with v. ′ Figure 13.3.20 shows the behavior for the case of ΔE0 = 0. At low v (Figure 13.3.20a), there is a single wave with a shape approaching that for the Er Er case. As the scan rate is increased, ′ the wave for the first E-step remains reversible and centered on E0 , but the slower second step results in a second wave splitting progressively away from the first (Figure 13.3.20c,d). For an Er Eq reaction where the second electron transfer occurs more easily than the first, the situation is somewhat different. Consider the example in Figure 13.3.21, where ΔE0 = 150 mV. When species B is formed, it is quickly reduced to C because the first electron transfer is already occurring at potentials well negative of E20 . Thus, at high scan rates (curves C and D), the cathodic wave is not split and reversibility is not seen at the potential of the first wave.

573

574

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

5 2.0

4

1.5

3

1.0 i/mA

2

0.5 i/mA

0.0

1 0

–0.5

–1

–1.0

–2

–1.5 300

200

100

–3 300

0 –100 –200 –300 E/V (a)

15

200

100

0 –100 –200 –300 E/V (b)

40 1

10

2

i/mA

5 i/mA

2 1

30 20 10

0

0

–5 –10 300

1

200

–10

1

2

2 100

0 –100 –200 –300 E/V (c)

–20 300

200

′

100

′

0 –100 –200 –300 E/V (d) ′

Figure 13.3.20 Representative behavior for an Er Eq reaction with ΔE 0 = 0, E10 = E20 = 0, n1 = n2 = 1, 𝛼 1 = 𝛼 2 = 0.5, k10 = 104 cm/s, k20 = 10−2 cm/ s, D = 10−5 cm2 /s, C * = 1 mM, A = 1 cm2 , T = 25 ∘ C, and scan rates, v, of (a) 1; (b) 10; (c) 100; (d) 1000 V/s. Rate constants for (13.3.33a) are assumed to be zero. Features related to couples 1 and 2 are distinguished in (c) and (d).

(d) Eq Er Reactions

When the first step is quasireversible and the second is reversible, the first is always ′ rate-determining. The parameters needed for description are k10 , 𝛼 1 , and ΔE0 . The result (Figure 13.3.22) is a shift in Epc to more negative values with increasing v, without splitting of the cathodic wave. The anodic wave splits at higher scan rates because the oxidation of species B to A occurs at more positive potentials. (e) General Treatment

A general treatment of the EE scheme involves all six parameters identified in (13.3.31)–(13.3.32) and is best approached by simulation. The problem is further complicated by the need to include homogeneous disproportionation and comproportionation, (13.3.33a). Thus, a seventh variable—the rate constant for disproportionation, k f —is required.16 Figure 13.3.23 illustrates the effect of this rate constant for an Er Eq reaction. It is helpful to refer also to Figure 13.3.20d, where the same conditions apply, except that (13.3.33a) is disregarded. Since Figure 13.3.23a [including (13.3.33a)] is practically identical to Figure 13.3.20d [disregarding (13.3.33a)], disproportionation clearly has little effect for k f ≤ 106 M/s. However, big changes are seen when k f is substantially larger. The disproportionation process, when effective, shortens the lifetime of B near the electrode surface, minimizing its ability to engage in the second E-step. The accompanying regeneration of A tends to channel a larger fraction 16 The comproportionation rate constant, k b , is not separately required because the equilibrium constant K ′

already defined by ΔE0 in (13.3.33b).

is

13.3 Survey of Behavior 8.0

2.5 2.0

6.0

1.5 i/mA

1.0

i/mA

0.5

4.0 2.0

0.0

0.0

–0.5 –2.0

–1.0 –1.5 400

200

0 E/mV (a)

–200

–4.0 400

–400

200

0 E/mV (b)

–200

–400

200

0 E/mV

–200

–400

50

20

40

15

30 i/mA

10 i/mA

5

20 10 0

0

–10 –5

–20

–10 400

200

0 E/mV

–200

–30 400

–400

(c)

(d)

Figure 13.3.21 Representative behavior for an Er Eq reaction. System with all parameters as in Figure 13.3.20, ′

except ΔE 0 = 150 mV. Scan rates of (a) 1; (b) 10; (c) 100; (d) 1000 V/s. Rate constants for (13.3.33a) are assumed to be zero. (Courtesy of Pine Research Instrumentation.)

of the faradaic current through the first E-step. If that step is reversible, then disproportionation causes the CV to appear more reversible. For the largest value of k f , the system in the example becomes essentially fully reversible (Figure 13.3.23c). Experimental results (49) for an Eq Eq system are provided in Figure 13.3.24, which relates to the oxidation and re-reduction of the dirhodium organometallic (Fv)Rh2 (CO)2 (μ-dppm), where Fv is fulvalene and dppm is 1,1-bis(diphenylphosphino)methane. The two Rh centers in this molecule (A) are oxidizable in two closely spaced steps leading to a stable dication featuring a Rh—Rh bond (Figure 13.3.24e), k10 ,𝛼1

+ −−−−−−−−− → A−e ← − A k20 ,𝛼2

2+ −−−−−−−−− → A+ − e ← − A

′

E10

(13.3.36)

′

E20

(13.3.37)

At the lowest scan rate (Figure 13.3.24a), the response nearly corresponds to a 2e reversible system, but at higher scan rates, ΔEp increases and the response broadens, especially for the forward (positive-going) scan. Comparative simulation (Section 13.2.5) provided a global fit for all four voltammograms with the parameters given in Figure 13.3.24. The removal of the first electron is slightly more ′ ′ difficult than removal of the second (E20 − E10 = −13 mV);17 and the kinetics of the first step are significantly slower (k10 ∕k20 < 0.2). The investigators interpreted the kinetic differences in ′

′

17 For an oxidation process, a negative value of E20 − E10 corresponds to

575

576

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

5 2.0

4

1.5

3

1.0 i/mA

2

0.5 i/mA

0.0

1 0

–0.5

–1

–1.0

–2

–1.5 300

14 12 10 8 6 4 i/mA 2 0 –2 –4 –6 300

200

100

–3 300

0 –100 –200 –300 E/mV (a)

200

100

0 –100 –200 –300 E/mV (b)

50

1,2

40

1,2

30 i/mA

20 10 0

1 200

2 100

0 –100 –200 –300 E/mV (c)

–10

1

–20 300

200

′

2 100

′

0 –100 –200 –300 E/mV (d) ′

Figure 13.3.22 Representative behavior for an Eq Er system with ΔE 0 = 0, E10 = E20 = 0, n1 = n2 = 1, 𝛼 1 = 𝛼 2 = 0.5, k10 = 10−2 cm/s, k20 = 104 cm/ s, D = 10−5 cm2 /s, C * = 1 mM, A = 1 cm2 , T = 25 ∘ C, and scan rates, v, of (a) 1; (b) 10; (c) 100; (d) 1000 V/s. Rate constants for (13.3.33a) are assumed to be zero. Features related to couples 1 and 2 are distinguished in (c) and (d).

terms of changes in interatomic spacings accompanying the electron transfers, especially along the Rh–Rh axis. The slower kinetics of the first step may indicate that it accomplishes most of the shortening of the Rh–Rh distance required for bond formation. 13.3.7

ECE/DISP Reactions

The general ECE reaction scheme (1, 7, 9, 50) is18 k10 ,𝛼1

−−−−−−−−− → A+e ← − B kf

B ⇌C kb

k20 ,𝛼2

−−−−−−−−− → C+e ← − D kd

B + C −−−→ A + D

E10

′

K = kf ∕kb ′

E20

(13.3.38) (13.3.39) (13.3.40) (13.3.41)

18 In much of the historic literature, including Savéant’s book (1) and earlier editions of this book, the rate constants for the forward and reverse reactions in (13.3.39) are oppositely labeled (as k b and k f , respectively), so that the equilibrium constant is K −1 , as we define it here. In this edition, the authors have elected to match the notation in this section to the standard practice of the chapter. This change simplifies the presentation of ECE/DISP systems and makes it consistent with current practice in the field. Readers of the

13.3 Survey of Behavior

Figure 13.3.23 Representative behavior for the Er Eq process as shown in Figure 13.3.20d, but including disproportionation and ′ comproportionation reactions. ΔE 0 = 0, ′ ′ 0 0 E1 = E2 = 0, n1 = n2 = 1, 𝛼 1 = 𝛼 2 = 0.5, k10 = 104 cm/s, k20 = 10−2 cm/s, D = 10−5 cm2 /s, C * = 1 mM, A = 1 cm2 , T = 25 ∘ C, and v = 1000 V/s. Disproportionation rate constant, kf , of (a) 106 ; (b) 108 ; (c) 1010 M−1 s−1 .

40 30 20 10 i/mA 0 –10 –20 300

200

100

0 –100 E/mV (a)

–200

–300

200

100

0 –100 E/mV (b)

–200

–300

200

100

0 –100 E/mV (c)

–200

–300

40 20 i/mA 0 –20 –40 300

60 40 20 i/mA 0 –20 –40 –60 300

A full treatment requires nine thermodynamic and kinetic parameters and is intricate but can be handled by simulation. However, extracting nine parameters from a set of experimental CV data is forbiddingly difficult, so one looks for simplifications. In almost all treatments, both E-steps are taken as reversible, so that k10 , k20 , 𝛼 1 , and 𝛼 2 , are not required, and the problem ′ ′ simplifies to five parameters: E10 , E20 , k f , k b (or K), and k d . The behavior depends on the relative positioning of the two formal potentials: ′

′

1) In the case of greatest interest, E20 is significantly more positive than E10 , so that species C is easier to reduce than species A and is, therefore, immediately reducible at the potential ′ ′ ′ used to initiate (13.3.38). Typically, ΔE0 = E20 − E10 ≥ 180 mV. 2) In the opposite limiting case, when the second E-step occurs at significantly more negative ′ potentials than the first (ΔE0 ≤ − 180 mV), there are two reduction waves. Only (13.3.38) and (13.3.39) take place in the first wave; hence, the electrode reaction is an EC sequence that can be analyzed as described in Sections 13.3.1–13.3.3. The second wave occurs when (13.3.40) becomes enabled at a more negative potential, and the full ECE sequence applies. Reaction (13.3.41) is included because species B can reduce species C homogeneously. In case ′ 1 above, with ΔE0 ≥ 180 mV, (13.3.41) can be taken as irreversible to the right. Because species B and C are at the same oxidation level, this reaction is often called a “disproportionation” reaction and schemes that include it are called ECE/DISP

577

578

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

(a)

(c) 2 μA

1 μA

Rh CO PPh2

Rh OC Ph2P

–2e 2+

(b)

(d)

2 μA

Rh OC Ph P 2

5 μA

Rh C O PPh2 (e)

–0.46

–0.74

–1.02

–0.46

–0.74

–1.02

E/V vs. Fc+/Fc

Figure 13.3.24 CV at a Pt disk for 0.71 mM (Fv)Rh2 (CO)2 (μ-dppm) in CH2 Cl2 + 0.1 M TBAPF6 . Scan rates: (a) 1, (b) 5, (c) 10, (d) 50 V/s. (e) Electrode reaction in structural form. Scans begin at the negative limit and first move ′ positively. Anodic currents are down. Points are experimental, curves are simulated for E10 = −0.762 V, ′ 0 0 0 E2 = −0.775 V, k1 = 0.035 cm/s, k2 > 0.2 cm/s, 𝛼 1 = 0.25, and 𝛼 2 = 0.50. [Adapted with permission from Chin, Geiger, and Rheingold (49). © 1996, American Chemical Society.]

In the remainder of this section, we will consider only systems involving reversible electron ′ transfers adhering to case 1 (ΔE0 ≥ 180 mV). (a) Er Ci Er Reactions

Let us first examine a further simplified system where k b = k d = 0, so that the only homogeneous chemistry is the irreversible conversion of B to C (an Er Ci Er sequence): ′

E10

kf

′

E20

−−−−−⇀ −−−−−⇀ A+e − ↽ −− B −−−→ C + e − ↽ −− D

(13.3.42)

The dimensionless kinetic parameter becomes 𝜆 = k f /fv. In the CV (Figure 13.3.25), only a single wave is observed on the forward scan (wave I, reduc′ tion of both A and C). It occurs near E10 , since the first E-step must happen before other chemistry can occur. If species C is formed, it is reducible to species D at this potential. If the C-step is not too rapid, or the scan is fast enough (Figure 13.3.25b), a reversal wave (II) is observed for oxidation of species B. As the reverse scan continues, a second oxidation wave (III) is seen for the heterogeneous conversion of species D to species C. Another reversal of the scan reveals a corresponding cathodic wave (IV) for the reduction of C to D. The relative sizes the waves depend upon k f /v (expressed as 𝜆). A useful diagnostic is the apparent number of electrons, napp , passed per molecule of A reduced in the forward peak. Using the following definition, one can calculate it from the height

13.3 Survey of Behavior

I

4 3

4 3

2 Current

I

1

–1

–2

–3

0.0

–0.2

–0.4 E/V (a) I

1

5

–0.6

–0.4 E/V (c) I

–0.6

–0.4 E/V (d)

–0.6

4 3

2 IV

2 IV

1 0

0 III

–1 –2

III –0.2

0.0

II

3 Current

0 –1

–2

IV

1

0

Current

Current

2

0.0

–0.2

II

–1

–0.4 E/V (b)

–0.6

–2

III 0.0

–0.2

′

′

Figure 13.3.25 Simulated CV for the Er Ci Er case where E10 = −0.44 V, E20 = −0.20 V, and n1 = n2 = 1. (a) 𝜆 = 0 (unperturbed nernstian reaction); (b) 𝜆 = 0.05; (c) 𝜆 = 0.40; (d) 𝜆 = 2, where 𝜆 = kf /fv. The sequence from (b) to (d) shows the effect of a 40-fold increase in kf at constant v or a 40-fold decrease in v at constant kf . The current scale is dimensionless and involves normalization by v1/2 .

of peak I, ipf (I): napp =

ipf (I) 1∕2

∗ v1∕2 0.4463Ff 1∕2 ADA CA

=

ipf (I) ipf,u (I)

(13.3.43)

where ipf, u (I) is the forward peak height of the unperturbed A/B voltammogram, given by (7.2.20). A plot of napp vs. log 𝜆 (Figure 13.3.26) furnishes a zone diagram for the system. When 𝜆 is small, species B is stable because the decay to species C does not have time to occur appreciably. The result is the unperturbed CV of the A/B couple (Figure 13.3.25a) and napp ≈ 1 (zone DO). In this region, waves III and IV are negligible. For large 𝜆 (zone KP), species B is fully converted to C, which immediately reacts to form D; therefore, napp ≈ 2. Waves III and IV are prominent, while wave II is absent (Figure 13.3.25d). Also presented in Figure 13.3.26 is |ipr (II)/ipf (I)|, which behaves qualitatively as in Case Er Ci , being essentially unity in zone DO and zero in zone KP. The quantitative behavior differs from that of Case Er Ci because the C-step augments ipf (I) in this instance. Zone KG is where 𝜏 obs = 1/fv and 𝜏 rxn = 1/k f are comparable. Both napp and |ipr (II)/ipf (I)| make their transitions between limiting values and offer the opportunity for measurement of k f . The analytical treatment of this scheme for CV has been described (1, 9, 51, 52), but, as we have commonly observed, comparative simulation (Section 13.2.5) is now the most effective way to test a mechanistic model and to extract parameters.

579

580

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

KG (or KI)

DO (or DP) 2.0

KP 1.0

ipr(II) ipf(I)

0.8

napp

napp

0.6 i (II) pr 1.5

0.4 ipf(I) 0.2 0.0

1.0 –3

–2

–1

0 log λ

1

2

3

Figure 13.3.26 Zone diagram for CV in an Er Ci Er system. napp is defined in (13.3.43). Peak labeling corresponds to Figure 13.3.25. 𝜆 = kf /fv. Zone labels at top. Historic alternative labels are shown for zones DO and KG.

(b) Er Cr Er Reactions

When reactions (13.3.38)–(13.3.40) are all reversible, the mechanism is Er Cr Er and the behavior depends on both K and 𝜆, where the latter is defined in terms of both rate constants for (13.3.39): 𝜆 = (kf + kb )∕fv

(13.3.44)

We continue to neglect disproportionation by assuming k d = 0; thus, the reaction scheme is ′

E10

′

E20

kf

−−−−−⇀ −−−−−⇀ A+e − ↽ −− B ⇄ C + e − ↽ −− D

(13.3.45a)

K = kf ∕kb

(13.3.45b)

kb

The zone diagram is shown in Figure 13.3.27 (53, 54).

–8

DI DO (or DP)

–4 log K

DE

KG (or KI)

0

KP

KE

4 0

5

10 log λ

15

20

Figure 13.3.27 Zone diagram for the Er Cr Er mechanism in (13.3.45a). K = kf /kb and 𝜆 = (kf + kb )/fv. Historic alternative labels are shown for zones DO and KG. [Adapted from Savéant, Andrieux, and Nadjo (53), with permission from Elsevier Science.]

13.3 Survey of Behavior

Several aspects are noteworthy: • When K is small, species B is always strongly favored and there is never an appreciable concentration of C or D. Thus, one sees the unperturbed 1e CV for the A/B couple (region DO). • Even when K is large, one can remain in zone DO if 𝜆 is small (high v or small k f + k b ). This is the case at the lower left of the diagram. • The opposite limit is on the far right, in zone DE. This is where K is moderate or large, so that the conversion of B to C is significant, and 𝜆 is large, so that there is enough time for (13.3.39) to equilibrate. In this zone, all steps in (13.3.45a) are always at equilibrium, so the net result is a 2e nernstian reaction A + 2e ⇌ D

(13.3.46)

producing a reversible 2e CV wave. By combining the relevant thermodynamic data, one 0 , to be finds the standard potential of (13.3.46), EA∕D 0 EA∕D = (E10 + E20 )∕2 + (RT∕2nF) ln K

(13.3.47)

The observed wave in the DE region occurs essentially at this potential. • For values of K and 𝜆 between zones DO and DE, one encounters regions where kinetics control the observed behavior. The special case of the Er Ci Er sequence is in the bottom right of this diagram where K is very large (e.g., log K = 4, making the C-step effectively unidirectional). One finds the same set of zones as in Figure 13.3.26. (c) DISP Reactions

In general treatments of ECE cases, one must take account of disproportionation, (13.3.41). It is readily practical to do so in simulations. We now focus on some limiting cases, called DISP systems (1, 7, 9, 17, 55), where this reaction plays such a strong role in the conversion of species A to species D that the second E-step, (13.3.40), simply does not occur. For the systems we have been considering, the second E-step is always thermodynamically enabled in the potential region where the reduction of species A occurs. However, it cannot occur if species C does not reach the electrode, which can be the situation, for example, when k f is small, so that C is mostly produced at a distance from the electrode, while k d is large, so that C diffusing toward the electrode mainly reacts homogeneously with B diffusing away from it. There are two DISP subcases: • DISP1, where the C-step, (13.3.39), is rate-determining. • DISP2, where the disproportionation, (13.3.41), is rate-determining, while the C-step is at equilibrium. Figure 13.3.28 is a simplified zone diagram (neglecting the intermediate regions), drawn in terms of the parameters K and p (7), where ( )1∕2 ∗ C C C )3∕2 kd CA 𝜏 𝜏 (𝜏rxn rxn p= 𝜆−1∕2 = rxn ⋅ = (13.3.48) d d (𝜏 1∕2 kf + kb 𝜏obs 𝜏rxn 𝜏rxn obs ) Here, 𝜆 is defined as in (13.3.44). The middle and right-hand representations of p in (13.3.48) C = 1∕(k + k ), the characare in terms of the characteristic time of reaction for the C-step, 𝜏rxn f b ∗ d teristic time of reaction for the disproportionation, 𝜏rxn = 1∕kd CA , and the characteristic time of observation, 𝜏 obs = 1/fv. Because p is a complex expression of timescales, it is not simple to interpret the zone diagram, but a few observations can help to convey its meaning: ∗. 1) For any fixed set of K, k f + k b , and v, positioning across the diagram is determined by kd CA Larger values of p correspond to faster, more important disproportionation.

581

582

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

–3

K

DISP2 kf + kb

–2

v kdCA*

G

H

E

F

B

C

Figure 13.3.28 Simplified zone diagram for ECE/DISP systems. Lettered points are discussed in text. Legend in upper right shows the effects of order-of-magnitude changes in the indicated parameters. [Adapted from Andrieux and Savéant (7), with permission.]

–1 log K

D

ECErev

DISP1

0 ECEirr 1 A

2

–3

–2

–1

log p

0

1

2

∗ is so small that dispropora) When K is large, the C-step is effectively irreversible. If kd CA tionation is negligible, then p is small and the system lies in zone ECEirr of Figure 13.3.28, perhaps at Point A. This is the Er Ci Er case examined above. ∗ is larger, then p is also larger and the system falls at a point further to the right, b) If kd CA but at the same K. It might lie at Point B, still in zone ECEirr , but corresponding to an Er Ci Er /DISP case, because disproportionation is not negligible. ∗ is so large that 𝜏 d ≪ 𝜏 C , then disproportionation can be so important that c) If kd CA rxn rxn species C is fully consumed in solution and never reaches the electrode. This is a DISP case in which the C-step is rate-determining, so the system resides in zone DISP1, perhaps at Point C. 2) For any given chemical system, K, k f + k b , and k d are fixed; hence, the value of p is deter∗ . If the concentration remains constant, larger values of p mined by the scan rate and CA correspond to lower values of v. A system with large K might reside at Point A at high v, then at Points B and C at successively lower scan rates. 3) If 0.1 < K < 1, the bidirectionality of the C-step is essential to an understanding of the system. That step favors species B, but there can be an appreciable formation rate for species C, which then might be reduced to D. a) If p is small, disproportionation is negligible and the system would lie in zone ECErev of Figure 13.3.28, perhaps at Point D. This situation corresponds to the Er Cr Er case considered above. ∗ or lower v, the system would lie b) For larger values of p, corresponding to greater kd CA further to the right. At point E, it would still be in zone ECErev , but as an Er Cr Er /DISP case. If disproportionation were to become very rapid, then species C would be entirely reduced homogeneously and the C-step would be rate-determining. The system would be in zone DISP1, perhaps at Point F. 4) If K is small, there is never a large enough concentration of species C in solution to support the second E-step, (13.3.40). The only way for species A to be converted to species D is by homogeneous reduction of C as it is formed in solution. a) If p is small, disproportionation is ineffective and can reduce only a small portion of C before it reverts to B; thus, (13.3.39) is at equilibrium and disproportionation is rate-limiting. This is the DISP2 case defined above. The system lies in zone DISP2 of Figure 13.3.28, perhaps at Point G.

13.3 Survey of Behavior

b) If p is large, disproportionation is highly effective, converting species C immediately upon formation; therefore, the C-step is rate-limiting. The system is in zone DISP1, perhaps at Point H. The general behavior of an ECE/DISP system is clearly complex. Differences among the cases ∗ and v is required for diagnosis and are subtle, so collection of CV data over a wide range of CA extraction of parameters. Theoretical voltammograms are readily obtainable by digital simula∗ , and v. In principle, the heterogeneous electron-transfer tion for the full set of ΔE0 , k f , k b , k d , CA kinetics can also be included; however, doing so requires additional parameters that may overwhelm the capacity for analysis of practical data. (d) An Experimental Case

Bulk reduction of Fe(CO)5 in THF gives the isolable product Fe2 (CO)2− , corresponding to 1e 8 per Fe(CO)5 . However, the reduction mechanism has been debated (56), in significant measure because of the complexity of the CV (Figure 13.3.29). For the moment, let us set aside the waves labeled O3 and R3 . At the higher scan rate (Figure 13.3.29b), the wave pattern resembles that in Figure 13.3.25d. The forward reduction, R1 , clearly yields a product showing a chemically reversible CV at a much more positive potential. The behavior is consistent with the following ECE sequence: Fe(CO)5 + e ⇌ Fe(CO)− 5

′

E10

(13.3.49)

Fe(CO)− → Fe(CO)− 4 + CO 5 2− Fe(CO)− 4 + e ⇌ Fe(CO)4

(13.3.50) E20

′

(13.3.51) 0′

Peak R1 would correspond to the whole process, initiated near E1 , while peaks O2 and R2 would ′ be assigned to the second couple at E20 . Strong confirmatory evidence for an ECE mechanism came from an independent measurement of napp = 1.95 ± 0.06 in process R1 (56). To obtain this result, the investigators employed a method based on chronoamperometry and SSV (57) similar to that described in Section 6.1.4(c). Figure 13.3.29 CV of 2 mM Fe(CO)5 in THF + 0.3 M TBABF4 at an Au disk (0.25 mm). T = 20 ∘ C. (a) v = 1 V/s. (b) v = 20 V/s. The sweep in each case begins at the origin mark and first moves negatively. [Reprinted with permission from Amatore, Krusic, Pedersen, and Verpeaux (56). © 1995, American Chemical Society.]

R1

R1

R2 R3

O3 –1.5

O2 O2 –2.0 (a)

–2.5

–3.0 –1.5 –2.0 E/ V vs. Ag/Ag+

–2.5 (b)

–3.0

583

584

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions ′

No reversal peak corresponding to Fe(CO)− could be seen near E10 , even at the highest stud5 ied scan rates; therefore, the lifetime of this intermediate [𝜏 rxn for (13.3.50)] was always much shorter than 𝜏 obs for the CV. However, the investigators were able to trap Fe(CO)− chemically 5 and estimated its lifetime to be 10–20 ns. On the timescale of the fastest CV employed (≥20 V/s), (13.3.49)–(13.3.51) adequately describe the system. The short lifetime of Fe(CO)− assures that K for the C-step is very large 5 and that the parameter p defined in (13.3.48) is very small; thus, the system would always be in zone ECEirr of Figure 13.3.28. Indeed, that short lifetime assures that there would never be any appreciable concentration of Fe(CO)− to engage in the disproportionation, 5 2− Fe(CO)− + Fe(CO)− 4 ⇌ Fe(CO)5 + Fe(CO)4 5

(13.3.52)

Accordingly, the system is on the left side of zone ECEirr , where disproportionation is always negligible. This is a true Er Ci Er case, and, since there is never a reversal peak for Fe(CO)− , the 5 system is always in zone KP of Figure 13.3.26. Many (probably most) real ECE sequences are followed by at least one additional homogeneous step, because a two-electron ECE product tends to be reactive. In this example, we have such a case. While the ECE sequence is enough to explain the behavior on short timescales, the picture is more complicated on longer timescales. Waves O3 and R3 grow into the CV and require explanation. Besides, other chemistry is needed to reach the known final product, Fe2 (CO)2− . For this system, the additional step was shown to be the dimerization of the 1e 8 intermediate, Fe(CO)− : 4 2− 2 Fe(CO)− 4 → Fe(CO)8

(13.3.53)

The investigators were able to confirm unambiguously that O2 and R2 arise from (13.3.51) and that and O3 and R3 are due to 2− Fe2 (CO)− 8 + e ⇌ Fe2 (CO)8

′

E30

(13.3.54)

Through various evaluative methods, including comparative simulation, the investigators ′ ′ ′ reported E10 , E20 , E30 , and the rate constant of (13.3.53) (56). 13.3.8

Concerted vs. Stepwise Reaction

Chemists have long understood that overall reactions involving multiple changes sometimes happen in a single step and sometimes serially. Much investigation and debate have been spent on factors determining the reaction path in such cases. This issue is also relevant in electrochemistry, especially for electrode reactions that involve bond breaking or bond formation (1, 4, 6, 7). The best-studied examples involve ejection of a leaving group after heterogeneous electron transfer, such as the elimination of Br− upon the 1e reduction of 9-bromoanthracene. For an electroreactant, RX, and a leaving group, X− , the mechanistic alternatives are simply expressed: 1) The bond breaks simultaneously with the entry of the electron into the molecule, so that the whole process is a concerted elementary reaction: k 0 ,𝛼

RX + e −−−−→ R∙ +X−

(13.3.55)

2) The entry of the electron and the breaking of the bond happen stepwise. The former triggers the latter, but not in the same moment. In this conception, the product of 1e transfer lives briefly before the leaving group departs: k 0 ,𝛼

k

f − −−−−−−− → RX + e ← − R−∙ −−−−→ R∙ +X

Thus, an intermediate exists in the stepwise process, but not in the concerted one.

(13.3.56)

13.3 Survey of Behavior

(a) Reaction Diagrams

Experimental investigations have clearly revealed that concerted mechanisms dominate in certain kinds of situations, and stepwise mechanisms in others (1, 4, 6, 7, 9). The reaction diagrams in Figure 13.3.30 will help us to see that both alternatives are conceivable for any system, but that specific chemical properties cause one to be favored over the other. Figure 13.3.30a is like the reaction diagrams we previously encountered in Figures 3.3.2 and 3.5.4. The darker curves, representing RX + e, make up the “reactant side.” Since the free energy of the electron on the electrode is linearly related to potential, these curves move up or down as the potential is made more negative or positive, respectively. The lower black curve is for the ′ reactants at E0 for the overall electrode reaction, and the upper curve represents the reactants

E

D

Standard free energy

Standard free energy

‡ ΔG E 0ʹ+η

B

RX + e (E 0ʹ+η)

RX + e (E 0ʹ)

Fʹ

Dʹ Bʹ Cʹ RX–·

‡

‡

ΔG E 0ʹ

ΔG E 0ʹ

R· + X–

A

Aʹ RX + e (E 0ʹ)

C Reaction coordinate

R· + X–

Eʹ

Reaction coordinate (b)

(a)

Gʺ Eʺ Fʺ – Bʺ RX · Standard free energy

‡

ΔG E 0ʹ+ ηʺ

Dʺ

RX + e (E 0ʹ + ηʺ)

‡

Aʺ

ΔG E 0ʹ+ηʹ

RX + e (E 0ʹ + ηʹ)

R· + X–

Cʺ

Reaction coordinate (c)

Figure 13.3.30 Diagrams for electrode reactions leading to elimination of a leaving group. (a) Concerted reaction. (b) Stepwise reaction. (c) Change from concerted to stepwise reaction upon increasing the driving force.

585

586

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions ′

at a more negative potential, E0 + 𝜂. The “product side” is a gray curve describing the repulsive separation of R∙ and X− . We can generally expect the reactant and product curves (representing multidimensional surfaces) to intersect. At that point, the system can pass from the reactant surface to the product surface (or vice versa); thus, the intersection is an activated complex or transition state. ′ If the electrode is at E0 , the reaction path begins at point A, rises up the reactant curve to transition state B, then falls down the product curve to point C and beyond. This reaction yields the final products directly, so it is the concerted process, (13.3.55). Because the reaction involves bond breaking, there is a sizable intrinsic barrier (1); thus, the free energy of activation, ΔG‡ 0′ , E is large, and one might expect a negligible corresponding rate of reduction. If a negative overpotential, 𝜂, is applied, the reactant curve shifts upward and there is a new reaction path starting at point D, proceeding through transition state E and then down to ′ point C. The free energy of activation, ΔG‡ 0′ , is smaller than at E0 , and the rate of reducE +𝜂

tion is exponentially enhanced by the differential. Accordingly, one might see current from the concerted reaction at this potential. There are three principal messages here: • The concerted reaction is conceptually simple and is always a possibility. ′ • It is likely to involve a substantial activation energy at any potential near E0 for the overall process. • A sizable overpotential is likely to be required for the observation of significant current (e.g., a CV peak). Now, we turn to Figure 13.3.30b. The reactant and product sides of this diagram are the same ′ as for E0 in Figure 13.3.30a, and the concerted reaction path exists exactly as before (through points now labeled A′ , F ′ , and E ′ ). The free energy of activation for the concerted process is ′ the same as for E0 in Figure 13.3.30a. The new element is that RX−∙ has a free-energy minimum below the transition state for the concerted process (point F ′ ). The energy surface for RX−∙ intersects both the reactant and product surfaces, opening a new reaction path. At potential ′ E0 , the system starts at A′ , rises to transition state B′ , crosses to the surface for RX−∙, lingers at C′ , rises to transition state D ′ , crosses to the product surface, and then falls to point E ′ . This path proceeds through the intermediate, RX−∙, and corresponds to the stepwise process in (13.3.56). Because RX−∙ has an energy minimum, it has a finite lifetime. The role of the intermediate is to facilitate the reduction by offering an energy minimum below the transition state for the concerted process. The free energy of activation, ΔG‡ 0′ , is E

′

much smaller than in Figure 13.3.30a, so one could expect significant current even at E0 , or perhaps with a small overpotential. An energy surface exists for RX−∙. The question is where it is situated on the energy scale, which is a matter of chemical properties. If it lies below the transition state for the concerted process, the system should favor the stepwise path. Otherwise, the concerted process should be favored. (b) Diagnostic Criteria

At first, one might think of both mechanisms as EC cases, so it is natural to wonder whether one can distinguish them experimentally. The answer is yes because they behave differently (1, 4, 6, 7). The stepwise process does fall in Case EC, but the concerted process is a one-step, one-electron heterogeneous reaction without coupled homogeneous chemistry. Our earlier discussion of Cases ECi (Sections 13.3.1 and 13.3.2) applies generally to systems in which an electron transfer triggers the stepwise loss of a leaving group. The zone diagram in

13.3 Survey of Behavior

Figure 13.3.5 includes reversible, quasireversible, and totally irreversible electron transfers, so it applies to all ECi cases, and is a useful reference for this discussion. The unmistakable signature of a stepwise mechanism is detection of the intermediate, RX−∙. In CV, the criterion is the appearance of a reversal peak at higher scan rates, which will occur if one can reach zones KO, KG, DO, or QR in Figure 13.3.5. If the C-step is too fast for a reversal peak to appear even at the highest accessible v, one still might be able to diagnose the mechanism from the shape of the forward CV peak and the way it shifts with v. If the E-step is reversible (true in many cases), the system would be in zone KE of Figure 13.3.5, where the variation of Epf with v is given by | 𝜕Epf | 2.303RT | | | |= | 𝜕 log v | 2F | |

(29.6 mV at 25 ∘ C)

(13.3.57)

The width of the peak should be independent of scan rate: |Ep∕2 − Epf | = 1.857

RT F

(47.7 mV at 25 ∘ C)

(13.3.58)

Uncompensated resistance affects measurements of these parameters, so the investigator must take care to minimize the error and, perhaps, to make corrections. If these criteria are fulfilled, the system can be diagnosed as proceeding stepwise. Figure 13.3.31 presents data for the reduction of 9-anthracenyldimethylsulfonium (58), which causes ejection of a methyl radical. The peak width remains within 2–5 mV of 47.7 mV over the studied range of v. Moreover, the peak shift with scan rate is near 29 mV per decade. The observed behavior is fully consistent with expectations for a stepwise mechanism. A concerted process is a totally irreversible 1e electrode reaction; therefore, the CV should be as described in Section 7.4. The shift of Epf with v is provided from (7.4.9) as | 𝜕Epf | 2.303RT | | | |= | 𝜕 log v | 2𝛼F | |

(29.6∕𝛼 mV at 25 ∘ C)

(13.3.59)

and from (7.4.10), the peak width should be |Ep∕2 − Epf | = 1.857

RT 𝛼F

(47.7∕𝛼 mV at 25 ∘ C)

(13.3.60)

Thus, a system proceeding by a concerted mechanism should show a broader voltammetric peak and a greater peak shift with scan rate than a system proceeding stepwise. One can

70

Me

Me S+

(Ep/2 – Ep)/mV

Figure 13.3.31 Peak width for reduction of 9-anthracenyldimethysulfonium at a glassy carbon disk in MeCN + 0.1 M TBABF4 . Dashed line is at 47.7 mV. Triangles are raw data; circles are corrected for uncompensated ohmic drop. [Adapted with permission from Andrieux, Robert, Saeva, and Savéant (58). © 1994, American Chemical Society.]

60

50

40 –2

–1

0 1 log v/ V s–1

2

3

587

588

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

express these diagnostics in terms of the parameter 𝛼. To diagnose a concerted mechanism, the extracted value should be in an appropriate range for a transfer coefficient. Indeed, Savéant argues that a signature for a concerted reaction is a small value of 𝛼 (1, 4, 59). This idea is based on the Marcus kinetic model, in which 𝛼 is a potential-dependent function ′ [Section 3.5.4(c)].19 At E0 , 𝛼 = 0.5, but it is expected to decline linearly toward more negative potentials. As noted above, the reduction peak in CV for a concerted reaction should lie at a significant negative overpotential; thus, the value of 𝛼 at the peak20 —the extracted value of 𝛼—should be notably less than 0.5. Only the Ei Ci stepwise case is indistinguishable by CV from a concerted mechanism. This kind of system would always be in zone IR of Figure 13.3.5. An intermediate cannot be detected voltammetrically, and the diagnostics based on wave position and shape are the same as for the concerted mechanism. It might be possible to detect an intermediate by non-electrochemical means, in which case the stepwise mechanism could be assigned. (c) Reductive and Oxidative Elimination Reactions

The ideas just presented were developed for reductions of RX to R ∙ and X− , but they are also applicable to other processes involving leaving groups, including RX+ + e → R ∙ + X∙

(13.3.61)

RX − e → R ∙ + X+

(13.3.62)

RX− − e → R ∙ + X∙

(13.3.63)

These processes are broadly labeled as reductive eliminations and oxidative eliminations (1, 4, 6, 7). For oxidative processes, such as (13.3.62) and (13.3.63), criteria (13.3.57)–(13.3.60) all apply, but with 1 − 𝛼 substituted for 𝛼. A small value of 1 − 𝛼 becomes the corresponding signature for a concerted oxidation. The reaction diagrams in Figure 13.3.30 also apply to oxidations, but the upward movement of the curves on the reactant side corresponds to a positive potential change. A great variety of chemistry has been explored and has been nicely summarized (1, 4, 6, 7). One can understand the mechanisms for most leaving-group eliminations in terms of • The standard potential for formation of the 1e product of the electroreactant (e.g., for the + RX∕RX−∙ or RX∕RX ∙ couple). • The strength of the R—X bond. The most broadly studied category is the reductive elimination of halides, for which there is a sharp division between aryl compounds, which generally react in a stepwise process, and aliphatic halides, which undergo concerted reduction and elimination. The difference is based on the ability of an aryl linkage to delocalize the added electron, stabilizing RX−∙ below the transition state for concerted reaction (as in Figure 13.3.30b). Aliphatic halides lack this capacity; hence, the standard potentials for the RX∕RX−∙ couples are much more negative, and the energy of RX−∙ is always well above the transition state for concerted reaction, as in Figure 13.3.30a. 19 This statement is in accord with Savéant’s own presentation in his book (1); however, he had earlier developed a model (59) using Morse curves (near the minimum on the reactant side, but only the repulsive portion on the product side). It predicted a quadratic dependence of activation energy on potential and a variation of 𝛼 resembling, but distinct from, the results from Marcus theory. 20 Savéant (1) uses the symbol 𝛼 p to distinguish this value of 𝛼.

13.3 Survey of Behavior

(d) Mechanistic Switching

An interesting possibility that we have not yet discussed is depicted in Figure 13.3.30c, where the energy of RX−∙ is neither far above nor significantly below the transition-state energy for concerted reaction. We need to define the case with more precision, because the transition-state energy for concerted reaction depends on the electrode potential, while the energy of RX−∙ is ′ fixed. In Figure 13.3.30a, for example, the transition state is at point B for E0 but is at point D ′ for the more negative potential, E0 + 𝜂. When we discussed Figure 13.3.30a above, we recog′ nized that the free energy of activation might be too great at E0 to see any appreciable current, ′ but that we might be able to see a voltammetric wave near E0 + 𝜂. Our focus now is on situations where • a wave can be observed and • in the potential range of the wave, the transition state for concerted reaction is close to the energy minimum for RX−∙. ′

Suppose the lower curve for the reactant side in Figure 13.3.1c is for a potential, E0 + 𝜂 ′ where a CV peak can be seen. That wave would arise from concerted reduction, following the path from A′′ through B′′ to C′′ . Since the reactant surface does not intersect the surface for RX−∙ below B′′ , there is no active stepwise path. ′ If we were to make the potential a little more negative—moving it to E0 + 𝜂 ′′ —the picture changes substantially. The reactant surface intersects the RX−∙ surface, creating a new transition state, and opening a stepwise path from D′′ through E′′ , F′′ , and G′′ to C′′ . At this potential, the system must favor the stepwise path because the transition state for concerted reaction is significantly higher in energy. Thus, it seems that one might find an experimental system in which such a mechanistic change could be observed. That idea has been explored in selected systems by observing behavioral details as the scan rate is elevated, which causes the CV peak to shift progressively to more extreme potentials. Figure 13.3.32 provides results for the first discovered example (1, 58), involving the reduction of benzylphenylmethylsulfonium (Figure 13.3.32c). At low v, this system shows a large peak width (Figure 13.3.32a) and a correspondingly small value of 𝛼, in keeping with expectations for a concerted mechanism. As v is increased, the peak shifts negatively; yet, it narrows and 𝛼 rises. This behavior is opposite the expectation (and normal observation) for a fully concerted process. It was interpreted as evidence of a switch toward stepwise reaction. If the switch were complete and the E-step remained reversible, the peak width should fall to a constant value near 50 mV, and the computed value of 𝛼 should reach unity. These limits are not reached, probably because this system is complicated by quasireversibility in the E-step (58), which causes the re-broadening of the peak at higher v. Additional cases of mechanistic switching have been documented (1), all based on fortuitous values of the standard potential for formation of the intermediate. (e) Proton-Coupled Electron Transfers

All of the ideas in this section also apply to reactions in which the two principal events are electron transfer and proton transfer. Proton-coupled electron transfer (PCET) happens both homogeneously and heterogeneously and is important in living systems, as well as in fundamental and technological electrochemistry (3, 60, 61). An intramolecular example is provided in (13.3.64), featuring oxidation of an aminophenol possessing an internal hydrogen bond between the acidic and basic moieties (3):

t Bu

O

H

N

tBu

•O

–e tBu

tBu

H

+N

589

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

130 (Ep/2 – Ep)/mV

590

(a)

120 H2 C

110

S +

H2 C

CH3 + e

100

S•

CH3

stepwise

90 (b)

concerted

0.45

S

CH3

CH2• +

α 0.40

(c) 0.35 –2

–1

0 1 2 log v/ V s–1

3

Figure 13.3.32 Behavior of the CV peak for reduction of benzylphenymethylsulfonium at a glassy carbon disk in MeCN + 0.1 M TBABF4 . (a) Measured peak width vs. scan rate. (b) Value of 𝛼 computed from |E p/2 − E pf | using (13.3.60). (c) Reaction scheme. [From Saveant (1), with permission. Original data from reference (58).]

The removal of an electron is accompanied by (in a concerted process) or triggers (in a stepwise process) the transfer of a proton to the amine functionality. Electrochemical PCET reactions can occur between pairs of reactants if they can reach the required transition state. Aqueous environments are especially supportive because H2 O is omnipresent at the electrode surface and can serve as both a proton acceptor and a proton donor. It also engages in hydrogen bonding, which is usually important to the transition state. This chemistry has a broad scope and has been reviewed (3, 60, 61). In Chapter 15, we will encounter very important PCET reactions involving adsorbates. 13.3.9

Elaboration of Reaction Schemes

Chemistry is very diverse, so a practitioner is bound to become interested in mechanistic patterns that we did not address. Accordingly, this chapter was designed to introduce and to exercise the principal elements required for the reader to take on new problems. Among them have been: 1) The primary dynamic components—varied heterogeneous and homogeneous steps, bidirectional processes, and homogeneous reactions linking redox couples. 2) The elementary mechanistic patterns—EC, EC′ , CE, EE, and ECE. 3) The principal diagnostic criteria for CV. 4) Analytical ideas of broad value, including characteristic times, zone diagrams, and the use of simulation to test mechanistic hypotheses and to extract parameters. Applying these elements to new problems is a matter of imagination, willingness, judgment, and practice.

13.4 Behavior with Other Electrochemical Methods

The availability and power of commercial simulation packages is especially liberating for any investigator interested in electrochemical mechanisms. With these tools, one is free to develop and to test hypotheses with reasonable confidence, but without the need to be diverted by the details of simulation. Thus, it has become easy to turn an elementary pattern, like EC, into something more complex, like Eq Cr C i . Many reactions that start with the ECE sequence are followed by an irreversible decay, so that they become ECEC. It is no problem with a simulator to add the additional step. Simulators can also treat patterns that we → − ← − recognized in Section 13.1 but did not cover, including E C E and square schemes. Some of these have also been addressed analytically, with theoretical results applied to actual systems (4, 5, 7, 62–65). For decades, the limit of complexity for a mechanistic hypothesis was determined by the theoretical ability to predict electrochemical responses. In current practice, the complexity of a model is rarely limited by simulation capacity. More commonly, it is confined by the experimenter’s ability to distinguish alternatives using real data, which always have natural imprecision and often also include interferences from background currents or waves from other components. More sophisticated models entail added chemical steps or added effects, each requiring at least one new parameter. Thus, the practical limit of complexity for a mechanistic model often translates into the number of parameters that can be realistically extracted from data. That number is, in turn, related to the size and complexity of the data set, but it rarely would be greater than five parameters, or, perhaps, six.21 One may have no choice but to limit the complexity of a model or to find independent experimental means for evaluating parameters. In the practical examples of this section, we saw cases in which the investigators had to • Elaborate an elementary mechanistic pattern by including quasireversibility or adding a chemical step [Sections 13.2.3, 13.3.2, 13.3.3, 13.3.5, and 13.3.7(d)]. • Shorten the list of fitted parameters by using an independent experimental means for evaluating one or more of the needed quantities (Section 13.3.5). • Stop short of completeness in the mechanistic model because greater elaboration could not be practically supported in the analysis of results (Section 13.3.3).

13.4 Behavior with Other Electrochemical Methods To give the reader a consistent context, the survey in Section 13.3 was confined to CV and LSV; however, many other electrochemical methods can be used for the diagnosis and quantification of electrode reactions coupled to homogeneous chemistry. The ideas developed above also apply to other methods, but one must recognize how the observational time, 𝜏 obs , varies from method to method. The definition of 𝜏 obs must be tailored for each technique (Table 13.4.1) because different variables control the timescale (e.g., v for CV, 𝜏 for double-step experiments, or 𝜔 for impedance). For CV or LSV, 𝜏 obs = 1/nfv, which is the time required to scan kT per electron (Section 13.2.2). For SSV, 𝜏 obs is governed by the electrode radius, r0 , and is ∼ r02 ∕D. The definitions used for 𝜏 obs in Table 13.4.1 are typical of the literature, but they are not necessarily unique. Sometimes, investigators have employed other definitions (of similar numeric order as those given in the table). 21 Mathematician John von Neumann is quoted as having said, “With four parameters, I can fit an elephant. With five, I can make him wiggle his trunk.”

591

592

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

Table 13.4.1 Observational Times and Time Windows for Experiments with Dissolved Species Technique

𝝉 obs

Practical Range(a)

Time Window/s(b)

CV, LSV(c)

1/fv

v = 0.01 – 3 × 106 V/s

10–8 – 3

SECM(d)

d2 /D

d = 10 nm − 10 μm

10–7 – 0.1

SSV (disk UME)

r02 ∕D

r0 = 10 nm – 25 μm

10–7 – 0.6

Impedance(e)

1/𝜔 = (2𝜋f )–1

f

= 10–2 – 106 s–1

Chronoamperometry(f )

2 × 10–7 – 20

𝜏

𝜏

= 10−6 – 10 s

Chronocoulometry(f )

10–6 – 10

𝜏

𝜏

= 10–6 – 10 s

Chronopotentiometry(g)

10–6 – 10

𝜏 or t 1 tp

t

= 10–6 – 50 s

10–6 – 50

t p = 5 × 10–5 – 0.1 s

5 × 10–5 – 0.1

NPV,

RPV(i)

𝜏 − 𝜏′

𝜏 − 𝜏′

10−4 − 0.2

RDE,

RRDE(j)

1/𝜔 = (2𝜋f )–1

𝜔 = 30 – 1000 s–1

2 × 10–4 – 5 × 10−3

t max

t max = 1 – 5 s

1–5

SWV(h)

dc Polarography(k) Thin-layer

coulometry(l)

= 10−4 − 0.2 s

t

t = 0.1 – 10 s

0.1 – 10 s

Bulk coulometry(l)

t

t = 100 – 3000 s

100 – 3000

Macroscale electrolysis(l)

t

t = 100 – 3000 s

100 – 3000

(a) Approximate range achieved in published literature for the controlling parameter in experiments with dissolved species. Experimental work on the shortest time or distance scales can be very demanding and far from routine. (b) Lower and upper limits of 𝜏 obs calculated from the practical range of controlling parameter. Where needed, D = 10−5 cm2 /s. (c) f = F/RT . (d) d = tip-substrate spacing. (e) f = frequency in Hz. (f ) Double-step mode, 𝜏 = forward step duration. (g) 𝜏 = transition time (no reversal); t 1 = switching time (with reversal). (h) t p = 1/2f = pulse duration. f = frequency (s−1 ). (i) (j) (k) (l)

𝜏 − 𝜏 ′ = pulse duration. f = rotation rate in s−1 (i.e., rps). t max = drop lifetime. t = electrolysis duration.

For a given method and a particular apparatus, 𝜏 obs can be varied over a certain range, sometimes called the time window. The shortest useful 𝜏 obs for transient methods is frequently determined by double-layer charging and instrumental response (which can be governed by the excitation apparatus, the measuring apparatus, the size of the working electrode, or the cell design; Section 16.7). The longest available 𝜏 obs may be governed by the onset of natural convection or changes in the electrode surface. Table 13.4.1 summarizes the achievable time windows in studies of dissolved solutes.22 The windows vary considerably among the techniques. To examine a coupled homogeneous reaction directly, one must be able to find conditions that place the characteristic time of reaction, 𝜏 rxn (Section 13.2.2), within the time window of the chosen technique. A consistent strategy is to vary the parameter determining 𝜏 obs (e.g., sweep rate, rotation rate, or applied current) and to observe the behavior of 22 The table applies to dissolved solutes to maintain comparability for this chapter, which deals with coupled homogeneous reactions. Coupled reactions can also be studied for surface-bound species. In that situation, the time windows for some of the techniques can be extended downward.

13.5 References

• Forward currents (e.g., ip /v1/2 C ∗ in CV, i𝜏 1/2 /C ∗ in chronopotentiometry, or il /𝜔1/2 C ∗ for voltammetry at an RDE). • Characteristic potentials (e.g., Ep and E1/2 ). • Reversal parameters (|ipr /ipf | in CV, −ir /if in chronoamperometry, Qd (2𝜏)/Qd (𝜏) in chronocoulometry, N at an RRDE). The directions and extents of variation can help toward diagnosis of the mechanism, and the measurements themselves provide data for evaluation of parameters. Comparative simulation is broadly applicable among different methods for testing mechanisms and quantification. Zone diagrams are available for many mechanisms and many methods. Often, they are essentially the same for other methods as for CV/LSV, although the dimensionless parameters must be expressed in terms of 𝜏 obs for the method employed (Problems 13.10 and 13.11). The 𝜆 parameter is consistently an expression of 𝜏 obs /𝜏 rxn . The second edition of this book reviews the effects of coupled homogeneous reactions on chronoamperometry and chronopotentiometry,23 hydrodynamic voltammetry at rotating electrodes,24 and controlled-potential coulometry.25

13.5 References 1 J.-M. Savéant, “Elements of Molecular and Biomolecular Electrochemistry,” Wiley, Hoboken, 2 3 4 5 6 7

8 9 10 11 12 13 14 15 16 17

NJ, 2006. J.-M. Savéant, ChemElectroChem, 3, 1967 (2016). C. Constentin, M. Robert, and J.-M. Savéant, Acc. Chem. Res., 43, 1019 (2010). J.-M. Savéant, Acc. Chem. Res., 26, 455 (1993). J.-M. Savéant, in “Advances in Physical Organic Chemistry,” Vol. 26, D. Bethell, Ed., Academic Press, New York, 1990, pp. 1–130. C. P. Andrieux, P. Hapiot, and J.-M. Savéant, Chem. Rev., 90, 723 (1990). C. P. Andrieux and J.-M. Savéant, in “Investigation of Rates and Mechanisms of Reactions,” 4th ed., Part II, Vol. 6, C. F. Bernasconi, Ed., Wiley-Interscience, New York, 1986, pp. 305–390. R. G. Compton, E. Kätelhön, K. R. Ward, and E. Laborda, “Understanding Voltammetry: Simulation of Electrode Processes,” World Scientific, London, 2020. R. G. Compton and C. E. Banks, “Understanding Voltammetry,” 3rd ed., World Scientific, London, 2018. Z. Galus, “Fundamentals of Electrochemical Analysis,” 2nd ed., Wiley, New York, 1994. D. H. Evans, Chem. Rev., 90, 739 (1990). D. D. Macdonald, “Transient Techniques in Electrochemistry,” Plenum, New York, 1977. P. Delahay, “New Instrumental Methods in Electrochemistry,” Wiley-Interscience, New York, 1954. I. M. Kolthoff and J. J. Lingane, “Polarography,” 2nd ed., Wiley-Interscience, New York, 1952. A. C. Testa and W. H. Reinmuth, Anal. Chem., 33, 1320 (1961). Q. Xie, E. Perez-Cordero, and L. Echegoyen, J. Am. Chem. Soc., 114, 3978 (1992). C. Amatore and J.-M. Savéant, J. Electroanal. Chem., 85, 27 (1977).

23 Second edition, Sections 12.2–12.3. Integrated with the treatment of effects on CV. 24 Second edition, Section 12.4. 25 Second edition, Section 12.7.

593

594

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

S. W. Feldberg, and L. Jeftic, J. Phys. Chem., 76, 2439 (1972). J. Jacq, J. Electroanal. Chem., 29, 149 (1971). W. H. Smith and A. J. Bard, J. Am. Chem. Soc., 97, 5203 (1975). R. S. Nicholson and I. Shain, Anal. Chem., 36, 706, (1964). O. Nekrassova, J. Kershaw, J. D. Wadhawan, N. S. Lawrence, and R. G. Compton, Phys. Chem. Chem. Phys., 6, 1316 (2004). D. Britz and J. Strutwolf, “Digital Simulation in Electrochemistry,” 4th ed., Springer, Switzerland, 2016. J. T. Maloy, in “Laboratory Techniques in Electroanalytical Chemistry,” P. T. Kissinger and W. R. Heineman, Eds., 2nd ed., Marcel Dekker, New York, 1996, Chap. 20. M. Rudolph, in “Physical Electrochemistry,” I. Rubinstein, Ed., Marcel Dekker, New York, 1995, Chap. 3. J.-M. Savéant and E. Vianello, Electrochim. Acta, 8, 905 (1965). J.-M. Savéant and E. Vianello, Electrochim. Acta, 12, 1545 (1967). R. S. Nicholson, Anal. Chem., 37, 667 (1965). M. L. Olmstead and R. S. Nicholson, Anal. Chem., 41, 851 (1969). M. L. Olmstead, R. T. Hamilton, and R. S. Nicholson, Anal. Chem., 41, 260 (1969). W. V. Childs, J. T. Maloy, C. P. Keszthelyi, and A. J. Bard, J. Electrochem. Soc., 118, 874 (1971). J.-M. Savéant and E. Vianello, Electrochim. Acta, 12, 629 (1967). C. P. Andrieux, L. Nadjo, and J.-M. Savéant, J. Electroanal. Chem., 26, 147 (1970). L. Nadjo and J.-M. Savéant, J. Electroanal. Chem., 48, 113 (1973). D. H. Evans, J. Phys. Chem., 76, 1160 (1972). C. L. Bentley, A. M. Bond, A. F. Hollenkamp, P. J. Mahon, and J. Zhang, J. Phys. Chem. C, 118, 22439 (2014). D. H. Evans, Acc. Chem. Res., 10, 313 (1977). C. P. Andrieux, C. Blocman, J.-M. Dumas-Bouchiat, and J.-M. Savéant, J. Am. Chem. Soc., 101, 3431 (1979). C. Costentin, M. Robert, and J.-M. Savéant, J. Phys. Chem., 104, 7492 (2000). J. M. Savéant and F. Xu, J. Electroanal. Chem., 208, 197 (1986). J. Orsini and W. E. Geiger, Organometallics, 18, 1854 (1999). D. S. Polcyn and I. Shain, Anal. Chem., 38, 370 (1966). D. H. Evans, Chem. Rev., 108, 2113 (2008). F. Ammar and J.-M. Savéant, J. Electroanal. Chem., 47, 215 (1973). J. B. Flanagan, S. Margel, A. J. Bard, and F. C. Anson, J. Am. Chem. Soc., 100, 4248 (1978). K. Itaya, A. J. Bard, and M. Szwarc, Z. Phys. Chem. N. F., 112, 1 (1978). J. Phelps and A. J. Bard, J. Electroanal. Chem., 68, 313 (1976). C. P. Andrieux and J.-M. Savéant, J. Electroanal. Chem., 28, 339 (1970). T. T. Chin, W. E. Geiger, and A. L. Rheingold, J. Am. Chem. Soc., 118, 5002 (1996). A. Molina, E. Laborda, J. M. Gómez-Gil, F. Martínez-Ortiz, and R. G. Compton, Electrochim. Acta, 195, 230 (2016). R. S. Nicholson and I. Shain, Anal. Chem., 37, 178 (1965). J.-M. Savéant, Electrochim. Acta, 12, 753 (1967). J.-M. Savéant, C. P. Andrieux, and L. Nadjo, J. Electroanal. Chem., 41, 137 (1973). M. Mastragostino, L. Nadjo, and J.-M. Savéant, Electrochim. Acta, 13, 721 (1968). C. Amatore and J.-M. Savéant, J. Electroanal. Chem., 102, 21 (1979).

13.6 Problems

56 C. Amatore, P. J. Krusic, S. U. Pedersen, and J.-N. Verpeaux, Organometallics, 14, 640

(1995). 57 C. Amatore, M. Azzabi, P. Calas, A. Jutand, C. Lefrou, and Y. Rollin, J. Electroanal. Chem.,

288, 45 (1990). 58 C. P. Andrieux, M. Robert, F. D. Saeva, and J.-M. Savéant, J. Am. Chem. Soc., 116, 7864

(1994). 59 J.-M. Savéant, J. Am. Chem. Soc., 109, 6788 (1987). 60 J. W. Darcy, B. Koronkiewicz, G. A. Parada, and J. W. Mayer, Acc. Chem. Res., 51, 2391

(2018). 61 J. J. Goings and S. Hammes-Schiffer, ACS Cent. Sci., 6, 1594 (2020). 62 B. W. Rossiter and J. F. Hamilton, Eds., “Physical Methods of Chemistry: Electrochemical 63 64 65 66

Methods,” 2nd ed., Vol. II, Wiley-Interscience, New York, 1986. J. Heinze, Angew. Chem. Int. Ed. Engl., 23, 831 (1984). J.-M. Savéant, Acc. Chem. Res., 13, 323 (1980). D. H. Evans and K. M. O’Connell, Electroanal. Chem., 14, 113 (1986). G. Costa, A. Puxeddu, and E. Reisenhofer, J. Chem. Soc. Dalton, 2034 (1973).

13.6 Problems 13.1 Consider the following system: A+e⇌B

′

E0 = −0.5 V vs. SCE

B→C C+e⇌D

′

E0 = −1.0 V vs. SCE

The half-life of B is 100 ms. Both charge-transfer reactions have large values of k 0 . Draw the expected cyclic voltammograms for scans beginning at 0.0 V vs. SCE and reversing at −1.2 V. Show curves for rates of 50 mV/s, 1 V/s, and 20 V/s. 13.2 Calculate 𝜏 rxn , 𝜏 obs , and 𝜆 values for Figure 13.3.1a–d. (a) Sketch the zone diagram in Figure 13.3.2 and mark the locations corresponding to each scan rate used in Figure 13.3.1a–d. (b) Discuss the value and effect of 𝜏 obs /𝜏 rxn in each case. (c) Calculate the value of 𝜏 rev at each scan rate? Discuss the value and effect of 𝜏 rev /𝜏 rxn in each case. (d) The characteristic time of observation in CV, 𝜏 obs , is defined as given in the text. Discuss the appropriateness of this definition with respect to determination of the shape of the forward peak vs. the reversal peak height. Is 𝜏 rev a better measure of the characteristic time of observation for the latter purpose? What is the weakness using 𝜏 rev ? 13.3 The data below were recorded from a series of CV experiments designed to elucidate the electroreduction of a certain compound. Formulate a mechanism to explain the behavior of the diagnostic functions, then briefly rationalize as many of the trends in the data as you can in terms of your mechanism. Each scan started at −0.8 V vs. SCE, reversed at −1.400 V vs. SCE, and returned to −0.8 V.

595

596

13 Electrode Reactions with Coupled Homogeneous Chemical Reactions

v

Epf

E pf

V/s

V vs. SCE

V vs. SCE

|ipr /ipf |

ipf /v 1/2 𝛍A s1/2 V−1/2

0.1

−1.253

−1.17

0.1

35

2.0

−1.260

−1.185

0.51

34.4

10

−1.265

–1.197

0.84

33.0

20

−1.270

−1.208

0.91

32.8

100

−1.271

−1.212

1.01

32.6

200

−1.270

−1.212

1.01

32.7

13.4 CV is often used to obtain information about standard potentials for correlative studies of molecular properties (e.g., ionization potentials, electron affinities, molecular orbital calculations; Section 15.1.2). How is a standard potential extracted? Are assumptions involved? What kind of error would appear if the electrode process were EC? What if it involved slow electron transfer to a chemically stable product? 13.5 The limiting current in LSV with an Er C′i system, iplateau , (13.3.17), can be derived by solving for the steady-state, where 𝜕C O (x,t)/𝜕t = 𝜕C R (x,t)/𝜕t = 0 and ∗ . Assume that D = D = D, and carry out the derivation. CO (x, t) + CR (x, t) = CO O R 13.6 In CV at a mercury electrode, Costa et al. (66) showed that the complex CoII (salen) can be reversibly reduced in DMF [where (salen) is the chelating ligand N,N ′ ethylenebis(salicylideneiminate)]. The reaction corresponds to CoII (salen) + e ⇌ CoI (salen)− When ethyl bromide (EtBr) is added, an irreversible homogeneous reaction can occur: k

CoI (salen)− + EtBr −−→ Et-CoIII (salen) + Br− where Et - CoIII (salen) is reduced at more negative potentials than CoII (salen). To measure k, the ratio |ipr /ipf | was determined at concentrations of EtBr where the following reaction was pseudo-first-order with k ′ = k[EtBr]. At 0 ∘ C with 13.3 mM EtBr, |ipr /ipf | was 0.7 when the time 𝜏 rev between E1/2 and the switching potential E𝜆 was 32 ms. (a) Using the working curve in Figure 13.3.3b, estimate k ′ . (b) Calculate and k and 𝜏 rxn . (c) What is the ratio 𝜏 rev /𝜏 rxn in this experiment? What value would it have if the scan were an order of magnitude slower? What would be the expected value of |ipr /ipf |? 13.7 Consider curve 1 in Figure 13.3.13, taken at v = 10 V/s. What is the approximate concentration of O at the start of the scan? Calculate ipf by assuming that the preceding reaction does not affect the behavior, and compare the result to the observed value of ipf . 13.8 The voltammograms in Figures 13.3.1 (EC), 13.3.9 (EC′ ), and 13.3.13 (CE) all involve nernstian electrode reactions with coupled chemical reactions. In each case, the total concentration of starting compound is 1 mM and D = 10−5 cm2 /s. From the data in these figures, prepare a plot of ipf vs. v1/2 for each mechanism, and include the line for

13.6 Problems

an uncomplicated nernstian electron transfer reaction. Justify the behavior in terms of the reactions occurring in each case. 13.9 Consider the example in Figure 13.3.14. The solution was prepared by adding the salt Rh(COD)+ PF− to produce a 1.4 mM solution. Assume that process (13.3.29) comes to 2 6 equilibrium. (a) Taking the definition of K assoc from footnote 13 and K = [Rh(COD)+ ]∕[Rh(COD) 2 + (acetone)2 ], derive the relationship between K assoc and K. (b) Calculate values for [COD] and K. (c) Calculate values for kf′ and 𝜆 for the CE mechanism in (13.3.29) and (13.3.30). (d) Find the operating point for the system of Figure 13.3.14 in the zone diagram of Figure 13.3.12. Does the pattern CV for the zone resemble the actual experimental result? 13.10 Using the concepts of Section 13.3.1(a), identify the parameter 𝜆 and draw rough, but quantitative, zone diagrams for Case Er Ci based on: (a) Collection efficiency, N, in hydrodynamic voltammetry at an RRDE. The collection efficiency for your RRDE is 0.45 for Fe(II) ⇌ Fe(III) + e. (b) The charge ratio Qd (2𝜏)/Qd (𝜏) in chronocoulometry at a disk electrode. (c) The ratio |ipr /ipf | for SWV at a disk electrode, where ipf and ipr are the peak heights of the forward- and reverse-sample voltammograms. (d) The transition time ratio 𝜏 2 /t 1 in chronopotentiometry at a disk electrode. In each case, identify zones DO, KO, and KP, and be specific about expected behavior in these zones. 13.11 A rule-of-thumb for an EC case is that the effect of the following decay will be about half manifested when 𝜏 obs = 𝜏 rxn . Suppose the following reaction has k f = 1000 s−1 . (a) At what value of 𝜆 is 𝜏 obs = 𝜏 rxn for CV? For other methods? (b) In CV, at roughly what value of v should one find |ipr /ipf | = 0.5. (c) In hydrodynamic voltammetry at an RRDE, what is the analogous effect? At what value of rotation rate (𝜔/2𝜋) should one expect it? (d) What is the analogous effect in SWV? At what value of pulse width should one expect it?

597

599

14 Double-Layer Structure and Adsorption In Chapter 1, we encountered some elementary ideas about the electrical double layer, and throughout the intervening text, we have seen references to its influence on electrode processes and measurements. It is now time to study the double layer in more detail. Our goals are to identify experiments that can illuminate and quantify its structure, to explore the important physical models, and to grasp their implications for electrode kinetics. A remarkable story in science concerns the discovery and rigorous effort leading to tested models of double-layer structure and behavior (1, 2). Early investigators used sophisticated thermodynamics to fully exploit the experimental methods available to them, which were remarkably simple as viewed from the present day. Clear, detailed models emerged from their work and remain central today. In earlier editions of this book1 and in other resources (3–7), the thermodynamic foundation is fully presented. In this edition, the authors have abbreviated it, to allow the reader to arrive more quickly at the widely used models.

14.1 Thermodynamics of the Double Layer Much of our knowledge about the double layer comes from measurements of macroscopic, equilibrium properties, such as interfacial capacitance and surface tension. The central concern is how these properties depend on potential and the composition of the electrolyte. When we focus on interfaces at equilibrium, we can expect thermodynamics to describe the system rigorously without any reference to a postulated model. This is an important point because it implies that one can obtain data that any successful model must rationalize. Let us begin by developing the Gibbs adsorption isotherm, which describes interfaces in general. From it, we obtain the electrocapillary equation, relating to the properties of a particular electrochemical interface. 14.1.1

The Gibbs Adsorption Isotherm

Suppose there is a planar interface of surface area A separating two phases, 𝛼 and 𝛽 (Figure 14.1.1). We can imagine two parallel planes, AA′ and BB′ , flanking and capturing the interfacial zone, which has special properties of interest to us. To the right of BB′ , pure phase 𝛽 exists, and to the left of AA′ , pure 𝛼. Near the actual interface between 𝛼 and 𝛽, intermolecular interactions are asymmetric; therefore, the composition can differ from that of a hypothetical system in which pure 𝛼 and pure 𝛽 extend right up to the interface from either side. 1 Second edition, Chapter 13; first edition, Chapter 12. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

600

14 Double-Layer Structure and Adsorption

A

Interface

B

Pure α

Pure β

A′

Interfacial zone

Figure 14.1.1 Schematic diagram of an interfacial region separating phases 𝛼 and 𝛽. The interface is a plane of area A, and the imaginary parallel planes, AA′ and BB′ , are of the same size. In a hypothetical reference system, pure 𝛼 and pure 𝛽 extend to the interface from either side. In the actual system, the composition near the interface can differ from that of either pure phase. AA′ and BB′ can be at any distance on either side of the interface, as long as they fully capture the region where the composition differs from that of a pure phase (Problem 14.1).

B′

Compositional differences can be described in terms of surface excess quantities. For example, the surface excess in the number of moles of any species, such as potassium ions or electrons, would be n𝜎j = nSj − nRj

(14.1.1)

where n𝜎j is the excess quantity and nSj and nRj are the numbers of moles of species j between AA′ and BB′ in the actual system and in the reference system, respectively, with the reference system being the hypothetical case mentioned above. In general, it is more convenient to speak of excesses per unit area of surface, so we divide by the area to produce the surface excess concentration, Γj = n𝜎j ∕A. Using thermodynamic relationships, one can treat the free-energy changes involved in assembling the interface. A most valuable outcome is the Gibbs adsorption isotherm,2 ∑ −d𝛾 = Γj d 𝜇 j (14.1.2) j

This equation links the surface tension, 𝛾—an important observable quantity—to the interfacial composition, expressed by the surface excess concentrations, Γj , of all components. The surface tension is defined as 𝜎

𝜕G (14.1.3) 𝜕A It is a measure of cost, in terms of electrochemical free energy, to expand the interface. It has units of energy/area (J/m2 in SI, but more commonly erg/cm2 in the literature). When 𝛾 is positive, the expansion of area requires energy; hence, the system acts spontaneously to minimize the area. This is the case for air/water interfaces, for example, where individual droplets coalesce. We will see below that mercury/electrolyte interfaces commonly have a surface tension in the range of 200–400 erg/cm2 .3 To see the experimental ramifications of the Gibbs adsorption isotherm, we need to specialize it to an electrochemical situation. That is our next job. 𝛾=

2 A complete derivation is available in the second edition, Section 13.1.1, or the first edition, Section 12.1.1. 3 It is more common to see surface tension expressed in units of dyn/cm. An erg is a dyn-cm; therefore, 1 dyn/cm is also 1 erg/cm2 . At the moment, we are stressing surface tension as the energy cost of expanding area, so we express in erg/cm2 .

14.1 Thermodynamics of the Double Layer

14.1.2

The Electrocapillary Equation

Let us now consider a specific electrochemical cell in which a mercury surface contacts an aqueous KCl solution. The potential of the mercury is controlled with respect to a reference electrode having no liquid junction with the test solution. Suppose also that the aqueous phase contains a neutral species, M. For example, the cell could be4 Cu′ ∕Ag∕AgCl∕K+ , Cl− , M∕Hg∕Ni∕Cu

(14.1.4)

We will focus on the interface between the mercury electrode and the aqueous solution. In writing the Gibbs adsorption isotherm for this case, it is useful to group terms relating to (a) components of the mercury electrode, (b) ionic components of the solution, and (c) neutral components of the solution. Since excesses of charge can exist on the electrode surface, we must consider a surface excess of electrons on the mercury, which may be positive or negative. Thus, Hg

−d𝛾 = (ΓHg d 𝜇 Hg + Γe d 𝜇 e ) + (ΓK+ d 𝜇 K+ + ΓCl− d 𝜇 Cl− ) + (ΓM d 𝜇M + ΓH

2

(14.1.5)

O d𝜇 H2 O )

Hg

where 𝜇 e refers to electrons in the mercury phase. Certain important linkages between electrochemical potentials allow simplification of this equation. In addition, we can identify the excess charge density on the metallic side of the interface as 𝜎 M = −FΓe

(14.1.6)

These considerations lead to the electrocapillary equation for our experimental system:5 −d𝛾 = 𝜎 M dE− + ΓK+ (H

2

O) d𝜇KCl

+ ΓM(H

2

O) d𝜇M

(14.1.7)

where E− is the potential of the mercury electrode with respect to the reference. We follow convention in attaching a negative subscript to signify that the reference electrode responds to an anionic component of our system. The quantities ΓK+ (H O) and ΓM(H O) are called relative 2 2 surface excesses and are amplified in Section 14.1.3. Every quantity in (14.1.7) is either controllable or measurable. This equation is the key to an experimental attack on double-layer structure. An electrocapillary equation is specific to a cell. Systems different from ours would have similar equations involving terms for their own components. More general statements of the electrocapillary equation are available in the specialized literature (5). 14.1.3

Relative Surface Excesses

In the derivation of the electrocapillary equation, one discovers that the surface excess concentrations on the solution side, ΓK+ , ΓM , and ΓH O , are not independently measurable. One of 2 them (usually the solvent) is chosen as the reference component. The measurables are the relative

4 Ni or W is needed in this cell as a contact between Cu and Hg because copper dissolves in mercury. The presence of this intermediary has no impact on the resulting electrocapillary equation. 5 A complete derivation is available in the second edition, Section 13.1.2, or the first edition, Section 12.1.2.

601

602

14 Double-Layer Structure and Adsorption

surface excesses of the other components, which, for our system, are X ΓK+ (H O) = ΓK+ − KCl ΓH O 2 XH O 2

(14.1.8)

2

ΓM(H

2

O)

= ΓM −

XM XH

O 2

ΓH

2

O

(14.1.9)

where X j are mole fractions in the bulk solution. A zero relative excess for K+ , for example, does not imply the lack of an excess presence of K+ , but only that K+ and H2 O have the same degree of interfacial excess. That is, K+ and H2 O are present at the interface in the same mole ratio that they have in the bulk electrolyte. A positive relative excess means that K+ is in excess to a greater degree than water, not in absolute molar quantities, but with respect to the amounts available in the bulk electrolyte. It is advantageous to select the solvent S as the reference component because one then does not have to be concerned with its activity. Also, one can sometimes argue that the quantities (X j /X S )ΓS are negligibly small, so that measured relative surface excesses can be regarded as absolute surface excesses. This assumption is not rigorous, but may be sound enough in many experimental situations involving dilute solutions.

14.2 Experimental Evaluations The thermodynamic relations bearing on interfacial structure (Section 14.1) highlight surface tension as a principal measurable. Since good measurements of this quantity are made far more conveniently at liquid metal electrodes, work with mercury and amalgams dominated research in this area for decades. Mercury offers other advantages too. It has a large hydrogen overpotential; hence, there is a wide potential range for which only nonfaradaic processes are significant. It is a liquid; therefore, surface features such as grain boundaries do not enter the picture. In some systems, the surface is renewed automatically, so that problems with progressive contamination of the working surface are minimized. Taken together, these advantages are quite favorable for probing interfacial structure. 14.2.1

Electrocapillarity

The name “electrocapillary equation” is a historic artifact derived from the use of such equations to interpret measurements of surface tension at mercury/electrolyte interfaces (1–9). The earliest such measurements were carried out by Lippmann, who invented a device called a capillary electrometer capable of great precision (10). Electrocapillary curves are plots of surface tension vs. potential. The DME (Figure 8.1.1) was later invented by Heyrovský (11) for the same purpose, although its utility ultimately far surpassed that intent (Section 8.1). The drop lifetime at a DME, t max , is directly proportional to 𝛾; hence, a plot of t max vs. E (also called an electrocapillary curve) has the same shape as 𝛾 vs. E. Let us now see how electrocapillary curves can reveal aspects of interfacial structure.

14.2 Experimental Evaluations

14.2.2

Excess Charge and Capacitance

We again take up the specific chemical system of Section 14.1.2. From its electrocapillary equation, (14.1.7), one can readily see that ( ) 𝜕𝛾 𝜎M = − 𝜕E− 𝜇

(14.2.1) ,𝜇 KCl M

hence, the excess charge on the electrode is the slope of the electrocapillary curve at any potential. Figure 14.2.1a is a plot of the drop time of a DME in 0.1 M KCl vs. potential. It has the nearly parabolic shape usually seen for these curves, although there are significant variations in the curves as the electrolyte is changed (Figure 14.2.1b). A feature common to all curves is the existence of an electrocapillary maximum (ECM), where 𝛾 has its greatest value. The slope of the curve is zero there; therefore, it occurs at the potential of zero charge (PZC, symbolized as Ez ), where 𝜎 M = 𝜎 S = 0. At more negative potentials than Ez , the electrode surface has a negative excess charge, and vice versa. The excess charges are mutually repulsive; hence, they lower the surface tension. Plots of surface charge can be made by differentiating electrocapillary curves (Figure 14.2.2). The interfacial capacitance quantifies the ability of the interface to store charge in response to a change in potential. The differential capacitance, Cd =

𝜕𝜎 M 𝜕E

(14.2.2)

is the slope of 𝜎 M vs. E at any point (Figure 14.2.3). One can see in Figures 14.2.2 and 14.2.3 that C d is not constant with potential, as it is for an ideal capacitor. One can also define an integral capacitance, C i (sometimes denoted K), which is the total charge density, 𝜎 M , at potential E divided by potential difference placing it there (Figure 14.2.3). That is, Ci =

𝜎M E − Ez

(14.2.3)

It is related to C d by E

Ci =

∫E

Cd dE z

E

∫E

E

= dE

1 C dE E − Ez ∫E d

(14.2.4)

z

z

Hence, it is an average of C d over the potential range from Ez to E. The differential capacitance is the more useful quantity because it is readily measurable. As we will see in Section 14.3, capacitance measurements are important to the formulation of structural models for the double layer.

603

3.0 420

2.8

Ca(NO3)2 380 γ/dyne cm–1

2.4

tmax/s

KOH

400

2.6

2.2 2.0

360 340 320

1.8

KI

300

1.6

280

1.4 1.2 0.0

NaCl KSCN NaBr

260 –0.5

–1.0 –1.5 E/V vs. SCE (a)

–2.0

–2.5

0.8

0.6

0.4

0.2

0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2 –1.4 (E – Ez)/V (b)

Figure 14.2.1 (a) Electrocapillary curve of drop time, tmax , vs. potential at a DME in 0.1 M KCl. [Data of Meites (12)]. (b) Electrocapillary curves of surface tension vs. potential for Hg in solutions of the indicated electrolytes at 18 ∘ C. Potential is expressed vs. the PZC for NaF. [Grahame (1).]

14.2 Experimental Evaluations

Figure 14.2.2 Charge density on Hg vs. potential in 1 M solutions of the indicated electrolytes at 25 ∘ C. Potentials are vs. the PZC for each electrolyte. [Grahame (1).]

–28 Na2SO4

–24 –20 –16

NaI

–12

Na2CO3 NaNO3

σ M/μC cm–2

–8 –4 0 4

NaF

8 NaNO3

12 16 20

Na2CO3 NaI Na2SO4

0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2 (E – Ez)/V

Cd at –1.0 V (Slope of tangent)

–30

σM/μC cm–2

–20

–10 Ci at –1.0 V (Slope of chord)

0

10

20 0.5

0.0

–0.5 (E – Ez)/V

–1.0

–1.5

Figure 14.2.3 Sketch of charge density vs. potential illustrating the definitions of the integral and differential capacitances.

Measurements of C d are largely equivalent to electrocapillary information. The capacitances are available from the electrocapillary curves by double differentiation, whereas the electrocapillary curves can be constructed from differential capacitances by double integration, if Ez is known (13, 14): 𝛾=

E

∫ ∫E

Cd dE z

(14.2.5)

605

606

14 Double-Layer Structure and Adsorption

Capacitances may be more generally useful primary data because the generation of 𝜎 M vs. E and 𝛾 vs. E from them involves integration, which averages random experimental variations. In contrast, differentiation of surface tension accentuates them. Moreover, capacitance measurements can be made readily at solid electrodes (Section 14.4.2), where 𝛾 is much less accessible. 14.2.3

Relative Surface Excesses

For the interface we have been considering, the electrocapillary equation, (14.1.7), provides the relative surface excess of K+ as ( ) 𝜕𝛾 ΓK+ (H O) = − (14.2.6) 2 𝜕𝜇KCl E ,𝜇 −

M

Since 0 𝜇KCl = 𝜇KCl + RT ln aKCl

we have ΓK+ (H

2

O)

−1 = RT

(

𝜕𝛾 𝜕 ln aKCl

(14.2.7) ) (14.2.8) E− ,𝜇M

Thus, we can evaluate ΓK+ (H O) at any potential E− by measuring the surface tension for several 2 KCl activities, while holding the activity of M constant. The relative surface excess of Cl− is then available from the excess charge because 𝜎 S = −𝜎 M = F(ΓK+ − ΓCl− )

(14.2.9)

By substitution from (14.1.8) and the analogous relationship for Cl− , we find that [ 𝜎 M = −F ΓK+ (H

] O) 2

− ΓCl− (H

(14.2.10)

O) 2

Since 𝜎 M and ΓK+ (H O) have already been evaluated, ΓCl− (H O) is calculable. 2 2 A relation like (14.2.8) can be readily derived for the neutral species, M; hence, ΓM(H O) can 2 be evaluated by the effect of aM on 𝛾. Figure 14.2.4 shows the relative surface excesses for the components of a 0.1 M KF solution in contact with mercury. At positive potentials vs. Ez , the surface excess of F− is positive, while that of K+ is negative. The opposite condition holds for potentials negative of Ez . The behavior of KF solutions conforms to what we would expect based on simple electrostatics. Contrasting behavior is found for Hg in contact with 0.1 M KBr (Figure 14.2.5), where ΓK+ (H O) remains positive 2

even for E > Ez (i.e., for positive 𝜎 M ). This interesting behavior relates to specific adsorption of Br− on mercury, to be discussed in Section 14.3.4.

14.3 Models for Double-Layer Structure Now that we have seen how charge density and relative molar excesses can be quantified for an interface, we would like to develop a picture of the way in which the excesses are physically distributed. Here, we will proceed through several models, adding refinements, until we reach a widely used concept (1–5, 7, 8, 17).

14.3 Models for Double-Layer Structure

12 10 K+

8

zj FΓj(H2O) /μC cm–2

6 4 2 0 –2

F–

Ez –0.5 K

E/ V vs. NCE

–1.0

+

–4 –6 –8

F–

0.4

0.2

0.0 –0.2 (E – Ez)/V

–0.4

–0.6

Figure 14.2.4 Surface excesses vs. potential for Hg in 0.1 M KF. Potential is referenced both to an NCE and to E z . [From data in Grahame and Soderberg (15).] Figure 14.2.5 Surface excesses vs. charge density, 𝜎 M , for Hg in 0.1 M KBr. [Reprinted with permission from Devanathan and Canagaratna (16). © 1963, Pergamon Press PLC.]

15

ΓK+(H2O)

zj FΓj(H2O) /μC cm–2

10 5 0 –5

–10 –15 ΓBr –(H2O) –20 –25 15

14.3.1

10

5

0 –5 σM/μC cm–2

–10 –15

The Helmholtz Model

Since the metallic electrode is a good conductor, it supports no electric fields within itself at equilibrium. In Chapter 2, we saw that, in consequence, any excess charge on a metallic phase resides strictly at the surface. Helmholtz, who was the first to think in detail about charge separation at interfaces, proposed that the countercharge in solution also resides at the surface.

607

14 Double-Layer Structure and Adsorption

Thus, there would be two sheets of charge of opposing polarity separated by a distance of molecular order. In fact, the name double layer arises from early writings (18–20), mainly by Helmholtz. Such a structure is equivalent to a parallel-plate capacitor, which has the following relationship between the stored charge density, 𝜎, and the voltage drop, V , between the plates (21): 𝜀𝜀0

𝜎=

d

(14.3.1)

V

where 𝜀 is the dielectric constant of the medium, 𝜀0 is the vacuum electric permittivity, and d is the interplate spacing (22).6 The predicted differential capacitance in the Helmholtz model, C H , Figure 14.3.1 Differential capacitance vs. potential at Hg in NaF solutions at 25 ∘ C. [Grahame (1).]

36 0.1 M 32 NaF

28

Cd /μF cm–2

608

24 20

1.0 M

0.01 M

16 0.001 M

12 8 4 0

0.8

0.4

0.0

–0.4 –0.8 –1.2 (E – Ez)/V

–1.6

6 Here and elsewhere in this book we use the electrical relations appropriate to SI units, which lead to the following definition of Coulomb’s law (22): F=

qq′ 4𝜋𝜀𝜀0 r2

The force, F, (in newtons) between two charges q and q′ (in coulombs) is related to the distance of charge separation r (in meters), the dielectric constant of the medium, 𝜀 (dimensionless), and the vacuum electric permittivity (also called the permittivity of free space), 𝜀0 . The last parameter is a measured constant equal to 8.85419 × 10−12 C2 N−1 m−2 . This system has the advantage that the electrical variables are measured in common units. An alternative is the electrostatic system, where Coulomb’s law is F=

qq′ . 𝜀r2

The force F (in dynes) is related in this case to the charges (in statcoulombs) by the dielectric constant 𝜀 and the separation distance (in cm). Equations for the electrostatic system can be converted to corresponding relations for SI units by replacing 𝜀 with 4𝜋𝜀𝜀0 , and vice versa. Many treatments of interfacial structure involve electrostatic units. They are recognizable by the absence of 𝜀0 and the appearance of multiples of 4𝜋 in the results. In some treatments, 𝜀𝜀0 is denoted as a single quantity, usually 𝜀, called the permittivity of the medium.

14.3 Models for Double-Layer Structure

is, therefore, 𝜀𝜀 𝜕𝜎 = CH = 0 𝜕V d

(14.3.2)

The weakness of this model is that C H is a constant. We already know from our earlier discussion that C d in real systems is not a constant. Figure 14.3.1 is a dramatic illustration for interfaces between mercury and sodium fluoride solutions of various concentrations. A more sophisticated model is clearly in order. 14.3.2

The Gouy–Chapman Theory

Even though the charge on the electrode is confined to the surface, the same is not necessarily true of the solution. Particularly at low concentrations of electrolyte, one has a phase with a relatively low density of charge carriers. Charge may have to be drawn from some significant thickness of solution to accumulate the excess needed to counterbalance 𝜎 M . A finite thickness would arise essentially because there is an interplay between the tendency of the charge on the metallic phase to attract or repel the carriers according to polarity and the tendency of thermal processes to randomize them. Thus, one arrives at the idea of a diffuse layer in the solution like that described in Section 1.6.3 (1–5, 7, 8, 17). The greatest concentration of excess charge would be adjacent to the electrode, where electrostatic forces are most able to overcome the thermal processes, while progressively lesser excess concentrations would be found at greater distances, as those forces become weaker. Thus, an average distance of charge separation would replace d in the capacitance (14.3.2). Also, one can expect the average distance to depend on potential and electrolyte concentration. As the electrode becomes more highly charged, the diffuse layer should become more compact and C d should rise. As the electrolyte concentration rises, there should be a similar compression of the diffuse layer and a consequent rise in capacitance. In fact, these qualitative trends are evident in the data of Figure 14.3.1. Gouy and Chapman independently proposed the idea of a diffuse layer and offered a statistical mechanical approach to its description (23–25). We outline the treatment here. Let us start by thinking of the solution as being subdivided into laminae parallel to the electrode, each with thickness dx, as shown in Figure 14.3.2. All laminae are in thermal equilibrium with each other. However, the ions of any species j are not at the same energy in different laminae because the electrostatic potential, 𝜙, varies. The laminae can be regarded as energy states with equivalent degeneracies; hence, the number concentrations of species in Laminae

dx

Reference lamina in bulk solution

Electrode

x Electrolyte

Figure 14.3.2 Representation of the solution near the electrode surface as a series of laminae.

609

610

14 Double-Layer Structure and Adsorption

two laminae have a ratio determined by a Boltzmann factor. If we take a reference lamina far from the electrode, where every ion is at its bulk number concentration n0j (cm−3 ), then the population in any other lamina at location x is ( ) −zj e𝜙 0 nj (x) = nj exp (14.3.3a) kT ( Cj (x) =

Cj∗ exp

−zj e𝜙

)

kT

(14.3.3b)

where 𝜙 is measured with respect to the bulk solution. The other quantities are the charge on the electron, e, the Boltzmann constant, k, the absolute temperature, T, and the (signed) charge, zj , on ion j. The version in (14.3.3b) is easily derived by recognizing that C j (x) = nj (x)/N A and Cj∗ = n0j (x)∕NA . Since any conversion factor will divide out of the right and left sides of (14.3.3b), the concentrations can be in any form of mol/volume. The total charge per unit volume in any lamina is then ( ) −zj e𝜙 ∑ ∑ 0 (14.3.4) 𝜌(x) = nj zj e = nj zj e exp kT j j where j runs over all ionic species. From electrostatics, we know that 𝜌(x) is related to the potential at distance x by the Poisson equation (26): 𝜌(x) = −𝜀𝜀0

d2 𝜙

(14.3.5) dx2 hence, (14.3.4) and (14.3.5) can be combined to yield the Poisson–Boltzmann equation, which describes our system: ( ) −zj e𝜙 d2 𝜙 e ∑ 0 (14.3.6) =− n z exp 𝜀𝜀0 j j j kT dx2 Equation 14.3.6 is treated by noting that ( )2 d2 𝜙 1 d d𝜙 = 2 d𝜙 dx dx2 hence, ( ) ( )2 −zj e𝜙 d𝜙 2e ∑ 0 d d𝜙 =− n z exp dx 𝜀𝜀0 j j j kT Integration gives ( ) ( )2 −zj e𝜙 d𝜙 2kT ∑ 0 + constant = n exp dx 𝜀𝜀0 j j kT

(14.3.7)

(14.3.8)

(14.3.9)

and the constant is evaluated by recognizing that at distances far from the electrode 𝜙 = 0 and d𝜙/dx = 0. Thus, [ ( ) ] ( )2 −zj e𝜙 d𝜙 2kT ∑ 0 −1 (14.3.10) = n exp dx 𝜀𝜀0 j j kT

14.3 Models for Double-Layer Structure

Now, it is useful to simplify the model to a system containing only a symmetrical electrolyte.7 Applying this limitation yields ( )1∕2 ( ) d𝜙 ze𝜙 8kTn0 =− sinh (14.3.11) dx 𝜀𝜀0 2kT The details of the transformation are left to Problem 14.2. In (14.3.11), n0 is the number concentration of each ion in the bulk and z is the magnitude of the charge on the ions. (a) The Potential Profile in the Diffuse Layer

Equation 14.3.11 can be rearranged and integrated in the following manner: ( )1∕2 x 𝜙 d𝜙 8kTn0 =− dx ∫𝜙 sinh(ze𝜙∕2kT) ∫0 𝜀𝜀0

(14.3.12)

0

where 𝜙0 is the potential at x = 0 relative to the bulk solution (i.e., the potential drop across the diffuse layer). The result is ] [ ( )1∕2 tanh(ze𝜙∕4kT) 8kTn0 2kT =− ln x (14.3.13) ze 𝜀𝜀0 tanh(ze𝜙0 ∕4kT) or, tanh(ze𝜙∕4kT) = e−𝜅x tanh(ze𝜙0 ∕4kT) where

( 𝜅=

2n0 z2 e2 𝜀𝜀0 kT

(14.3.14)

)1∕2 (14.3.15)

For dilute aqueous solutions (𝜀 = 78.49) at 25 ∘ C, this equation can be expressed as 𝜅 = (3.29 × 107 )zC ∗1∕2

(14.3.16)

where C * is the bulk z:z electrolyte concentration in mol/L and 𝜅 is given in cm−1 . Equation 14.3.14 describes the potential profile in the diffuse layer in a general way, and in Figure 14.3.3, one sees calculated profiles for several different values of 𝜙0 . The potential always decays away from the surface. At large 𝜙0 (a highly charged electrode), the drop is precipitous because the diffuse layer is relatively compact. As 𝜙0 becomes smaller, the decline is more gradual. In the limit of small 𝜙0 , the potential profile becomes exponential, as one can readily show for ze𝜙0 ∕4kT < 0.5. In this range, tanh(ze𝜙∕4kT) ≈ ze𝜙∕4kT everywhere, and 𝜙 = 𝜙0 e−𝜅x

(14.3.17) ∘ This relation is a good approximation for 𝜙0 ≤ 50/z mV at 25 C. The reciprocal of 𝜅 has units of distance and determines the spatial decay of potential. It can be regarded as a characteristic thickness of the diffuse layer and is often called the Debye length. Table 14.3.1 provides values of 𝜅 −1 for several concentrations of a 1:1 electrolyte. The diffuse layer is quite thin by comparison to the distance scale encountered for diffusion layers in typical faradaic experiments. It becomes thicker as the concentration of electrolyte falls, as we anticipated in the qualitative discussion above. 7 An electrolyte having only one cationic species and one anionic species, both with charge magnitude z. Sometimes symmetrical electrolytes, for example, NaCl, HCl, and CaSO4 , are called “z:z electroytes.”

611

612

14 Double-Layer Structure and Adsorption

1.0

0.8 ϕ0 = 10 mV (limiting exponential form)

0.6 ϕ/ϕ0 0.4 ϕ0 = 100 mV

0.2 ϕ0 = 1000 mV

0.0

0.0

1.0

2.0

3.0

4.0 x/nm

5.0

6.0

7.0

8.0

Figure 14.3.3 Potential profiles through the diffuse layer in the Gouy–Chapman model. Calculated for a 10−2 M aqueous solution of a 1:1 electrolyte at 25 ∘ C. 1/𝜅 = 3.04 nm. Table 14.3.1 Characteristic Thickness of the Diffuse Layer(a) C * /M(b)

1

0.1

0.01

0.001

0.0001

𝜅 −1 /nm

0.30

0.96

3.04

9.61

30.4

(a) For a 1:1 electrolyte at 25 ∘ C in water. (b) C * = n0 /N A , where N A is the Avogadro constant.

(b) The Relation Between 𝝈 M and 𝝓0

Suppose we now imagine a Gaussian surface in the shape of a box placed in our system as shown in Figure 14.3.4. One end is at the interface. The sides are perpendicular to this end and extend far enough into the solution that the field strength −d𝜙/dx is essentially zero on the remote face parallel to the electrode. The box, therefore, contains all of the charge in the diffuse layer opposite the portion of the electrode surface adjacent to the end. In a dielectric phase, such as an aqueous solution, this charge is given by the Gauss law (Section 2.2.1) as q = 𝜀𝜀0

∮surface

E ⋅ dS

(14.3.18)

Since the field strength, E, is zero at all points on the surface except the end at the interface [where it is (−d𝜙/dx)x = 0 at every point], and since the unit vector, S, points negatively on that end of the surface, we obtain ( ) ( ) d𝜙 d𝜙 q = 𝜀𝜀0 dS = 𝜀𝜀0 A (14.3.19) ∫ end dx x=0 dx x=0 surface

Substituting from (14.3.11) and recognizing that q/A is the solution phase charge density, 𝜎 S , we obtain ( ) ze𝜙0 M S 0 1∕2 𝜎 = −𝜎 = (8kT𝜀𝜀0 n ) sinh (14.3.20) 2kT

14.3 Models for Double-Layer Structure

Electrode surface Gaussian enclosure

End surface Area = A dϕ/dx = 0

Surface against electrode

Figure 14.3.4 A Gaussian box enclosing the charge in the diffuse layer opposite an area, A, of the electrode surface.

For dilute aqueous solutions at 25 ∘ C, the constants can be evaluated to give 𝜎 M = 11.7C ∗1∕2 sinh(19.5z𝜙0 )

(14.3.21)

where C * is in mol/L when 𝜎 M is in μC/cm2 . Note that 𝜙0 is related monotonically to the charge density on the electrode. (c) Differential Capacitance

Now, we are in a position to predict the differential capacitance for the Gouy–Chapman model, C D , simply by differentiating (14.3.20): ( )1∕2 ( ) ze𝜙0 2z2 e2 𝜀𝜀0 n0 d𝜎 M CD = = cosh (14.3.22) d𝜙0 kT 2kT For dilute aqueous solutions at 25 ∘ C, this equation can be written, CD = 228zC ∗1∕2 cosh(19.5z𝜙0 )

(14.3.23)

where C D is in μF/cm2 when the bulk electrolyte concentration, C * , is in mol/L. Figure 14.3.5 shows the dependence of C D with potential according to (14.3.23). There is a minimum at the PZC and a steep rise on either side. The predicted V-shaped capacitance function does resemble the observed behavior in NaF at low concentrations and at potentials not too far from the PZC (Figure 14.3.1). However, the actual system shows a flattening of capacitance at more extreme potentials, and the valley at the PZC disappears altogether at high electrolyte concentrations. Moreover, the actual capacitance is usually much lower than the predicted value. The partial success of the Gouy–Chapman theory suggests that it has elements of truth, but its failures indicate major defects.

613

14 Double-Layer Structure and Adsorption

500

400 CD /μF cm–2

614

300 1M 200

0.1 M 0.01 M

100

0 150

100

50

0 –50 (E – Ez)/mV

–100

–150

Figure 14.3.5 Predicted differential capacitances from the Gouy–Chapman theory. Calculated from (14.3.23) for the indicated concentrations of a 1:1 electrolyte in water at 25 ∘ C. The vertical and horizontal scales are greatly expanded vs. those of Figure 14.3.1 because the predicted capacitance rises very rapidly from E z .

14.3.3

Stern’s Modification

The reason for the unlimited rise in differential capacitance with 𝜙0 in the Gouy–Chapman model is that the ions are treated as point charges, able to approach the surface arbitrarily closely. At high polarization, the effective separation between the metallic- and solution-phase charge layers decreases continuously toward zero. This concept is not realistic. The ions have a finite size and cannot approach the surface any closer than the ionic radius. If they remain solvated, the thickness of the primary solution sheath would have to be added to that radius. Still another increment might be necessary to account for a layer of solvent on the electrode surface. In general, then, we can envision a plane of closest approach at some distance x2 (Figure 1.6.3). In systems with low electrolyte concentration, this restriction would have little impact on the predicted capacitance for potentials near the PZC because the thickness of the diffuse layer is large compared to x2 . However, at larger polarizations or with more concentrated electrolytes, the charge in solution becomes more tightly compressed against the boundary at x2 , and the whole system begins to resemble the Helmholtz model. Then, we can expect a corresponding leveling of the differential capacitance. The plane at x2 is an important concept and is called the outer Helmholtz plane (OHP). This interfacial model (1–5, 7, 8, 17), first suggested by Stern (27), can be treated by extending the considerations of Section 14.3.2. The Poisson–Boltzmann equation, (14.3.6), and its solutions, (14.3.10) and (14.3.11), still apply at distance x ≥ x2 . Now, the potential profile in the diffuse layer of a z:z electrolyte is given by ( )1∕2 x 𝜙 d𝜙 8kTn0 =− dx (14.3.24) ∫𝜙 sinh(ze𝜙∕2kT) ∫x 𝜀𝜀0 2

2

or tanh(ze𝜙∕4kT) = e−𝜅(x−x2 ) tanh(ze𝜙2 ∕4kT)

(14.3.25)

where 𝜙2 is the potential at x2 with respect to the bulk solution, and 𝜅 is defined by (14.3.15).

14.3 Models for Double-Layer Structure

The field strength at x2 is given from (14.3.11): ( ) ( )1∕2 ( ) ze𝜙2 d𝜙 8kTn0 − = sinh dx x=x 𝜀𝜀0 2kT

(14.3.26)

2

Since the charge density at any point from the electrode surface to the OHP is zero, we know from (14.3.5) that this same field strength applies throughout that interval. Thus, the potential profile in the compact layer is linear. Figure 14.3.6b is a summary of the situation. Now, we find the total potential drop across the double layer to be ( ) d𝜙 𝜙0 = 𝜙2 − x (14.3.27) dx x=x 2 2

All of the charge on the solution side resides in the diffuse layer, and its magnitude can be related to 𝜙2 by considering a Gaussian box exactly as we did above.8 ( ) ( ) ze𝜙2 d𝜙 𝜎 M = −𝜎 S = −𝜀𝜀0 = (8kT𝜀𝜀0 n0 )1∕2 sinh (14.3.28) dx x=x 2kT 2

To find the interfacial capacitance, we substitute for 𝜙2 by (14.3.27): [ ( )] Mx 𝜎 ze 2 𝜙0 − 𝜎 M = (8kT𝜀𝜀0 n0 )1∕2 sinh 𝜀𝜀0 2kT Figure 14.3.6 (a) A representation of the differential capacitance in the Gouy–Chapman–Stern (GCS) model as a series combination of Helmholtz-layer and diffuse-layer capacitances. (b) Potential profile through the solution side of the double layer according to GCS theory. Calculated from (14.3.25) for 10−2 M 1:1 electrolyte in water at 25 ∘ C.

CH

(14.3.29)

CD

CGCS (a)

140 Linear profile to x2 120

x2 ϕ2 = 100 mV

100

80 Compact layer

ϕ/mV 60

40

20

0

Diffuse layer

0

1.0

2.0

x/nm (b)

8 See (14.3.16, 14.3.21, 14.3.23) for evaluations of the constants for aqueous solutions at 25 ∘ C.

3.0

4.0

5.0

615

14 Double-Layer Structure and Adsorption

Differentiation and rearrangement (Problem 14.4) give the predicted differential capacitance for the Gouy–Chapman–Stern (GCS) model, CGCS =

(2𝜀𝜀0 z2 e2 n0 ∕kT)1∕2 cosh(ze𝜙2 ∕2kT) d𝜎 M = d𝜙0 1 + (x2 ∕𝜀𝜀0 )(2𝜀𝜀0 z2 e2 n0 ∕kT)1∕2 cosh(ze𝜙2 ∕2kT)

(14.3.30)

which is more simply stated as the inverse: 1 CGCS

=

x2 𝜀𝜀0

+

1 (2𝜀𝜀0

z2 e2 n0 ∕kT)1∕2

(14.3.31)

cosh(ze𝜙2 ∕2kT)

By comparing the first term with (14.3.2), one can easily identify it as 1/C H from the Helmholtz model, corresponding to the inverse capacitance of the charges held at the OHP (d = x2 ). Likewise, one can see by comparison with (14.3.22) that the second term in (14.3.31) is essentially 1/C D from the Gouy–Chapman model, corresponding to the inverse capacitance of the truly diffuse charge.9 Thus, C CGS is made up of two components that can be separated in the reciprocal, exactly as one would find for two capacitors in series (Figure 14.3.6a): 1 1 1 = + CGCS CH CD

(14.3.32)

The value of C H is independent of potential, but C D varies in the V-shaped fashion we found in Section 14.3.2(c). The composite capacitance C GCS shows a complex behavior and is governed by the smaller of the two components. Near the PZC in systems with low electrolyte concentration, we expect to see the V-shaped function characteristic of C D . At larger electrolyte concentrations, or even at large polarizations in dilute media, C D becomes so large that it no longer contributes to C GCS and one sees only the constant capacitance of C H . Figure 14.3.7 is a schematic picture of this behavior.

High electrolyte concentration

CGCS /μF cm–2

616

Minima at E = Ez

Dip due to CD

Low electrolyte concentration

CH

(+)

0 (E – Ez)/V

(–)

Figure 14.3.7 Expected behavior of differential capacitance according to GCS theory as the electrolyte concentration changes. 9 The second term in (14.3.31) has 𝜙2 in the argument of the hyperbolic cosine, while (14.3.22) has 𝜙0 . This difference occurs because the diffuse layer is bounded at the OHP in the GCS model, but extends to x = 0 in the Gouy–Chapman model.

14.3 Models for Double-Layer Structure

The CGS model gives predictions that account for the gross features of behavior in real systems. There are still discrepancies, in that C H is not truly independent of potential. Figure 14.3.1 is a plain illustration. This aspect must be handled by refinements to the GCS theory that take into account the structure of the dielectric in the compact layer, saturation (i.e., full polarization) of that dielectric in the strong interfacial field, differences in x2 for anionic and cationic excesses, and other similar matters (1–5, 7, 8, 17, 28). The theory also neglects ion pairing (or ion–ion correlation) effects in the double layer and strong nonspecific interactions of the ions with the surface charge on the electrode. The latter effect can be described in terms of “ion condensation” in the electrical double layer and can be treated by a model in which the surface charge is considered as an “effective surface charge,” smaller than the actual charge on the electrode because of the condensed ionic countercharge (28, 29). Another point of interest is the idea that the electron distribution in the metal is not confined to the “metallic side” of the interface, but extends into the electrolyte. Theoretical work addressing this concept, called electron spillover, has appeared and is relevant to some anomalies in capacitance data (30–32). It also underlies electron transfer by tunneling between an electrode and a redox center spaced away from the surface [Section 3.5.2(a)]. Computational methods, especially density functional theory (DFT), are now being used to investigate interfacial structure on the molecular scale (33), providing insight into effects like those just mentioned. The finer details of interfacial structure are not easily probed by measurements of capacitance or surface tension but may be addressed by alternative methods of studying the interface, as discussed in Chapters 18 and 21. 14.3.4

Specific Adsorption

In constructing models for interfacial structure, we have so far considered only long-range electrostatic effects as the basis for creating the excesses of charge found in the solution phase. Aside from the magnitude of the charges on the ions, and possibly their radii, we have been able to ignore their chemical identities. They are said to be nonspecifically adsorbed.10 However, there is far more to the picture. Consider the data in Figure 14.2.1b. At potentials more negative than the PZC, the surface tension follows the kind of decline we have come to expect and the decline is the same regardless of the composition of the system. This result is predictable from the GCS theory. On the other hand, the curves at potentials more positive than the PZC diverge markedly from each other. Since the divergences occur at potentials where anions must be in excess, we suspect that some sort of specific adsorption of anions takes place on mercury (1–5, 7, 8, 34, 35). Specific interactions would have to be very short range in nature; hence, we gather that specifically adsorbed species are bound by a chemical interaction with the electrode surface in the manner depicted in Figure 1.6.3. The locus of their centers is the inner Helmholtz plane (IHP), at distance x1 from the surface. To detect and to quantify specific adsorption, one can focus on the relationships between the relative surface excesses and the surface charge density. Turning now to Figure 14.2.5, we note several peculiarities. First, there are positive relative excesses of bromide at potentials negative of the PZC and of potassium at potentials more positive than the PZC. At the PZC itself, positive excesses of both species are found. None of these features is accountable within an electrostatic model, such as the basic GCS theory. 10 Nonspecifically adsorbed species are not really adsorbed at all within the usual meaning of the word “adsorption.” There is no close-range interaction. Surface excesses are created only by attraction of ions to an oppositely charged electrode.

617

618

14 Double-Layer Structure and Adsorption

The identity of the specifically adsorbed ion is revealed by considering the slopes of zj FΓj(H vs. 𝜎 M in key regions. It is always true that 𝜎 M = −[FΓK+ (H

2

O)

− FΓBr− (H

2

2

O)

(14.3.33)

O) ]

In the absence of specific adsorption, the charge on the electrode is counterbalanced by an excess of one ion and a deficiency of the other, as we see in the negative region of Figure 14.2.5. If the electrode becomes even more negative, the excess charge is accommodated by a growth in both the excess and the deficiency. As a consequence, FΓK+ (H O) does not grow as fast as 𝜎 M . 2

That is, the slope of FΓK+ (H

2

M O) vs. 𝜎 in the negative region should have a magnitude no greater

than unity. By similar reasoning, we conclude that the slope magnitude of −FΓBr− (H O) vs. 𝜎 M 2 should also be less than or equal to unity in the positive region. The data of Figure 14.2.5 show that the system is well-behaved in this respect at potentials much more negative than the PZC. However, in the positive region, there is superequivalent adsorption of bromide, meaning that the slope d(−FΓBr− (H O) )∕d𝜎 M exceeds unity in mag2 nitude. Thus, a change in charge on the electrode is countered by more than an equivalent amount of Br− . This evidence points strongly to specific adsorption of bromide at potentials more positive than the PZC. The positive excess of K+ in the same region is explained by the necessity for partial compensation of the superequivalence of adsorbed bromide. Also, the forces responsible for specific adsorption are apparently strong enough to withstand an opposing coulombic field in at least part of the negative region, as one may infer from the positive excess of bromide in the zone of small negative 𝜎 M . Another indicator of specific adsorption of charged species is the Esin–Markov effect, which is a shift in the PZC with a change in electrolyte concentration (36), as seen in Table 14.3.2. The magnitude of the shift is usually linear with the logarithm of electrolyte activity, and the slope of the linear plot is the Esin–Markov coefficient for the condition of 𝜎 M = 0. Similar results are obtained at nonzero, but constant, electrode charge densities; hence, the Esin–Markov coefficient can be written generally as 1 RT

(

𝜕E± 𝜕 ln asalt

)

( =

𝜎M

𝜕E± 𝜕𝜇salt

) (14.3.34) 𝜎M

Nonspecific adsorption provides no mechanism for the electrode potential to depend on the concentration of the electrolyte, so the Esin–Markov coefficient should be zero in the absence of specific adsorption. However, the situation is different when the anion, for example, is specifically adsorbed. If the electrode is being held at the PZC, and we introduce more of the same electrolyte, additional anions will be adsorbed; hence, 𝜎 S becomes nonzero and must be balanced. Since the electrode is more polarizable than the solution, the countercharge is induced there. To regain the condition 𝜎 M = 0, the potential must be shifted to a more negative value, so that the charge excess of specifically adsorbed anions is exactly counterbalanced by an opposing excess charge in the diffuse layer. Thus, specific adsorption of anions is indicated by a negative shift in potential at constant charge density. Conversely, specific cationic adsorption is revealed by a positive shift, as the electrolyte concentration is elevated. From the data in Table 14.3.2, we see that chloride, bromide, and iodide all appear to be specifically adsorbed, but fluoride is not. It is clear now why sodium and potassium fluoride solutions in contact with mercury are the standard systems for testing the GCS theory of nonspecific adsorption.

14.4 Studies at Solid Electrodes

Table 14.3.2 Potentials of Zero Charge in Various Electrolytes(a) Electrolyte

NaF

NaCl

KBr

KI

Concentration, M

Ez /V vs. NCE(b)

1.0

−0.472

0.1

−0.472

0.01

−0.480

0.001

−0.482

1.0

−0.556

0.3

−0.524

0.1

−0.505

1.0

−0.65

0.1

−0.58

0.01

−0.54

1.0

−0.82

0.1

−0.72

0.01

−0.66

0.001

−0.59

(a) Grahame (1). (b) NCE = normal calomel electrode, (2.1.55).

Since specific adsorption must alter the interfacial capacitance, it should also be detectable by the study of C d . We will see some examples in Section 14.4.2. Specific adsorption of ionic species can markedly alter the potential profile in the interfacial zone. Figure 14.3.8 is a set of curves presented by Grahame (1) for a mercury interface with 0.3 M NaCl. Note particularly the traces for the most positive potentials. Such potential profiles can influence electrode kinetics by mechanisms considered in Section 14.7 below. Neutral molecules are also interesting as adsorbates because they can influence or participate in faradaic processes (1–5, 7, 8, 35, 37). They can be detected and studied by the methods we have outlined above (Problem 14.8). An interesting aspect of their behavior is that adsorption from aqueous solutions is often effective only at potentials relatively near the PZC. The usual rationale rests on the idea that adsorption of a neutral molecule requires the displacement of water molecules from the surface. When the interface is strongly polarized, the water is tightly bound and its displacement by a less dipolar substance is energetically unfavorable. Adsorption can take place only near the PZC, where the water can be removed more easily. The applicability of this rationale in any given case must depend on the electrical properties of the specific neutral species at hand.

14.4 Studies at Solid Electrodes Virtually all experimental work discussed in Sections 14.2 and 14.3 involved mercury electrodes for the practical reasons identified at the opening of Section 14.2. However, electrochemists are much more interested in interfacial structure at solid electrodes because nearly all electrochemical research and technology involves such electrodes (e.g., Pt, Au, C, Li, Fe, Ni, or various

619

620

14 Double-Layer Structure and Adsorption

Figure 14.3.8 Calculated potential profiles according to the GCS model for Hg in aqueous 0.3 M NaCl at 25 ∘ C. Potentials vs. the PZC in NaF. At positive potentials, the profile has a sharp minimum because chloride is specifically adsorbed. Electron spillover would tend to round off the sharp break at the electrode surface. This and other effects would also tend to modify the potential profile out to the IHP so that it is not strictly linear. [Grahame (1).]

0.4

0.2

0.0 Outer Helmholtz plane

–0.2 E/V –0.4

Inner Helmholtz plane –0.6

–0.8

–1.0 Mercury surface 0.0

0.2

0.4 0.6 x/nm

0.8

1.0

semiconductors). The rigorous study of solid surfaces is challenging because they are difficult to reproduce and to keep clean. Impurities can adsorb over time, significantly altering interfacial properties. Moreover, the surfaces of solids, unlike those of mercury, are not atomically smooth, but have defects with a density of at least 105 to 107 cm−2 . In comparison, a typical metal surface has about 1015 atoms/cm2 . Despite the challenges, much progress has been made toward understanding interfacial behavior at solid electrodes. And because of the challenges, experimental methods tend to be more varied and complex than those historically used with mercury. 14.4.1

Well-Defined Single-Crystal Electrode Surfaces

Most reported electrochemistry with solid electrodes involves polycrystalline materials, which consist of small domains offering different crystal faces to the electrolyte. These faces have distinct properties (e.g., PZC or work function); hence, the behavior of a polycrystalline electrode represents an average over all exposed crystal planes and defect sites. The desire for simpler experimental situations and clearer insights has led to extensive research on electrodes with carefully prepared surfaces of known structure (3, 38–43). (a) Metals

Single crystals of metals can be grown by zone refining and are commercially available. Many of the metals used as electrodes (e.g., Pt, Pd, Ag, Ni, Cu) form face-centered cubic (FCC) crystal structures. After careful orientation with an X-ray or laser beam, these can be cut to expose different crystal faces (index planes, Figure 14.4.1). The low-index planes [(100), (110), and (111)] are shown in this figure and are also the surfaces most frequently used as electrodes. They tend to be stable and can be polished to yield fairly smooth, uniform surfaces. High-index

14.4 Studies at Solid Electrodes

Atop site Bridge site 4-fold hollow site

(100)

z

(110) 1-fold atop site 2-fold bridge site 3-fold hollow site

x

y (111) Face-centered cubic (FCC) structure

Figure 14.4.1 The atomic structure of the low-Miller-index surfaces—(100), (110), and (111)—of a face-centered cubic crystal, as obtained by cutting along the planes shown on the left. The Miller indices are obtained by noting the intersection of the plane of interest with the principal axes (x, y, z) shown in the lower left diagram. The Miller indices (hkl) are the smallest integers h, k, and l such that h : k : l = (1/p) : (1/q) : (1/r), where p, q, and r are the coordinates of the intersections with the x, y, and z axes, respectively. For example, if a plane intersects the axes at x = 2, y = 2, z = 2, then h : k : l = (1/2) : (1/2) : (1/2) = 1 : 1 : 1, so this is the (111) plane. The atomic arrangements for the indicated planes are shown on the right. The shaded atoms in the (110) face are in the plane below the unshaded (surface) atoms. The names used for different sites on the surfaces are also indicated.

planes, exemplified in Figure 14.4.2, have smaller atomically smooth terraces, with many more exposed edges and kink sites (44). Alternative approaches to obtaining well-defined crystal faces include flame annealing or vacuum evaporation. In the flame annealing method (45, 46), a wire (e.g., Pt, Pd, Au, Ag) is melted in a hydrogen–oxygen flame and then cooled to produce a small metal bead with eight clear (111) facets in an octahedral configuration. Vacuum evaporation under carefully controlled conditions on the proper substrate (e.g., Au on mica or glass) can also produce (111) regions of atomic smoothness. Surfaces can be characterized by low-energy electron diffraction (LEED) and other techniques described in Chapters 18 and 21. Even the most carefully prepared surfaces are not atomically smooth over areas larger than a few square micrometers, they inevitably show step edges and defect sites (as seen, for example, by scanning tunneling microscopy). Well-defined surfaces are not necessarily stable. They may undergo spontaneous structural changes in a process called reconstruction. When a solid is cleaved or interfacial conditions change in some different way, the surface atoms are subjected to altered local forces and may reconfigure to minimize the surface energy. Reconstruction often accompanies changes of electrical potential or the extent of specific adsorption [Sections 14.4.2(d) and 14.5.1].

621

622

14 Double-Layer Structure and Adsorption

FCC (977)

FCC (443)

FCC (14, 11, 10)

FCC (755)

FCC (332)

FCC (10, 8, 7)

FCC (533)

FCC (331)

FCC (13, 11, 9)

Figure 14.4.2 The atomic structure of several high-Miller-index surfaces, showing terraces, step edges, and kink sites. [From Somorjai (44), with permission.]

(b) Carbon

Carbon is a very widely used electrode material. A well-defined bulk form is highly oriented pyrolytic graphite (HOPG), which is polycrystalline, but ordered, so that similar faces of the graphite grains are aligned (47). Each grain consists of graphene sheets, in which the hexagonally close-packed (HCP) carbon atoms are strongly bonded into an expansive, planar, aromatic structure. These planes are stacked as shown in Figure 14.4.3, and the basal plane of each grain is an exposed face of graphene. Bonding between graphene sheets involves van der Waals forces and is much weaker than within a sheet; thus, HOPG is easily cleaved along its basal plane to expose a fresh, relatively smooth, HCP surface. Graphene sheets are also available for use as an electrode material (49, 50); however, they are mostly in the form of powders of small particle size. These powders are of high interest in electrochemical devices, but do not readily lend themselves to careful studies of the double layer because they are usually prepared for electrochemical purposes as composites involving polymeric binders. Individual exfoliated graphene flakes with A = 0.001 − 0.01 cm2 are available but are difficult to manipulate.

14.4 Studies at Solid Electrodes

Basal plane Graphene sheet a Interplanar distance b Edge plane

C0

a

Figure 14.4.3 Structure of graphite. The interplanar distance between the a and b graphene layers is 0.335 nm, but the unit cell distance, c0 , is 0.670 nm because of the abab … stacking. [Adapted from Bard (48).]

Carbon nanotubes (CNT) (51) are essentially graphene sheets in a tubular form, with the sheet edge conceptually rolled over to bind to the opposite sheet edge, so that the wall of the tube becomes a single aromatic molecule. Like graphene, this material is available in powders made up of individual nanotubes, but it is simpler to use without a binder. Double-layer studies have been carried out using nanotube electrodes (52). Diamond, the ideal crystalline form of carbon, can be rendered semiconducting with doping by boron (53–55). Double-layer studies have been carried out at boron-doped diamond (BDD) electrodes (56). 14.4.2

The Double Layer at Solids

(a) Approaches to Measurement

We noted in Section 14.2.2 that primary information about the double layer can come, largely equivalently, from measurements of either surface tension or differential capacitance. With solid electrodes, there is necessarily a reliance on differential capacitance (3–6, 43, 57). Measurements of surface tension or surface stress are not very convenient with solids, although attempts have been made to determine values of the PZC for solid electrodes attached to piezoelectric materials (58, 59). Surface excesses of electroactive species are frequently measured at solid electrodes by methods based on faradaic reactions of the adsorbates (Sections 17.2 and 17.3). Cyclic voltammetry, chronocoulometry, and thin-layer methods are all useful. In addition, spectroscopic and microscopic methods (e.g., surface-enhanced Raman spectroscopy, infrared spectroscopy, and various forms of microscopy) have been used to probe the composition and structure of electrode surfaces (Chapters 18 and 21). Some of this work has been aimed at metal surfaces in UHV on which water and other species have been allowed to coadsorb as a model for the double layer (60). (b) Properties of the Double Layer at Well-Defined Solid Surfaces

To derive surface charges and relative excesses from capacitance data, one must know the PZC, which is usually identified at a solid electrode as the potential of minimum differential capacitance in dilute electrolyte. This linkage rests on the GCS theory (Section 14.3).

623

14 Double-Layer Structure and Adsorption

Cd/µF cm–2

50

0 –1.5

Cd/µF cm–2

624

–1.0

E/V vs. SCE (a) KPF6

–0.5

50

0 –1.5

–1.0

E/V vs. SCE (b) NaF

–0.5

Figure 14.4.4 Capacitance curves for Ag(100) in aqueous solutions of (a) KPF6 and (b) NaF for different concentrations of electrolyte: Top to bottom, 100, 40, 20, 10, and 5 mM. v = 5 mV/s. 𝜔/2𝜋 = 20 Hz. [Reprinted from Valette (61), with permission from Elsevier Science.]

Capacitance curves for Ag(100) at different concentrations of two electrolytes (61), KPF6 and NaF, are shown as examples in Figure 14.4.4. In both electrolytes, a minimum is observed and the capacitance at the minimum increases with increasing electrolyte concentration. Both findings are in good accord with the GCS model (Figure 14.3.7). The capacitance curve in the presence of KPF6 , Figure 14.4.4a, is symmetrical about the minimum, indicating that neither PF− nor K+ specifically adsorbs at Ag(100). Thus, the capaci6 tance minimum corresponds to the PZC in the absence of specific adsorption, −0.865 ± 0.005 V vs. SCE. In NaF electrolyte (Figure 14.4.4b), the differential capacitance at negative potentials vs. the PZC is essentially identical to that observed in KPF6 solution, but it is significantly larger at positive potentials. These results are attributed to the weak adsorption of F− . Furthermore, while the PZC in KPF6 solutions is independent of electrolyte concentration, it shifts negatively by 22 mV in NaF solutions as the concentration is elevated from 0.005 to 0.1 M. This Esin–Markov effect (Section 14.3.4) confirms adsorption of F− . Figure 14.4.4b also demonstrates that the Helmholtz or inner-layer capacitance is not constant, as predicted in the Gouy–Chapman–Stern model (Figure 14.3.7), but decreases as the potential is moved a few hundred millivolts positively or negatively from the PZC. This potential dependent inner-layer capacitance has been shown to result primarily from orientation of solvent dipoles at the electrode surface in response to high electric fields in these potential regions.

14.4 Studies at Solid Electrodes

Figure 14.4.5 Capacitance curves for the (111), (100), and (110) faces of Ag in 10 mM NaF. v = 5 mV/s. 𝜔/2𝜋 = 20 Hz. [Adapted from Valette and Hamelin (62), with permission.]

Cd /µF cm–2

60

40

20

0

(110) (100) (111)

0.0

–0.5 E/V vs. SCE

–1.0

(c) Variations with Crystal Face

Studies with single-crystal electrodes and other well-defined surfaces have clearly demonstrated that interfacial properties such as PZC and work function depend on the index plane presented to the solution. Figure 14.4.5 illustrates such variations for a given electrolyte at different crystal faces of Ag (43, 62). If the capacitance minimum marks the PZC, then the PZC clearly depends upon the exposed crystal face. At Ag(111), Ez = − 0.69 V vs. SCE, but at Ag(110), Ez = − 0.98 V vs. SCE. Since a polycrystalline electrode presents a variety of crystal faces to a solution, different portions of the same surface can carry different charges. For example, on polycrystalline Ag held at −0.8 V vs. SCE, the (111) domains are more negative than their PZC and will carry a negative charge, while the (110) domains will be positively charged (57). The catalytic and adsorptive properties of solid surfaces can also vary with the crystal face. A striking illustration (63) is seen in the cyclic voltammograms for reductive adsorption and oxidative desorption of hydrogen on the low-index surfaces of Pt (Figure 14.4.6). (d) Reconstruction

Another complication with solid electrodes is the possibility of surface reconstruction during an investigation, or even during a potential scan (65). Consider, for example, the results in Figure 14.4.7, relating to an Au(100) electrode (66). When this surface is heated in a flame, it undergoes reconstruction to form a surface layer of the close-packed form, a slightly buckled (111) arrangement often called a (5 × 20) surface from its LEED pattern. This surface structure is maintained when the electrode is immersed in a solution of 0.01 M HClO4 . If the potential of the electrode is cycled to more positive potentials, the surface structure does not change if the potential does not exceed +0.5 V vs. SCE. The capacitance during the return sweep (curve 2) continues to represent the (5 × 20) surface. However, if the potential is taken beyond +0.5 V, the surface converts to the native (100) structure [with a (1 × 1) LEED pattern]. The original reconstruction is said to have been lifted. Accompanying this change is a large alteration in the capacitance curve, suggesting a shift in

625

14 Double-Layer Structure and Adsorption

100

Pt (100)

j/µA cm–2

µC cm–2 20

(QO)des = 212 (QH)des = 205

0.1

0.5

1.0

E/V vs. NHE Pt (111)

j/µA cm–2

60 20

(QO)des = 95 (QH)des = 240

0.1

0.5

1.0

E/V vs. NHE 160

Pt (110) (QO)des = 195

j/µA cm–2

626

40

(QH)des = 200 20 60 0.1

0.5

1.0

E/V vs. NHE

Figure 14.4.6 First cyclic voltammograms at flame-treated Pt of different orientations in 0.5 M H2 SO4 at 50 mV/s. Positive potentials are plotted to the right, and anodic currents up. Compare these results to the voltammogram of a polycrystalline electrode, Figure 14.6.1. QH and QO represent the areas under the hydrogen and oxygen desorption peaks, respectively. The values for (QH )des were later shown to involve charge from both desorption of atomic hydrogen and adsorption of the electrolyte anion (64). [Reprinted with permission from Clavilier (63). © 1988, American Chemical Society.]

the PZC. Similar effects have been seen with other surfaces (67), clearly indicating that the electrode surface structure commonly changes during potential sweeps. (e) A Note about Practical Electrodes

This section has emphasized behavior on clean, well-defined surfaces; however, most practical electrochemistry with solid electrodes involves polycrystalline surfaces, often bearing

14.5 Extent and Rate of Specific Adsorption

60 Au(100): (5 × 20) → (1 × 1) 0.01 M HCIO4

3

Cd/µF cm–2

50

40

2 4

30

20 1, 3

10 –0.4

–0.2

–0.0

0.2

0.4

0.6

E/V vs. SCE

Figure 14.4.7 Capacitance curves for an Au(100) electrode with a reconstructed (5 × 20) surface in 0.01 M HClO4 at v = 10 mV/s. Numbers refer to the sequence of scans. The (5 × 20) reconstructed surface is maintained in scans 1 and 2 to a potential of about 0.4 V vs. SCE. However, on scan 3, beyond 0.6 V, the surface reverts to the original Ag(100) (1 × 1) surface. [Reprinted from Kolb and Schneider (66), with permission from Elsevier Science.]

intentional or adventitious adsorbates (Section 14.6). In such systems, the strong potential dependence expected for C d (especially near the PZC) usually becomes washed out, so that C d becomes fairly flat over significant potential ranges.

14.5 Extent and Rate of Specific Adsorption In Section 14.3.4, we drew a contrast between specific and nonspecific adsorption. Let us emphasize the difference once more: • Nonspecific adsorption is a physical effect, analogous to the attraction of an ion into the ionic atmosphere of an oppositely charged species in solution (e.g., as modeled by the Debye–Hückel theory; Section 2.1.5).11 Nonspecific adsorption can affect an electrochemical response by altering the concentration of the electroactive species or the potential distribution near the electrode (Section 14.7). • Specific adsorption is a chemical effect, analogous to the formation of a bond between two reactants, as in a coordination step. It can have large impact on electrochemical behavior because it alters the chemistry of the interface.

11 The Gouy–Chapman and Debye–Hückel theories have a common theoretical basis. Both rest on solutions of the Poisson–Boltzmann equation, but in linear coordinates for the former and in polar coordinates for the latter.

627

628

14 Double-Layer Structure and Adsorption

If an adsorbate is also an electroreactant, the theoretical treatment of the faradaic response must be modified to account for its increased presence at the electrode surface. Moreover, specific adsorption often changes the energetics of a reaction. For example, adsorbed O is generally more difficult to reduce than dissolved O. Such effects are addressed in Section 17.2. Sometimes, the changes in energetics are great enough to promote electrocatalysis (enhancement of the heterogeneous kinetics) or passivation (inhibition of the heterogeneous kinetics) (Chapter 15). Specific adsorption of an electroinactive species can also alter the electrochemical response, perhaps most obviously by forming a blocking layer on the electrode surface. However, adsorption can also increase the reactivity of a species, for example, by causing dissociation of an unreactive material into reactive fragments, such as in the adsorption of aliphatic hydrocarbons on a Pt electrode, leading to electrocatalysis. 14.5.1

Nature and Extent of Specific Adsorption

For electroreactants and electroproducts, one can use a faradaic response to determine amounts adsorbed. Sections 17.2 and 17.3 provide a detailed discussion of the methods. Nonelectrochemical methods have often been used to measure surface excesses of both electroactive and electroinactive species. For example, one can immerse a large-area electrode at a given potential and monitor the change in the solution concentration of the species adsorbed. A sensitive analytical technique (e.g., fluorimetry, chemiluminescence, radioactivity) can thereby provide a direct measurement of the amount of substance that has left the bulk solution upon adsorption (8, 68, 69). Any such approach requires high sensitivity and precision because the change in bulk concentration caused by adsorption is usually small (Problem 14.9). Measurements of radioactivity can also be applied to electrodes removed from the solution, with suitable corrections for bulk solution still wetting the electrode (69). The amount of material in a monolayer of adsorbate depends on the size of the adsorbing molecule and its orientation on the electrode surface. If the adsorbate follows a pattern in precise correspondence with the atomic arrangement of the underlying surface, the adsorption is said to be commensurate. For example, there are 1.2 × 1015 Au atoms/cm2 on the surface of Au(100), with an interatomic spacing of 0.29 nm (Figure 14.5.1a). If adsorbed atoms take atop sites (Figure 14.4.1) on each gold atom (an arrangement called a 1 × 1 superlattice), the surface coverage would be 2.0 × 10−9 mol/cm2 . For the most part, the molecular dimensions of adsorbates are too large to pack in this way and they are more widely spaced. Figure 14.5.1a provides an illustration from a study (70) of I− adsorption on Au(100) carried out by in-situ scanning tunneling microscopy (STM), which provides atomic-level images. In this experiment, the well-defined gold surface was held initially at −0.25 V vs. SCE in 10 mM KI solution and then was stepped to −0.1 V vs. SCE, while the STM image in Figure 14.5.1a was being recorded from bottom to top. The lower third of the image shows the electrode surface at the initial potential, and the upper two-thirds show it after the step. At −0.25 V, iodide is not specifically adsorbed, and the lower part shows a square array of atoms separated by 0.29 nm, corresponding to Au(100). After the step to −0.1 V—in the upper part the figure—the structure has become converted to a hexagonal array aligned along the diagonal of the Au lattice. Nearest-neighbor spacings have enlarged to 0.42–0.46 nm. This result is consistent with electrochemical evidence for specific adsorption of I− at E ≥ − 0.2 vs. SCE. The results are interpreted in terms of Figure 14.5.1b, showing a model in which a hexagonal I− (or I atom) adlattice is oriented to produce rows of ions (or atoms) along diagonals of the underlying Au(100). The adlattice is registered to the Au(100) in rectangles having sides that are

14.5 Extent and Rate of Specific Adsorption

Adsorbed I–

Au (100) substrate atom

2√ 2

2 √ 11 2 rows

11 rows

(a)

(b)

Figure 14.5.1 (a) In-situ STM image of a potentiostatically controlled Au(100) surface in 10 mM KI. As the image was being rastered from bottom to top, the potential was stepped from −0.25 to −0.10 V at the time corresponding to the discontinuity in the image. For the bottom third, the potential was −0.25 V and the image corresponds to the square array for Au(100). The unit cell (white square at lower left) was 0.29 nm on a side. For the top two-thirds, the potential was −0.10 V vs. SCE and the image shows a hexagonal array with nearest-neighbor spacing of 0.42–0.46 nm (white hexagon, upper middle). This array features rows aligned with diagonals of the Au(100) lattice (white line). (b) Proposed adlattice structure corresponding to the upper part of (a). The array of white circles represents the Au(100) substrate. The gray circles represent the adlattice, with darker shading marking adatoms on atop sites. [Adapted with permission from Gao et al. (70). © 1994, American Chemical Society.]

√ √ 2 2 and 11 2 times the Au(100) unit-cell length. These rectangles are most easily discerned by picking out the locations where the adsorbed I− occupies an atop site (darker shading for the corner √ atoms√in the rectangle in Figure 14.5.1b). Because of its periodicity, this layer is called a “2 2 × 11 2 adlattice.” At 0.0 V vs. SCE—just 100 mV more positive than used to obtain Figure 14.5.1a—the system adopts a quite different structure in which the surface density of I− √ √ is higher, the packing is tighter, and the adlattice repeats in much smaller units of 2 2 × 2. The density of adsorbed I− on Au(100) is ∼8 × 10−10 mol/cm2 , or about 40% of the density of the underlying Au atoms (70). For low-molecular-weight substances on well-defined metal surfaces, coverages typically lie in the range of 10−9 to 10−10 mol/cm2 , representing an easily measurable amount of charge (>10 μC/cm2 ). Electrochemical studies of adsorbed electroactive materials can readily detect fractions of monolayers (Sections 17.2 and 17.3). The coverages quoted above are for an atomically smooth surface. Because the surfaces of essentially all solid electrodes, including single-crystal surfaces, have steps, plateaus, and defects, they usually show larger coverages per unit projected area. The ratio of the actual area of an electrode to the projected area (the area assuming the electrode is perfectly smooth) is called the roughness factor (Section 6.1.5). Even apparently smooth and polished solid electrodes have roughness factors of 1.5 to 2 or more. 14.5.2

Electrosorption Valency

As interest has grown in the nature of adsorbed species as they exist on the electrode, the concept of electrosorption valency has become more frequently exploited (3, 71). The idea

629

630

14 Double-Layer Structure and Adsorption

originated in the view (72) that adsorption might sometimes involve partial charge transfer between an adsorbate, j, and the electrode, with a concomitant change in the chemical nature of adsorbed j vs. unadsorbed j. The term became quantified (73) as the variation in charge density on the electrode, 𝜎 M , vs. the variation in the surface excess of j, Γj . Thus, the electrosorption valency, l, is defined as l = −(𝜕𝜎 M ∕𝜕Γj )E ∕F

(14.5.1)

This is a dimensionless quantity, scaled in units of electronic charge; hence, l = − 1 implies that a single adsorbate ion or molecule makes the electrode more positive by one electronic unit. To illustrate, let us consider again the adsorption of I− on Ag. In the simplest view, one expects that each adsorbed ion will be neutralized at the interface by a positive change of one electronic unit on the Ag. In effect, the result of adsorption has been to transfer one unit of negative charge per adsorbed I− to the electrode because the charge arriving on the I− has been pushed coulombically into the metal, and perhaps even into an external circuit. This concept is the basis of the oft-made statement that the electrosorption valency is “the charge transferred to the electrode” in a single adsorption. From this discussion, we expect l = − 1 for I− on Ag. The reality is, however, that l varies with potential in that system, always being more positive than −1 (71). There must be more to the story than the simple coulombic picture that we have just conceived. Indeed, electrosorption valency reflects any factor that influences the charge density at an electrode surface, so it is sensitive to • The charge carried by the adsorbate. • Any partial charge transfer (74) that might be involved in the electrode-adsorbate interaction. • The orientation (or any change in the orientation) of a dipole associated with the adsorbate. Measurements of electrosorption valency provide structural and chemical information about an adsorbate layer that is difficult to obtain by other means. Relevant theory and experimental practice have been reviewed in detail (3, 71). 14.5.3

Adsorption Isotherms

For species j at a given temperature, an equation called the adsorption isotherm links the amount adsorbed per unit area, Γj , to both the activity in bulk solution, abj , and the electrical state of the system as given by E or qM . The isotherm describes equilibrium and is derived from the corresponding electrochemical potentials (Section 2.2.4). When species j adsorbs, it occupies a site, L, on the electrode, disrupting prior interactions (e.g., with the solvent) at that site. Using superscripts A and b for adsorbed j and for j in the bulk, respectively, we can write the process as ⇀ jb + L − ↽ − jA

(14.5.2a)

𝜇bj

(14.5.2b)

+ 𝜇L =

𝜇A j

Thus, 𝜇j0,b + RT ln abj + zj F𝜙b + 𝜇L0 + RT ln aL = 𝜇j0,A + RT ln aA + zj F𝜙A j

(14.5.3)

where the 𝜇j0 is standard chemical potentials and 𝜙A and 𝜙b are the electric potentials at the plane of adsorption and in the bulk, respectively. The electrochemical potential of the site L is

14.5 Extent and Rate of Specific Adsorption

assumed to be unaffected by 𝜙A . Rearrangement gives −RT ln

aA j abj aL

= (𝜇j0,A + zj F𝜙A ) − (𝜇j0,b + zj F𝜙b ) − 𝜇L0

(14.5.4) 0

The standard electrochemical free energy of adsorption, ΔGj , is the set of terms on the right side of (14.5.4), which can regrouped as 0

ΔGj = (𝜇j0,A − 𝜇j0,b − 𝜇L0 ) + zj F(𝜙A − 𝜙b )

(14.5.5)

This equation shows that the electrochemical free energy of adsorption has two parts, the first expressing a chemical energy change and the second expressing electrical work. The electrode potential clearly affects the second part through 𝜙A − 𝜙b . It may also affect the first by altering chemical factors, such as physical orientation or bond strength. 0

Using ΔGj from (14.5.5), we can rewrite (14.5.4) as aA j aL

0

= abj e

−ΔGj ∕RT

= 𝛽j abj

(14.5.6)

where 0

𝛽j = e

−ΔGj ∕RT

(14.5.7)

Equation (14.5.6) is a general adsorption isotherm, with aA ∕aL being expressed in terms of abj j and 𝛽 j . Different specific isotherms result from particular assumptions or models, and several have been proposed (3–5, 8, 34, 75). We will identify three that are widely used. (a) Lack of Lateral Interaction

The Langmuir isotherm is derived from three assumptions: • Homogeneity of the surface. • No interactions between absorbed species. • Saturation of the electrode by adsorbate at high bulk activities (e.g., to form a monolayer). This is equivalent to assuming a fixed number of adsorption sites. When all sites become occupied, the system has reached the saturation coverage, Γs . It is convenient to use unit mole fraction as the standard state for species on the surface, where mole fraction is measured in terms of occupancy of the available sites. Thus, aA = 𝛾jA Γj ∕Γs and j aL = 𝛾 L (Γs − Γj )/Γs . Since there are no interactions among adsorbates, we can take 𝛾jA = 𝛾L = 1 and (14.5.6) becomes Γj Γ s − Γj

= 𝛽j abj

𝜃 = 𝛽j abj 1−𝜃

(14.5.8a)

(14.5.8b)

The version in (14.5.8b) is expressed in terms of the fractional coverage of the surface, 𝜃 = Γj /Γs , which is a commonly used variable.

631

632

14 Double-Layer Structure and Adsorption

The Langmuir isotherm can be written in terms of the concentration of species j in solution by including the remaining activity coefficient and standard-state concentration (Section 2.1.5) in 𝛽 j . Thus, Γj =

Γs 𝛽j Cj

(14.5.9)

1 + 𝛽j Cj

Although we have not changed symbols, the 𝛽 j in (14.5.9) has units of cm3 /mol, while that in (14.5.8) is unitless. If two species, j and k, are adsorbed competitively, the appropriate Langmuir isotherms are Γj = Γk =

Γj,s 𝛽j Cj

(14.5.10a)

1 + 𝛽j Cj + 𝛽k Ck Γk,s 𝛽k Ck

(14.5.10b)

1 + 𝛽j Cj + 𝛽k Ck

where Γj, s and Γk, s represent the saturation coverages of j and k, respectively. These equations can be derived from a kinetic model (Problem 14.12) by assuming independent coverages of 𝜃 j and 𝜃 k , with the rate of adsorption of each species proportional to the free area, 1 − 𝜃 j − 𝜃 k , and the solution concentrations, C j and C k . The individual rates of desorption are assumed to be proportional to 𝜃 j and 𝜃 k . (b) Including Lateral Interactions

Interactions among adsorbed species complicate the problem by making the energy of adsorption a function of surface coverage. The Frumkin isotherm is the most prominent embodiment of this possibility, ( ) Γj 2gΓj b 𝛽 j aj = exp − (14.5.11) Γ s − Γj RT It is derived from a saturation model like that used for the Langmuir isotherm, but with the assumption that the electrochemical free energy of adsorption, (14.5.5), is linearly related to Γj : 0

0

ΔGj (Frumkin) = ΔGj (Langmuir) − 2gΓj

(14.5.12)

The parameter g typically has units of J/mol per mol/cm2 . If g is positive, the interactions between neighboring adsorbed molecules on the surface are attractive; and if g is negative, they are repulsive. As g → 0, the Frumkin isotherm approaches the Langmuir isotherm. This isotherm can also be written in the form, 𝜃 𝛽j Cj = exp(−g ′ 𝜃) (14.5.13) 1−𝜃 where g ′ = 2gΓs /RT and 𝛽 j now includes 𝛾jb and the standard state concentration for j in solution. Often g ′ is treated as a constant in the range –2 ≤ g ′ ≤ 2, but sometimes it is handled as a function of potential (37). An alternative treatment of lateral interactions (37) leads to the logarithmic Temkin isotherm: Γj =

RT ln(𝛽j abj ) 2g

(0.2 < 𝜃 < 0.8)

(14.5.14)

14.5 Extent and Rate of Specific Adsorption

14.5.4

Rate of Adsorption

The adsorption of species j upon creation of a fresh electrode surface (e.g., at a fresh mercury drop at a DME) shows kinetic behavior analogous to that of an electrode reaction. If the kinetics of adsorption are rapid, equilibrium is established at the electrode surface, and the amount of substance adsorbed at a given time, Γj (t), is linked to the surface concentration of the adsorbate, C j (0, t), according to the appropriate isotherm. The buildup of the adsorbed layer to its equilibrium value, Γj , is then governed by the rate of mass transfer to the electrode surface. This situation has been treated in a general way for mass transfer by diffusion and convection by using the diffusion-layer approximation and a linearized isotherm (76). When 𝛽 j C j ≪ 1, the isotherm in (14.5.9) can be expressed in a linear form to yield (Problem 14.10) Γj = Γs 𝛽j Cj = bj Cj

(14.5.15)

where bj = 𝛽 j Γs . In our usual notation for diffusion problems, this equation becomes (14.5.16)

Γj (t) = bj Cj (0, t)

For the case in which the adsorbate layer builds up under diffusion control. We can use (14.5.16) in a solution of Fick’s second law for j. Also applicable are the initial condition, Cj (x, 0) = Cj∗ , and the semi-infinite condition, lim Cj (x, t) = Cj∗ . The amount of material x→∞ adsorbed at any time, t, is the integral of the arriving flux at the electrode surface. For linear diffusion to a stationary plane electrode, [ ] t 𝜕Cj (x, t) Γj (t) = D dt (14.5.17) ∫0 j 𝜕x x=0

The solution is (76)

[ ] ⎛ Dj t ⎞ (Dj t)1∕2 x x = 1 − exp ⎜ + 2 ⎟ erfc + ⎜ bj bj Cj∗ 2(Dj t)1∕2 bj ⎟ ⎝ ⎠ [ ] ⎛D t ⎞ Γj (t) Cj (0, t) (Dj t)1∕2 j = = 1 − exp ⎜ 2 ⎟ erfc ⎜b ⎟ Γj bj Cj∗ ⎝ j ⎠

Cj (x, t)

(14.5.18)

(14.5.19)

Figure 14.5.2 shows (14.5.19) dimensionlessly. 1.0

Гj(t) Гj

0.5

0.0 0

1

2

3

4

Dt/bj2

Figure 14.5.2 Attainment of equilibrium coverage, Γj , for diffusion-controlled adsorption under conditions of a linearized isotherm as given by (14.5.19). bj = 𝛽 j Γs .

633

634

14 Double-Layer Structure and Adsorption

When the linearized isotherm applies, Γj (t)/Γj is independent of Cj∗ and, for realistic values of Dj and bi , a rather long time is required to attain equilibrium coverage [i.e., for Γj (t)/Γj → 1; Problem 14.11]. Equilibrium may not be achieved at a DME or during an LSV sweep at a stationary electrode from an initial potential where adsorption does not occur. The assumption of a linear isotherm is, of course, valid only over a limited concentration range. The use of the full adsorption isotherm may require numerical solution of the problem. The results of such treatments are in qualitative agreement with that for the linearized isotherm (77, 78); however, the rate of attainment of equilibrium depends on Cj∗ . The rate of adsorption can be increased by stirring the solution. For the linearized isotherm in stirred solution (76), ( ) Γj (t) −mj t = 1 − exp (14.5.20) Γj bj where mj is the mass-transfer coefficient for j. Other treatments of mass-transfer-controlled adsorption kinetics have been reviewed (79). The case where the rate of adsorption is governed by the adsorption process itself has also been treated under the assumptions of a logarithmic Temkin isotherm and Temkin kinetics (5, 37, 80). The results have not been widely applied, although measurements of adsorption rates have been attempted (81). Delahay (37) concludes that the kinetics of adsorption, at least on mercury from aqueous solution, are usually facile, so that the overall adsorption rate is normally governed by mass transfer.

14.6 Practical Aspects of Adsorption Largely because of technological implications, a great number of investigations have been dedicated to the effects of adsorption on solid electrodes (3, 5, 8, 9, 34, 37, 39–43, 57, 65). Of special interest are noble metals used as electrodes or electrocatalysts in fuel cells or other applications (82). In aqueous solutions, many noble metals form layers of adsorbed hydrogen or adsorbed oxygen (or, equivalently, oxide film monolayers) and these layers can affect the electrochemical behavior, as shown in Figure 14.4.6, even apart from the involvement of other adsorbates. The current-potential curve for a polycrystalline platinum electrode in an aqueous solution (Figure 14.6.1) shows peaks for the formation and stripping of both adsorbed hydrogen and adsorbed oxygen (39, 40, 45, 46, 63, 82, 83). Polycrystalline electrodes are made up of microscopic crystalline grains with varied orientations; thus, different crystal faces are exposed at the surface when the material is prepared as an electrode. The observed i − E response is the summation of individual i − E responses on the individual faces, each type of which exhibits unique adsorption and capacitive behavior (e.g., Figure 14.4.6). Measurement of the areas under the hydrogen and oxygen adsorption peaks, assuming they represent monolayer coverage, has been widely used as a means of determining the “true” or “microscopic” (as opposed to “geometric” or “projected”) area of the electrode [(84), Section 6.1.5]. For polycrystalline Pt, the area under the hydrogen desorption peaks (shaded zone, Figure 14.6.1) corresponds to ∼210 μC/cm2 ; (82). This is the most commonly used basis for estimation of the microscopic area of polycrystalline Pt electrodes. Foreign substances (e.g., compounds of mercury and arsenic, carbon monoxide, or many organic species) can adsorb on a platinum electrode and inhibit the hydrogen electrode

14.6 Practical Aspects of Adsorption

Oxygen region

60

Hydrogen region

Doublelayer region Oc

40 Hc

1

20 j/µA cm–2 0

–20 2

Oa Ha

–40 1.2

1.0

0.8

0.6

0.4

0.2

0.0

E/V vs. NHE

Figure 14.6.1 Cyclic voltammogram for a smooth platinum electrode in 0.5 M H2 SO4 . Peaks Hc : formation of adsorbed hydrogen. Peaks Ha : oxidation of adsorbed hydrogen. Peaks Oa : formation of adsorbed oxygen or a platinum oxide layer. Peak Oc : reduction of the oxide layer. Point 1: start of bulk hydrogen evolution. Point 2: start of bulk oxygen evolution. The shape, number, and size of the peaks for adsorbed hydrogen depend on the crystal faces of platinum exposed (83), pretreatment of electrode, solution impurities, and supporting electrolyte. See also Figure 14.4.6. Shading shows the zone used for measurement of microscopic area.

reactions. Evidence for this effect is the decrease in the area under the adsorbed hydrogen region of the i − E curve when a competitive adsorbate is present. An adsorbed oxygen (or oxide) layer on platinum often inhibits other electrode processes (e.g., the oxidations of hydrogen, oxalic acid, hydrazine, and many organic substances). However, the oxide layer on Pt can also act as an intermediate in electrocatalytic reactions, for example, the oxidation and reduction of H2 O2 [(85); Section 15.3.7]. Scanning electrochemical microscopy operated in surface interrogation mode (Section 18.5) can be used to quantify chemical reactions of oxidants and reductants with the surface oxide. Adsorption of electroinactive substances plays an important positive role in electrodeposition, where they can act as “brighteners” (Section 12.2.4). Some adsorbed organic molecules (such as acridine or quinoline derivatives) can act as corrosion inhibitors by decreasing the rates of electrode reactions at a metal surface (e.g., metal dissolution or oxygen reduction). A common experience in the laboratory is to encounter effects from an unknown adsorbate (often called “getting crap on the electrode”). This substance may inhibit (or “poison”) an electrode reaction (e.g., by formation of an impervious layer that blocks a portion of the electrode surface), or it may accelerate the electrode reaction [e.g., by double-layer effects (Section 14.7)

635

636

14 Double-Layer Structure and Adsorption

or by the anion-induced adsorption of metal ions (Section 3.5.1)]. With solid electrodes, one may see a slow change in behavior with time, perhaps reflecting the buildup of adsorbed impurities at rates limited by their transport from the bulk solution. Reproducible behavior can sometimes be promoted at a solid electrode by “activating” the electrode surface (86) before carrying out the experimental sequence of interest. The activation process involves subjecting the working electrode to a program of potential steps to values where desorption of impurities occurs or where oxide films are formed and then re-reduced. Related reviews have appeared (37, 87–91). The cleaning of Pt electrodes is discussed in Section 6.8.1. The goal is to achieve a surface that gives voltammetric behavior comparable to that of Figure 14.6.1, which is a reference for a clean Pt surface.

14.7 Double-Layer Effects on Electrode Reaction Rates 14.7.1

Introduction and Principles

As early as 1933 (92), the structure of the double layer was recognized as affecting the kinetics of electrode reactions. The consequences often appear as anomalies. For example, • The rate constant, k 0 , for a heterogeneous electron transfer might depend on the identity or the concentration of supporting electrolyte, even when the electrolyte ions seem uninvolved in specific processes such as complexation or ion pairing. • Highly nonlinear Tafel plots [Section 3.4.3(d)] may be observed. • Rather spectacular effects may be seen in voltammograms, such as a sudden decrease in current with increasing driving force. The reduction of persulfate, S2 O2− , furnishes an example of the latter. This process is very 8 sensitive to the free charge on the surface of a metal electrode and has been extensively investigated at different metals. Consider the CV for 1 mM S2 O2− at Pt(111) in 0.1 M aqueous HClO4 8 (Figure 14.7.1) (93). In the negative-going scan, the onset of S2 O2− reduction occurs at ∼0.65 V 8 vs RHE. The current increases rapidly until reaching a maximum at ∼0.5 V, but it is suddenly inhibited, returning to the background level at ∼0.30 V. Persulfate reduction occurs within the double-layer region of Pt(111), and inhibition of the current happens when the applied potential becomes negative relative to the PZC of Pt(111), resulting in electrostatic repulsion between the electrode and the S2 O2− (94). Figure 14.7.1 is in accord with Frumkin’s first observation of 8 the system at a RDE (92). Effects like those in the bulleted list above can be understood and interpreted in terms of the variation of concentration and potential in the double-layer region, as discussed in Section 14.3. The basic concepts were proposed by Frumkin (8, 92), and this phenomenon is sometimes called the Frumkin effect.12 Suppose a species Oz undergoes a one-step, one-electron reaction −⇀ Oz + e ↽ − Rz−1

(14.7.1)

If Oz is not specifically adsorbed, then its position of closest approach to the electrode is the OHP (x = x2 ; Section 14.3.3). The potential at the OHP, 𝜙2 , is not equal to the potential in bulk solution, 𝜙S , because of the potential drop through the diffuse layer (and possibly because some ions are specifically adsorbed). Potential differences in the double layer (Figures 14.3.6 and 14.3.8) can affect the electrode reaction kinetics in two ways. 12 Sometimes also called “the 𝜙2 effect” or, in the Russian literature, “the Ψ effect.”

14.7 Double-Layer Effects on Electrode Reaction Rates

100 (a)

50

j/µA cm–2

Figure 14.7.1 CV for the reduction of S2 O2− on 8 Pt(111). Scans begin at 0.9 V and first move negatively. Cathodic current is down. (a) Blank 0.1 M HClO4 ; (b) 0.1 M HClO4 with 1 mM K2 S2 O8 . v = 50 mV/s. In both voltammograms, the current between 0.0 and 0.35 V is due to adsorption and desorption of hydrogen on the Pt surface [Sections 14.4.2(c) and 14.6]. In the blank, the current between 0.6 and 0.9 V results from the adsorption and desorption of oxygen-containing species. Between these zones, the Pt electrode is said to be “in the double-layer region.” [From V. Climent et al. (93), with permission.]

0

–50

–100

–150

–200

(b)

0.0

0.2

0.4 0.6 E/V RHE

0.8

1.0

1) If z ≠ 0, the concentration of Oz at x2 will be different from that immediately outside the b , which for our purpose here can be regarded as the concentration “at the diffuse layer, CO electrode surface.”13 From (14.3.3b), b −zF𝜙2 ∕RT CO (x2 , t) = CO e

(14.7.2)

When the electrode has a positive charge (i.e., qM > 0), then 𝜙2 > 0. Anions (e.g., z = − 1) will be attracted to the electrode surface, while cations (e.g., z = + 1) will be repelled. For qM < 0, b. the opposite effect applies. At the PZC, qM = 0, 𝜙2 = 0, and CO (x2 , t) = CO 2) The potential difference driving the electrode reaction is not 𝜙M − 𝜙S , but instead 𝜙M − 𝜙S − 𝜙2 ; thus, the effective electrode potential is E − 𝜙2 . Now consider the rate equation for a totally irreversible one-step, one-electron reaction (Section 3.3): 0′ i = k 0 CO (0, t)e−𝛼f (E−E ) (14.7.3) FA Let us now apply both the correction in (14.7.2) and that for E. The equation written in terms of the true rate constant, kt0 , is then ′ i b e−zf 𝜙2 e−𝛼f (E−𝜙2 −E0 ) = kt0 CO FA

(14.7.4)

13 In this section, we must distinguish two separate meanings of “the concentration at the electrode surface.” For electrode kinetics, the relevant distance scale is a fraction of a nanometer; thus, “x = 0” is very near the interface, and C O (0, t) must be understood essentially as C O (x2 , t). When the focus is on diffusion, “x = 0” is the inner boundary of the diffusion layer. Since the diffuse layer thickness (∼1/𝜅; Table 14.3.1) is normally much smaller than the diffusion layer thickness (typically micrometers even for dilute solutions and rather short experimental durations), this plane is operationally quite close to the electrode. Thus, in equations derived from diffusion theory, C O (0, t) is what we mean b. here as CO

637

638

14 Double-Layer Structure and Adsorption

or ′ i b −𝛼f (E−E0 ) = kt0 e(𝛼−z)f 𝜙2 CO e FA b ≈ C (0, t), we find By comparison of (14.7.3) and (14.7.5), noting that CO O [ ] −(𝛼 − z)F𝜙2 kt0 = k 0 exp RT

(14.7.5)

(14.7.6)

In this important relationship, the exponential factor is sometimes called the Frumkin correction because it allows the calculation of the true (or corrected) standard rate constant kt0 from the apparent one, k 0 . In a similar way, a true exchange current, i0,t , can be defined as in (3.4.6): (1-𝛼) ∗𝛼 CR

∗ i0,t = FAk 0t CO

[ i0,t = i0 exp

−(𝛼 − z)F𝜙2

(14.7.7) ]

RT

(14.7.8)

Alternative derivations of (14.7.6) and (14.7.8) can be obtained using an approach based on electrochemical potentials (37, 95), as outlined in the first edition.14 The overall effect of the double layer on kinetics is that the apparent quantities, k 0 and i0 , become functions of potential through the variation of 𝜙2 with E − Ez . They also depend on the supporting electrolyte concentration since 𝜙2 varies with it. The correction of apparent rate data to find the potential- and concentration-independent kt0 or i0,t , therefore, requires a value of 𝜙2 for the given experimental conditions based on some model for the double-layer structure (Section 14.3). 14.7.2

Double-Layer Effects Without Specific Adsorption of Electrolyte

In the absence of specific adsorption, 𝜙2 can be calculated by assuming the GCS model and using (14.3.28). Table 14.7.1 presents Frumkin corrections for the reduction of Zn(II) at a Zn(Hg) electrode in aqueous solution (96) and for the reduction of several aromatic compounds in DMF (97). For the Zn(II) reduction, where z = 2 and 𝛼 = 0.6, the i0 value is larger than i0,t because the negative charge on the electrode (corresponding to 𝜙2 < 0) attracts the positively charged zinc ion. This concentration effect outweighs the inhibiting kinetic effect of the potential drop in the diffuse double layer. In contrast, for the reductions of uncharged aromatic compounds, the concentration effect is absent because z = 0; however, kt0 is larger than k 0 because of correction for the kinetically inhibiting 𝜙2 < 0. Correction factors are shown in Table 14.7.2 for actual 𝜙2 values observed for a mercury electrode in NaF (98). Note that the corrections can become quite large, especially at low concentrations of supporting electrolyte and at potentials distant from Ez . While the corrections we have just discussed are useful in explaining supporting electrolyte effects on rate constants, one must be aware of several limitations in the treatment: • The complete absence of specific adsorption in a system is a rarity, rather than the rule. • The limitations of the GCS model, as well as the lack of a single “plane of closest approach” when the electrolyte contains a number of different ions, lead to uncertainties in the best values for 𝜙2 and x2 . Indeed, these uncertainties can become so large as to hinder the comparison of measured rate constants with predictions of different theories of electron transfer (97). 14 First edition, Sections 3.4 and 12.7.1.

14.7 Double-Layer Effects on Electrode Reaction Rates

• The GCS model involves average potentials in the vicinity of the electrode and ignores the discrete nature of charges in solution. Such “discreteness of charge effects” have been treated and invoked to account for failures in the usual double-layer corrections (99).

Table 14.7.1 Typical Experimental Results Showing Corrections of Heterogeneous Electron-transfer Rate Data for Double-Layer Effects A. Zn(II) Reduction at Zn(Hg)(a) Supporting Electrolyte/M

𝝓2 /mV

i0 /mA cm−2

i0,t /mA cm−2

0.025

−63.0

12.0

0.39

0.05

−56.8

9.0

0.41

0.125

−46.3

4.7

0.38

0.25

−41.1

2.7

0.29

B. Reduction of Aromatics at Mercury in 0.5 M TBAP in DMF (b) Compound

E1/2 /V vs. SCE

𝜶

𝝓2 /mV

k0 /cm s−1

kt0 /cm s−1

Benzonitrile

−2.17

0.64

−83

0.61

4.9

Phthalonitrile

−1.57

0.60

−71

1.4

7.5

Anthracene

−1.82

0.55

−76

5

p-Dinitrobenzene

−0.55

0.61

−36

0.93

26 2.2

(a) Data from reference (96), T = 26 ± 1 ∘ C, C Zn(II) = 2 mM, C Zn(Hg) = 0.048 M. Supporting electrolyte, Mg(ClO4 )2 . Exchange currents determined by the galvanostatic method. 𝛼 = 0.60. Final column calculated from (14.7.8). In the original literature, this case was analyzed under the assumption that it had a 2e RDS, so the authors found 𝛼 = 0.3. Treating it as a multistep process (Section 3.7), one finds best agreement with a mechanism having the first electron transfer upon reduction as the RDS with 𝛼 = 0.6. (b) From reference (97), T = 22 ± 2 ∘ C, concentration of compounds ∼1 mM. Rate constants measured by ac impedance method. Final column calculated from (14.7.6).

14.7.3

Double-Layer Effects with Specific Adsorption

When an ion from the supporting electrolyte (e.g., Cl− or I− ) is specifically adsorbed, 𝜙2 is perturbed from the value calculated strictly from diffuse double-layer corrections. Specific adsorption of an anion will cause 𝜙2 to be more negative, while specific adsorption of a cation will cause 𝜙2 to be more positive. In principle, these effects can be taken into account using the Frumkin correction; however, the location of the plane of closest approach for the reacting species and the actual potential at the OHP often cannot be defined. Qualitative, rather than quantitative, explanations of these effects are, therefore, given. Specific adsorption of an ion may also physically block the electrode surface, as discussed in Section 14.6, and may inhibit the reaction independently of the 𝜙2 effect. Although studies of the effects of double-layer structure on reaction rates are frequently complicated (3, 4, 32, 33), they can provide information about mechanistic details, the location of the reacting species, and the nature of the reacting site. See, for example, studies on the electroreduction of complex ions at a mercury electrode (100).

639

640

14 Double-Layer Structure and Adsorption

Table 14.7.2 Double-Layer Data and Frumkin Correction Factors for a Mercury Electrode in NaF(a) Frumkin correction factors (𝜶 = 0.5)(b) (E − E z )/V

𝝈 M /𝛍C cm−2

𝝓2 /V

z=0

z=1

z = −1

0.010 M NaF (Ez = − 0.480 V vs. NCE) −1.4

−23.2

−0.189

0.025

39.5

1.6 × 10−5

−1.0

−16.0

−0.170

0.037

27.3

4.9 × 10−5

−0.5

−8.0

−0.135

0.072

13.8

3.8 × 10−4

0

1.0

1.0

1.0

0.153

19.6

0.051

7.5 × 103

0

0

+0.5

11.5

0.10 M NaF (Ez = − 0.472 V vs. NCE) −1.4

−24.4

−0.133

0.075

13.3

4.3 × 10−4

−1.0

−17.0

−0.114

0.11

9.2

1.3 × 10−3

−0.5

−8.9

−0.083

0.20

5.0

7.9 × 10−3

0

1.0

1.0

1.0

0.102

7.3

0.14

3.8 × 102

0

0

+0.5

13.2

1 M NaF (Ez = − 0.472 vs. NCE) −1.4

−25.7

−0.078

0.22

4.6

1.1 × 10−2

−1.0

−18.0

−0.062

0.30

3.3

2.6 × 10−2

−0.5

−9.8

−0.039

0.47

2.1

0.10

0

1.0

1.0

1.0

0.054

2.9

0.35

23

0

0

+0.5

14.9

(a) 𝜎 M and 𝜙2 data taken from compilation (98) based on Grahame’s data. (b) Correction factor = exp[(𝛼 – z)f𝜙2 ].

14.7.4

Diffuse Double-Layer Effects on Mass Transport

The Gouy–Chapman–Stern model of the diffuse double layer was developed in Section 14.3 on the basis of the Poisson–Boltzmann equation under equilibrium conditions. The model involves no electrode reactions and no net flux of ions or molecules in the solution. A more general theory allows for electrode reactions and ion fluxes. Consider, for example, a solution containing only FeSO4 . At potentials where Fe2+ is not oxidized, the distributions of Fe2+ and SO2− within the diffuse double layer are established in 4 accord with GCS theory. However, the situation becomes more complex when the potential is moved to a value where Fe2+ is oxidized to Fe3+ because the electron-transfer reaction depletes Fe2+ and generates Fe3+ , resulting in a new set of non-equilibrium diffuse-double-layer distributions of Fe3+ , Fe2+ , and SO2− Additionally, there are net fluxes of all three species across 4 . the diffuse double layer. A diffuse double layer is still maintained at the electrode interface, but it now has a potential-dependent structure that is no longer described by the GCS theory. The diffuse double-layer structure in the presence of ion fluxes has been called the dynamic diffuse layer (101).

14.7 Double-Layer Effects on Electrode Reaction Rates

A general model allowing for faradaic reactions and ion fluxes is developed from the Poisson equation, 𝜌(x) = −𝜀𝜀0

d2 𝜙

(14.7.9)

dx2

and a set of Nernst–Planck equations, (4.1.13), each describing the contribution to the total flux by diffusion and migration of a participating ion or molecule, j, Jj = −Dj ∇Cj −

zj F RT

Dj Cj ∇𝜙

(14.7.10)

The Poisson equation links the charge density, 𝜌(x), to the electric potential, 𝜙, and embodies the electrostatic interactions between solute ions, as well as the interactions between the solute ions and charged electrode surface. This set of equations, along with the general continuity equation, 𝜕Cj 𝜕t

+ 𝛁 ⋅ Jj = 0

(14.7.11)

comprises a system of partial nonlinear differential equations called the Poisson–Nernst–Planck (PNP) model. The PNP equations, as applied to electrochemical measurements, are conveniently solved using finite-element numerical simulations. At equilibrium, the diffusional fluxes of ionic species within the diffuse double layer are precisely balanced by migration fluxes in the opposite direction, and Jj = 0 everywhere. In this limit, one can readily show that rearrangement of the Nernst–Planck equation yields the Boltzmann expression, (14.3.3b). Thus, at equilibrium, the PNP model for the diffuse layer is equivalent to the Gouy–Chapman model. Determining the structure of the diffuse double layer while a faradaic reaction is occurring is not generally amenable to the experimental methods that are used to probe equilibrium double-layer structure (e.g., measurements of capacitance or surface tension). Rather, the interplay of the diffuse layer with the transport of reactant or product ions is inferred from the dependence of the current on the electrode potential and electrolyte concentration. For instance, consider again the oxidation of Fe2+ to Fe3+ in a solution containing only FeSO4 . At a Pt electrode, oxidation of Fe2+ occurs at potentials positive of the PZC of Pt; thus, the electrode surface is positively charged as this reaction proceeds. Electrostatic repulsion between the surface and the Fe2+ is anticipated to cause migration of Fe2+ away from the electrode surface, resulting in a reduced overall flux of Fe2+ toward the surface and a decrease in the observed faradaic current. Conversely, electrostatic attraction between a positively charge redox reactant and a negatively charged electrode would result in an increase in the faradaic current. These electrostatic effects on transport rates arise from the electric fields generated by the electrode surface charge and have a different origin than the migrational effects in low ionic strength solutions discussed in Section 5.7. The latter are caused by the electric field associated with the ohmic potential drop within the solution. In general, migration of ions resulting from both sources of electric field occurs simultaneously. Whether the double layer inhibits or enhances the flux of a redox reactant is determined by the relative thicknesses of the diffuse double layer and depletion layer. The former is characterized by the Debye length, 𝜅 −1 , while the latter is characterized by the depletion layer thickness, 𝛿. These two quantities have different dependencies on experimental conditions. The Debye length is proportional to the inverse square root of the concentration of ionic species but is essentially independent of electrode size, electrode geometry, and time. In contrast, we have

641

642

14 Double-Layer Structure and Adsorption

seen in Chapters 1, 5, and 10 that 𝛿 is a function of the electrode size and may increase with time but is not a function of the ionic concentration in solution. In experiments using macroscopic electrodes in solutions containing a high concentration of supporting electrolyte, 𝛿 is many orders of magnitude larger than 𝜅 −1 , and the diffuse double layer has little or no effect on the overall mass transport resistance of the redox species. Accordingly, the diffuse double layer has no measurable consequence on ion fluxes in the vast majority of conventional electrochemical measurements. However, diffuse double-layer effects on mass transport do become important when 𝛿 is comparable to 𝜅 −1 . There are three types of experiments in which this situation has been encountered: • In SSV at a UME when the critical dimension of the electrode and, thus, 𝛿 is reduced to values comparable to 𝜅 −1 (102–108). • In potential step experiments at very short times (109) or in cyclic voltammetry at very high sweep rates (110, 111), such that the time-dependent 𝛿 is comparable to 𝜅 −1 . • In two electrode systems where the distance between the electrodes defines a transport length 𝛿 that is comparable to 𝜅 −1 (112–118). As a specific example, consider Figure 14.7.2, which shows the relative dimensions of 𝜅 −1 and 𝛿 expected at a 10-nm-radius hemispherical electrode at which the faradaic reaction O+ + e → O is occurring at steady state in a 1-mM aqueous solution of a 1:1 electrolyte. The value of 𝜅 −1 for a 1-mM ionic solution is 9.6 nm (Table 14.3.1), and the distance where the diffuse-layer potential decays by 95% of 𝜙2 − 𝜙S (∼3𝜅 −1 ) is ∼30 nm. The value of 𝛿 is also straightforward to calculate. For a purely diffusion-controlled process, the concentration of reactant reaches 90% of its bulk value at a distance of ∼10 radii from the center of the spherical electrode, i.e., 𝛿 ∼ 9r0 . Thus, for the example shown in Figure 14.7.2, 𝛿 ∼ 90 nm. The 30-nm-thick diffuse layer overlaps significantly with the 90-nm-thick depletion layer established by the faradaic reaction; consequently, an electric force is exerted on the molecule O+ as it moves toward the electrode, the sign of which depends on whether the electrode is positively or negatively charged. This force produces a migration flux of O+ away from or toward the electrode surface, respectively, countering or augmenting the diffusive flux and resulting in a decrease or increase in the current. For a macroscopic electrode placed in a solution containing excess supporting electrolyte, 𝜅 −1 is an insignificant fraction of 𝛿 and the electric field has an insignificant effect on transport of ions to the electrode surface. 50 nm

O+

δ 3 κ–1

Figure 14.7.2 Diagram (drawn to scale) of the relative thicknesses of the diffuse double layer (∼3𝜅 −1 ) and the depletion layer 𝛿 ∼ 9r0 surrounding a 10-nm radius hemispherical electrode immersed in an aqueous solution containing 10−3 M 1:1 electrolyte (𝜅 −1 = 9.6 nm). [Based on Norton, White, and Feldberg (102).]

14.7 Double-Layer Effects on Electrode Reaction Rates

–3

0.0

–2 –1 0

i/id

+1 +2 0.5

+3 Diffusion

1.0

–1.5

–1.0

–0.5 (E –

0.0

E0′)/mV

Figure 14.7.3 Simulated voltammetric response at a nanodisk electrode (r0 = 5 nm) for the 1e reduction of charged reactants (Oz + e ⇌ Rz − 1 , with z = − 3 to +3). The solution contains 5 mM Oz and 0.5 M 1:1 electrolyte. Simulations are based on the PNP model. The PZC corresponds to E = 0, which is set equal to the potential of the bulk solution. A 0.56-nm-wide Helmholtz layer with a spatially dependent dielectric constant is included in the simulations, while the electron-transfer reaction is assumed to be governed by Marcus kinetics with k0 = 1 cm/s and 𝜆 = 100 kJ/mol. The simulation model also allows for long-range electron transfer [𝛽 = 10 nm−1 , Section 3.5.2(a)] resulting an effective electrode radius somewhat larger than r0 . All currents are normalized to the expected diffusion-limited current at a disk in the absence of double layer effects and long-range electron transfer (id = 4nFDCr0 ). [Adapted with permission from Liu, He, Zhang, and Chen (104). © 2010, American Chemical Society.]

Figure 14.7.3 shows finite-element simulations of i − E curves based on the PNP model for the steady-state 1e reduction of Oz (z = − 3 to +3) at a 5-nm-radius disk in an aqueous solution. The model also considers the potential drop across a 0.56-nm-wide Helmholtz layer with a dielectric less than that of bulk water, which is necessary to accurately compute the potential distribution and ion fluxes within the diffuse layer. The solution was assumed to contain 5 mM Oz and 500 mM 1:1 electrolyte. In the absence of any double-layer effects (i.e., z = 0), the i − E curve has a sigmoidal shape with a potential-independent limiting current. However, the effects of the diffuse double layer on the limiting current are clearly seen when the effect of surface charge is included in computing the i − E curve, especially for the reduction of highly charged reactants. Transport of Oz is enhanced when z is positive, and the electrode potential is negative of the PZC, but inhibited when the sign of z and the electrode are both negative. For ′ the example in Figure 14.7.3, the PZC is assumed to be equal to E0 , so the electrode is negatively charged over the entire voltammetric wave. For negatively charged Oz , the inhibition of the overall flux caused by migration becomes progressively more severe as the electrode becomes more negatively charged; consequently, the steady-state voltammogram exhibits a current maximum. Dramatic demonstrations of the effect of the diffuse double layer on i − E behavior have been reported in systems where two electrodes are separated by an electrolyte solution of thickness comparable to 𝜅 −1 . One electrode is maintained at a potential where O is reduced to R,

643

14 Double-Layer Structure and Adsorption

while the second electrode held at a potential to regenerate O from R, resulting in steady-state redox cycling (Sections 5.6.3, 12.6.3, and 19.6). In this configuration, at steady state, the distance between the two electrodes defines the diffusional distance, d. Systems where d is between 10 and 200 nm are readily realized using lithographic fabrication methods (112, 113), or by employing an SECM to position a metal tip close to an electrode surface (114) (Section 18.4). Figure 14.7.4b shows the i − E response corresponding to steady-state redox cycling of the ′ FcTMA2+ /FcTMA+ system (FcTMA2+ + e ⇌ FcTMA+ , E0 ∼ 0.4 V vs. Ag/AgCl) in lithographically prepared twin-electrode cells (Figure 14.7.4a) comprising two parallel Pt electrodes separated by distances of 53-, 74-, and 210-nm thickness (112). When the potential of the bottom electrode is scanned to potentials positive of 0.4 V, FcTMA+ is oxidized to FcTMA2+ , which is then transported back to the top electrode (held at 0 V), where it is reduced. The FcTMA+ is then transported back to the bottom electrode, completing one redox cycle. This process repeats with high frequency and offers large current enhancements, as compared to a single Pt electrode of the same size in a solution of the same composition (Section 19.6). In the presence of an excess supporting electrolyte (200 mM TBAPF6 ), these cells provide sigmoidal voltammograms with a limiting current consistent with the cell thickness, d. However, at TBAPF6 concentrations of 10 mM and lower, the redox cycling currents are attenuated and the i − E responses become peak-shaped, reaching a maximum at potentials ∼100 mV positive ′ of E0 . The i − E response retraces itself on the reverse scan, indicating that the responses are essentially steady state and not due to transient phenomena. The attenuated currents and peak-shaped behavior are most apparent at lower electrolyte concentrations and for thinner Experiment 30

(1)

Simulation

200 mM

(2)

20 10

53 nm

10 2.5 0.0

(3)

200

74 nm

10 2.5 0.0

(5)

200

0

Ag/AgCl

10 6 µm 0

SiNx/SiO2

Pt d

+ + + + + + + + + + 0 –1 V

20 µm

Pt

50 – 200 nm

0V e O e R

8

(6)

10 4 210 nm

2.5 0.0

0 0.0

(a)

(4)

i/nA

20

~800 nm

644

0.5

1.0

0.0 E/V vs. Ag/AgCl

0.5

1.0

(b)

Figure 14.7.4 (a) Schematic of the dual-electrode cell for investigating the effect of the diffuse double layer on redox cycling between two parallel electrodes. The cell has an opening in the upper electrode to allow exchange of ions between the bulk solution and the electrolyte held between the electrodes. The spacing between the electrodes is not drawn to scale and is much smaller than shown in the schematic. (b) Experimental (1, 3, 5) and simulated (2, 4, 6) voltammograms. The bulk acetonitrile solution contained 50 μM FcTMAPF6 with TBAPF6 at 0, 2.5, 10, and 200 mM. The cell thicknesses were, respectively (1, 2) 53 nm, (3, 4) 74 nm, and (5, 6) 210 nm. Scan rate = 10 mV/s. The top electrode was held at 0 V (where FcTMA2+ is reduced at the transport-limited rate), while the bottom electrode was scanned between 0 and 1 V vs. Ag/AgCl. A 0.6-nm wide Helmholtz layer with 𝜀 = 6 is assumed in the simulation. [Adapted with permission from Xiong, Chen, Edwards, and White (112). © 2015, American Chemical Society.]

14.8 References

cells (i.e., smaller d). They are interpreted as manifesting electrostatic interactions between the charged redox species (FcTMA+ and FcTMA2+ ) and the Pt electrodes. As the bottom electrode is scanned positively, repulsive electrostatic forces between the positively charged bottom electrode and FcTMA+ result in a decrease in the flux of this ion to the bottom electrode where it is oxidized during redox cycling; additionally, the repulsive forces between the bottom electrode and both FcTMA+ and FcTMA2+ result in partial expulsion of these charged redox species from the very narrow solution confined between the two Pt electrodes. Both effects combine to yield a dramatic decrease in the steady-state current. In the presence of 200 mM TBAPF6 , sufficient ions are present to screen the electrode surface charge, resulting in a diffusion-limited current plateau. In Figure 14.7.4b, the experimental results are compared to finite-element simulations based on a PNP model of the transport of TBA+ , PF− , FcTMA+ , 6 and FcTMA2+ within the diffuse double layer. Excellent agreement is observed between the simulations and experiment. Diffuse double-layer effects can produce very large increases in the rate of redox cycling between two electrodes when attractive electrostatic forces operate between the electrodes 3+∕2+ and the redox species, e.g., during redox cycling of Ru(NH3 )6 between two negatively charged electrodes (113, 116). The primary reason for the increased current in this situation is the enhanced concentration of redox ions in the electric field between the two electrodes. This phenomenon has been strategically used to improve analytical sensitivity, with redox cycling enhancements as large as 500 reported in the literature (116).

14.8 References 1 D. C. Grahame, Chem. Rev., 41, 441 (1947). 2 D. C. Grahame, Annu. Rev. Phys. Chem., 6, 337 (1955). 3 W. Schmickler and E. Santos, “Interfacial Electrochemistry,” 2nd ed., Springer, Heidelberg,

2010. 4 J. Goodisman, “Electrochemistry: Theoretical Foundations, Quantum and Statistical

Mechanics, Thermodynamics, The Solid State,” Wiley, New York, 1987, Chaps. 4–6. 5 D. M. Mohilner, Electroanal. Chem., 1, 241 (1966). 6 P. Delahay, “Double Layer and Electrode Kinetics,” Wiley-Interscience, New York, 1965,

Chap. 2. 7 R. Parsons, Mod. Asp. Electrochem., 1, 103 (1954). 8 B. E. Conway, “Theory and Principles of Electrode Processes,” Ronald, New York, 1965,

Chaps. 4 and 5. 9 R. Payne, in “Techniques of Electrochemistry,” Vol. 1, E. Yeager and A. J. Salkind, Eds., 10 11 12 13 14 15 16 17 18

Wiley-Interscience, New York, 1972, pp. 43ff. G. Lippmann, Compt. Rend., 76, 1407 (1873). J. Heyrovský, Chem. Listy, 16, 246 (1922). L. Meites, J. Am. Chem. Soc., 73, 2035 (1951). J. Lawrence and D. M. Mohilner, J. Electrochem. Soc., 118, 259, 1596 (1971). D. M. Mohilner, J. C. Kreuser, H. Nakadomari, and P. R. Mohilner, J. Electrochem. Soc., 123, 359 (1975). D. C. Grahame and B. A. Soderberg, J. Chem. Phys., 22, 449 (1954). M. A. V. Devanathan and S. G. Canagaratna, Electrochim. Acta, 8, 77 (1963). P. Delahay, op. cit., Chap. 3. H. L. F. von Helmholtz, Ann. Phys., 89, 211 (1853).

645

646

14 Double-Layer Structure and Adsorption

19 G. Quincke, Pogg. Ann., 113, 513 (1861). 20 H. L. F. von Helmhotz, Ann. Phys., 7, 337 (1879). 21 J. Walker, D. Halliday, and R. Resnick, “Fundamentals of Physics,” 10th ed., Wiley,

Hoboken, NJ, 2014. 22 E. M. Pugh and E. W. Pugh, “Principles of Electricity and Magnetism,” 2nd ed., Addison 23 24 25 26 27 28

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

52 53 54 55 56

Wesley, Reading, MA, 1970, Chap. 1. G. Gouy, J. Phys. Radium, 9, 457 (1910). G. Gouy, Compt. Rend., 149, 654 (1910). D. L. Chapman, Philos. Mag., 25, 475 (1913). E. M. Pugh and E. W. Pugh, op. cit., pp. 69, 146. O. Stern, Z. Elektrochem., 30, 508 (1924). (a) S. L. Carnie and G. M. Torrie in “Advances in Chemical Physics,” Vol. 56, I. Prigogine and S. A. Rice, Eds., Wiley-Interscience, New York, 1984, pp. 141–253; (b) L. Blum in “Advances in Chemical Physics,” Vol. 78, I. Prigogine and S. A. Rice, Eds., Wiley-Interscience, New York, 1990, 171–222; (c) P. Attard in “Advances in Chemical Physics,” Vol. 92, I. Prigogine and S. A. Rice, Eds., Wiley-Interscience, New York, 1996, 1–159. P. Attard, J. Phys. Chem., 99, 14174 (1995). J. P. Badiali, M. L. Rosenberg, and J. Goodisman, J. Electroanal. Chem., 130, 31 (1981). A. A. Kornyshev, W. Schmickler, and M. A. Vorotynsev, Phys. Rev. B, 25, 5244 (1982). W. Schmickler, Chem. Phys. Lett., 99, 135 (1983). W. Schmickler, J. Solid State Chem., 24, 2175 (2020). P. Delahay, op. cit., Chap. 4. F. C. Anson, Acc. Chem. Res., 8, 400 (1975). O. A. Esin and B. F. Markov, Acta Physicochem. USSR, 10, 353 (1939). P. Delahay, op. cit., Chap. 5. R. M. Ishikawa and A. T. Hubbard, J. Electroanal. Chem., 69, 317 (1976). V. Climent and J. M. Feliu, Adv. Electrochem. Sci. Eng., 17, 1 (2017). C. Korzeniewski, V. Climent, and J. M. Feliu, Electroanal Chem., 24, 75 (2012). A. T. Hubbard, E. Y. Cao, and D. A. Stern, in “Physical Electrochemistry,” I. Rubinstein, Ed., Marcel Dekker, New York, 1995, Chap. 10. A. T. Hubbard, Chem. Rev., 88, 633 (1988). A. Hamelin, Mod. Asp. Electrochem., 16, 1 (1985). G. A. Somorjai, in “Photocatalysis—Fundamentals and Applications,” N. Serpone and E. Pelizzetti, Eds., Wiley, New York, 1989, p. 265. J. Clavilier, R. Fauré, G. Guinet, and D. Durand, J. Electroanal. Chem., 107, 205 (1980). J. Clavilier, D. El Achi, and A. Rodes, Chem. Phys., 141, 1 (1990). A. G. Güell, S.-Y. Tan, P. R. Unwin, and G. Zhang, Adv. Electrochem. Sci. Eng., 16, 68 (2015). A. J. Bard, “Integrated Chemical Systems,” Wiley, New York, 1994, p. 132. H. V. Patten, M. Velický, and R. A. W. Dryfe, Adv. Electrochem. Sci. Eng., 16, 397 (2015). A. Ambrosi, C. K. Chua, A. Bonanni, and M. Pumera, Chem. Rev., 114, 7150 (2014). E. N. Primo, F. Gutiérrez, M. D. Rubianes, N. F. Ferreyra, M. C. Rodríguez, M. L. Pedano, A. Gasnier, A. Gutierrez, M. Eguílaz, P. Dalmasso, G. Luque, S. Bollo, C. Parrado, and G. A. Rivas, Adv. Electrochem. Sci. Eng., 16, 136 (2015). J. Li, P. H. Q. Pham, W. Zhou, T. D. Pham, and P. J. Burke, ACS Nano, 12, 9763 (2018). J. V. Macpherson, Adv. Electrochem. Sci. Eng., 16, 398 (2015). G. M. Swain, Electroanal. Chem., 22, 181 (2003). Yu. V. Pleskov, Adv. Electrochem. Sci. Eng., 8, 209 (2003). A. J. Lucio, S. K. Shaw, J. Zhang, and A. M. Bond, J. Phys. Chem. C, 122, 11777 (2018).

14.8 References

57 58 59 60 61 62 63 64

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

R. Parsons, Chem. Rev., 90, 813 (1990). A. V. Gokhshtein, Russ. Chem. Rev., 44, 921 (1975). R. E. Malpas, R. A Fredlein, and A. J. Bard, J. Electroanal. Chem., 98, 339 (1979). F. T. Wagner, in “Structure of Electrified Interfaces,” J. Lipkowski and P. N. Ross, Eds., VCH, New York, 1993, Chap. 9. G. Valette, J. Electroanal. Chem., 138, 37 (1982). G. Valette and A. Hamelin, J. Electroanal. Chem., 45, 301 (1973). J. Clavilier, in “Electrochemical Surface Science: Molecular Phenomena at Electrode Surfaces,” M. Soriaga, Ed., ACS Books, Washington, DC, 1988, p. 205. (a) J. Clavilier, R. Albalat, R. Gomez, J. M. Orts, J. M. Feliu, and A. Aldaz, J. Electroanal. Chem., 330, 489 (1992); (b) J. M. Feliu, J. M. Orts, R. Gomez, A. Aldaz, and J. Clavilier. J. Electroanal. Chem., 372, 265 (1994). D. M. Kolb, in “Structure of Electrified Interfaces,” J. Lipkowski and P. N. Ross, Eds., VCH, New York, 1993, Chap. 3. D. M. Kolb and J. Schneider, Electrochim. Acta, 31, 929 (1986). A. Hamelin, M. J. Sottomayor, F. Silva, S. C. Chang, and M. J. Weaver, J. Electroanal. Chem., 295, 291 (1990). B. E. Conway, T. Zawidzki, and R. G. Barradas, J. Phys. Chem., 62, 676 (1958). N. A. Balashova and V. E. Kazarinov, Electroanal. Chem., 3, 135 (1969). X. Gao, G. J. Edens, F.-C. Liu, A. Hamelin, and M. J. Weaver, J. Phys. Chem., 98, 8086 (1994). R. Guidelli and W. Schmickler, Mod. Asp. Electrochem., 38, 303 (2005). W. Lorenz and G. Salié, Z. Phys. Chem., 218, 259 (1961). K. J. Vetter and J. W. Schultze, Ber. Bunsenges. Phys. Chem., 76, 920, 927 (1972). W. Schmickler and R. Guidelli, Electrochim. Acta, 127, 489 (2014). R. Parsons, Trans. Faraday Soc., 55, 999 (1959); J. Electroanal. Chem., 7, 136 (1964). P. Delahay and I. Trachtenberg, J. Am. Chem. Soc., 79, 2355 (1957). P. Delahay and C. T. Fike, J. Am. Chem. Soc., 80, 2628 (1958). W. H. Reinmuth, J. Phys. Chem., 65, 473 (1961). R. Parsons, Adv. Electrochem. Electrochem. Eng., 1, 1 (1961). P. Delahay and D. M. Mohilner, J. Am. Chem. Soc., 84, 4247 (1962). W. Lorenz, Z. Elektrochem., 62, 192 (1958). R. Woods, Electroanal. Chem., 9, 1 (1976). P. N. Ross, Jr., J. Electrochem. Soc., 126, 67 (1979). J. M. Doña Rodríguez, J. A. Herrera Melián, and J. Pérez Peña, J. Chem. Ed., 77, 1195 (2000). I. Katsounaros, W. B. Schneider, J. C. Meier, U. Benedikt, P. U. Biedermann, A. A. Auer, and K. J. J. Mayrhofer, Phys. Chem. Chem. Phys., 14, 7384 (2012). S. Gilman, Electroanal. Chem., 2, 111 (1967). J. Heyrovský and J. Kuta, “Principles of Polarography,” Academic, New York, 1966. C. N. Reilley and W. Stumm, in “Progress in Polarography,” Vol. 1, P. Zuman and I. M. Kolthoff, Eds., Wiley-Interscience, New York, 1962, 81–121. H. W. Nurnberg and M. von Stackelberg, J. Electroanal. Chem., 4, 1 (1962). A. N. Frumkin, Dokl Akad. Nauk. S.S.S.R., 85, 373 (1952); Electrochim. Acta, 9, 465 (1964). R. Parsons, J. Electroanal. Chem., 21, 35 (1969). A. N. Frumkin, Z. Physik. Chem., 164A, 121 (1933). V. Climent, M. D. Maciá, E. Herrero, J. M. Feliu, and O. A. Petrii, J. Electroanal. Chem., 612, 269 (2008). R. Martínez-Hincapié, V. Climent, and J. M. Feliu, Electrochem. Commun., 88, 43 (2018).

647

648

14 Double-Layer Structure and Adsorption

95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

D. M. Mohilner and P. Delahay, J. Phys. Chem., 67, 588 (1963). A. Aramata and P. Delahay, J. Phys. Chem., 68, 880 (1964). H. Kojima and A. J. Bard, J. Am. Chem. Soc., 97, 6317 (1975). C. D. Russell, J. Electroanal. Chem., 6, 486 (1963). W. R. Fawcett and S. Levine, J. Electroanal. Chem., 43, 175 (1973). M. J. Weaver and T. L. Satterberg. J. Phys. Chem., 81, 1772 (1977). V. G. Levich, “Physicochemical Hydrodynamics,” Prentice-Hall, Englewood Cliffs, NJ, 1962. J. D. Norton, H. S. White, and S. W. Feldberg, J. Phys. Chem., 94, 6772 (1990). C. P. Smith and H. S. White, Anal. Chem., 65, 3343 (1993). Y. Liu, R. He, Q. Zhang, and S. Chen, J. Phys. Chem. C, 114, 10812 (2010). Y. Liu and S. Chen, J. Phys. Chem. C, 116, 13594 (2012). E. J. F. Dickinson and R. G. Compton, J. Phys. Chem. C, 113, 17585 (2009). E. J. F. Dickinson and R. G. Compton, J. Electroanal. Chem., 661, 198 (2011). D. Krapf, B. M. Quinn, M.-Y. Wu, H. W. Zandbergen, C. Dekker, and S. G. Lemay, Nano Lett., 6, 2531 (2006). I. Streeter and R. G. Compton, J. Phys. Chem. C, 112, 13716 (2008). C. Amatore and C. Lefrou, J. Electroanal. Chem., 296, 335 (1990). C. Lee and F. C. Anson, J. Electroanal. Chem., 323, 381 (1992). J. Xiong, Q. Chen, M. A. Edwards, and H. S. White, ACS Nano, 9, 8520 (2015). Q. Chen, K. McKelvey, M. A. Edwards, and H. S. White, J. Phys. Chem. C, 120, 17251 (2016). J. H. Bae, Y. Yu, and M. V. Mirkin, J. Phys. Chem. Lett., 8, 1338 (2017). L. Fan, Y. Liu, J. Xiong, H. S. White, and S. Chen, ACS Nano, 8, 10426 (2014). K. Fu, D. Han, C. Ma, and P. W. Bohn, Nanoscale, 9, 5164 (2017). C. Ma, W. Xu, W. R. A. Wichert, and P. W. Bohn, ACS Nano, 10, 3658 (2016). J. Lu and B. Zhang, Anal. Chem. 89, 2739 (2017). G. Gouy, Ann. Chim. Phys., 8, 291 (1906). K. Asada, P. Delahay, and A. K. Sundaram, J. Am. Chem. Soc., 83, 3396 (1961).

14.9 Problems 14.1 Show that surface excesses are independent of the position of the dividing surfaces AA′ and BB′ in Figure 14.1.1, when these surfaces are in pure phase 𝛼 and pure phase 𝛽, respectively. 14.2 Derive the special case (14.3.11) from (14.3.10). 14.3 Present an argument, based only on Gaussian boxes, for a linear potential profile inside the compact layer. 14.4 Obtain (14.3.31) from (14.3.29). 14.5 The Debye length, 𝜅 −1 , characterizes the spatial decay of the electric potential. In the limit of small 𝜙0 , show that the capacitance associated with the diffuse double layer is equivalent to that of an ideal parallel plate capacitor of thickness 𝜅 −1 . 14.6 Calculate the values of 𝜎 M corresponding to various values of 𝜙2 (from −0.2 to +0.2 V) for a mercury electrode in 0.01 M NaF based on the GCS model.

14.9 Problems

649

(a) Plot 𝜙2 vs. 𝜎 M . (b) From the variation of 𝜎 M with E − Ez shown in Table 14.7.2, prepare a plot of 𝜙2 vs. E − Ez . 14.7 Why do we view adsorbed neutral species as being intimately bound to the electrode surface, rather than being collected in the diffuse layer? 14.8 Interpret the data in Figure 14.9.1. How do the traces in Figure 14.9.1b relate to those in Figure 14.9.1a? What implications can be derived from the flat region in the electrocapillary curves in the presence of n-heptyl alcohol? Construct a chemical model to explain the very low differential capacitance from −0.4 to −1.4 V in the presence of n-heptyl alcohol. Can you provide a mathematical rationale for the sharp peaks in C d ? Can you rationalize them chemically? 14.9 A solution containing a certain organic compound, Z, at 1.00 × 10−4 M shows a UV absorbance, A, of 0.500 when measured at 330 nm in a spectrophotometric cell of path length 1.00 cm. Into 50 cm3 of this solution, a Pt electrode with A = 100 cm2 is immersed. If the amount of Z adsorbed corresponds to 1.0 × 10−9 mol/cm2 , what will be the absorbance of the solution after adsorption equilibrium occurs? The adsorption of substance X follows a Langmuir isotherm. Saturation coverage is 8 × 10−10 mol/cm2 and 𝛽 = 5 × 107 cm3 /mol (with aj ∕Cj0 included in 𝛽). At what concentration of X will the electrode surface be half covered (i.e., 𝜃 = 0.5)? Sketch the 90 440

80

420

70

Na2SO4

60 Cd/µF cm–2

400 γ/dyn cm–1

14.10

Na2SO4 + C7H15OH

380

360

50 40 30 Na2SO4 20

340

Na2SO4 + C7H15OH

10 320 0.0

–0.4

–0.8

–1.2

E/V vs. NCE (a)

–1.6

–2.0

0 0.0

–0.4

–0.8

–1.2

–1.6

–2.0

E/V vs. NCE (b)

Figure 14.9.1 (a) Electrocapillary curves for Hg in 0.5 M Na2 SO4 in the presence and absence of n-heptanol. [Data from Gouy (119).] (b) Differential capacitance curves corresponding to (a). [Grahame (1).]

–2.4

650

14 Double-Layer Structure and Adsorption

adsorption isotherm of X. For which concentration range will the linearized isotherm be valid to ∼1%? 14.11 For the substance X in Problem 14.10, how long after immersion will be required for the surface of a planar electrode to attain half of the equilibrium coverage (see Figure 14.5.2)? Assume linearized conditions and D = 1 × 10−5 cm2 /s. How long will be required to attain half of equilibrium coverage if the solution is stirred and mX = 10−2 cm/s? 14.12 Using a kinetic model, derive the Langmuir isotherms, (14.5.10a,b), for the simultaneous adsorption of two species, j and k. 14.13 Prepare a spreadsheet to calculate and plot 𝜃 vs. C j for the Frumkin isotherm, (14.5.11), at g ′ = − 2, 0, and 2. Discuss how attractive (g ′ = 2) and repulsive (g ′ = − 2) interactions affect the isotherm. 0

14.14 The potential dependence of adsorption can be treated by expanding ΔGj , as is usually done for electrochemical potentials, into a standard free energy of adsorption at E = 0 (against an arbitrary reference electrode) and the potential dependence, zj F(𝜙A – 𝜙b ). (See equation (14.5.5).) For the Langmuir isotherm, this procedure yields 𝜃 0 = abj exp(−ΔGads ∕RT) exp(−zj FE∕RT) 1−𝜃

(14.9.1)

which is sometimes written 𝜃 = Cj Kj,ads exp(−zj FE∕RT) 1−𝜃

(14.9.2)

where K j, ads is the “equilibrium constant for adsorption.” Derive these equations. What do they predict about the effect of potential on the adsorption of anions and cations? What is neglected in this model (e.g., to account for the behavior of neutral species)? Derive the equivalent expression for the Frumkin isotherm. 14.15 The Frumkin effect causes Tafel plots to become nonlinear, with the following varying slope in the cathodic region: [ ( )] 𝜕𝜙2 f −𝛼 + (𝛼 − z) (14.9.3) 𝜕𝜂 (a) Derive this relationship for BV kinetics. (b) Asada, Delahay, and Sundaram (120) suggested that a plot of ln[i exp(zF𝜙2 /RT)] against 𝜙2 − 𝜂 (a “corrected Tafel plot”) would be linear with a slope of 𝛼F/RT. Show that this is so. 14.16 Consider the i − E or i − t behavior for a Nernstian, outer-sphere redox reaction involving freely diffusing molecules, e.g., the redox chemistry of ferrocene/ferrocenium −⇀ (Fc+ + e ↽ − Fc) in acetonitrile. (a) Using the definition of the electrochemical potential, (2.2.9), show that the activities of Fc and Fc+ are functions of electric potential, 𝜙, within the electric double layer. Assume that electron transfer occurs when Fc and Fc+ are located within

14.9 Problems

tunneling distance of the electrode, corresponding to a distance from the electrode surface where the electric potential, 𝜙, is different from the bulk solution potential, 𝜙S . (b) Assuming that equilibrium is maintained by diffusion between the redox molecules within the electric double layer and in the bulk solution, demonstrate that the electric potential distribution within the double layer will have no effect on the i − E or i − t behavior for a nernstian reaction. (c) How large can the sweep rate, v, be in a cyclic voltammetric experiment before the diffusional equilibrium described in (b) no longer applies? 14.17

Derive the Poisson equation (14.3.5), by applying the Gauss law (14.3.18), to the volume shown in Figure 14.9.2. Assume that the volume extends infinitely in the y and z directions.

z

E(x + dx) E(x)

q

dx

Figure 14.9.2

y

651

653

15 Inner-Sphere Electrode Reactions and Electrocatalysis Most elementary electrochemical steps described to this point have been 1e outer-sphere reactions. This choice was made because of the simplicity of processes where all reactants and products are in the solution phase and the electrode material is of minor importance. Even for more complex cases, like EC and EE reactions (Chapter 13), the electron-transfer steps were treated as 1e outer-sphere reactions, so that we could employ kinetic models like those of Chapter 3. However, the electrode reactions of many important species—for example, the reduction of H+ or N2 , the oxidation of CO or methanol, or the deposition of Cu—do not occur by outer-sphere paths. Valuable electrochemical processes, such as those occurring in fuel cells or during water electrolysis, often involve electrode reactions in which the adsorption of intermediates is important and the electrode material plays an enormous role. Such processes are called inner-sphere reactions. An electrode surface that can accelerate the kinetics of a reaction without being chemically changed is said to exhibit electrocatalysis, which is a principal focus of this chapter.

15.1 Inner-Sphere Heterogenous Electron-Transfer Reactions 15.1.1

The Role of the Electrode Surface

In an outer-sphere reaction (Section 3.5.1), the reactants, products, and intermediates do not interact strongly with the electrode material, and electron transfer occurs by tunneling across at least a monolayer of solvent. In contrast, a heterogeneous inner-sphere reaction involves a strong interaction of a reactant, a product, or an intermediate with the electrode surface. Usually, at least one participant is specifically adsorbed. Inner- and outer-sphere electrode reactions display very different kinetic sensitivity to the electrode material. For instance, the CV response for the oxidation of ferrocene dissolved in an aprotic solvent is nearly identical at Pt, Au, and C electrodes because the reaction is a simple 1e transfer in which neither Fc nor its oxidation product, Fc+ , is adsorbed. It is typical of outer-sphere reactions to behave nearly independently of the electrode material. In contrast, the rate of reduction of H+ varies widely at these same electrode materials. The exchange current density for proton reduction, as determined by Tafel analyses (Section 15.2.2), is reported to be ∼9 orders of magnitude smaller at an Hg electrode than on noble metal electrodes such as Pt or Ir (1). This difference in kinetics reflects the strength of adsorption of the hydrogen atom, an intermediate in the overall reduction of H+ to H2 . A similar, vast dependence of reaction rates on the nature of the electrode is observed for many other electrocatalytic reactions, including the reduction of O2 . Extreme differences in reaction rates Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

15 Inner-Sphere Electrode Reactions and Electrocatalysis

Propane 4.0

E1/2 / V vs. SCE

654

MeOH NH3

3.0

Benzene

CH4, H2O, CO, CO2, H2

2.0

1.0

TMPD 0.0 6.0

7.0

8.0

9.0

10.0

11.0

IE/eV

Figure 15.1.1 Filled circles: Experimental values of reversible E 1/2 vs. ionization energy, IE, for 1e outer-sphere oxidations of various solutes (2, 3). The line is computed only from the filled circles (slope = 0.76). Open squares: Points for various small molecules, placed on the line extension according to their IE values. As IE becomes more positive, more energy is needed to remove an electron from a molecule. The shaded area is not electrochemically accessible with any conventional electrolyte. Points for CH4 (IE = 12.61 eV), H2 O (12.62 eV), CO2 (13.78 eV), CO (14.01 eV), and H2 (15.43 eV) lie off the chart to the upper right.

reflect large variations in the chemical interactions of reactants, intermediates, or products with the electrode surface. 15.1.2

Energetics of 1e Electron-Transfer Reactions

In considering why some electron-transfer reactions proceed by an outer-sphere pathway and others by an inner-sphere pathway, it is useful to think first about the energy needed to remove or to add a single electron to the reactant. When this action is energetically prohibitive within the available solvent window, oxidation or reduction can proceed only if there is a mechanistic pathway that modifies the reaction energetics. Catalytic electrode surfaces that promote chemical interactions can create that possibility. ′ For outer-sphere 1e oxidation reactions in aprotic solvents, values of E0 [or, alternatively, the “reversible” E1/2 , (6.2.13)] are often correlated with the ionization energy (IE),1 which is the energy required to create a cation by removing an electron from the neutral molecule in ′ the gas phase (4–10). Plots of E0 or E1/2 vs. IE often show linear correlations within a class of structurally similar compounds (even though the E1/2 data are also influenced by the solvation energies of the neutral parent and the cationic product in solution).2 Figure 15.1.13 provides an example for oxidations of a series of aromatic hydrocarbons, ranging from TMPD (an easily 1 Historically called the ionization potential, IP. 2 The solvation energy of a species is the free energy change upon moving that species from vacuum into a specified solution. In general, solvation energies are negative and vary with molecular size and polarity. 3 This figure follows a presentation devised in reference (9) from related plots in references (6–8). This graph was drawn using only data from reversible systems. Values are (a) for TMPD, TTF, and TH from Table C.3, (b) for perylene, benzo[ghi]perylene, and pyrene from reference (2), (c) for alkylated benzenes from reference (6), and (d) for bicyclic peroxides from reference (7). IE values were taken from references (6) or (7), or from the NIST’s “evaluated” result or most precise citation (3).

15.1 Inner-Sphere Heterogenous Electron-Transfer Reactions

oxidized molecule with a low IE, Figure 1) to benzene (oxidized near the positive background limit for MeCN and with IE ∼3 eV more positive than that of TMPD). The slope of the correlation, ∼0.8, is less than the value of unity expected in the absence of solvation effects (10). The solvent window of acetonitrile extends positively to about 2.7 V vs. SCE, corresponding to the lower edge of the shaded area in Figure 15.1.1. The line of correlation between E1/2 and IE can be used to predict potentials for the reversible 1e oxidations of other species based on their IE values; however, 1e redox reactions that occur within the shaded area cannot be experimentally observed because their values of E1/2 lie beyond the positive background limit. Methanol, for instance, is expected to be in this group. It has an IE of 10.8 eV, which corresponds to E1/2 ≈ 3.8 V vs. SCE; thus, the simple 1e oxidation of methanol, +

CH3 OH → CH3 OH ∙ + e

(15.1.1)

should not be experimentally accessible. Despite that prediction, methanol is of interest as the fuel in the direct methanol fuel cell because it does undergo a 6e oxidation in acidic aqueous solution (11), CH3 OH + H2 O → CO2 + 6H+ + 6e

(15.1.2)

Intermetallic Pt/Ru nanoparticles are effective electrocatalysts for the reaction (Section 15.3.3). + The outer-sphere 1e oxidation of methanol to produce CH3 OH ∙ requires potentials much more positive than the thermodynamic standard potential of the overall 6e reaction, E0 = 0.030 V vs. NHE (−0.214 V vs. SCE). Thus, the oxidation MeOH must occur through an inner-sphere mechanism—a route involving specific adsorption of participants on an electrocatalytic surface, to provide a lower-energy path for removal of the first electron. Reaction (15.1.2) must also be a multi-step process involving intermediate chemical species, given the 6e overall change. The details of this and other electrocatalytic reactions are sought in much experimental and theoretical research. The method of correlating E1/2 with IE to determine whether a species can undergo a 1e outer-sphere reaction is only approximate because solvation effects can be very important, especially for small molecules. For example, consider the hydrogen atom, H ∙, which has an IE of 13.69 eV in vacuum and, thus, would be predicted to be reversibly oxidized to H+ at E1/2 ≈ 5 V vs. SCE based on the correlation in Figure 15.1.1. However, the large solvation energy of H+ in water (∼11.4 eV) (12, 13) moves the potential for oxidation to a much less positive range. Such effects are less important with larger molecules, where the solvation energies of reactants and products are similar in magnitude to those of the molecules used to construct the correlation. A parallel correlation strategy applies to 1e reductions. Figure 15.1.2 shows that E1/2 values for reductions of aromatic hydrocarbons are correlated with the gas-phase electron affinities (EAs) of the molecules (4, 5, 9, 10), where −EA is the energy change upon adding an electron to the molecule in the gas phase.4 Most molecules in the figure have positive EA, meaning that they acquire an electron spontaneously. As EA becomes less positive, less energy is released upon adding an electron. Negative values denote molecules that spontaneously lose an electron in the gas phase. Species with EA and E1/2 outside the shaded box in Figure 15.1.2 can be reduced in aprotic solvents by an outer-sphere 1e reaction before reaching the negative potential window of the solvent (∼−2.7 V vs. SCE), while those with EA and E1/2 inside the box cannot. Accordingly, the electrochemical reduction of N2 to NH3 is unlikely to begin by a 1e outer-sphere reaction yielding N2 ∙ dissolved in solution because this reaction is predicted from Figure 15.1.2 to occur 4 E1/2 values from reference (14), if available, otherwise from Table C.3. EA values were generally the average of figures listed by the NIST (3).

655

15 Inner-Sphere Electrode Reactions and Electrocatalysis

0.0 O2 (MeCN) –1.0

E1/2 /V vs. SCE

656

FA

CO2 (DMF)

–2.0

NP –3.0

NB AB BP

BQ

TCNQ

CP[cd]Py B[ghi]FA

An Py Ch

CO2 (from EA)

H2O CO

–4.0 N2 –5.0 –3.0

–2.0

–1.0

0.0 EA/eV

1.0

2.0

3.0

Figure 15.1.2 Filled circles: Experimental values of reversible E 1/2 in MeCN vs. electron affinity, EA, for 1e outer-sphere reductions of various solutes. The regression line was computed using only the filled circles (slope = 1.03). Open squares: Points placed on the line extension according to the EA values of small molecules of broad interest. Shaded area is not electrochemically accessible with any conventional electrolyte. Filled diamonds: Experimental results for CO2 and O2 , which are both reduced much more easily than would be predicted from the regression line. Their displacement from the line is attributed to strong solvation of their reduction products, which feature much greater charge density than the reduction products of the much larger species used to obtain the filled circles. AB = azobenzene, An = anthracene, B[ghi]FA = benzo[ghi] fluoranthene, BP = benzophenone; BQ = p-benzoquinone, Ch = chrysene; CP[cd]Py = cyclopenta[cd]pyrene, FA = fluoranthene, NB = nitrobenzene, NP = naphthalene, Py = pyrene; TCNQ = 7,7,8,8-tetracyanoquinodimethane. (Structures in Figure 1).

well beyond the negative background limit (∼−4.7 V). However, the 6e reduction of N2 to NH3 does occur on electrocatalytic surfaces (e.g., Pt) (15), N2 + 6H2 O + 6e → 2NH3 + 6OH−

E0 = −0.77 V vs. NHE5

(15.1.3)

As for the oxidation of methanol, the predicted potential for the outer-sphere 1e reaction is much more negative than the thermodynamic potential of the overall multi-electron reaction, suggesting that N2 reduction must also occur by an inner-sphere mechanism. While this discussion emphasizes the role of energetics in determining whether a reaction proceeds by an outer- or inner-sphere mechanism, the nature of the solvent and other species present in solution can also influence the pathway. For instance, depending on the solvent and proton availability, O2 can be reduced in a simple, outer-sphere reaction or by a complex inner-sphere pathway (16): • In an aprotic medium, the reversible 1e reduction of O2 to superoxide, O2 ∙, is observed (16). Because O2 ∙ is strongly solvated, the potential for this reduction (−0.82 V vs. SCE in acetonitrile) is considerably less negative than the approximate value of −2 V predicted from 5 Based on standard chemical potentials at pH 14 (15).

15.2 Electrocatalytic Reaction Mechanisms

the EA of O2 (0.45 eV) (Figure 15.1.2). O2 + e ⇌ O2 ∙

E0 = −0.82 Vvs. SCE (in MeCN)

(15.1.4)

• In aqueous media, the 4e reduction of oxygen to water O2 + 4H+ + 4e ⇌ 2H2 O

E0 = +1.229 V vs. NHE

(15.1.5)

occurs via a complex inner-sphere reaction that is shifted positive by ∼2 V relative to the formation of O2 ∙, largely due to the stability of the product, H2 O. However, in very alkaline aqueous solutions (>6 M NaOH), the 1e reduction of O2 is again observed, a result that stresses the role of H+ in forming H2 O (17). O2 + e ⇌ O2 ∙

E0 = −0.284 V vs. NHE

(15.1.6)

The 4e reduction of O2 is discussed in more detail in Sections 15.3.1 and 15.4.3. 15.1.3

Adsorption Energies

The adsorption energy of a chemical species involved in an inner-sphere reaction can significantly reduce a kinetic barrier, or even provide a large driving force, allowing the overall process to follow a surface-based pathway. For example, while the weak solvation of a hydro∙ , the large negative gen atom prohibits the reduction of H+ to a solvated hydrogen atom, Haq ∙ on a metal such as Pt allows the reductive free energy associated with chemisorption of Haq formation of the surface-bound Pt–H species, Hads .6 The free energy of adsorption of H ∙ at a metal electrode affects both the rate and the mechanism of proton reduction; thus, correlations exist between electrocatalytic rates and adsorption energies on different metals (Section 15.5). Adsorbed species are key intermediates in all heterogeneous inner-sphere reactions and are discussed throughout the remainder of the chapter.

15.2 Electrocatalytic Reaction Mechanisms In Section 15.1, we introduced the idea that many interesting reactions, e.g., N2 reduction and methanol oxidation, proceed through mechanisms involving adsorbed intermediates. Although one can propose reaction sequences for each of these electrocatalytic processes, the true mechanism and the identities of any adsorbed intermediates are frequently not well defined by experimental measurements. In this section, the reduction of H+ to H2 —a relatively simple, but still not completely understood, inner-sphere reaction—is used to illustrate the importance of adsorption in electrocatalysis and to provide a framework for how electrocatalytic reactions are treated and investigated. 15.2.1

Hydrogen Evolution Reaction

Electrocatalysis generally requires that a reactant or intermediate be adsorbed at the electrode. Once this step has occurred, the reaction mechanism proceeds through other steps involving the breaking and formation of chemical bonds, followed by desorption of the product. Here, we will use principles outlined in Chapter 14 to consider how adsorption processes can influence 6 The general practice in this book is to designate an odd-electron species in solution by including a dot in the symbol (e.g., H ∙ ). If there is a desire to emphasize the solvation, a subscript may be added (e.g., Haq ∙ ). If such a species is adsorbed, the subscript becomes “ads,” and the dot is omitted (e.g., Hads ) in recognition that the odd electron is likely to participate in bonding to the surface.

657

658

15 Inner-Sphere Electrode Reactions and Electrocatalysis

the rates of inner-sphere reactions. The energetics of adsorption turn out to be critical in determining the overall rate. One can readily perceive two limits: A reactant that is very weakly adsorbed may not be able to undergo even the initial step of a reaction, while a very strongly adsorbed intermediate may tie up an active site on the electrode by desorbing only very slowly. Thus, we can anticipate the existence of an ideal strength of adsorption, heavily influenced by the identity of the electrode material. One of the best-studied inner-sphere reactions is the reduction of protons to generate hydrogen, usually called the hydrogen evolution reaction (HER): 0 EH + ∕H = 0.0 V vs. NHE

2H+ + 2e ⇌ H2

(15.2.1)

2

0 + While EH + ∕H is determined solely by the free energies of formation of H and H2 [and is 2

independent of the choice of metal; Section 2.2.4(e)], the reaction rate varies by orders of magnitude, being very fast on metals like Pt and Pd and exceedingly slow on Hg. We will now explore mechanistic possibilities for these differences. (a) Reductive Adsorption of Hydrogen Atoms

The proposed mechanisms for generating H2 from H+ include at least two elementary steps, the first being the reductive adsorption of H+ to form an adsorbed H atom, Hads : H+ + L + e ⇌ Hads

(15.2.2)

where L is a vacant site on the electrode. This reaction is understood as a concerted process, involving simultaneous electron transfer and adsorption. To appreciate the role of adsorption energetics in determining the rate of H2 generation, one can think about (15.2.2) as occurring in two steps: an initial electron transfer to create a ∙ , followed by adsorption of Haq ∙ at the metal surface. hydrogen atom in solution, Haq ∙ H+ + e ⇌ Haq

(15.2.3)

∙ +L⇌H Haq ads

(15.2.4a)

aH

ads

∕aL = 𝛽H aH∙

aq

(15.2.4b)

∙ (Section 14.5.3). The standard free energy of where (15.2.4b) is the adsorption isotherm of Haq ∙ , ΔG0 , has been reported (13) as 223 ± 3 kJ/mol, based on the measured formation for Haq H∙ aq

standard free energy of formation of H ∙ in the gas phase (203.278 kJ/mol), plus a theoretically 0 estimated energy of solvation of about 20 kJ/mol. Using ΔG0 = − nFE0 , the reported ΔGH ∙ 0 leads to EH + ∕H

aq

∙ aq

= −2.31 V vs. NHE (13).

Figure 15.2.1 shows the free energy diagram that one might anticipate for this system. The ∙ (without adsorption) would follow the path from curve 1 to 3. The 1e reduction of H+ to Haq transition state for that step occurs at the intersection of these curves, and the corresponding activation free energy, ΔG‡ , is the energy change in going from H+ to that activated complex. ∙ is weakly solvated in most solvents, including H O. This value is large because Haq 2 ∙ is spontaneously adsorbed at a metal electrode, the free energy curve for H Because Haq ads ∙ . The effect is to open a path to reductive adsorp(curve 2) lies at lower values than that for Haq tion, (15.2.2), with a transition state at the intersection between curves 1 and 2, and with a ‡ smaller activation energy, ΔGads . The reaction then proceeds through the second transition state to curve 4, producing H2 . ∙ adsorption are expressed in the adsorption isotherm, (15.2.4b), where The energetics of Haq 0 ∕RT) and ΔG0 is the standard free energy of adsorption (Section 14.5.3). 𝛽H = exp(−ΔGads ads

15.2 Electrocatalytic Reaction Mechanisms

3

Standard free energy/kJ mol–1

H•aq 223

2 ΔG‡ ΔG0ads

4

1 ΔG‡ads 0

Hads

H+

H2 Reaction coordinate

Figure 15.2.1 Simplified drawing of the standard free energy, G0 , vs. reaction coordinate for 2H+ + 2e ⇌ H2 . Reductive adsorption of H+ to form the intermediate species Hads lowers the activation barrier for the electron transfer step from ΔG‡ to ΔG‡ads .

The adsorbed hydrogen acting as an intermediate in the HER (Hads ) is not necessarily equivalent to the adsorbed hydrogen found on electrocatalytic metals at potentials positive of 0 EH + ∕H . The latter is commonly known as underpotential-deposited (UPD) hydrogen and is 2

visible in acidic media through reversible CV peaks, indicative of relatively strong adsorption. Figures 14.4.6 and 14.6.1 provide examples at Pt. In contrast, the intermediate Hads involved in 0 HER (at E < EH + ∕H ) is assumed to correspond to a more weakly bound species. Experimental 2

studies and discussion of the relationship between UPD Hads and the HER intermediate Hads have been presented (18, 19). (b) Volmer–Tafel and Volmer–Heyrovský Mechanisms

Following the reductive adsorption of H+ , two neighboring Hads may react to form H2 , which rapidly desorbs from the surface. 2Hads ⇌ H2 + 2L

(15.2.5a)

Keq = aH a2L ∕a2H

(15.2.5b)

2

ads

The sequence of (15.2.2) plus (15.2.5a) is called the Volmer–Tafel mechanism. To analyze it, we employ the dissected equivalent, (15.2.3), (15.2.4a), and (15.2.5a). Starting with the Nernst equation for (15.2.3), 0 E = EH + ∕H + ∙ aq

a + RT ln H F aH ∙

(15.2.6)

aq

and then using the equilibrium expressions (15.2.4b) and (15.2.5b) to eliminate the activities of ∙ , H , and L, we obtain Haq ads 0 E = EH + ∕H + ∙ aq

2 RT RT aH+ 1∕2 ln 𝛽H Keq + ln F 2F aH 2

(15.2.7)

659

660

15 Inner-Sphere Electrode Reactions and Electrocatalysis

By comparing this result with the Nernst equation for (15.2.1), we find that 0 0 EH + ∕H = EH+ ∕H + 2

∙ aq

RT 1∕2 ln 𝛽H Keq F

(15.2.8)

0 0 which provides the connection between EH + ∕H and EH+ ∕H in terms of the individual thermo2

∙ aq

dynamic parameters 𝛽 H and K eq . The numeric difference between these two standard potentials 1∕2

defines the product 𝛽H Keq as 1.1 × 1039 , representing the overall driving force for recombina∙ to form H . Even though 𝛽 and K tion of two Haq 2 H eq are both individually dependent on the interfacial structure (i.e., electrode material, crystallographic orientation, solvent, electrolyte) 1∕2 and may be individually potential dependent, the product 𝛽H Keq is independent of both inter0 0 facial structure and electrode potential. This is evident by noting that EH + ∕H and EH+ ∕H 2

∙ aq

in (15.2.8) are thermodynamic constants independent of the electrode. Even so, 𝛽 H and K eq determine the surface coverage of Hads and influence the overall HER rate. We will see evidence in Section 15.5, which presents correlations of current density with the adsorption energy of H∙ on different metals (e.g., in “volcano plots”). For metal electrodes at which the HER is nearly reversible (e.g., Pt), the thermodynamic parameters 𝛽 H and K eq do not affect the position or shape of the i − E response, since overall reversibility requires that all intermediate steps are also be reversible. An alternative explanation for the HER is the Volmer–Heyrovský mechanism. As in the case of the Volmer–Tafel mechanism, the process begins with the reductive adsorption of H+ to form Hads , (15.2.2). However, in the Volmer–Heyrovský mechanism, a second H+ undergoes concerted reduction and reaction with Hads to form H2 , Hads + H+ + e ⇌ H2 + L

(15.2.9)

For either mechanism, the rate-determining step could be either the initial reductive adsorption, producing Hads , or the following formation of H2 . Inspection of reactions (15.2.2), (15.2.5a), and (15.2.9) suggests that the Volmer–Tafel and Volmer–Heyrovský mechanisms would have different dependencies on the surface coverage of Hads and on electrode potential. One must also be prepared to consider that the two reaction mechanisms operate in parallel, but in varying fractions depending on the electrode material, potential, and solution conditions. 15.2.2

Tafel Plot Analysis of HER Kinetics

Tafel analysis is one of the more common methods of using electrochemical data to identify the rate-determining step of an inner-sphere reaction (1, 20). Recall from Section 3.4.3(d) that a plot of overpotential, 𝜂 = E − Eeq , vs. log i, known as a Tafel plot, is a historic method for evaluating kinetic parameters. In fact, Tafel discovered his empirical relationship 𝜂 = a + b log i

(15.2.10)

while studying the reduction of H+ at Hg, Pt, and other metal electrodes (21). As detailed in Section 3.4.3(c), the Butler–Volmer equation can be approximated in the form of (15.2.10) when a sufficiently large overpotential, 𝜂, is applied, so that either the cathodic or anodic branch of the current can be ignored. Quantitatively, the Tafel expression is expected to be valid when the back reaction contributes EV + 59 log CH+ ∕C 0 + 59 log 9 mV (25 ∘ C). In this potential range, the second term in the denominator of (15.2.26) is always much larger than the first, so (15.2.26) reduces to 𝜃=

CH+ ∕C 0

(15.2.27)

′

e

0 ) f (E−EV ′

0 − 3 mV. For 0.1 M H+ , the range for 𝜃 ≤ 0.1 corresponds to E ≥ EV 8 The activity of H+ is always CH+ = 𝛾H+ CH+ ∕C 0 , where C 0 is the standard-state concentration. If CH+ is in molar units, then C 0 = 1 M. In this book, the concentration of species j is often in mol/cm3 . Then, C 0 = 10−3 mol/cm3 . Usually, C 0 is gathered into the related rate constant or equilibrium constant, giving that constant the appropriate scale and units. Here, it is left expressed, so that the concentration of H+ is expressed as the dimensionless ratio CH+ ∕C 0 .

663

664

15 Inner-Sphere Electrode Reactions and Electrocatalysis ′

0 )] ≤ (1∕9)C ∕C 0 , show2) High 𝜃: When 𝜃 ≥ 0.9, (15.2.26) can be rearranged to exp[f (E − EV H+ ′ 0 0 ing that the approximation applies when E ≤ E + 59 log C + ∕C + 59 log(1∕9) mV (25 ∘ C). V

′

H

0 < −116 mV. For 0.1 M H+ , the range for 𝜃 ≥ 0.9 corresponds to E − EV

The width of the transition zone between these two cases is the difference between the boundaries given above, which is (2 × 59) log 9 = 113 mV at 25 ∘ C, regardless of the concentration of H+ . When the Heyrovský step is rate limiting, one obtains the current for the HER at low or high 𝜃 from (15.2.24) as follows: • At low 𝜃, (15.2.27) describes 𝜃 and substitution into (15.2.24) provides i=

nFAk0H

0′

0′

2 −𝛼H f (E−EH )−f (E−EV ) CH +e

C0 Taking the logarithm of both sides yields, 𝜂HER = E − Eeq

) ( nFAk0H 2 2.303 = log CH+ (1 + 𝛼H )f C0 𝛼H ′ ′ 1 2.303 + E0 + E0 − Eeq − log i (1 + 𝛼H ) H (1 + 𝛼H ) V (1 + 𝛼H )f

(15.2.28)

(15.2.29)

For 𝛼 H = 0.5 and T = 25 ∘ C, the corresponding Tafel plot has a slope of 2.303/(1 + 𝛼 H )f = 0.039 V per decade change in i. • At high coverage by Hads , we simply apply the limit of 𝜃 → 1, so that 0′

i = nFAk0H CH+ e−𝛼H f (E−EH )

(15.2.30)

which rearranges to the corresponding Tafel equation 2.303 2.303 0′ log(nFAk0H CH+ ) + EH − Eeq − log i 𝛼H f 𝛼H f

𝜂HER = E − Eeq =

(15.2.31)

∘

For 𝛼 H = 0.5 and T = 25 C, the Tafel plot has a slope of 2.303/𝛼 H f = 0.118 V per decade change in i. (c) Rate-Limiting Tafel Reaction

If one assumes a slow Tafel dimerization of two adsorbed hydrogen atoms, followed by fast desorption of H2 , the analysis of 𝜂 HER is somewhat different than in Sections (a) and (b) above because this elementary reaction does not involve an electron transfer. Thus, the rate constant, k T , in (15.2.17) is independent of the electrode potential. However, the Tafel reaction rate remains potential-dependent because 𝜃 depends on E, as described by (15.2.26). Combination of (15.2.17), (15.2.18), and (15.2.26) gives the general expression for the current, 2

⎛ ⎞ CH+ ∕C 0 ⎟ i = nFAkT ⎜ 0′ ) ⎟ ⎜ f (E−EV 0 ⎝ CH+ ∕C + e ⎠

• At small 𝜃, the parenthesized factor is given by (15.2.27); hence, ( )2 CH+ ∕C 0 i = nFAkT 0′ ef (E−EV )

(15.2.32)

(15.2.33)

15.2 Electrocatalytic Reaction Mechanisms

The resulting Tafel expression is 𝜂HER = E − Eeq

( ) CH+ 2 2.303 2.303 0′ = log nFAkT + EV − Eeq − log i 0 2f 2f C

(15.2.34)

For a rate-limiting Tafel step at low 𝜃, the Tafel plot should have a slope of 2.303/2f = 0.0296 V per decade change in i. The predicted slope in this case is not a function of 𝛼, which is a consequence of the Tafel step not involving electron transfer. • At high coverage by Hads , i.e., 𝜃 → 1, the current is no longer a function of E and is given by (15.2.35)

i = nFAkT

In this limiting condition, the slope of a Tafel plot would approach infinity. The physical reason is that once there is full coverage by Hads , a further negative change in the electrode potential cannot increase the rate of the Tafel step. (d) Rate-Limiting Mass Transport of H+

If both steps of the overall HER are reversible, the current is limited by transport of H+ to the electrode surface. For this case, a nernstian i − E response is observed [e.g., (1.3.15)] and a Tafel plot of E or 𝜂 HER vs. log i would be nonlinear. The HER at Pt at low H+ concentrations closely approaches the nernstian response, implying that each intermediate step is microscopically reversible. (e) Kinetic Analysis Based on Tafel Slopes

In Sections (a)–(d), we have emphasized that the potential dependence of the HER rate is sensitive to the mechanistic path; consequently, one should be able to extract mechanistic and kinetic information from Tafel plots under varied experimental conditions. Table 15.2.1 summarizes the Tafel slopes derived in this section for the different HER mechanisms and rate-limiting steps. To extract a valid Tafel slope, the plot of 𝜂 HER vs. log i should correspond to currents that extend over at least 2 orders of magnitude. For metals at which the HER is very slow, e.g., Hg, this is readily accomplished because the currents can be sufficiently low that mass transfer does not impose any limitation. On electrocatalytic surfaces, e.g., Pt, where the HER is fast, the potential region for employing the Tafel equation [no mass-transfer effects, but a sufficient overpotential to prevent any reverse electron-transfer reaction (i.e., 𝜂 HER > 118 mV)] is limited (22). Often an overall reaction transfer coefficient, 𝛼 R , is reported in the literature and used as a characteristic kinetic parameter to describe the reaction kinetics. Unlike the parameters 𝛼 V and 𝛼 H , which correspond to the transfer coefficients for the Volmer and Heyrovský 1e steps and are measures of symmetry of the energy barriers associated with those reactions, 𝛼 R is simply Table 15.2.1 Expected Tafel Slopes for the HER with Different Rate-Limiting Steps(a) Expected Tafel Slope/mV(b) Surface Coverage

Volmer Step

Heyrovský Step

Tafel Step

Mass Transfer

Low, 𝜃 ≈ 0

118

40

30

Nonlinear

High, 𝜃 → 1

N/A

118

∞

Nonlinear

(a) At 25 ∘ C. (b) Where the named step for each column is the RDS.

665

15 Inner-Sphere Electrode Reactions and Electrocatalysis

Table 15.2.2 Expected Overall Reaction Transfer Coefficients, 𝛼 R Expected 𝜶 R (a) Surface Coverage

Volmer Step

Heyrovský Step

Tafel Step

Low, 𝜃 ≈ 0

𝛼V

1 + 𝛼H

2

𝛼H

0

High, 𝜃 → 1

𝛼V

(a) Where the named step for each column is the RDS.

0.3

0

j/mA cm–2

666

MoS2/RGO

η/V

Pt

b = 41

0.1

–20

–30

MoS2/RGO

0.2

–10

–0.1 0.0 E/V vs. RHE (a)

0.1

Pt

b = 30 m

0.0 –0.2

cade

mV/de

1

j/mA cm–2 (b)

e V/decad

10

Figure 15.2.2 (a) i − E curves obtained in 0.5 M H2 SO4 for HER at glassy carbon electrodes on which equal amounts of Pt and MoS2 /RGO nanoparticle catalysts were deposited. (b) Corresponding Tafel plots in terms of current density, j. Overpotential, 𝜂, was measured vs. E eq at Pt. The Pt nanoparticles correspond to a commercial 20 wt% Pt on Vulcan carbon black. [Adapted with permission from Li et al. (24). © 2011, American Chemical Society.]

an alternate expression of the Tafel slope, 𝛼 R = (2.303/f )(Tafel slope)−1 . Table 15.2.2 presents expected values of 𝛼 R for the HER in different mechanistic cases. A key limitation of Tafel analysis for the HER is the assumption of limiting surface coverages (𝜃 ≈ 0 or 𝜃 → 1) in obtaining the values listed in Tables 15.2.1 and 15.2.2. It is likely that 𝜃 not only has values between these limits, but that it is also dependent on the electrode potential. As discussed in Section 15.2.1(b), both the adsorption isotherm 𝛽 H for H∙ and the equilibrium constant K eq for H2 formation and desorption are likely potential-dependent quantities. Thus, the mix of HER mechanisms, and even the predominant mechanism, may vary with E, resulting in nonlinear Tafel plots (1, 20, 23). Figure 15.2.2 shows Tafel analyses for the HER at two electrocatalytic materials: Pt nanoparticles and hybrid MoS2 /RGO nanoparticles, where RGO corresponds to reduced graphene oxide. Molybdenum sulfide is a layered crystalline material, and the MoS2 particles synthesized on RGO have a flat structure with a thickness corresponding to a few layers, exposing electrocatalytic edges of the particles to the solution. The i − E curve in Figure 15.2.2a shows that the onset of HER at the Pt nanoparticles occurs slightly negative of 0.0 V vs. RHE, indicating a very small overpotential on Pt. The onset of HER for an equivalent mass loading of MoS2 /RGO nanoparticles is shifted about 0.1 V more negative, indicating slower kinetics on the non-noble-metal electrocatalyst (corresponding to an i0 for HER that is ∼1.5 orders of magnitude smaller).

15.3 Additional Examples of Inner-Sphere Reactions

The Tafel plots of 𝜂 HER vs. log j (i.e., in terms of current density) for the two electrocatalysts are shown in Figure 15.2.2b. For Pt, the Tafel slope at low 𝜂 HER has a value of 30 mV per decade change in current, consistent with a Volmer–Tafel mechanism having the recombination of Hads to H2 , (15.2.14), as the RDS. This finding is in agreement with prior studies at Pt electrodes, where it is reported that the Volmer–Tafel mechanism is dominant at low Hads coverages. At more negative potentials, 𝜃 increases and the Volmer–Heyrovský mechanism becomes dominant, as indicated by a change in the Tafel slope to ∼118 mV/decade (not shown, see Refs. (1) and (18)). In contrast, the Tafel slope for HER on MoS2 /RGO at low overpotentials and, thus, small 𝜃, is 41 mV/decade, suggesting that the Volmer–Heyrovský mechanism is operative and that H2 desorption, (15.2.13), is rate determining for this electrocatalysis. Hydrogen is very weakly adsorbed on Hg and, thus, 𝜃 ≈ 0 under all conditions. The experimental Tafel plot at Hg has a slope of ∼118 mV/decade, indicating that the Volmer–Heyrovský mechanism is operative with hydrogen adsorption, (15.2.12), being rate determining. Tafel analyses for other inner-sphere reactions, e.g., O2 reduction, can found in the literature or can be readily developed by methods like those used above for the HER. Insights into the RDS of an electrochemical reaction can be obtained from Tafel plots by comparing predicted slopes with experimental values. A detailed overview of this approach, as applied to electrocatalytic reactions of interest in energy conversion, is available (23). In any application of Tafel analysis for investigation of complex electrode reaction mechanisms, it is important to recall the underlying assumptions: • • • • • •

The overall rate is limited by a single RDS. Butler–Volmer kinetics apply to intermediate 1e redox steps. Any rate-determining electron-transfer reaction is treated as totally irreversible. 𝛼 is assumed to be 0.5 for any step for which this parameter is unknown. Limiting surface coverages are generally assumed, i.e., 𝜃 ≈ 0 or 𝜃 → 1. Mass-transfer limitations, including effects of iR drop in solution, are negligible (or the data have been corrected for such effects).

If the formal potential for the RDS is unknown (a common situation), one cannot extract an i0 for the microscopic rate-determining step from a Tafel plot. For example, in the case of HER, if 0′ is required to determine the Volmer step is assumed to be rate determining, then a value of EV i0 from a Tafel plot analysis based on (15.2.22). In the absence of H2 , the value of Eeq in 15.2.22 is also undefined; thus, i0 is often reported arbitrarily as the current density measured at the RHE potential (25).

15.3 Additional Examples of Inner-Sphere Reactions Inner-sphere reaction mechanisms are important in a wide spectrum of inorganic and organic redox reactions underlying energy technologies [e.g., fuel cells (26)] or electrosynthesis of new compounds. Much research is focused on the molecular-level understanding of mechanisms and the factors that control reaction rates, with the goal of developing of new electrocatalysts for carrying out these reactions in energy-efficient processes. This section provides a brief overview of several important inner-sphere reactions. 15.3.1

Oxygen Reduction Reaction

The electroreduction of O2 , i.e., the oxygen reduction reaction (ORR), is of special interest as the cathodic process in fuel cells and for its broad role in biological systems. In electrochemical

667

668

15 Inner-Sphere Electrode Reactions and Electrocatalysis

cells, the ORR mechanism depends strongly on the electrolyte composition and the electrode material. For example, the quasireversible 1e reduction of O2 to superoxide, O2 ∙, is observed in aprotic solvents and alkaline aqueous solutions, while the 2e electroreduction to hydrogen peroxide and the 4e electroreduction to H2 O occur in neutral and acidic aqueous electrolytes. Fundamental understanding of the ORR is important to the discovery of new, inexpensive, durable, and efficient electrocatalysts. Advances relevant to fuel cell applications have been reviewed (27–34). This section is a brief overview of mechanistic hypotheses for important ORR pathways, with a focus on reactions in aqueous environments. The 2e and 4e reductions of O2 to H2 O2 and H2 O are very complex processes, involving both electron and proton transfers, as well as the breaking and forming of chemical bonds. As discussed in Section 15.1, these reactions can occur only by inner-sphere mechanisms involving adsorbed intermediates. The overall 4e reduction of O2 in acidic solutions is O2 + 4H+ + 4e ⇌ 2H2 O

E0 = 1.229 Vvs. NHE

(15.3.1)

Thus, O2 is ideally reducible in aqueous 1 M acid at ∼1.23 V vs. NHE; however, a large overpotential is needed to drive O2 reduction, even on the best catalytic electrodes.9 At Pt, significant reduction does not occur unless the electrode is more negative than about 0.88 V vs. NHE, representing an overpotential of ∼0.35 V. In contrast to the HER, which is nearly thermodynamically reversible at Pt, the ORR is always regarded as a thermodynamically irreversible reaction. The slow electron-transfer kinetics associated with the ORR represent a key limitation in current fuel cell technologies because the overpotential translates to a sizable loss of electrical energy deliverable by the cell. Several different reactions have been proposed as the initial elementary step in the ORR. The simplest is the 1e reduction of O2 to superoxide, O2 + e ⇌ O2 ∙

E0 = 0.284 Vvs. NHE

(15.3.2)

which is observed in aprotic solvents, as well as in strongly basic aqueous solutions (>6 M NaOH), where the absence of protons and the low activity of H2 O increase the stability of O2 ∙. Under these conditions, O2 ∙ does not undergo further chemical reactions. In acidic aqueous solutions, electrogenerated O2 ∙ reacts rapidly with H+ (EC mechanism) and is not observed electrochemically. At an active site, L, on a metal electrode, O2 can be reduced to adsorbed superoxide, O− , which would be rapidly protonated to form the 2,ads adsorbed peroxide radical, HO2, ads : O2 + e + L ⇌ O− 2,ads

(15.3.3)

O− + H+ → HO2,ads 2,ads

(15.3.4)

The HO2, ads then can be expected to undergo three sequential proton and electron transfers to yield H2 O: HO2,ads + H+ + e → H2 O + Oads

(15.3.5)

Oads + H+ + e → HOads

(15.3.6)

HOads + H+ + e → H2 O + L

(15.3.7)

9 The 4e ORR can occur with minimal overpotential at enzyme-modified electrodes (Section 17.8.2).

15.3 Additional Examples of Inner-Sphere Reactions

Summation of (15.3.3)–(15.3.7) yields the overall 4e ORR, (15.3.1). Because the O—O bond is not broken during the initial adsorption step, this reaction sequence is called the associative mechanism for ORR. A characteristic intermediate of the associative mechanism is the peroxide radical HO 2, ads formed in (15.3.4), which might be able to accept an electron to form H2 O2 , which would rapidly desorb: HO2,ads + H+ + e → H2 O2 + L

(15.3.8)

If significant, this reaction would be competitive with (15.3.5) and would lead, instead, to the overall 2e ORR: O2 + 2H+ + 2e ⇌ H2 O2

E0 = 0.695 Vvs. NHE

(15.3.9)

In fuel cells, the production of H2 O2 by the 2e pathway is generally problematic because it reduces the energy efficiency and creates reactive H2 O2 , which can oxidize the carbon support for the electrocatalyst. A different way of writing the initial steps, (15.3.3) and (15.3.4), is to assume that molecular oxygen, rather than superoxide, is initially adsorbed at the electrode, followed by a concerted 1e/1H+ transfer10 to form adsorbed peroxide radical, O2 + L ⇌ O2,ads

(15.3.10)

O2,ads + H+ + e → HO2,ads

(15.3.11)

The HO2, ads can then undergo (15.3.5)–(15.5.7), as before, to yield H2 O. Together, (15.3.10) and (15.3.11) are energetically equivalent to (15.3.3) and (15.3.4) but represent different mechanisms, perhaps leading to different kinetic limitations. If O2 dissociates upon adsorption at the electrode surface, the ORR mechanism follows an entirely different pathway, called the dissociative mechanism. In acidic media, this reaction sequence is usually written as: O2 + 2L ⇌ 2Oads

(15.3.12)

2 × (Oads + H+ + e → HOads )

(15.3.13)

2 × (HOads + H+ + e → H2 O + L)

(15.3.14)

Combining (15.3.12)–(15.3.14) yields the overall 4e electroreduction of O2 , (15.3.1). Whether ORR occurs by a dissociative or associative mechanism at a specific metal electrode likely depends on the adsorption energetics of the intermediates. For instance, O2 is known to dissociate on metals such as Ni and Co, but not on Au; thus, the dissociative mechanism may be favored on the former. While reaction products from the ORR can be experimentally detected (e.g., H2 O2 ), there are few direct experimental verifications of the adsorbed intermediates and it is difficult to experimentally identify which of the elementary steps is rate-determining. Much progress has been made in theoretically computing the energies of adsorbed intermediates as a function of the applied potential, which allows predictions about energetic bottlenecks within a proposed mechanism (Section 15.4). 10 The concerted addition of 1e and 1H+ to a surface-absorbed species occurs frequently in inner-sphere mechanisms and is often referred to as hydrogenation. Conversely, dehydrogenation refers to the loss of 1e and 1H+ . This meaning is different from the more common usage of this term in chemistry to describe addition/loss of H 2 to/from a molecule.

669

670

15 Inner-Sphere Electrode Reactions and Electrocatalysis

15.3.2

Chlorine Evolution

The oxidation of Cl− to produce Cl2 at a metal electrode, e.g., Pt, can be written as Cl− + L → Clads + e

(15.3.15)

Clads + Cl− → Cl2 + L + e

(15.3.16)

where Clads represents a chlorine atom adsorbed on the metal surface. Mechanistically, (15.3.15) and (15.3.16) are completely analogous to the Volmer–Heyrovský mechanism for the HER [(15.2.12) and (15.2.13)]; thus, the kinetic analysis of chlorine evolution closely follows the presentation of Section 15.2.2. Tafel expressions for limiting cases in which either (15.3.15) or (15.3.16) is rate determining are readily derived (Problem 15.3) or can be found in the literature (1). Because of the highly corrosive nature of Cl2 , dimensionally stable anodes [DSA, Section 20.1.5(a)] are employed in the industrial production of Cl2 . 15.3.3

Methanol Oxidation

The oxidations of methanol and formic acid are of interest due to their application at the anode in the direct methanol fuel cell (DMFC) or the direct formic acid fuel cell (11, 35–37). The oxidation of methanol, CH3 OH + H2 O ⇌ CO2 + 6H+ + 6e

E0 = 0.030 V vs. NHE

(15.3.17)

can be schematically represented by a series of steps (Figure 15.3.1) (38), in which each step represents the loss of 1e accompanied by the loss of an H+ or uptake of an OH− . For instance, the reaction CH2 O → CHO represents a 1e/1H+ dehydrogenation. Regardless of the path taken through the different possible intermediates, at least six steps are required to convert the initial reactant, CH3 OH, to the desired final product, CO2 , corresponding to the 6e indicated in (15.3.17). Species CH2 O (formaldehyde) and HCOOH (formic acid) are thermodynamically stable in solution; therefore, the effectiveness of converting CH3 OH all the way to the 6e product, CO2 , depends on the rates at which these intermediate species are formed and then further oxidized, relative to their rates of desorption into the solution (36, 39). Proposed mechanisms for the electrocatalytic oxidation of methanol are complex and generally feature dual pathways involving (a) strongly adsorbed carbon monoxide, COads , and (b) weakly adsorbed or dissolved intermediates, such as formaldehyde (H2 CO), formic acid (HCOOH), or formate (HCOO− ). The CO resulting from partial oxidation of CH3 OH (Figure 15.3.1) tenaciously adsorbs at catalytic electrodes such as Pt and acts as a “poison” by blocking surface sites necessary for adsorption of other intermediates on the path to CO2 . Adsorbed CO can be oxidized to CO2 , but this reaction requires an oxygen donor, as CH3OH

CH2OH

CHOH

COH

CH2O

CHO

CO

HCOOH

COOH

CO2

Figure 15.3.1 Schematic depicting adsorbed intermediates and mechanistic pathways for CH3 OH oxidation. The bold species are thermodynamically stable in the solution. [Adapted from Bagotzky, Vassiliev, and Khazova (38).]

15.3 Additional Examples of Inner-Sphere Reactions

indicated in Figure 15.3.1. The latter is provided through adsorption of OH ∙ on the Pt surface by activation of H2 O, H2 O + L ⇌ OHads + H+ + e

(15.3.18)

In a following reaction, OHads then reacts with adsorbed CO to generate CO2 , which rapidly desorbs to produce a clean Pt surface. COads + OHads → CO2 + H+ + 2L + e

(15.3.19)

This sequence, called the Langmuir–Hinshelwood mechanism for CO electrooxidation (40), notably includes a bimolecular reaction between two molecules adsorbed on neighboring surface sites, (15.3.19). Once the strongly adsorbed CO is removed from the surface, other intermediates may be adsorbed on the Pt surface, opening additional sites for CO2 generation. Figure 15.3.2a shows the voltammetric behavior of methanol oxidation at Pt in a solution containing 0.5 M CH3 OH and 0.1 M HClO4 . On the positive-going scan, an anodic current corresponding to CH3 OH oxidation is observed beginning near 0.5 V vs. RHE. This current increases as the positive scan proceeds, reaches a maximum near 1.0 V, and then decreases essentially to background levels. A similar anodic peak is observed on the reverse scan; the current rises sharply at 0.95 V, reaches a maximum, and then drops to nearly zero near 0.5 V. Comparison of the onset potential for CH3 OH oxidation with E0 = 0.03 V for the reaction makes it immediately clear that a significant overpotential is required to oxidize CH3 OH on Pt. This overpotential is largely associated with the adsorption of CO, which can be directly observed using vibrational spectroscopy (41, 42). Figure 15.3.2b shows in situ surface-enhanced IR absorption spectra (SEIRAS; Section 21.4.1) recorded simultaneously with the voltammetry (43). SEIRAS is highly surface-selective and allows identification of adsorbed species during the voltammetric scan. At the beginning of the scan, and well before the onset of CH3 OH oxidation, bands are observed at 2060 and 1860 cm−1 , corresponding to CO molecules linearly and bridge-bonded to the Pt surface, respectively (Figure 15.3.3). The intensities of these bands remain constant up to ∼0.5 V (Figure 15.3.2c), corresponding to the onset of anodic current from CH3 OH oxidation. Concurrent with the decrease in the CO bands, a new band appears at 1230 cm−1 , resulting from adsorbed formate. The increase in the intensity of this band parallels the increase in voltammetric current, reaching a maximum at ∼1.0 before falling to background levels due to the formation of Pt oxide. On the reverse scan, a similar parallel dependence is observed for the formate IR band at 1230 cm−1 and the voltammetric current. As summarized structurally in Figure 15.3.3, the combined in-situ IR and voltammetric measurements demonstrate that CH3 OH oxidation proceeds by a dual pathway, involving the oxidation of COads and HCOO− to CO2 . Alternative pathways involving additional adsorbed ads (e.g., formaldehyde) and dissolved species have been proposed for CH3 OH (35) and formic acid oxidation (44). It is apparent from Figure 15.3.2 why adsorbed CO is considered a “poison” in methanol oxidation. The IR data clearly demonstrate that this species blocks surface sites for adsorption and other potential intermediates and that it is not oxidized and removed until of HCOO− ads large overpotentials are reached. The adsorption of CO is an issue in the oxidation of many small organic molecules (45). A review of CO adsorption and oxidation on different metal surfaces is available (40). Bimetallic Pt/Ru electrocatalysts display a higher tolerance than pure Pt for CO adsorption at lower overpotentials, an effect that results from the bifunctional nature of the mixed catalyst (35, 46). Formation of OHads on exposed Ru surface sites occurs in parallel with CO adsorption on Pt sites, and these two adsorbates react according to (15.3.19), producing a decrease

671

15 Inner-Sphere Electrode Reactions and Electrocatalysis

1.2

j/mA cm–2

0.9 0.6 ×50

0.3 0.0

–0.3 –0.6 0.2

0.4

0.100

0.6 0.8 1.0 E/V vs. RHE (a)

1.2

COB

COL

Formate

0.075

0.05

1.3

E/V

0.025

.R

HE

0.050

vs

Absorbance

0.000

0.05

2200

2000

1800

1600

1400

1200

Wavenumber/cm–1 (b) 1.0 0.8 Intensity/a.u.

672

COL Formate

0.6

×6

0.4 0.2 0.0

0.2

0.4

0.6 0.8 E/V vs. RHE (c)

1.0

1.2

Figure 15.3.2 (a) Cyclic voltammograms of Pt in 0.1 M HClO4 (dashed line) and 0.1 M HClO4 + 0.5 M CH3 OH (solid line). Scan rate: 5 mV/s. (b) SEIRAS during CH3 OH oxidation at Pt in 0.1 M HClO4 + 0.5 M CH3 OH. The absorbances labeled COL and COB correspond to linear and bridging geometries for adsorbed CO, respectively. (c) SEIRAS absorption intensities as a function of the electrode potential. [Adapted with permission from Chen et al. (43). © 2003, American Chemical Society.]

15.3 Additional Examples of Inner-Sphere Reactions

Figure 15.3.3 Reaction paths during the oxidation of CH3 OH at Pt. [Chen et al. (43).]

H O

C–

O

Pt

CH3OH

CO2

O

O

C

C Pt

in COads at less positive potentials than on a pure Pt surface, thus, improving the catalytic activity. Extensive experimental and theoretical studies exploring the mechanisms of methanol and formic acid oxidation on Pt/Ru (46) and other bimetallic catalysts, e.g., Pt/Au (47), have been reported (48). 15.3.4

CO2 Reduction

In Section 15.3.3, the 6e oxidation of CH3 OH to CO2 was discussed in context of DMFCs for the generation of electricity. The reverse reaction—the electroreduction of CO2 —is a potential route for using electricity to convert CO2 to high-energy chemicals, e.g., hydrocarbons and alcohols (49, 50); however, the generation of even a small organic molecule by this process involves a complex series of reaction steps and absorbed intermediates. Negative potentials are always required, and a general obstacle is the competing reduction of H+ , which lowers the faradaic efficiency for producing the desired organic compound. Consider, for instance, the 8e reduction of CO2 to methane: CO2 + 8H+ + 8e ⇌ CH4 + 2H2 O

E0 = 0.169 Vvs. NHE

(15.3.20)

While this reaction has a thermodynamic potential positive of E0 (H+ /H2 ), the generation of CH4 requires significant overpotentials, and the HER always competes. Copper is the only metal that yields significant amounts of CH4 or other hydrocarbons during the electroreduction of CO2 , but at the cost of nearly a 1-V overpotential. Figure 15.3.4 shows a product distribution for CO2 electrolysis at Cu, demonstrating the general ineffectiveness of the reaction (51). The dominant product is H2 until the potential reaches about −0.8 V, where modest amounts of CH4 and C2 H4 are evolved. Theoretical investigations based on density functional theory (DFT; Section 15.4.1) suggest that the critical steps in the formation of CH4 are the formation of COads , followed by its hydrogenation to produce HCOads (52). The reaction then proceeds through a series of electron-transfer and protonation steps involving adsorbed formaldehyde (H2 COads ) and methoxy (CH3 Oads ), eventually resulting in methane. The hydrogenation of COads to HCOads is proposed to be strongly dependent upon the binding strength of CO to the metal; therefore, CH4 production is expected to be dependent on the choice of metal (52). For metals that bind CO very weakly (e.g., Au), CO desorbs from the surface as the dominant product before the formation of the necessary intermediate HCOads can occur. Conversely, for metals that bind CO very strongly (e.g., Pt), the hydrogenation of COads to HCOads is thermodynamically prohibited, preventing the formation of CH4 . Theoretical calculations suggest that the binding of CO to Cu is of intermediate strength, allowing the formation of the intermediate HCOads (52). The reduction of CO2 using molecular redox catalysts is also extensively studied (53). An example is the use of cobalt phthalocyanine (Figure 1, MPc, M = Co) immobilized on carbon

673

j/mA cm–2

15 Inner-Sphere Electrode Reactions and Electrocatalysis

5.0

(a)

1.0 0.5 0.1 H2

80

(b)

60 Faradaic yield/%

674

Figure 15.3.4 (a) Current density and (b, c) faradaic yields of products in the controlled potential reduction of CO2 at a Cu electrode in 0.1 M KHCO3 (pH 6.8) saturated with CO2 . T = 19 ∘ C. The faradaic yields were calculated on the basis of the number of electrons required for the formation of products from the reduction of CO2 in H2 O: 8 for CH4 , 12 for C2 H4 , 2 for CO, 2 for HCOO− , and 2 for H2 . The amounts of electrogenerated products were determined using gas and ion chromatography. [Adapted from Hori, Murata, and Takahashi (51), with permission.]

40 20

HCOO–

0 40

(c)

CH4

30

C2H4

20 10 0

CO –0.8

–1.0 –1.2 E/V vs. NHE

–1.4

nanotubes, which has been shown to catalyze the 6e reduction of CO2 to methanol with a faradaic efficiency of 40% (54). This is an example of an electrode that has been chemically modified with a thin layer of catalyst to perform a reaction that would be slow on the bare electrode. Reaction dynamics at modified electrodes are discussed in Section 17.5. A key issue for molecular redox catalysts in applications is their long-term stability. 15.3.5

Oxidation of NH3 to N2

A proposed renewable pathway to the production of H2 involves the electrooxidation of NH3 to N2 in alkaline solutions (55): 2NH3 + 6OH− ⇌ N2 + 6H2 O + 6e

E0 = −0.77 Vvs. NHE

(15.3.21)

When (15.3.21) is coupled with the HER in alkaline solutions, 6H2 O + 6e ⇌ 3H2 + 6OH−

E0 = −0.83 Vvs. NHE

(15.3.22)

0 Ecell = 0.06 V

(15.3.23)

the net reaction is 2NH3 ⇌ N2 + 3H2

which is the cell reaction for electrolysis of ammonia. This process is attractive for energy storage and conversion because it requires a low voltage [∼5% of the input energy required to drive 0 = 1.23 V)], and the only products are N and H . H2 production via H2 O electrolysis (Ecell 2 2 Although ammonia electrolysis presents a significant technological opportunity, the oxidation of NH3 to N2 is kinetically slow, and there are nitrogen-containing side products. The most active metal surfaces for (15.3.21) are Pt and Ir, but, even on these metals, high overpotentials are required to drive (15.3.21) at significant rates. Figure 15.3.5 shows a proposed mechanism for NH3 oxidation at Pt(100), including the various adsorbed N species proposed as reaction intermediates (15). Although the thermodynamically stable product, N2 requires a 6e/6H+ transformation, the intermediate steps, all involving the loss of 1e and 1H+ , allow the reaction to proceed stepwise. Experimental and theoretical evidence suggests that the dimerization

15.3 Additional Examples of Inner-Sphere Reactions + H2O

+ OHads H+ + e

NH3, ads

H+ + e

NH2, ads

NHads + NHads

H+ + e

+ OHads

Slow

NOHads

Nads

NOads

Fast H+ + e

N2H2, ads

H+ + e

N2Hads

N2

Figure 15.3.5 Proposed scheme for the most feasible steps during the electrochemical oxidation of NH3 on Pt(100). [Adapted from Katsounaros et al. (15), with permission.] 1.2 1.0 Ox2

j/mA cm–2

0.8 0.6

Ox3

0.4 Ox1 0.2 0.0

–0.2 –0.2

0.0

0.2

0.4

0.6 0.8 1.0 E/V vs. RHE

1.2

1.4

1.6

Figure 15.3.6 CV in a stagnant solution on Pt(100) in 0.1 M KOH and 1 mM NH4 ClO4 . The solid and dashed curves represent voltammograms with different upper potential limits. [Adapted from Katsounaros et al. (15), with permission.]

of NHads to N2 H2, ads is the key step toward forming an N2 -containing species. As indicated in Figure 15.3.5, N2 H2, ads is then converted to N2 by two sequential steps, each involving 1e-oxidation and loss of a proton. Figure 15.3.6 shows the voltammetric response of a Pt(100) electrode in aqueous solution containing 0.1 M KOH and 1 mM NH4 ClO4 . On the forward (positive-going) scan, a reversible wave (Ox1) is observed, corresponding to adsorption of NH3 , 2Hads + NH3 ⇌ NH3,ads + 2H+ + 2e

(15.3.24)

The adsorption of one NH3 molecule requires the oxidative desorption of two H atoms, giving rise to a voltammetric current. As the potential is scanned further positive, a large irreversible peak is observed beginning at c. 0.5 V vs. RHE (Ox2). This wave is assigned to a combination of processes, including the dehydrogenation of NH3, ads , the recombination of NHads intermediates, and the continued adsorption of NH3 . The association of peak Ox2 with the generation of N2 is confirmed by online electrochemical mass spectrometry of the dissolved products during the voltammetric experiment (Figure 15.3.7), which indicates that N2 generation commences

675

15 Inner-Sphere Electrode Reactions and Electrocatalysis

6 N2 (m/z: 28) 5 Ion current/nA

676

4

3

2 N2O (m/z: 44)

1

NO (m/z: 30) 0 0.0

0.2

0.4

0.6 0.8 E/V vs. RHE

1.0

1.2

1.4

Figure 15.3.7 Mass spectrometry of volatile products during a positive-going scan (1 mV s−1 ) in 0.1 M KOH + 1 mM NH3 . [Adapted from Katsounaros et al. (15), with permission.]

at the foot of wave Ox2. Other experimental evidence shows that the production of N2 at potentials positive of ∼0.7 V decreases due to the formation of spectator species that block sites for NH3 adsorption (15). These species may include OHads or Oads , as well as NOads produced by the oxidation of NHads by OH− . Mass spectrometry (Figure 15.3.7) confirms the production of both NO and N2 O when the electrode potential is scanned above 0.8 V. The oxidation of NO to N2 O corresponds to the peak labeled Ox3 in Figure 15.3.6.

15.3.6

Organic Halide Reduction

Inner-sphere reactions are not limited to oxidations and reductions of small molecules. For instance, the reduction of organic halides, RX (where X represents a halogen atom and R is the remainder of an organic molecule), has applications in electrochemical synthesis. Halide ions are good leaving groups, making the carbon–halogen bond a good target for substitution. Although multiple mechanisms have been proposed for organic halide reductions (Section 13.3.8), they generally begin with the activation of the carbon–halogen bond by 1e reduction of the parent, RX (56, 57). One proposed mechanism follows a route in which RX is reduced to the radical ion, RX−∙, which rapidly undergoes C—X bond cleavage to yield R∙ and X− : RX + e → RX−∙

(15.3.25)

RX−∙ → R∙ +X−

(15.3.26)

These steps are followed by either chemical reaction of R∙ (e.g., dimerization) or acceptance of a second electron to form the carbanion, R− , R∙ +e → R−

(15.3.27)

15.3 Additional Examples of Inner-Sphere Reactions

This sequence constitutes a non-catalytic mechanism, in that the electron-transfer steps, (15.3.25) and (15.3.27), are both outer-sphere reactions. Experimental evidence shows that the rates of electroreduction of many organic halides are often highly dependent on the choice of the metal electrode (58). For instance, the reductions of benzyl bromide and halothane (CF3 − CHBrCl) occur on Ag at potentials that are hundreds of millivolts less negative than on Pt. On carbon-based electrodes (e.g., GC or BDD), the overpotentials are even larger than on Pt. This dependence of overpotential on the choice of electrode material suggests that the overall reactions involve an inner-sphere mechanism that includes adsorbed organic halide. It has been proposed that this reaction may proceed through a concerted pathway in which the electron transfer step occurs simultaneously with C—X bond cleavage (57) RX + L → L − (RX) + e → R∙ +X− + L

(15.3.28)

where L represents an adsorption site on the Ag electrode. The overpotential for halothane reduction is correlated with the bromine–metal bond strength, suggesting that RX is adsorbed through the bromine atom (59).

15.3.7

Hydrogen Peroxide Oxidation and Reduction

Hydrogen peroxide, H2 O2 , displays unusual redox behavior in that it can be both oxidized and reduced within the same potential range. From the following reactions and standard potentials, H2 O2 + 2H+ + 2e ⇌ 2H2 O

E0 = 1.763 V vs. NHE

(15.3.29)

O2 + 2H+ + 2e ⇌ H2 O2

E0 = 0.695 V vs. NHE

(15.3.30)

one can see that both the reduction and oxidation of H2 O2 are spontaneous when 0.695 V < E < 1.763 V. Addition of (15.3.29) and the reverse of (15.3.30) yields the over0 = 1.068 V). Thus, hydrogen peroxide all spontaneous reaction, 2H2 O2 ⇌ O2 + 2H2 O (Erxn is thermodynamically unstable at all potentials, and the fact that dilute H2 O2 solutions can be stored without decomposition is related to its kinetic stability toward oxidation or reduction. A catalyst, such as Pt powder or hemoglobin, greatly lowers the activation barrier to decomposition, leading to O2 evolution when it is added to an H2 O2 solution. The electrochemical reduction and oxidation of H2 O2 have been studied in detail at Pt and other metals, significantly because the parasitic 2e reduction of O2 to H2 O2 is detrimental to the overall efficiency of the ORR in fuel-cell applications (60). Voltammetry at a Pt RDE demonstrates that H2 O2 is oxidized at potentials corresponding to the formation of Pt oxide (∼0.95 V vs. RHE), while H2 O2 is reduced at potentials corresponding to the reduction of the oxide (∼0.8 V) (Figure 15.3.8). These correspondences are attributed to distinct chemical reactions of H2 O2 with bare Pt and Pt oxide. The oxidation is proposed to follow a two-step mechanism, 2Pt(OH) + H2 O2 → 2Pt + O2 + 2H2 O

(15.3.31)

2Pt + 2H2 O → 2Pt(OH) + 2H+ + 2e

(15.3.32)

where (15.3.31) represents the chemical reduction of Pt(OH) by H2 O2 , which is followed in (15.3.32) by regeneration of the Pt(OH) surface in an electron-transfer step. Addition of these two reactions yields the overall H2 O2 oxidation, (15.3.30).

677

15 Inner-Sphere Electrode Reactions and Electrocatalysis

0.4 Oxide formation

0.0 Oxide stripping

–0.4

j/mA cm–2

678

–0.8

(a) 4 3 2 1

2.0

0.0

–2.0

0.0

1 2 3 4

(b) 0.2

0.4

0.6

0.8 1.0 E/V vs. RHE

1.2

1.4

1.6

Figure 15.3.8 (a) CV of polycrystalline Pt in Ar-saturated 0.1 M HClO4 . (b) RDE voltammetry of the same Pt electrode in Ar-saturated 0.1 M HClO4 and 1 mM H2 O2 . (1) 400 rpm, (2) 900 rpm, (3) 1600 rpm, (4) 2500 rpm. Scan rate = 0.1 V/s. [Adapted from Katsounaros et al. (60), with permission.]

The reduction of H2 O2 is proposed to follow an analogous two-step process, but the active surface is bare Pt: 2Pt + H2 O2 → 2Pt(OH)

(15.3.33)

2Pt(OH) + 2H+ + 2e → 2Pt + 2H2 O

(15.3.34)

where (15.3.33) represents the chemical oxidation of Pt by H2 O2 and (15.3.34) is the regeneration of Pt in the electron-transfer step. Addition of these two equations yields the overall H2 O2 reduction, 15.3.29. In general, oxide layers on metals play an important role in electrocatalysis, as they can block the adsorption of reactants or displace intermediate species. An electrochemical method based on SECM that allows measurement of the rates of chemical reactions of dissolved molecular species with oxide layers is presented in Section 18.5.

15.4 Computational Analyses of Inner-Sphere Electron-Transfer Reactions In Chapter 3, the Marcus microscopic model was shown to describe the activation barriers, ΔG‡ , and rate constants, k f , of simple one-electron outer-sphere reactions (e.g., O + e ⇌ R), in which neither the reactant, O, nor the product, R, interacts with the electrode, other than being near enough to the surface to allow electron tunneling. In the Marcus model, ΔG‡ is expressed in terms of a reorganization energy, 𝜆, describing the transformation of the nuclear configuration of the reactant (including solvation sphere) into that of the product

15.4 Computational Analyses of Inner-Sphere Electron-Transfer Reactions

[(Section 3.5.3(c)]. We saw that 𝜆 is independent of the electrode properties and potential, but does relate to the structures of O and R and the physical properties of the solvent. The Marcus model and related methods allow ΔG‡ to be estimated from the reorganization energies and the properties of the molecules and solvents, allowing predictions of trends in outer-sphere heterogeneous electron-transfer kinetics that have been widely discussed and experimentally examined (61–65). Electrocatalytic reactions are not so simple, as they generally involve multistep mechanisms featuring adsorption, bond breaking, bond formation, and charge transfer between reactant molecules and the electrode. A different approach is necessary to describe such complex processes. Any microscopic kinetic model for electrocatalytic reactions necessarily requires consideration of the roles that the atomic structure and electronic properties of the electrode play in determining the energetics of adsorbates, as well as inclusion of the interactions of absorbates with the solvent and ions. To address this new complexity, we now consider alternatives using ab initio quantum chemical methods. 15.4.1

Density Functional Theory Analysis of Electrocatalytic Reactions

Our general goal is to compute, from first principles, the energy of all electrons for a set of positions of nuclei fully defining the complete electrode/electrolyte interface, including adsorbed species, solvent, ions, and electrode. This is an electronic structure problem where the traditional approach involves a many-electron Schrödinger equation, in which the system’s energy is minimized over all particle wave functions. However, methods based on wave functions are limited in terms of the size of the systems that can be addressed because the number of computational steps scales as ∼N 6 , where N is the number of electrons. The large number of atoms required to model an electrode/electrolyte interface makes this approach computationally prohibitive. A principal alternative is density functional theory (DFT), in which the electron density, rather than the wave function, is employed as the fundamental variable in computing the system energy. DFT-based calculations have much better computational efficiency, scaling as N 3 . Advances in DFT methods and computational efficiency have made possible realistic calculations of energetics of electrochemical interfacial structure, including determining the energies of molecular intermediates that are postulated to be key intermediates of inner-sphere electron-transfer reactions (66–71). Because DFT can be used to calculate the adsorption energy of an intermediate, the method is frequently used to investigate the elementary kinetics of electrocatalytic systems. An often-used simplifying strategy is to assume that the activation energy of an intermediate step is proportional to the difference in ground state energies of the initial and final states of that step.11 We encountered this concept in Figure 15.2.1, which demonstrated schematically how the reductive adsorption of H+ to form an energetically favored intermediate species, Hads , lowers the ‡ activation barrier from ΔG‡ to ΔGads , facilitating the overall reduction of H+ to H2 . Accurately ‡ determining how much ΔG is reduced due to the adsorption of H ∙ requires detailed calculations of the energy of the transition state or assumptions about the shapes of the free energy curves for H+ and Hads . 15.4.2

Hydrogen Evolution Reaction

The focus of DFT in electrocatalysis is the calculation of relative equilibrium free energies of reaction participants, including adsorbed intermediates. The microscopic step with the largest 11 Similar to the classical relationship between ΔG0 and ΔG‡ proposed in the Bronsted–Evans–Polanyi relationship (72, 73), ΔG‡ = 𝛼ΔG0 , where 0 < 𝛼 90∘ , resulting in a flattened nucleus with reduced surface area between the nucleus and solution. Conversely, a weak interaction of the gas with the electrode corresponds to 𝛾 ls < 𝛾 gs and 𝜃 < 90∘ , resulting in a more spherical nucleus and a larger surface area between the nucleus and solution. As before, the free energy of the nucleus is obtained by combining the free energy contributions from the interfaces and bulk. This can be written as: ΔG = Agl 𝛾 + Ags (𝛾gs − 𝛾gl ) + Vsc ΔGV

(15.6.11)

where Agl and Ags are, respectively, the areas of the gas/liquid and gas/solid interfaces, 1 + cos 𝜃 Asphere 2 1 − cos2 𝜃 Ags = 𝜋a2 = 𝜋r2 sin2 𝜃 = Asphere 4 and V sc is the volume of the spherical cap, Agl = 2𝜋r2 (1 + cos 𝜃) =

Vsc =

(2 − cos 𝜃)(1 + cos 𝜃)2 𝜋r3 (2 − cos 𝜃)(1 + cos 𝜃)2 = Vsphere 3 4

(15.6.12) (15.6.13)

(15.6.14)

In (15.6.12)–(15.6.14), Asphere = 4𝜋r2 and V sphere = (4/3)𝜋r3 are, respectively, the surface area and volume of a sphere of radius r. Combining (15.6.11)–(15.6.14) with Young’s equation, (15.6.10) yields ΔG = Φ(𝜃)(Asphere 𝛾 + Vsphere ΔGv )

(15.6.15)

where Φ(𝜃) =

(2 − cos 𝜃)(1 + cos 𝜃)2 4

(15.6.16)

The function Φ(𝜃), plotted in Figure 15.6.4, describes how the shape of the nucleus is determined by the strength of the interactions between constituents of the new phase and the surface. In the limit 𝜃 → 0 (weak interaction), Φ(𝜃) → 1 and (15.6.15) reduces to (15.6.4). Equation (15.6.15) is more general and corresponds to heterogenous nucleation because it includes the surface energies 𝛾 ls and 𝛾 gs . While these parameters do not explicitly appear in (15.6.16), they affect the value of Φ(𝜃), which is computed from Young’s equation, (15.6.10). 16 The angle 𝜃 is often defined in the literature as the “angle into the bubble or droplet,” in contrast to “the angle into the solution” depicted in Figure 15.6.3. Both definitions of 𝜃 yield Young’s equation (15.6.10), using the identity cos 𝜃 = −cos(180∘ − 𝜃).

15.6 Electrochemical Phase Transformations

Weak interaction γls < γgs

1.0 0.8 0.6 Φ(θ) 0.4

Strong interaction γls > γgs

0.2 0.0 0

90 θ/deg

180

Figure 15.6.4 Plot of Φ(𝜃). The angle 𝜃 is defined in Figure 15.6.3.

Setting the derivative of (15.6.15) to zero yields rc = −

2𝛾 ΔGV

(15.6.17)

Thus, rc is independent of 𝜃, as expected, as it represents the characteristic thermodynamic radius [Section 15.6.2(b)]. Substitution of rc into (15.6.15) yields ΔG‡ =

16𝜋𝛾 3 Φ(𝜃)

(15.6.18)

2 3ΔGV

which indicates that ΔG‡ for heterogeneous nucleation [Φ(𝜃) < 1] may be significantly less than for homogenous nucleation [Φ(𝜃) = 1]. The rate of heterogeneous nucleation, J n , is given by substitution of (15.6.18) into (15.6.3): Jn = Jn,0 exp(−

16𝜋𝛾 3 Φ(𝜃) 2 3kTΔGV

)

(15.6.19)

Expressions 15.6.17–15.6.19 are completely general and apply to nucleation of both gas and solid phases. Inspection of (15.6.19) shows that the nucleation rate is determined by the prefactor, J n,0 , and the four energies: ΔGV , 𝛾, 𝛾 gs , and 𝛾 ls [the latter two being implicitly included in Φ(𝜃)]. Typically, J n,0 and the three surface tensions remain relative constant during an experiment.17 The volumetric free energy, ΔGV , is variable and is often of primary concern because it can be readily controlled by adjusting the applied electrode potential or applied current. Expressions for ΔGV can be found in the literature and depend upon the type of phase transformation being considered. 17 We saw in Section 14.1 that surface excesses and surface tension depend on the electrode potential, E. Thus, 𝛾 gs and 𝛾 ls must also be functions of E. However, we will see below that nucleation rates vary by many orders of 2 . The values of 𝛾 and 𝛾 magnitude over a very small range of E (tens of mV) due to the dependence of ΔG‡ on ΔGV gs ls remain relatively constant over such a small potential range. Furthermore, the function Φ(𝜃) dampens the effect of changes in 𝛾 gs and 𝛾 ls on ΔG‡ .

695

696

15 Inner-Sphere Electrode Reactions and Electrocatalysis

The limitations and modifications of CNT have been reviewed (109, 112), and molecular level simulations (113), and DFT (112) have been used to investigate the dynamics and energetics of nucleation phenomena. Applied to gas nucleation, CNT gives good agreement with experimental results (Section 15.6.4). In the simplified form presented above, the theory is more limited in the quantitative treatment of electrodeposition of metals because the nucleation of a solid phase is influenced by crystallographic features and surface diffusion of adsorbed atoms (Section 15.6.3). Treatment, therefore, requires a more atomic view of chemical interactions than is provided by consideration of thermodynamic surface tensions. Surface features (e.g., step edges, polishing scratches, and impurities) can act to greatly lower the activation energy for both gas and solid nucleation. Despite these limitations, CNT provides a useful framework for understanding many electrochemical nucleation phenomena. (d) Estimation of 𝚫GV for 3D Solid Nucleation

For the electrodeposition of an incompressible solid (e.g., an Ag nanoparticle deposited by reduction of Ag+ ), the change in electrochemical free energy per mole of Mn+ deposited in the reaction Mn+ + ne → M is S M ΔG = 𝜇M M − 𝜇 Mn+ − n𝜇 e

(15.6.20)

M where 𝜇SMn+ , 𝜇M M , and 𝜇 e are the electrochemical potentials of the participants, and the superscripts S and M refer to the solution and the metal, respectively. Equation 15.6.20 is expanded as usual (Section 2.2.4), 0M − 𝜇 0S − RT ln a S 0M M (15.6.21) ΔG = 𝜇M Mn+ − nF𝜙 − n𝜇e + nF𝜙 Mn+ F which is regrouped as ( ) 0M − 𝜇 0S − n𝜇 0M − RT ln a ΔG = 𝜇M + nF(𝜙M − 𝜙S ) (15.6.22) n+ e M Mn+ F The first parenthesized group is the chemical free energy change, ΔG, and the last term is the electrical contribution. Equation 15.6.22 applies at any potential, E, which differs from 𝜙M − 𝜙S only by an additive constant, b. Thus, for E = Eeq , (15.6.22) becomes

ΔG(Eeq ) = ΔG + nF(Eeq − b)

(15.6.23)

and when an overpotential, 𝜂, is applied, it is ΔG(Eeq + 𝜂) = ΔG + nF(Eeq + 𝜂 − b)

(15.6.24)

Thus, the overpotential causes an incremental change in free energy per mole of electrodeposited atoms equal to the difference between (15.6.24) and (15.6.23): ΔG(Eeq + 𝜂) − ΔG(Eeq ) = nF𝜂

(15.6.25)

The overpotential for a reductive process is always negative, so the free energy differential is negative, as it must be for spontaneous nucleation. It follows that the free energy differential per atom is nF𝜂/N A = ne𝜂; therefore, the free energy change per volume is ΔGV = ne𝜂∕Va

(15.6.26)

where V a is the atomic volume of electrodeposited M. Since 𝜂 is negative, ΔGV is also negative. Equation (15.6.26) is the key relationship between the driving force for nucleation, ΔGV , and the overpotential, 𝜂. At Eeq , ΔGV = 0, consistent with experimental observations that reductive nucleation of a bulk phase always occurs at E < Eeq .

15.6 Electrochemical Phase Transformations

Substitution of (15.6.26) into (15.6.5) yields ΔG in terms of 𝜂, ΔG = 4𝜋r2 𝛾 + 4𝜋r3 ne𝜂∕3Va

(15.6.27)

For small values of 𝜂, ΔG is positive when the surface energy is larger in magnitude than the volumetric energy (Figure 15.6.2). Because the metal deposit becomes stable and grows when the critical radius is reached, it immediately follows that 𝜂 = 𝜂 n at r = rc . Thus, substituting (15.6.26) into (15.6.7) yields rc =

−2𝛾Va ne𝜂n

(15.6.28)

Following the procedure outlined for heterogeneous nucleation in Section 15.6.2(c), it also immediately develops that ΔG‡ is a function of the electrode overpotential, rapidly decreasing in proportion to 𝜂 −2 . ΔG‡ =

16𝜋𝛾 3 Va2 Φ(𝜃) 3n2 e2 𝜂 2

(15.6.29)

Finally, the nucleation rate for electrodeposition is obtained from substitution of (15.6.29) into the Arrhenius expression, (15.6.3). [ ] 16𝜋𝛾 3 Va2 Φ(𝜃) Jn = Jn,0 exp − (15.6.30) 3n2 e2 𝜂 2 kT The treatment leading to the relationship between ΔGV and 𝜂—and, thus, also the subsequent expressions for rc and J n —is greatly simplified relative to more exact treatments of nucleation during metal electrodeposition (109). The effect of the crystallographic features and surface interactions in determining the nucleus shape has been ignored. Nevertheless, this approximate treatment provides the essential conceptual framework for three-dimensional nucleation. In particular: • Equation (15.6.30) indicates that J n increases in proportion to exp(−B/𝜂 2 ), where B is a constant. Plots of ln J n vs. 𝜂 −2 for deposition of quasi-spherical particles are frequently linear, in agreement with theory, and provide values of nucleus size and activation energy that are in reasonable agreement with expectations (109). • The ln J n vs. 𝜂 −2 dependence also suggests that J n increases rapidly over small changes in 𝜂. This is confirmed by experiments that show that J n increases by several orders of magnitude over a change in 𝜂 of several tens of mV, as seen in Figure 15.6.5 for the deposition of Hg droplets on Pt. Experimentally, J n is obtained by counting the number of Hg droplets under a microscope after a set period of electrolysis at different values of 𝜂 and assuming that every nucleation event resulted in a droplet. The strong dependence of J n on 𝜂 is also qualitatively reflected in the LSV response when a new phase is nucleated on the electrode surface. As 𝜂 is increased during the voltammetric scan, the nucleation and growth of particles are frequently indicated by a sudden increase in current when 𝜂 approaches 𝜂 n and the potential-dependent timescale for nucleation (𝜏n = Jn−1 ) suddenly becomes smaller than the observational timescale for LSV (𝜏 obs = RT/nFv). An example is seen in Figure 15.6.1a for the electrodeposition of Ag particles on HOPG. • Equation (15.6.28) provides an approximate means to evaluate rc . For instance, the measured value of 𝜂 n = 140 mV for Ag deposition on GC (Figure 15.6.1a) gives rc = 1.1 nm using 𝛾 = 0.72 J/m2 (114) and V a = 1.7 × 10−29 m3 /atom for Ag. It is important to recall that rc is the

697

698

15 Inner-Sphere Electrode Reactions and Electrocatalysis

7

6 log Jn/s–1 5

4 90

100

110 120 η–2/ V–2

130

140

Figure 15.6.5 Log Jn vs. 𝜂 −2 for the electrodeposition of Hg droplets on Pt. Each value of Jn corresponds to the steady-state nucleation rate following a potential step to a given overpotential. The range of 𝜂 −2 corresponds to −106 mV < 𝜂 < − 84 mV. The nucleation rate increases by more than 103 over this 22-mV range! Measurements were made in aqueous 0.5 M Hg2 (NO3 )2 at a hemispherical Pt single crystal electrode (5 × 10−4 cm). [Toschev and Markov (115), with permission.]

thermodynamic radius of the nucleus. The actual geometry and number of atoms contained within the critical nucleus require experimental evaluation of the contact angle 𝜃.18 The latter would be obtained from measurements of the slope of a plot of ln J n vs. 𝜂 −2 . Electrodeposition at high overpotentials results in critical nuclei comprising a small number of atoms. For such situations, it is questionable to employ bulk values for 𝛾 and ΔGV and to assume a spherical geometry. “Small cluster” nucleation models address these issues (109). Electrochemical nucleation of 2D (i.e., UPD) metal layers occurs by the formation of a critical 2D cluster of adsorbed atoms. The initial 2D cluster grows to a critical size by depositing additional atoms at the terrace edge of the cluster (Figure 21.1.3 for an example). Equations analogous to those presented above for 3D nucleation rate have been derived for 2D nucleation (109). The key difference is that the critical cluster size and nucleation rate for 2D deposition are expressed in terms of energy per unit length of the edge of the cluster, in contrast to using the energy per unit area, 𝛾, in the case of 3D nucleation. Two-dimensional nucleation is proportional to exp(𝜂 −1 ), rather than exp(𝜂 −2 ), as found for 3D nucleation (109). (e) Estimation of 𝚫GV for Bubble Nucleation

For a nucleation process involving formation of a gas bubble, ΔGV can be obtained by writing the minimum work required for reversibly transferring gas from the liquid phase into a bubble of volume V , including pressure–volume work. The expression is (111, 116, 117) Wmin = A𝛾 − V (Pi − Pe ) + (N∕NA )(𝜇V − 𝜇L )

(15.6.31)

In this equation, Pi is the gas pressure within the bubble, Pe is the hydrostatic pressure on the surrounding liquid, 𝜇V and 𝜇L are the chemical potentials of the transferred species in the gas and liquid phases, respectively, N is the number of molecules transferred from the liquid to the 18 A spherical Ag particle of 1.1-nm radius contains ∼320 Ag atoms. Nuclei for Ag particle deposition on HOPG and Pt electrodes are reported to contain between 1 and 6 atoms, corresponding to 143∘ < 𝜃 < 165∘ . This range of 𝜃 corresponds to a sphere cap geometry where the height of the sphere cap is much smaller than rc (Figure 15.6.4).

15.6 Electrochemical Phase Transformations

gas phase, and N A is the Avogadro constant. The term V (Pi − Pe ) must be included because the new phase is compressible. If the nucleus is assumed to be spherical, (15.6.31) becomes Wmin = 4𝜋r2 𝛾 − (4∕3)𝜋r3 (Pi − Pe ) + (N∕NA )(𝜇V − 𝜇L )

(15.6.32)

The value of W min at the critical radius, rc , is identified as ΔG‡ . At this size, the probability that the bubble will expand is equal to the probability that it will shrink; thus, there is no net transfer of molecules between the gas and liquid phases. Accordingly, 𝜇V,c = 𝜇L,c , where the added subscript “c” denotes values at the critical radius (116) The derivative of (15.6.32) with respect to r is zero at r = rc ; thus, one finds that rc =

2𝛾 Pi − Pe

(15.6.33)

Substitution of rc into (15.6.18) yields the activation energy: ΔG‡ =

16𝜋𝛾 3 Φ(𝜃) 3(Pi − Pe )2

(15.6.34)

Comparison of (15.6.33) and (15.6.34) with (15.6.17) and (15.6.18) allows one to identify ΔGV as the difference in pressure between the interior of the bubble and the solution,19 ΔGV = −(Pi − Pe ) With ΔGV in this form, the nucleation rate of a gas bubble is given by [ ] 16𝜋𝛾 3 Φ(𝜃) Jn = Jn,0 exp − 3kT(Pi − Pe )2

(15.6.35)

(15.6.36)

This expression of J n is convenient in experimental analysis because the value of Pi is readily related through Henry’s law to the concentration of dissolved gas, which can be determined experimentally from the current at the electrode (Section 15.6.4). The behavior here is similar to that in the electrodeposition of metals, where we saw in (15.6.30) that J n varies by orders of magnitude with a small change in 𝜂 [J n proportional to exp(−B/𝜂 2 ), where B is a constant]. An analogous strong dependence of J n on (Pi − Pe ) is clear from inspection of (15.6.36). For a gas-evolving reaction, a small increase in (Pi − Pe ) can increase J n by orders of magnitude. We will see an example of this extreme behavior in Section 15.6.4. 15.6.3

Electrodeposition

The electrochemical deposition of metals is one of the earliest applications of electrochemistry and remains of great importance in many technologies. Electrodeposition provides a means for coating materials with metal and metal oxide layers to improve corrosion resistance, surface hardness, electrical properties, catalytic activity, and visual appearance. For example, the circuitry on printed-circuit boards is electroplated with Au, Ni, and other metals to prevent corrosion and improve function. The reversibility of Li-metal anodes used in batteries depends upon the ability of Li+ to be uniformly electrodeposited upon battery charging. Electrodeposition is also used in the synthesis of nanoparticles and nanowires, epitaxial layers, organic 19 This result is consistent with the definition of ΔGV as a free energy change per volume. Pressure times volume has units of energy.

699

700

15 Inner-Sphere Electrode Reactions and Electrocatalysis

and inorganic crystals, multilayer polymer electrolytes, as well as in the electrowinning and electrorefining of metals (e.g., Cu). This section provides an overview of fundamental issues associated with electrodeposition of metals. A detailed introduction to the theory and literature of metal electrodeposition is available in reference (109). The simplest form of electrodeposition occurs when a metal ion, Mn+ , in solution is reduced at electrode made of the same metal, M: Mn+ + ne ⇌ M

(15.6.37)

The equilibrium potential, Eeq , depends on the solution composition and can be expressed in terms of the formal potential for Mn+ /M and the concentration of Mn+ . ′ RT Eeq = E0 + ln CMn+ (15.6.38) nF Electroreduction of Mn+ occurs when the applied potential is more negative than Eeq (E < Eeq ), and the opposite process, electrodissolution, occurs when E > Eeq . As for other electrochemical reactions, the rate of electrodeposition or electrodissolution may be limited either by interfacial kinetics or by the transport of Mn+ . The kinetics of (15.6.37) are very fast for some Mn+ /M systems. For instance, the standard exchange current density for the Ag+ /Ag system is reported to be 24 A/cm2 (118), comparable in magnitude to values for fast outer-sphere reactions in which both O and R are dissolved. Thus, Ag+ reduction occurs readily on Ag with minimal overpotential. The reduction of Mn+ differs significantly from an outer-sphere reaction in that the solvation sphere of Mn+ is removed during the process, and the resulting M atom is stabilized by its interaction with other lattice atoms of the M electrode. In general, surface sites with the strongest interactions correspond to kinks and steps. Atoms electrodeposited on flat terraces may diffuse across the electrode surface to these locations if surface diffusion is fast. The crystallinity and surface structure of electrodeposited metals depend on many factors, including the deposition overpotential, current density, temperature, solvent and electrolyte, and intentional use of small-molecule adsorbates to control grain size and surface smoothness (often referred to as “brighteners” and “levelers” in the electroplating industry). Mass-transfer-limited electrodeposition at large overpotentials often results in the formation of dendrites. ′ The electroreduction of Mn+ on a different metal, M , is more complex than on M itself because it requires the nucleation of a new phase. Equation (15.6.37) still describes the redox process, but the reaction thermodynamics are now influenced by the adsorption energy of M ′ ′ atoms on the electrode M . The chemical interactions between an M adatom20 and an M surface can be quite different than for an M adatom on an M surface, and their character can significantly influence the structure of the new electrodeposited phase: ′

• When M is strongly adsorbed on M , it becomes thermodynamically easier to reduce Mn+ ′ ′ on an M surface than on M itself; therefore, the potential for deposition on M is more pos′ itive than E0 (Mn+ ∕M) [Sections 5.3.2(c) and 17.2.4(a)]. This effect is called underpotential deposition (UPD) and often results in well-ordered two-dimensional films on the electrode surface. ′ • Conversely, when M is weakly adsorbed on M , it is thermodynamically harder to reduce Mn+ ′ ′ on M than on M; and the potential for deposition shifts more negative than E0 (Mn+ ∕M). This is overpotential deposition (OPD), resulting in the formation of three-dimensional particles and islands. 20 Adatom refers to an adsorbed atom on the electrode surface that results from the oxidation or reduction of a chemical species in solution. Similarly, adlayer refers to a layer of adatoms.

15.6 Electrochemical Phase Transformations

(a)

(b)

(c)

Figure 15.6.6 (a) Volmer–Weber, (b) Frank–van der Merwe, and (c) Stranski–Krastanov modes of electrodeposition. The arrows show the direction of growth of the electrodeposit. [Adapted from Budevski, Staikov, and Lorenz (109), with permission.]

Figure 15.6.6 illustrates three electrodeposition modes distinguished on the basis of adsorp′ tion energies and crystallographic structures of M and M (109): ′

1) Weak interaction between M and M . In this case, the adsorption energy of an M adatom on ′ M is larger (more negative) that that of an M adatom on metal M ; thus, the electrodeposition ′ of a 3D structure for deposited M is energetically favored to minimize M–M interactions (Figure 15.6.6a). Often, particles or islands of M are formed on the electrode surface. This case is known as Volmer–Weber deposition, an example of which is the deposition of Ag and other metals on the basal plane of HOPG (119–121).

701

702

15 Inner-Sphere Electrode Reactions and Electrocatalysis ′

2) Strong interaction between M and M with crystallographic alignment. A strong interaction between the M adatoms and a foreign surface results in the formation of a 2D adlayer of ′ ′ M on M (Figure 15.6.6b). If the crystallographic structures of M and M are sufficiently similar, epitaxial growth occurs, known as Frank–van der Merwe deposition. An example is the deposition of Ag on Au single crystal electrodes, where the similar lattice spacings of Ag and Au allow epitaxial growth of Ag films (122). ′ 3) Strong interaction between M and M with crystallographic misalignment. As in case 2, the strong interaction of M with the surface results in the initial formation of one or two ′ monolayers of M; however, if the crystallographic structures of M and M are mismatched, the resulting 2D adlayer is energetically strained. The deposition of 3D particles or islands (Volmer–Weber deposition) then becomes favored due to the strain (Figure 15.6.6c). This growth mode is called Stranski–Krastanov deposition, an example or which is the deposition of Pb on Ag single-crystal electrodes (123). (a) Underpotential Deposition (UPD) of Monolayer Adsorbates

The UPD of one (and sometimes two) monolayer of a metal on a foreign metal electrode is of fundamental interest and has been extensively investigated (109, 124). Many different examples have been discovered and documented [e.g., Hg on Au(111) (125), Tl on Ag(100) (126), and Cu on Pt(111) (127)]. Single-crystal electrodes are frequently used in these investigations to provide a well-characterized surface orientation. Because the energetics of adlayer formation are highly dependent on the exposed surface [e.g., Au(111) vs. Au (110)], the voltammetric features associated with UPD depend on the orientation used, providing detailed insights into how adsorption energies are influenced by the atomic arrangement of the underlying electrode. Figure 15.6.7 shows the UPD of Pb at a 300-nm-thick Ag(111) film electrode (A = 0.22 cm2 ) prepared by thermal vapor deposition of Ag onto a molecularly smooth mica surface (128). The deposition of one monolayer of Pb from an aqueous solution occurs near −300 mV vs. Ag/AgCl, corresponding to an underpotential of ∼150 mV relative to the formal potential of ′ the Pb2+ /Pb couple (E0 = −440 mV vs. Ag/AgCl). On as-prepared electrodes (dashed curve), the negative potential scan results in the UPD of one monolayer of Pb (Pb2+ + 2e → Pb) in A2

10 μA

i

A3

C3

A1

C1

C2 –400

–300 –200 E/mV vs. Ag/AgCl

Figure 15.6.7 UPD of a Pb monolayer on an Ag(111) film electrode (deposited on mica) in a solution containing 0.005 M Pb(ClO4 )2 , 0.1 M NaClO4 , and 0.01 M HClO4 . Positive potentials are to the right, and anodic currents are up. Scans start at the positive limit and first move negatively. v = 10 mV/s. Dashed curve: CV of Pb2+ /Pb on the as-deposited Ag film. Solid curve: CV after the Ag film was annealed for 12 h at 300 ∘ C under vacuum (10−6 torr). Peak labels are discussed in text. [Stevenson, Hatchett, and White (128). © 1996, American Chemical Society.]

15.6 Electrochemical Phase Transformations

three voltammetric peaks (C1 , C2 , and C3 ). On the reverse scan, the Pb atoms are oxidatively desorbed (Pb → Pb2+ + 2e) in three peaks (A3 , A2 , and A1 ). These peaks make up three reversible pairs. The largest, C2 /A2 , corresponds to UPD of Pb on atomically flat Ag terraces, and the two side-pairs, C1 /A1 and C3 /A3 , correspond to UPD at surface defects. Thermal annealing of the Ag(111) film at 300 ∘ C yields a more ordered surface, as confirmed by in-situ STM and ex-situ X-ray diffraction. In parallel, the side peaks, C1 /A1 and C3 /A3 , disappear. In the annealed system, the integrated charge associated with peak C2 is 318 𝜇C/cm2 , corresponding to a surface concentration of ∼1.0 × 1015 Pb atoms/cm2 . In-situ surface X-ray scattering shows that the Pb monolayer is incommensurate with the Ag(111) surface and undergoes a potential-dependent compression as the electrode potential is scanned negative between the potentials for UPD and bulk Pb deposition (129). Overpotential deposition of additional Pb on top of the Pb monolayer results in the formation of Pb islands (128). In addition to the UPD voltammetric signatures being dependent on the electrode crystal orientation and surface structure, they are also strongly influenced by the choice of the supporting electrolyte anion. The UPD of Cu on Au(111) displays sharp voltammetric features at potenor a halide is used as the electrolyte anion. These results are tials that depend on whether SO2− 4 interpreted as reflecting the coadsorption of the anion during UPD (130). Potential-dependent structural reordering of a UPD layer is also frequently observed. UPD is not limited to the deposition of metal monolayers. The reversible voltammetric ′ peaks associated with hydrogen UPD in acidic media occur at E > E0 (H+ ∕H2 ), indicating relatively strong adsorption. (See Figures 14.4.6 and 14.6.1 for example at Pt.) Moreover, the electrooxidation of a metal in the presence of strongly complexing ions can result in the formation of a UPD monolayer. For example, the oxidative adsorption of HS− at Ag single crystal electrodes results in the formation of an S adlayer at underpotentials that are 0.5 V or ′ more negative of E0 for the deposition of bulk Ag2 S (−0.88 V vs. Ag/AgCl). The multipeak voltammetric response for UPD of S on Ag is strongly dependent on the crystallographic orientation of the surface exposed to the solution [i.e., (111), (110), or (100)] (131). While electrochemical methods such as voltammetry and coulometry are invaluable for evaluating thermodynamic properties (e.g., adsorption energies) and the amount of electrical charge associated with a UPD redox transition, they provide minimal structural information. A myriad of in-situ techniques based on vibrational spectroscopy, x-ray diffraction, scanned probe microscopies (e.g., STM and AFM), and the measurement of deposited mass by QCM have proven invaluable for characterizing the structure of electrodeposited films. These methods are discussed in Chapter 21. (b) Electrochemical Growth of a New Phase

Following the nucleation of a stable phase on an electrode, the growth of that phase and the resulting structure are determined by the number of nuclei that have been deposited, the structure of the electrodeposit (i.e., a 2D monolayer, or a 3D particle), and the applied potential. The rate of deposition may be limited by the interfacial deposition kinetics or, at high overpotentials, by transport. Analysis of nucleation and growth rates is commonly performed by chronoamperometry. The current is monitored for a finite time following a potential step to an overpotential where nucleation and growth occurs. The resulting i − t transient is then analyzed based on mathematical models that incorporate assumed geometric considerations (3D vs. 2D growth), nucleation rates, and kinetic or transport limitations. In the case of electrodeposition of 3D particles or islands, interactions of particles due to overlapping diffusional fields or Ostwald ripening [Section 15.6.3(c)] are also included as appropriate. Most often, the chronoamperometric analysis is greatly simplified by accompanying structural imaging using

703

15 Inner-Sphere Electrode Reactions and Electrocatalysis

j/mA cm–2

0.8

–500 mV

0.6 0.4 –250 mV

0.2

–100 mV 0.0 0

10

20 30 40 t/ms (a)

50

60 Å 40

0.8

30

–500 mV j/mA cm–2

704

20

0.6

10 0

0.4 –250 mV

0.2

0.0

–100 mV 0.0 0

2

4 t1/2/ms1/2 (b)

6

0.2

0.4

0.6

0.8

1.0 μm

(c) 8

Figure 15.6.8 Current transients for deposition of Ag on HOPG. (a) j vs. t; (b) j vs. t1/2 . (c) AFM image. [Zoval, Stiger, Biernacki, and Penner (120). © 1996, American Chemical Society.]

STM or AFM, which informs the experimentalist of the number of nucleation sites and the structure of the new phase. As an example, consider the electrodeposition of Ag nanoparticles on HOPG (120). Figure 15.6.8 shows results from potentiostatic deposition experiments in which the potential of the HOPG electrode was pulsed for 50 ms to a specific overpotential (−100, −250, and ′ −500 mV vs. E0 (Ag+ ∕Ag)). The i − t responses (Figure 15.6.8a) display two distinct behaviors. At short times (5 ms), the current rises slowly, displaying a peak at the highest overpotential. As noted above, deposition of Ag on HOPG follows a Volmer–Weber growth mode, which was verified in this case by imaging Ag nanoparticles deposited on the HOPG surface using non-contact AFM (Figure 15.6.8c). The particles were disk-shaped with diameters between 20 and 60 nm and heights of heights between 1.5 and 5 nm. Nucleation of the particles is generally expected to be very fast for the large overpotentials employed in these experiments, and, on the timescale of these experiments, it can be treated as “instantaneous,” allowing the i − t response to be interpreted solely in terms of the diffusion-limited flux of Ag+ to the growing particles. If the diffusional fields of the particles are non-overlapping, the i − t response is given by (120, 132) i(t) =

F𝜋(2DAg+ C ∗

1∕2

Ag+

)3∕2 MAg Nt 1∕2

1∕2

dAg

(15.6.39)

15.6 Electrochemical Phase Transformations

where MAg is the atomic mass of Ag, dAg is the density of Ag, and N is the number of nuclei that lead to stable Ag particles. The slope of a plot of i vs. t 1/2 yields the value of N, which can be compared to the areal density of Ag particles determined from AFM images. For the experiments in Figure 15.6.8, N is 2.7 × 109 at −500-mV overpotential, 4.3 × 108 at −250 mV, and 4.2 × 107 at −100 mV. The values at overpotentials of −500 and −250 are in excellent agreement with results from AFM imaging, while the AFM-determined value at the lowest overpotential is too small by an order of magnitude. Other analyses of Ag deposition on HOPG suggest that aggregation and detachment of particles occur during deposition, perhaps accounting for the discrepancy at −100 mV (121). The assumption of non-overlapping diffusional fields, made in employing (15.6.39), is clearly approximate from inspection of Figure 15.6.8c, where the distance between some particles is on the order of the particle diameter. Overlapping diffusional fields has been shown in simulations to result in a larger particle size dispersion. Much narrower particle size ranges are predicted to be obtained as the deposition overpotential is decreased (133). (c) Dependence of E0 on Particle Size

It is interesting to consider how many atoms or what particle size is needed to produce “bulk metal” and whether the standard (or formal) potential is a function of particle size. These questions have been addressed (134–140). For clusters of n atoms, the value of En0 indeed turns out to be different from the value for the bulk metal, E0 . Consider, for example, silver clusters, Agn . For a silver atom (n = 1), E10 can be related to E0 through a thermodynamic cycle involving the ionization potential of Ag and the hydration energy of Ag and Ag+ . This process yields Ag+ (aq) + e ⇌ Ag1 (aq)

E10 = −1.8 V vs. NHE

(15.6.40)

which is 2.6 V more negative than for the corresponding reaction involving bulk Ag. Thus, it is much easier energetically to remove an electron from a single, isolated Ag atom than to remove an electron from a lattice of Ag atoms. Experimental work carried out with larger silver clusters shows that as the cluster size increases, En0 moves toward the value for the bulk metal. Surface atoms are bonded to fewer neighbors than atoms within a crystal; thus, an extra surface free energy is required to create additional surface area by subdivision of a metal. Conversely, the total energy of a system can be minimized by decreasing the surface area, such as by taking on a spherical shape or by fusing small particles into larger ones. Ostwald ripening of colloidal particles to form precipitates is a manifestation of the latter effect. The dependence of E0 on particle size has important consequences in the electrodeposition of metals, especially at low overpotentials. If E0 for a metal cluster is more negative than the nucleation potential, then the cluster is thermodynamically unstable and will oxidize and dissolve back into solution. This concept is consistent with the earlier discussion of CNT (Section 15.6.2), where we found that an overpotential, 𝜂 n , is required to deposit a nucleus with a critical radius that can continue to grow into a stable phase. At overpotentials of lesser magnitude, metal atoms and clusters form, but dissolve before growing further. Based on similar reasoning, larger overpotentials are expected to result in smaller critical radii. Experimentally, critical nuclei as small as 1–10 atoms are often reported. For example, the 3D electrodeposition of Cu on Au(111) at overpotentials between 71 and 82 mV is reported to correspond to nuclei containing 2 or 3 Cu atoms. Similarly, for the deposition of Hg on Pt, the slope of the straight line in Figure 15.6.5 indicates that only 3–10 Hg atoms make up the critical nuclei. The validity of using CNT for such small clusters, where one assumes a bulk value of the surface tension, has been questioned; hence, more atomistic models have been proposed (109). While “small cluster” nucleation models are based on more physically realistic assumptions, analyses of the critical nucleus size generally provide estimates in line with CNT.

705

15 Inner-Sphere Electrode Reactions and Electrocatalysis

Section 19.5 describes experiments where deposition of Pt is performed in solutions containing such a low concentration of the ionic Pt precursor that one statistically expects the deposition of quite small clusters, Ptn , where n = 1, 2, 3, 4, …. These experiments are performed at very large deposition overpotentials, where the resulting Pt atom is stable and can be used as an electrocatalyst. The influence of surface energy on E0 is quantitatively expressed by the Gibbs–Thomson relationship. For the reversible Mn+ /M couple, the standard redox potential for a spherical particle of radius r, E0 (r), is given by E0 (r) = E0 −

𝛾ΩM nF

∙

2 r

(15.6.41)

where E0 corresponds to a macroscopic electrode (r → ∞), 𝛾 is the surface tension of the metal particle/electrolyte interface (J/m2 ), and ΩM is the molar volume of M (m3 /mol). Figure 15.6.9 shows an experimental case in which (15.6.41) was applied to the dissolution of Pt nanoparticles (Pt → Pt2+ + 2e) (141). Experimental values of the particle dissolution potential, Ed , were determined as a function of r by using in-situ electrochemical scanning tunneling microscopy (EC-STM) to observe a collection of Pt nanoparticles (r = 0.58 − 1.43 nm) on an Au electrode surface in 0.1 M H2 SO4 . Beginning at an electrode potential where all of the nanoparticles are stable (0.60 V vs. NHE), the potential was increased in 50-mV steps, while simultaneously imaging the particles by EC-STM. In this way, the potential at which each particle dissolved could be estimated. The data points in Figure 15.6.9 show the measured values of Ed vs. 2/r. Approximating E0 (r) in (15.6.41) by the experimental Ed values allows one to compare the measured data with the prediction from with the Gibbs–Thomson relationship. The solid line in Figure 15.6.9 is computed from (15.6.41) using 𝛾 = 2.35 J/m2 (142), E0 = 1.01 V vs. NHE for Pt in 0.1 M H2 SO4 , and the bulk value of ΩM for Pt (9.1 × 10−6 m3 /mol). No adjustable parameters are used. Very good agreement is observed between the experimental data and the Gibbs–Thomson relationship, demonstrating the anticipated thermodynamic instability as the particle size is decreased. If the dependence of E0 (r) on particle size can be established for a given system, it can then be used as an analytical tool to measure particle sizes (143). A convenient approach to calibration is to determine the dissolution potentials for several samples, each consisting of many particles of uniform size, as determined by SEM. Since the particle size is known for each sample, a calibration plot of E0 vs. r can be drawn. Using this method, the dissolution potentials of Au nanoparticles of known size [immobilized on glass/indium tin-oxide electrodes modified by (3-aminopropyl)triethoxysilane] were measured in 10 mM Br− + 0.1 M KClO4 using anodic Figure 15.6.9 Dependence of dissolution potential, E d , on the Pt particle size (2/r). Experimental data points are for Pt particles on Au in 0.1 M H2 SO4 .The downward vertical bar corresponds to the 50-mV incremental step between ECSTM observations of particle dissolution. The horizontal error bars indicate 8% error associated with measurement of the particle size. The solid line is the Gibbs–Thomson equation, (15.6.41), for the system. [Tang et al. (141). © 2010, American Chemical Society.]

1.2 1.0 E/ V vs. NHE

706

0.8 0.6 0.4 0.2 0

1

2

3 4 2r –1/nm–1

5

6

15.6 Electrochemical Phase Transformations

Figure 15.6.10 Anodic stripping voltammograms (ASVs) of citrate-coated 7.5-nm-radius Au nanoparticles after Ostwald ripening at 0.45 V for 0, 35, 70, 105, or 140 minutes in 10 mM KBr + 0.1 M KClO4 . The ASVs were recorded in the same solution at a scan rate of 0.01 V/s. [Pattadar and Zamborini (143). © 2019, American Chemical Society.]

2.0 0.0

Ripening time

–2.0

Ripening at 0.45 V 0 min 35 min 70 min 105 min 140 min

i/μA –4.0 –6.0 1.2

1.0

0.8 0.6 0.4 E/V vs. Ag/AgCl

0.2

0.0

stripping voltammetry (ASV, Section 12.7). In bromide media, gold undergoes both 3e and 1e dissolution processes, yielding AuBr− and AuBr− : 4 2 0 − AuBr− 4 + 3e ⇌ Au + 4Br

E0 = 0.85 Vvs. NHE

(15.6.42)

0 − AuBr− 2 + e ⇌ Au + 2Br

E0 = 0.96 V vs.NHE

(15.6.43)

For three particle samples, the ASV peak, Ep , shifted negatively as the SEM-determined radius decreased in the order: 7.5 nm (0.77 V); 2 nm (0.69 V); and 0.8 nm (0.45 V vs. NHE), consistent with (15.6.41). These results established the relationship between Ep and r, which was used subsequently in additional experiments aimed at electrochemically induced Ostwald ripening. The general phenomenon of Ostwald ripening refers to an instability in an ensemble of particles causing some particles to become larger, while others become smaller or disappear altogether. The process is driven by the dependence of particle stability on its size, analogous to the dependence of E0 on r in (15.6.41). Electrochemical Ostwald ripening of Au nanoparticles held at a given potential results from the oxidative dissolution of smaller particles [having the least positive E0 (r) values], followed by reduction of the oxidation products AuBr− and AuBr− 4 2 0 at the larger particles [with more positive E (r) values]. Thus, the larger particles increase in size at the expense of the smaller particles. For example, Figure 15.6.10 shows ASV curves for electrodes coated with 7.5 nm-radius Au nanoparticles held at E = 0.45 V for periods of varying length. The wave at Ep = 0.77 V, associated with the 7.5-nm-radius particles, clearly decreases with ripening time, and a new ASV peak appears at Ep = 0.95 V. The latter corresponds to growth of some particles to a radius of ∼10 nm. 15.6.4

Gas Evolution

Electrochemical reactions sometimes result in the spontaneous formation of a gas phase (i.e., bubbles) at the electrode surface. The liquid-to-gas transformation occurs because product molecules that are stable as gases under ambient conditions often have very limited solubility in the electrolyte. For instance, the solubility of H2 in water at 25 ∘ C under 1 bar of H2 gas is ∼0.8 mM, far below the concentrations of H2 (>100 mM) that can be readily generated at an electrode by the reduction of protons in a strongly acidic solution (2H+ + 2e ⇌ H2 ). When the local concentration of dissolved H2 at the electrode surface exceeds the saturation concentration, gaseous H2 becomes thermodynamically favored. As for electrodeposition of metals (Section 15.6.3), the nucleation of a stable gas phase is an activated process, requiring a significant overpotential.

707

708

15 Inner-Sphere Electrode Reactions and Electrocatalysis

Electrochemical gas evolution is commonly associated with electrolysis reactions used in the production of H2 , O2 , and Cl2 , but also occurs in many other situations. For instance, the oxidation of hydrazine, N2 H4 , is used in direct hydrazine fuel cells: N2 H4 → N2 + 4H+ + 4e

(15.6.44)

This process generates N2 (144), which has only modest solubility in aqueous solutions (∼0.7 mM). Many reactants and products of the electrocatalytic reactions discussed in Section 15.3 are stable gases at room temperature (e.g., N2 O, NO, NH3 , CH4 , CO2 , and C2 H4 ). Sometimes, one encounters unwanted gas evolution, as can occur in Li-ion batteries from decomposition of the electrolyte at the electrodes, resulting in production of H2 and flammable organic products. In general, the formation of gas bubbles is deleterious to the efficiency of an electrochemical reaction. Bubbles that remain attached to the electrode reduce the amount of active surface available for the reaction, while bubbles that detach from the electrode impede the transport of reactants, products, and ions, resulting in increased mass-transport and ohmic resistance overpotentials. Gas formation can also cause mechanical damage to the electrode and other cell components. These issues have been extensively studied and reviewed (145, 146). The focus of this section is on the application of electrochemical methods to investigate the thermodynamics and kinetics of gas-phase transformations at the electrode/electrolyte interface. Voltammetric and chronoamperometric measurements of gas-evolving reactions are often complicated by the appearance of “noise” in the electrical current reflecting the stochastic nucleation, growth, and detachment of bubbles at many locations on the electrode surface. The superposition of these individual signals reflects the complex dynamics of multiple bubble events and generally prevents a quantitative analysis of the underlying physics and chemistry of bubble formation. A strategy that circumvents this problem is to employ an electrode of sufficiently small size that the probability of nucleating more than a single bubble during the measurement is essentially zero (147). Figure 15.6.11 shows an example involving N2 nucleation during N2 H4 oxidation, (15.6.44). The voltammogram in Figure 15.6.11a is relatively straightforward to interpret using the schematic representation of bubble formation shown in Figure 15.6.12 as a guide. Beginning at −0.7 V vs. SCE—close to the thermodynamic potential for N2 H4 oxidation (148)—the current rises as the potential is swept positively (Figure 15.6.12a). The corresponding rate of reaction 12

ip , Ep

30

8 i/nA

20 ip/nA

4

10

ir 0 –0.8

–0.4

0.0 0.4 E/V vs. SCE (a)

0.8

0

0

20

40 60 r0/nm (b)

80

100

Figure 15.6.11 (a) i − E response corresponding to N2 bubble formation at a 32-nm-radius Pt nanoelectrode in 1.0 M N2 H4 at v = 200 mV/s. Two voltammetric cycles are shown. Initial potential is −0.80 V. (b) ip vs. Pt nanoelectrode radius in 1.0 M N2 H4 [Chen, Wiedenroth, German, and White (149). © 2015, American Chemical Society.]

15.6 Electrochemical Phase Transformations

N2(aq) N2H4

N2(aq)

N2H4

N2(aq)

N2H4

N2H4

Nucleus

Pt

Nucleation

(a)

Pt

(b)

Growth

Pt

(c)

Figure 15.6.12 Cross section of a Pt nanodisk electrode depicting formation of an N2 nanobubble. (a) Reduction of N2 H4 to produce dissolved N2 ; (b) formation of a single bubble nucleus; (c) stable N2 nanobubble following growth of the nucleus. A single nanobubble is generated at disks with r0 ≤ 100 nm. [Adapted from Edwards, White, and Ren (150). © 2019, American Chemical Society.]

(15.6.44) is controlled in this region by N2 H4 transport and electron-transfer kinetics. As the current continues to increase, the solution adjacent to the electrode surface becomes increasingly supersaturated with dissolved N2 , eventually reaching a critical value where the N2 gas phase is nucleated (Figure 15.6.12b). The sudden drop in current at potential Ep signals the rapid growth of an individual N2 nanobubble that covers the electrode surface, creating a three-phase (solid/gas/liquid) boundary defined by the UME circumference (Figure 15.6.12c).21 The residual current, ir , after formation of the bubble is nearly independent of the electrode potential and results from a dynamic balance between the rate of N2 electrogeneration at the three-phase boundary and the diffusive outflux of N2 from the bubble, through the gas/liquid interface, into the solution (149). On the reverse scan, the current smoothly returns to the baseline value when the potential is sufficiently negative to cease oxidation of N2 H4 . Without continued generation of N2 , the bubble shrinks and disappears. The voltammetric response in Figure 15.6.11a is also characteristic of the nucleation and growth of bubbles of other gases at nanoscale UMEs, including CO2 generated by oxidation of formic acid (151), O2 generated by oxidation of H2 O2 (152, 153), and H2 generated by proton reduction (147, 154). Results for the latter are previously shown in Figure 15.6.1b. Much can be learned about the thermodynamics of bubble formation from the voltammetric response (155). The peak current, ip , can be used to calculate the concentration of dissolved N2 at the electrode surface at the moment of nucleation. At any potential prior to the formation of the bubble, the current at the disk-shaped UME can be expressed in terms of the steady-state diffusional flux of dissolved N2 away from a disk electrode of radius r0 , i = 4nFC g Dg r0

(15.6.45)

where Dg and C g represent the diffusivity and concentration of dissolved gas at the electrode surface, respectively. At the moment of nucleation, C g corresponds to the supersaturation concentration required to nucleate the new phase, CgS . CgS = ip ∕4nFDg r0

(15.6.46)

From CgS , one can compute the supersaturation ratio, S, required for gas nucleation, defined as: S = CgS ∕Cg (1 bar) − 1

(15.6.47)

21 Mathematically, the three-phase boundary at the UME disk is defined as a circular line of infinitesimal width. The physical width has been estimated from the residual current (147) and from molecular dynamic simulations (113) to be on the order of the size of a single Pt atom.

709

710

15 Inner-Sphere Electrode Reactions and Electrocatalysis

where C g (1 bar) is the saturation concentration of dissolved gas in equilibrium with 1 bar of gas over the solution. For instance, using ip = 10 nA from Figure 15.6.11a, r0 = 32 nm, and Dg = 2.1 × 10−5 cm2 /s for N2 , CgS is found to be 94 mM. The corresponding supersaturation ratio determined from the voltammogram, S = (94 mM/0.67 mM) − 1 = 139, is in good agreement with values of 136 ≥ S ≥ 90 obtained using classical methods, where a solution is equilibrated with N2 at known high pressures and then suddenly decompressed to release bubbles. Figure 15.6.11b shows that ip is proportional to r0 , demonstrating that CgS and S are independent of the electrode size for r0 < 100 nm. The voltammetric peak associated with nucleation of the N2 bubble appears only in solutions containing at least 0.3 M N2 H4 . Below this concentration, a sigmoidal wave is observed that is characteristic of a diffusion-limited response, with no indication of bubble formation. Above this threshold concentration, ip is independent of the concentration of N2 H4 (up to 2 M), indicating that bubble nucleation is determined by the supersaturation concentration, CgS , of reaction product, N2 , rather than the concentration of the reactant, N2 H4 . If equilibrium applies across the gas/liquid interface, the partial pressure of dissolved gas is related to C g through Henry’s law Cg = KH Pg

(15.6.48)

where K H is the Henry’s law constant. Accordingly, the pressure associated with the critical nucleus, PgS , is defined by, CgS = KH PgS

(15.6.49)

For N2 , PgS = 94 mM∕(0.67 mM bar−1 ) = 140 bar. The radius of the critical nucleus, rc , can be estimated by noting that the internal pressure of a bubble, Pi , can be expressed in terms of the ambient pressure, Pe , and the Laplace pressure, PL , the latter associated with the surface tension of the gas/liquid interface, Pi = PL + Pe

(15.6.50)

where PL is given by the Young–Laplace equation. PL = 2𝛾∕rc.

(15.6.51)

For a subcritical bubble, r < rc , and the large value of PL drives gas molecules out of the bubble, resulting in its collapse. The total pressure acting on the bubble to shrink it, PL + Pe , is countered by the partial pressure of dissolved gas, Pg , which is equal to the bubble’s internal pressure, i.e., Pg = Pi . When the bubble nucleus is at its critical size, Pi = PgS . Combining this result with (15.6.48)–(15.6.51) yields the critical nucleus size: rc = 2𝛾∕(PgS − Pe )

(15.6.52)

Noting that 1 bar = 105 N/m2 = 105 Pa, we find for N2 gas that rc = 2 × 0.073 N m−1 /139 × 105 N m−2 = 1 × 10−8 m = 10 nm. In summary, the single voltammogram in Figure 15.6.11a allows one to determine rc (10 nm) and the pressure of the critical nucleus (PgS = 140 bar) generated by electrochemical supersaturation of the solution with N2 . Values computed here were based on reported figures for 𝛾 at 1 bar and K H for gas dissolution into pure water. More accurate results (rc = 7.9 nm and PgS = 160 bar) can be obtained by accounting for the variation of 𝛾 at high pressure and the reduced solubility of gases in electrolyte solutions (155).

15.6 Electrochemical Phase Transformations

CNT for gas evolution [Section 15.6.2(e)] predicts that the rate of bubble nucleation, J n , increases very rapidly as the partial pressure of the dissolved gas, Pg , increases. Substitution of (15.6.46) and (15.6.49) into (15.6.36) yields J n in terms of the current: ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ 3 Φ(𝜃) 16𝜋𝛾 ⎥ Jn = Jn,0 exp ⎢− ( ) 2⎥ ⎢ i ⎢ 3kT − Pe ⎥⎥ ⎢ K 4nFD r H g 0 ⎣ ⎦

(15.6.53)

This relationship suggests that the preexponential factor, J n, 0 , and activation energy, ΔG‡ , can be obtained from a plot of ln J n vs. [(i/K H 4nFDg r0 ) – Pe ]−2 . Because the rising part of the voltammetric wave corresponds to a steady-state response, one can experimentally measure J n as a function of i by applying a constant E and then measuring the time required to observe nucleation (as indicated by the sudden decrease in current caused by bubble formation on the electrode surface). More precise control of the current is obtained, however, by performing a galvanotstatic experiment in which a strictly constant current is applied, and the potential is measured. A sudden shift in the electrode potential (on the order of 0.5 V) signals gas-phase nucleation at a time, t n , measured from application of the current. An example of this experiment is shown in Figure 15.6.13 for the nucleation of H2 nanobubbles at a 41-nm-radius Pt disk electrode (154). As discussed earlier in connection with Figure 15.6.1b, the reduction of H+ results in nucleation of a H2 nanobubble, causing voltammetric behavior similar to that discussed above for N2 bubble formation during 0 –10

i/nA

–20 Accessible rates –0.8

–0.4 E/ V vs. Ag/AgCl (a)

–30 0.0 8 ln Jn/s–1

i/nA –30

3

–32 –34 –36 –0.62

–0.52 E/ V vs. Ag/AgCl (b)

–2

0 0.8 1.6 1015(i/KH4nFDgr0 – Pe)–2/Pa–2 (c)

Figure 15.6.13 Analysis of nucleation rates for H2 bubble formation at a 41-nm-radius Pt nanodisk electrode in 0.5 M H2 SO4 + 0.05 M KCl. (a) CV beginning at 0.0 V and first scanning negatively. Cathodic currents are down. (b) Enlargement of the i − E curve near the voltammetric peak, also showing values of applied currents (30–36 nA) used to measure Jn . These currents cover a 25-mV potential range. (c) Plot of ln Jn vs. [(i/K H 4nFDg r0 ) – Pe ]−2 . 1 Pascal (Pa) =10−5 bar. [German, Edwards, Ren, and White (154). © 2018, American Chemical Society.]

711

712

15 Inner-Sphere Electrode Reactions and Electrocatalysis

oxidation of N2 H4 (Figure 15.6.11). The measurement of J n is performed by applying a current corresponding to a value on the rising part of the voltammetric wave, as indicated in Figure 15.6.13b. Nucleation is a first-order stochastic process; hence, a different value of t n is obtained each time the experiment is repeated at the same applied current. Since J n is constant for a given current, nucleation can be described as a Poisson process, for which t n from many repeated experiments is described by an exponential distribution. The cumulative probability of nucleation at time t, P(t), is defined as N + /N T , where N + is the number of trials in which a bubble has formed by time t, and N T is the total number of trials at the given applied current. In the limit where N T becomes very large, P(t) becomes a continuous function, P(t) = 1 − e−Jn t

(15.6.54)

Values of J n at each applied current are obtained by fitting (15.6.54) to plots of experimental values of P(t) vs. t (154). Figure 15.6.13c shows the plot of ln J n vs. [(i/K H 4nFDg r0 ) – Pext ]−2 for H+ reduction at the 41-nm Pt disk. The value of Φ(𝜃) is extracted from the slope of the line [= −16𝜋𝛾 3 Φ(𝜃)∕3kT = −2.08 × 1016 Pa2 ], which, employing (15.6.16), yields a nucleus contact angle 𝜃 of 150∘ ± 1∘ . The intercept corresponds to the pre-exponential factor, J n, 0 ∼ 1012 s−1 . The finding that 𝜃 = 150∘ indicates that H2 bubble nucleation is a heterogeneous process (Figure 15.6.4) and that the bubble nucleus has a flattened sphere-cap geometry with a circular footprint of 2.7 nm radius and height of 0.8 nm (Figure 15.6.14). The number of H2 molecules, nc , in this critical nucleus can be estimated using the ideal gas law, the thermodynamic radius of the nucleus, rc (∼5 nm), and the volume of the spherical cap [= 4𝜋rc3 Φ(𝜃); Section 15.6.2(e)]. nc = Pg 4𝜋rc3 Φ(𝜃)∕3kT

(15.6.55)

For the experiments shown in Figure 15.6.13 using the 41-nm radius Pt UME, nc decreases from 55 to 35 H2 molecules as the current increases from −30 to −36 nA, reflecting the increase in PgS and decrease in rc at higher supersaturation (i.e., larger current and higher surface concentration of dissolved H2 ). Table 15.6.1 summarizes parameters for electrochemical nucleation of H2 , N2 , and O2 obtained from galvanostatic (153, 154) and voltammetric measurements (150). An interesting finding is that the size, geometry, and number of molecules defining the nuclei are comparable for these three gases. In each case, the intermolecular distances between the gas molecules in the nucleus are significantly smaller than the corresponding distances between molecules in solution at the critical saturation concentration, CgS , existing just prior to nucleation. For instance, H2 molecules are separated by 0.5 nm within the gas nucleus (at PL = 305 bar), but by 1.9 nm in solution (at CgS ≈ 0.24 M) just prior to nucleation. This finding indicates that h = 0.8 nm θ = 146°

H2 Pt

rfootprint = 2.7 nm

Figure 15.6.14 Cross-sectional geometry of an H2 nucleus at the peak potential, E p , of the voltammogram. The nucleus contains ∼50 H2 molecules (equation (15.6.55)). The curved solid line represents the time-average gas/liquid interface. [Adapted from Edwards, White, and Ren (150). © 2019, American Chemical Society.]

15.7 References

Table 15.6.1 Parameters Describing Electrochemical Nucleation of H2 , N2 , and O2 Phases Gas

Reaction

H2

2H+ + 2e → H2

N2 O2

N2 H4 → N2

C g S /M

PL /bar

rc /nm

𝜽/∘

nc

0.24

305 ± 5

5±1

146 ± 1

61 ± 4

+ 4H+ + 4e

0.11

177 ± 6

8±3

158 ± 1

39 ± 5

H2 O2 → O2 + 2H+ + 2e

0.21

167 ± 3

9±2

154 ± 1

84 ± 6

Modified from Edwards et al. (150).

the dynamics of nucleation represent the expulsion of water and an inward transient flux of gas molecules. The similarity among parameters describing the nucleus size and shape also indicate that the mechanism and dynamics of nucleation are nearly independent of gas type. Given the weak intermolecular forces between molecules of each type (e.g., interaction between two H2 molecules), the results suggest that nucleation is driven primarily by the disruption of the water structure by the supersaturated dissolved gas, rather than by the attraction between the gas molecules. Molecular dynamics simulations of electrochemical gas bubble formation at Pt nanodisks give results that are consistent with the experimental findings and provide additional information about the dynamical shape of the nucleus and stable bubble (156). Activation energies for electrochemical nucleation of single bubbles, measured using different Pt UMEs are variable, ranging, for instance, between 8 and 28 kT for H2 (Problem 15.8). This is not surprising, as heterogenous nucleation is known to be highly dependent on the nature of active surface sites. Each Pt UME has different surface defect structures.

15.7 References 1 E. Gileadi, “Electrode Kinetics for Chemists, Chemical Engineers, and Materials Scientists,”

Wiley-VCH, New York, 1993, pp. 161–169. 2 V. D. Parker, J. Am. Chem. Soc., 98, 98 (1976). 3 NIST Chemistry Webbook, https://webbook.nist.gov/chemistry/, updated 2018. 4 A. Streitwieser, Jr., “Molecular Orbital Theory for Organic Chemists,” Wiley, New York,

1961, Chap. 7. 5 M. E. Peover, Electroanal. Chem., 2, 41 (1967). 6 J. O. Howell, J. M. Goncalves, C. Amatore, L. Klasinc, R. M. Wightman, and J. K. Kochi,

J. Am. Chem. Soc., 106, 3968 (1984). 7 S. F. Nelsen, M. F. Teasley, A. J. Bloodworth, and H. J. Eggelte, J. Org. Chem., 50, 3299

(1985). 8 S. F. Nelsen, J. A. Thompson-Colon, B. Kirste, A. Rosenhouse, and M. Kaftory, J. Am.

Chem. Soc., 109, 7128 (1987). 9 A. J. Bard, J. Am. Chem. Soc., 132, 7559 (2010). 10 D. H. Evans, Chem. Rev., 108, 2113 (2008). 11 J.-M. Leger, C. Coutanceau, and C. Lamy, in “Fuel Cell Catalysis,” M. T. M. Koper, Ed.,

Wiley, Hoboken, NJ, 2009, Chap. 11. 12 M. D. Tissandier, K. A. Cowen, W. Y. Feng, E. Gundlach, M. H. Cohen, A. D. Earhart,

J. V. Coe, and T. R. Tuttle, J. Phys. Chem. A, 102, 7787 (1998). 13 D. A. Armstrong, R. E. Huie, W. H. Koppenol, S. V. Lymar, G. Merényi, P. Neta, B. Ruscic,

D. M. Stanbury, S. Steenken, and P. Wardman, Pure Appl. Chem., 87, 1139 (2015), including Supplementary Material.

713

714

15 Inner-Sphere Electrode Reactions and Electrocatalysis

14 C. Koper, M. Sarobe, and L. W. Jenneskens, Phys. Chem. Chem. Phys., 6, 319 (2004). 15 I. Katsounaros, M. C. Figueiredo, F. Calle-Vallejo, H. Li, A. A. Gewirth, N. M. Markovic,

and M. T. M. Koper, J. Catal., 359, 82 (2018). 16 D. T. Sawyer, G. Chiericato, C. T. Angelis, E. J. Nanni, and T. Tsuchiya, Anal. Chem., 54, 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

1720 (1982). C. Zhang, F.-R. F. Fan, and A. J. Bard, J. Am. Chem. Soc., 131, 177 (2009). B. E. Conway and B. V. Tilak, Electrochim. Acta, 47 3571 (2002). G. Jerkiewicz, Prog. Surf. Sci., 57, 137 (1998). E. Gileadi and E. Kirowa-Eisner, Corros. Sci., 47, 3068 (2005). J. Tafel, Z. Phys. Chem., 50, 641 (1905). R. R. Adži´c, M. D. Spasojevi´c, and A. R. Despi´c, Electrochim. Acta, 24, 569 (1979). T. Shinagawa, A. T. Garcia-Esparza, and K. Takanabe, Sci. Rep., 5, 13801 (2015). Y. Li, H. Wang, L. Xie, Y. Liang, G. Hong, and H. Dai, J. Am. Chem. Soc., 133, 7296 (2011). R. Gómez, A. Fernandez-Vega, J. M. Feliu, and A. Aldaz, J. Phys. Chem., 97, 4769 (1993). M. T. M. Koper, Ed. “Fuel Cell Catalysis,” Wiley, Hoboken, NJ, 2009. M. Shao, Q. Chang, J.-P. Dodelet, and R. Chenitz, Chem. Rev., 116, 3594 (2016). Y. Xu, M. Shao, M. Mavrikakis, and R. R. Adži´c, in “Fuel Cell Catalysis,” M. T. M. Koper, Ed., Wiley, Hoboken, NJ, 2009, Chap. 9. S. Gottesfeld, in “Fuel Cell Catalysis,” M. T. M. Koper, Ed., Wiley, Hoboken, NJ, 2009, Chap. 1. A. Kulkarni, S. Siahrostami, A. Patel, and J. K. Nørskov, Chem. Rev., 118, 2302 (2018). J. Stacy, Y. N. Regmi, B. Leonard, and M. Fan, Renew. Sustain. Energy Rev., 69, 401 (2017). Y. Li, Q. Li, H. Wang, L. Zhang, D. P. Wilkinson, and J. Zhang, Electrochem. Energy Rev., 2, 518 (2019). S. Sui, X. Wang, X. Zhou, Y. Su, S. Riffat, and C. Liu, J. Mater. Chem. A, 5, 1808 (2017). X. Wang, Z. Li, Y. Qu, T. Yuan, W. Wang, Y. Wu, and Y. Li, Chem., 5, 1486 (2019). M. Neurock, M. Janik, and A. Wieckowski, Faraday Discuss., 140, 363 (2009). Z. Juys and R. J. Behm, in “Fuel Cell Catalysis,” M. T. M. Koper, Ed., Wiley, Hoboken, NJ, 2009, Chap. 13. W. Huang, H. Wang, J. Zhou, J. Wang, P. N. Duchesne, D. Muir, P. Zhang, N. Han, F. Zhao, M. Zeng, J. Zhong, C. Jin, Y. Li, S.-T. Lee, and H. Dai, Nat. Commun., 6, 10035 (2015). V. S. Bagotzky, Y. B. Vassiliev, and O. A. Khazova, J. Electroanal. Chem., 81, 229 (1977). C. Korzeniewski and C. L. Childers, J. Phys. Chem. B, 102, 489 (1998). M. T. M. Koper, S. C. S. Lai, and E. Herrero, in “Fuel Cell Catalysis,” M. T. M. Koper, Ed., Wiley, Hoboken, NJ, 2009, Chap. 6. J. K. Foley, C. Korzeniewski, J. L. Daschbach, and S. Pons, Electroanal. Chem., 14, 309 (1986). D. C. Corrigan, E. S. Brandt, and M. J. Weaver, J. Electroanal. Chem., 235, 327 (1987). Y. X. Chen, A. Miki, S. Ye, H. Sakai, and M. Osawa, J. Am. Chem. Soc., 125, 3680 (2003). J. Joo, T. Uchida, A. Cuesta, M. T. M. Koper, and M. Osawa, J. Am. Chem. Soc., 135, 9991 (2013). R. Parsons and T. VanderNoot, J. Electroanal. Chem., 257, 9 (1988). H. E. Hoster and R. J. Behm, in “Fuel Cell Catalysis,” M. T. M. Koper, Ed., Wiley, Hoboken, NJ, 2009, Chap. 14. J.-H. Choi, K.-J. Jeong, Y. Dong, J. Han, T.-H. Lim, J.-S. Lee, and Y.-E. Sung, J. Power Sources, 163, 71 (2006). Z. A. C. Ramli and S. K. Kamarudin, Nanoscale Res. Lett., 13, 410 (2018).

15.7 References

49 S. Nitopi, E. Bertheussen, S. B. Scott, X. Liu, A. K. Engstfeld, S. Horch, B. Seger, I. E. L.

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Stephens, K. Chan, C. Hahn, J. K. Nørskov, T. F. Jaramillo, and I. Chorkendorff, Chem. Rev., 119, 7610 (2019). M. Umeda, Y. Niitsuma, T. Horikawa, S. Matsuda, and M. Osawa, ACS Appl. Energy Mater., 3, 1119 (2020). Y. Hori, A. Murata, and R. Takahashi, J. Chem. Soc., Faraday Trans. 1, 85, 2309 (1989). A. A. Peterson, F. Abild-Pedersen, F. Studt, J. Rossmeisl, and J. K. Nørskov, Energy Environ. Sci., 3, 1311 (2010). E. Boutin, L. Merakeb, B. Ma, B. Boudy, M. Wang, J. Bonin, E. Anxolabéhère-Mallart, and M. Robert, Chem. Soc. Rev., 49, 5772 (2020). Y. Wu, Z. Jiang, X. Lu, Y. Liang, and H. Wang, Nature, 575, 639 (2019). F. Vitse, M. Cooper, and G. G. Botte, J. Power Sources, 142, 18 (2005). A. A. Isse, A. De Giusti, A. Gennaro, L. Falciola, and P. R. Mussini, Electrochim. Acta, 51, 4956 (2006). A. A. Isse, P. R. Mussini, and A. Gennaro, J. Phys. Chem. C, 113, 14983 (2009). C. Bellomunno, D. Bonanomi, L. Falciola, M. Longhi, P. R. Mussini, L. M. Doubova, and G. Di Silvestro, Electrochim. Acta, 50, 2331 (2005). J. Langmaier and Z. Samec, J. Electroanal. Chem., 402, 107 (1996). I. Katsounaros, W. B. Schneider, J. C. Meier, U. Benedikt, P. U. Biedermann, A. A. Auer, and K. J. J. Mayrhofer, Phys. Chem. Chem. Phys., 14, 7384 (2012). C. E. D. Chidsey, Science, 251, 919 (1991). H. O. Finklea and D. D. Hanshew, J. Am. Chem. Soc., 114, 3173 (1992). L. Tender, M. T. Carter, and R. W. Murray, Anal. Chem., 66, 3173 (1994). M. Velický, S. Hu, C. R. Woods, P. S. Tóth, V. Zólyomi, A. K. Geim, H. D. Abruña, K. S. Novoselov, and R. Dryfe, ACS Nano 14, 993 (2020). A. D. Clegg, N. V. Rees, O. V. Klymenko, B. A. Coles, and R. G. Compton, J. Am. Chem Soc., 126, 6185 (2004). M. J. Janik, S. A. Wasileski, C. D. Taylor, and M. Neurock, in “Fuel Cell Catalysis,” M. T. M. Koper, Ed., Wiley, Hoboken, NJ, 2009, Chap. 4. J. Rossmeisl, E. Skúlason, M. E. Björketun, V. Tripkovic, and J. K. Nørskov, Chem. Phys. Lett., 466, 68 (2008). E. Skúlason, Procedia Comput. Sci., 51, 1887 (2015). A. Roldan, Curr. Opin. Electrochem., 10, 1 (2018). C. D. Taylor, S. A. Wasileski, J.-S. Filhol, and M. Neurock, Phys. Rev. B, 73, 165402 (2006). M. J. Janik, C. D. Taylor, and M. Neurock, J. Electrochem. Soc., 156, B126 (2009). J. N. Brønsted, Chem. Rev., 5, 231 (1928). M. G. Evans and M. Polanyi, Trans. Faraday Soc., 34, 11 (1938). E. Skúlason, G. S. Karlberg, J. Rossmeisl, T. Bligaard, J. Greeley, H. Jónsson, and J.K. Nørskov, Phys. Chem. Chem. Phys., 9, 3241 (2007). E. Skúlason, V. Tripkovic, M. E. Björketun, S. Gudmundsdóttir, G. Karlberg, J. Rossmeisl, T. Bligaard, H. Jónsson, and J. K. Nørskov, J. Phys. Chem. C, 114, 18182 (2010). J. K. Nørskov, J. Rossmeisl, A. Logadóttir, L. Lindqvist, J. R. Kitchin, T. Bligaard, and H. Jónsson, J. Phys. Chem. B., 108, 17886 (2004). D. Cao, G.-Q. Lu, A. Wieckowski, S. A. Wasileski, and M. Neurock, J. Phys. Chem. B., 109, 11622 (2005). V. R. Stamenkovic, B. S. Mun, M. Arenz, K. J. J. Mayrhofer, C. A. Lucas, G. Wang, P. N. Ross, and N. M. Markovic, Nat. Mater., 6, 241 (2007). J. Greeley, I. E. L. Stephens, A. S. Bondarenko, T. P. Johansson, H. A. Hansen, T. F. Jaramillo, J. Rossmeisl, I. Chorkendorff, and J. K. Nørskov, Nat. Chem., 1, 552 (2009).

715

716

15 Inner-Sphere Electrode Reactions and Electrocatalysis

80 I. E. L. Stephens, A. S. Bondarenko, F. J. Perez-Alonso, F. Calle-Vallejo, L. Bech, T. P.

81 82 83

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114

Johansson, A. K. Jepsen, R. Frydendal, B. P. Knudsen, J. Rossmeisl, and I. Chorkendorff, J. Am. Chem. Soc., 133, 5485 (2011). M. J. Eslamibidgoli, J. Huang, T. Kadyk, A. Malek, and M. Eikerling, Nano Energy, 29, 334 (2016). D. R. Alfonso, D. N. Tafen, and D. R. Kauffmann, Catalysts, 8, 424 (2018). H. Oberhofer, in “Handbook of Materials Modeling”, W. Andreoni, S. Yip, Eds., Springer Link, Cham, Switzerland, 2018. https://www.scribd.com/document/476221415/handbookof-materials-modeling-2020-pdf. N. Bonnet, T. Morishita, O. Sugino, and M. Otani, Phys. Rev. Lett., 109, 266101 (2012). R. Jinnouchi and A. B. Anderson, J. Phys. Chem. C, 112, 8747 (2008). G. Fisicaro, L. Genovese, O. Andreussi, N. Marzari, and S. Goedecker, J. Chem. Phys., 144, 014103 (2016). Y. Ping, R. J. Nielsen, and W. A. Goddard, J. Am. Chem. Soc., 139, 149 (2017). J. D. Goodpaster, A. T. Bell, and M. Head-Gordon, J. Phys. Chem. Lett., 7, 1471 (2016). N. G. Hörmann, O. Andreussi, and N. Marzari, J. Chem. Phys., 150, 041730 (2019). J. A. Gauthier, S. Ringe, C. F. Dickens, A. J. Garza, A. T. Bell, M. Head-Gordon, J. K. Nørskov, and K. Chan, ACS Catal., 9, 920 (2019). G. Henkelman, B. P. Uberuaga, and H. Jónsson, J. Chem. Phys., 113, 9901 (2000). M. J. Janik and M. Neurock, Electrochim. Acta, 52, 5517 (2007). J. K. Nørskov, F. Studt, F. Abild-Pedersen, and H. Bligaard, “Fundamental Concepts in Heterogeneous Catalysis,” Wiley, Hoboken, NJ, 2014. R. Parsons, Trans. Faraday Soc., 54, 1053 (1958). H. Gerischer, Bull. Soc. Chim. Belg., 67, 506 (1958). M. T. M. Koper, J. Solid State Electrochem., 17, 339 (2013). M. T. M. Koper, J. Solid State Electrochem., 20, 895 (2016). A. R. Zeradjanin, J.-P. Grote, G. Polymeros, and K. J. J. Mayrhofer, Electroanalysis, 28, 2256 (2016). P. Quaino, F. Juarez, E. Santos, and W. Schmickler, Beilstein J. Nanotechnol., 5, 846 (2014). N. M. Markovic, B. N. Grgur, and P. N. Ross, J. Phys. Chem. B, 101, 5405 (1997). J. Zhou, Y. Zu, and A. J. Bard, J. Electroanal. Chem., 491, 22 (2000). C. G. Zoski, J. Phys. Chem. B, 107, 6401 (2003). K. C. Leonard and A. J. Bard, J. Am. Chem. Soc., 135, 15885 (2013). M. S. Whittingham, Chem. Rev., 114, 11414 (2014). M. Volmer and A. Z. Weber, Z. Phys. Chem., 119, 277 (1926). L. Farkas, Z. Phys. Chem., 125, 236 (1927). A. A. Isse, S. Gottardello, C. Maccato, and A. Gennaro, Electrochem. Commun., 8, 1707 (2006). Y. Liu, M. A. Edwards, S. R. German, Q. Chen, and H. S. White, Langmuir, 33, 1845 (2017). E. Budevski, G. Staikov, and W. J. Lorenz, “Electrochemical Phase Formation and Growth,” VCH, Weinheim, 1996. H. Vehkamäki, A. Määttänen, A. Lauri, I. Napari, and M. Kulmala, Atmos. Chem. Phys., 7, 309 (2007). M. Blander and J. L. Katz, AIChE J., 21, 833 (1975). S. Karthika, T. K. Radhakrishnan, and P. Kalaichelvi, Cryst. Growth Des., 16, 6663 (2016). Y. A. Perez Sirkin, E. D. Gadea, D. A. Scherlis, and V. Molinero, J. Am. Chem. Soc., 141, 10801 (2019). S.F. Chernov, Y. Fedorov, and V. N. Zakharov, J. Phys. Chem. Solids, 54, 963 (1993).

15.7 References

115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152

S. Toschev and I. Markov, Ber. Bunsenges. Phys. Chem., 73, 184 (1969). C. F. Delale, J. Hruby, and F. Marsik, J. Chem. Phys., 118, 792 (2003). P. G. Bowers, K. Bar-Eli, and R. M. Noyes, J. Chem. Soc., Faraday Trans., 92, 2843 (1996). D. Larkin and N. Hackerman, J. Electrochem. Soc., 124, 360 (1977). R. T. Pötzschke, C. A. Gervasi, S. Vinzelberg, G. Staikov, and W. J. Lorenz, Electrochim. Acta, 40, 1469 (1995). J. V. Zoval, R. M. Stiger, P. R. Biernacki, and R. M. Penner, J. Phys. Chem., 100, 837 (1996). S. C. S. Lai, R. A. Lazenby, P. M. Kirkman, and P. R. Unwin, Chem. Sci., 6, 1126 (2015). G. Staikov, K. Jüttner, and W. J. Lorenz, Electrochim. Acta, 39, 1019 (1994). W. Obretenov, U. Schmidt, W. J. Lorenz, G. Staikov, E. Budevski, D. Carnal, U. Müller, H. Siegenthaler, and E. Schmidt, Faraday Discuss., 94, 107 (1992). E. Herrero, L. J. Buller, and H. D. Abruña, Chem. Rev., 101, 1897 (2001). J. Li and H. D. Abruña, J. Phys. Chem. B, 101, 2907 (1997). A. Bewick and J. Thomas, J. Electroanal. Chem., 70, 239 (1976). N. Markovic and P. N. Ross, Langmuir, 9, 580 (1993). K. J. Stevenson, D. W. Hatchett, and H. S. White, Langmuir, 12, 494 (1996). M. F. Toney, J. G. Gordon, M. G. Samant, G. L. Borges, O. R. Melroy, D. Yee, and L. B. Sorenson, J. Phys. Chem., 99, 4733 (1995). M. F. Toney, J. N. Howard, J. Richer, G. L. Borges, J. G. Gordon, O. R. Melroy, D. Yee, and L. B. Sorensen. Phys. Rev. Lett., 75, 4472 (1995). D. W. Hatchett and H. S. White, J. Phys. Chem., 100, 9854 (1996). G. Gunawardena, G. Hills, and I. Montenegro, J. Electroanal. Chem., 184, 357 (1985). R. M. Penner, J. Phys. Chem. B, 105, 8672 (2001). A. Henglein, Ber. Bunsenges. Phys. Chem., 94, 600 (1990). A. Henglein, Top. Curr. Chem., 143, 113 (1988). A. Henglein, Acc. Chem. Res., 9, 1861 (1989). J. Zhang, Z. Li, Q. Fu, Y. Xue, and Z. Cui, J. Electrochem. Soc., 164, H828 (2017). W. J. Plieth, J. Phys. Chem., 86, 3166 (1982). L. Tang, X. Li, R. C. Cammarata, C. Friesen, and K. Sieradzki, J. Am. Chem. Soc., 132, 11722 (2010). D. K. Pattadar, J. N. Sharma, B. P. Mainali, and F. P. Zamborini, Curr. Opin. Electrochem., 13, 147 (2019). L. Tang, B. Han, K. Persson, C. Friesen, T. He, K. Sieradzki, and G. Ceder, J. Am. Chem. Soc., 132, 596 (2010). J. L. F. Da Silva, C. Stampfl, and M. Scheffler, Surf. Sci., 600, 703 (2006). D. K. Pattadar and F. P. Zamborini, Langmuir, 35, 16416 (2019). A. Serov and C. Kwak, Appl. Catal., B: Environ. 98, 1 (2010). X. Zhao, H. Ren, and L. Luo, Langmuir, 35, 5392 (2019). A. Angulo, P. van der Linde, H. Gardeniers, M. Modestino, and D. Fernández Rivas, Joule, 4, 555 (2020). L. Luo and H. S. White, Langmuir, 29, 11169 (2013). J. T. Maloy, in “Standard Potentials in Aqueous Solutions,” A. J. Bard, R. Parsons, and J. Jordan, Eds., Marcel Dekker, New York, 1985, pp. 127–139. Q. Chen, H. S. Wiedenroth, S. R. German, and H. S. White, J. Am. Chem. Soc., 137, 12064 (2015). M. A. Edwards, H. S. White, and H. Ren, ACS Nano, 13, 6330 (2019). H. Ren, M. A. Edwards, Y. Wang, and H. S. White, J. Phys. Chem. Lett., 11, 1291 (2020). H. Ren, S. R. German, M. A. Edwards, Q. Chen, and H. S. White, J. Phys. Chem. Lett., 8, 2450 (2017).

717

718

15 Inner-Sphere Electrode Reactions and Electrocatalysis

153 Á. M. Soto, S. R. German, H. Ren, D. van der Meer, D. Lohse, M. A. Edwards, and

H. S. White, Langmuir, 34, 7309 (2018). 154 S. R. German, M. A. Edwards, H. Ren, and H. S. White, J. Am. Chem. Soc., 140, 4047

(2018). 155 S. R. German, M. A. Edwards, Q. Chen, Y. Liu, L. Luo, and H. S. White, Faraday Discuss.,

193, 223 (2016). 156 E. D. Gadea, Y. A. Perez Sirkin, V. Molinero, and D. A. Scherlis, J. Phys. Chem. Lett., 11,

6573 (2020). 157 J. Tafel, Zeit. Physik. Chem., 34, 187 (1900). 158 C. Korzeniewski, V. Climent, and J. M. Feliu, Electroanal. Chem., 24, 75 (2011). 159 E. Herrero, K. Franaszczuk, and A. Wieckowski, J. Phys. Chem., 98, 5074 (1994).

15.8 Problems 15.1 In 1900, Tafel proposed (157). that the HER proceeds by the initial reductive adsorption of H+ , (15.2.12), followed by recombination of two adsorbed hydrogen atoms to produce H2 , (15.2.14). This combination is now known as the Volmer–Tafel mechanism. Five years later (21), he reported rates of H2 evolution at various metals and proposed the well-known Tafel equation, 𝜂 = a + b log i The original data reported for Hg at 26.4 ∘ C in 1 M H2 SO4 are shown below. Note that Tafel reported current densities, j, and used the opposite sign notation for E than is currently used. Assume that his values of E are measured vs. the electrode equilibrium potential.

E/V

j/A cm−2

E/V

j/A cm−2

E/V

j/A cm−2

1.665

0.004

1.824

0.01

1.940

0.10

1.713

0.001

1.858

0.02

1.963

0.14

1.7465

0.002

1.878

0.03

1.989

0.30

1.7665

0.003

1.891

0.04

1.777

0.004

1.912

0.06

(a) Plot E vs. log j and demonstrate that the Tafel equation does indeed describe the i − E behavior for the HER at Hg. (b) Use the slope of the plot in (a) to determine whether Tafel’s proposal for the mechanism of the HER is in agreement with his observations. (c) Determine the exchange current density for the HER and compare it to the value reported in Figure 15.5.2. 0 15.2 Equation (15.2.8) expresses the standard potential of the HER, EH + ∕H , in terms of 2

0 EH + ∕H , plus 𝛽 H and K eq for the Volmer–Tafel mechanism. Derive the equivalent ∙ aq

expression for the Volmer–Heyrovský mechanism.

15.8 Problems

15.3 Based on the mechanism for Cl2 evolution given by (15.3.15) and (15.3.16), compute the Tafel slopes assuming that (a) (15.3.15) is rate-determining. (b) (15.3.16) is rate-determining, at both low (𝜃 < 0.1) and high (𝜃 > 0.9) coverages of Clads . 15.4 Figure 15.8.1 presents CV responses of the three low-index faces of Pt in 0.2 M MeOH and 0.5 M H2 SO4 . Provide an explanation for: (a) The shape of the anodic waves observed on the forward and reverse scans. (b) The absence of hydrogen adsorption waves in the potential region between −0.25 and 0.0 V vs. Ag/AgCl. (c) The dependence of the onset potential of the anodic wave on the crystallographic orientation of the Pt surfaces. Figure 15.8.1 CV at Pt(111), Pt(100) and Pt(110) electrodes in 0.2 M MeOH and 0.5 M aqueous H2 SO4 . v = 50 mV/s. [From Korzeniewski, Climent, and Feliu (158), with permission. Originally from a figure in (40), based on data from Herrero et al. (159).]

–0.4 –0.2 1.0

E/V vs. Ag/AgCl 0.0 0.2 0.4

0.6

0.8

0.6

0.8

j/mA cm–2

Pt(111) 0.5

0.0

j/mA cm–2

2.0 1.5

Pt(100)

1.0 0.5 0.0 25

j/mA cm–2

20

Pt(110)

15 10 5 0 –0.4 –0.2

0 15.5 Derive (15.5.8), expressing i0 vs. ΔGH

0.0 0.2 0.4 E/V vs. Ag/AgCl

for the HER, assuming the Volmer–Tafel mechads

anism. Begin with the Langmuir adsorption isotherm, (15.5.7), and the expression for the exchange current in terms of 𝜃, (15.5.5).

719

720

15 Inner-Sphere Electrode Reactions and Electrocatalysis

15.6 Following the approach used in Problem 15.5 to derive (15.5.8), derive the expression 0 for i0 vs. ΔGH assuming the Volmer–Heyrovský mechanism for the HER. ads

15.7 Using (15.5.8) for the HER assuming a Volmer–Tafel mechanism, show that 0 0 (a) i0 → 0 when ΔGH ≫ 0 or when ΔGH ≪ 0. ads

0 (b) The maximum i0 occurs when ΔGH

ads

∼ 0. ads

15.8 Figure 15.6.13c shows a plot of ln J n vs. [(i/K H 4nFDg r0 ) – Pe ]−2 for the nucleation of H2 bubbles at a 41-nm-radius Pt disk electrode for applied currents between 30 and 36 nA. The slope of the plot, S, is −2.08 × 1016 Pa2 . (a) By comparison of (15.6.3) and (15.6.53), derive the expression for ΔG‡ for nucleation of a H2 nanobubble in terms of S. (b) Using your value of S, compute values of ΔG‡ for i = 30, 32, 34, and 36 nA. Assume K H = 0.78 mM/bar, DH = 4.5 × 10−5 cm2 /s, and n = 2. 2

(c) Your results from part (b) should show that ΔG‡ decreases as i increases. Provide a qualitative explanation of this finding using Figure 15.6.2 and (15.6.8) as guides. (d) Compare the magnitude of ΔG‡ for nucleation to typical values of ΔG‡ for molecular diffusion in aqueous solutions. Comment on how the physical processes of nucleation and diffusion are consistent with the difference in the activation energies.

721

16 Electrochemical Instrumentation This chapter addresses the design, behavior, and limitations of electrochemical instruments. A typical measurement system includes • A potentiostat, for controlling the potential of an electrode, or a galvanostat, for controlling the current through a cell. • A function generator, for creating the desired perturbation at the working electrode. • A recording and display system for measuring and presenting results. Although these functions can be implemented separately, many electrochemical instruments are now fully integrated, with all elements contained in a single package, essentially as described in Figure 5.1.1. Such systems are sometimes called electrochemical workstations or automated potentiostats. While most laboratory work can be conducted with commercial workstations, research in some areas makes demands beyond their capabilities. Very small currents (1 A), and very short timescales ( bR ), the wave is displaced negatively from E0 . If R is ′ adsorbed more strongly (bO < bR ), the wave occurs at more positive potentials than E0 . (b) Nonideal Behavior

The wave shape observed in real systems depends on the actual isotherm and rarely shows the ideal shape of Figure 17.2.1. Nonideal behavior has been treated historically in terms of lateral interactions among adsorbate molecules, which, in effect, cause the binding strength to vary with the surface coverage and composition of the adsorbed layer. For example, if a Frumkin

17.2 Cyclic Voltammetry of Adsorbed Layers

isotherm (23, 28, 29) is assumed, the analog of (17.2.10) is [ ] 𝜃 nF 0′ exp (E − Eads ) = O exp[2𝜈𝜃O (aOR − aO ) + 2𝜈𝜃R (aR − aOR )] RT 𝜃R

(17.2.16)

where aOR , aO , and aR are the O–R, O–O, and R–R interaction parameters (aj > 0 for an attractive interaction and aj < 0 for a repulsive one), respectively. The fractional coverages are 𝜃 O = ΓO /Γm and 𝜃 R = ΓR /Γm , respectively, with Γm being the larger of the saturation concentrations, ΓO, s or ΓR, s . The parameter 𝜈 is the number of water molecules displaced from the surface by adsorption of one O or R. From (17.2.16), one can derive (26, 30) [ ] f (1 − f ) n2 F 2 AvΓ∗ i= (17.2.17) RT 1 − 2𝜈g𝜃T f (1 − f ) where f = 𝜃 O /𝜃 T , 1 − f = 𝜃 R /𝜃 T , 𝜃 T = 𝜃 O + 𝜃 R , g = aO + aR – 2aOR , and Γ* = ΓO + ΓR . The variation of i vs. E in this result is determined by the variation of f with E, which can be worked out in terms of 𝜃 O or 𝜃 R from (17.2.16). Figure 17.2.2a shows i − E curves based on (17.2.17). When the interaction parameter, 𝜈g𝜃 T , is zero, the Langmuir case (Figure 17.2.1) applies and ΔEp,1/2 = 90.6/n mV (T = 25 ∘ C). When 𝜈g𝜃 T > 0, ΔEp,1/2 < 90.6/n mV. Conversely, when 𝜈g𝜃 T < 0, ΔEp,1/2 > 90.6/n mV. Figure 17.2.2b provides example of an experimental voltammogram compared to a theoretical treatment involving interaction parameters. Equations 17.2.16 and 17.2.17 apply to a random distribution of O and R sites in the film. If the film is structured (as in an organized monolayer deposited by the LB technique), there will be an i (n2F2/RT)vAΓ* 1.25 0.75

1.0

0 i/μA 0.1 0.2

–0.1

–2 –0.2 n(E – E 0′ads)/V

0.0

–1.0

0.3

0.2

0.1

E/V vs. SCE (a)

(b)

Figure 17.2.2 (a) Effect of interactions on CV wave shape for an electroactive layer governed by Frumkin isotherms. Values of 𝜈g𝜃 T are shown for each curve. [Adapted from Laviron (30), with permission from Elsevier Science.] (b) CV for reduction and reoxidation of 9,10-phenanthrenequinone irreversibly adsorbed on a basal-plane graphite electrode. A = 0.2 cm2 , Γ* = 1.9 × 10–10 mol/cm2 ; v = 50 mV/s; in 1 M HClO4 . Solid curve: Experimental. Dashed curve: Langmuir case, (17.2.13). Points: Langmuir case with nonideality parameters. [Reprinted with permission from Brown and Anson (31). © 1977, American Chemical Society.]

761

762

17 Electroactive Layers and Modified Electrodes

(a)

ϕM ϕPET

ϕS

Electrode surface

PET

Figure 17.2.3 Profile of electric potential, 𝜙, near an electrode surface modified by an irreversibly adsorbed or covalently attached layer of redox-active species. Dark curves are 𝜙 vs. distance from the electrode surface, x. The locus of the redox centers (shaded circles) is the PET, at distance x PET . The zig-zag pattern denotes an intervening dielectric, e.g., alkyl linking moieties. In (a), the redox centers are in immediate contact with the solution. In (b), they are separated from the solution by an additional dielectric zone. [Reprinted with permission from Smith and White (33). © 1992, American Chemical Society.]

(b)

ϕM ϕPET

ϕS xPET

ordered distribution of the sites, and a statistical mechanical approach is needed to account for the interactions and to find the i − E curve (32). For negative values of the interaction parameter in a structured film, a double wave results, even for a single electrode reaction, while a random distribution produces only a single broadened wave. The use of interaction parameters is always empirical. The parameters are not predicted from fundamental properties of the substances involved but are simply adjusted to provide the best match between theory and the observed voltammetric peak shapes. A more fundamental approach is based on explicit treatment of the electric potential distribution in the double layer, which changes as the electrode potential is swept through the wave (33, 34). The concept, known as the Smith–White model, is illustrated in Figure 17.2.3, showing irreversibly adsorbed or covalently bound redox centers in a plane of electron transfer (PET) at a distance, xPET , from the electrode surface. At the PET, the electric potential is 𝜙PET . In simple theory, electron transfer is driven by the potential difference 𝜙M − 𝜙S , which is a linear fraction of the electrode potential, E. However, the actual driver is 𝜙M − 𝜙S − (𝜙PET − 𝜙S ), and the effective potential is E − (𝜙PET − 𝜙S ). Due to the difference in the electric charges of O and R, 𝜙PET is not only a function of E, but also of the oxidation state of the film, which changes ′ most rapidly at potentials near E0 for the O/R couple. The treatment resembles that of Section 14.7.1. We recognized there that a diffusing species reacts at the OHP, where it experiences an electric potential 𝜙2 , leading to an effective electrode potential E − (𝜙2 − 𝜙S ). In double-layer theory, the scale for 𝜙 is chosen so that 𝜙S = 0; hence, the effective potential simplifies to E − 𝜙2 . The Frumkin correction was developed in Section 14.7.1 to account for the effect of 𝜙2 on the electrode kinetics of dissolved species. In the present situation, the choice of 𝜙S = 0 is also made; hence, the effective potential becomes E − 𝜙PET . In a reversible system, as considered here, the effective potential determines the nernstian balance according to ′

E − 𝜙PET = E0 +

RT ΓO ln nF ΓR

(17.2.18)

By assuming a particular O/R ratio, one can calculate E − 𝜙PET from (17.2.18). Moreover, it is practical to compute 𝜙PET using the methods developed in Section 14.3. With both E − 𝜙PET and 𝜙PET in hand, one also has the value of E corresponding to the assumed O/R ratio.

17.2 Cyclic Voltammetry of Adsorbed Layers

Repeating this process for a series of O/R ratios defines ΓO /ΓR vs. E, then, with application of (7.2.7) and (7.2.12), a predicted voltammogram is obtained. This method accounts explicitly for the location of the redox molecule, xPET , the dielectric constants of the solution and film, the concentration of supporting electrolyte, and the surface concentration of adsorbate. It predicts the shapes of CV curves without the need for adjustable interaction parameters. The Smith–White model successfully accounts for the observed voltammetry in various specific cases, including the shift in peak potential and wave broadening seen for ferrocene-terminated SAMs of alkane thiols (33) and ferrocenylalkanethiolate monolayers (35). This electrostatic treatment has been extended by incorporating the effects of discreteness of charge (36), ion pairing (34, 37), and slow electron transfer (38) on the voltammetric wave shape. Lateral interactions that are chemical in nature, e.g., strong interactions between moieties of different molecules or even bond formation, can also occur in electroactive surface layers. Sometimes, a surface layer can even undergo a two-dimensional structural transformation such as reorientation of the molecules in the layer or their condensation into an ordered phase. Such an event causes a discontinuity in the voltammetry (often a current spike). An example (39) is seen in Figure 17.2.4, which concerns a strongly adsorbed layer of anthraquinone-2,6-disulfonate (2,6-AQDS; Figure 1) on a mercury electrode in 0.1 M HNO3 . The system undergoes the electrode reaction, 2,6-AQDS + 2H+ + 2e ⇌ 2,6-DHADS

(17.2.19)

where 2,6-DHADS is 9,10-dihydroxyanthracene-2,6-disulfonate (Figure 1 and Figure 17.2.4c). At a low bulk concentration of 2,6-AQDS (Figure 17.2.4a), the CV shows only a response for the adsorbate layer and the behavior is almost ideal. The forward and reverse peaks are only 1 mV apart, and their respective values of ΔEp, 1/2 are 48 and 45.6 mV. The peak currents differ by only about 10%. At the bulk concentration used for Figure 17.2.4a, the isotherm indicates that the surface is essentially saturated. The surface area per molecule, 𝜎, is 1.8 nm2 /molecule (180 Å2 /molecule), consistent with a flat orientation. When the bulk concentration of 2,6-AQDS is elevated 100-fold, the adsorption wave (near 0.0 V in Figure 17.2.4b) becomes much more complicated. In the forward scan, the wave begins as in Figure 17.2.4a, but a very narrow spike appears just after the peak is passed. The current falls sharply after the spike, cutting off the tail of the wave. This behavior, appearing only at high concentrations of both 2,6-AQDS and HNO3 , is an indication of reorganization into a new two-dimensional phase (23). The reorganization was ascribed (39) to the capacity of the reduced species, 2,6-DHADS, to organize into large networks by hydrogen bonding, as illustrated in Figure 17.2.4c. While a sizable body of observation is consistent with this hypothesis, the structural condensation may depend on other factors, including the composition and charge array in the diffuse layer. Electrochemical evidence alone is rarely sufficient to confirm a structural model.

17.2.3

Irreversible Adsorbate Couples

For Oads reduced in a totally irreversible one-step, one-electron reaction (23, 25, 26), one can write k 0 ,𝛼

Oads + e −−−−→ Rads

′

0 ) (Eads

(17.2.20)

where the standard rate constant, k 0 , has units of s−1 . The langmuirian–nernstian boundary condition (17.2.10) is then replaced by a kinetic condition resembling (7.4.1), which applies to

763

17 Electroactive Layers and Modified Electrodes

Anodic Cathodic

0.5 μA

0.15 0.0

E/V vs. Ag/AgCl/KCl (1 M)

–0.5

OH SO3– –O S 3

H OH

(a)

O H SO3–

(c)

–O S 3

OH

Anodic Cathodic

764

2 μA

0.2

0.0

–0.5 –1.0 E/V vs. Ag/AgCl/KCl (1 M)

(b)

Figure 17.2.4 CV of 2,6-AQDS at an HMDE in 0.1 M HNO3 . Scans begin at the positive limit and first move negatively. v = 200 mV/s. (a) 1 × 10−5 M 2,6-ADQS; (b) 1 × 10−3 M 2,6-ADQS; (c) structural diagram of 2,6-DHADS showing capacity for network development based on hydrogen bonding. In (b), the adsorption wave is the small feature near 0.0 V in the forward scan. The large cathodic wave at more negative potentials, as well as most of the anodic response, is due to diffusing 2,6-AQDS and 2,6-DHADS [Section 17.2.4(a)]. The peak current for the adsorption peak in (b) (before the spike) is similar to the forward peak current in (a). Current scale in (b) is about 8 times larger than in (a). [Adapted with permission from He, Crooks, and Faulkner (39). © 1990, American Chemical Society.]

Osoln + e → Rsoln : i = kf ΓO (t) FA In (17.2.21), k f is still given in the form of (3.3.7a). For LSV, E = Ei – vt; hence, kf = k 0 e

′

0 ) −𝛼f (Ei −Eads 𝛼fvt

e

= kfi e𝛼fvt

(17.2.21)

(17.2.22)

where k fi is the rate constant at Ei . By combining (17.2.12), (17.2.21), and (17.2.22), we obtain dΓO (t)

= −kfi e𝛼fvt ΓO (t) dt With the initial condition that ΓO (0) = Γ* , (17.2.23) is solved to give ΓO (t) = Γ∗ eki ∕𝛼fv e−kf ∕𝛼fv

(17.2.23)

(17.2.24)

17.2 Cyclic Voltammetry of Adsorbed Layers

i (F2/RT)αvAΓ*

0.4 N

N

0.3

trans-1,2-di(4-pyridyl)ethylene (c) 0.2 0.1 μA 0.1

150

100

50

0

–50

–0.4

–0.5

–0.6

α(E – Ep)/mV

E/V

(a)

(b)

Figure 17.2.5 LSV for a system where Oads is irreversibly reduced. (a) Theoretical curve from (17.2.23). (b) Experimental result at an Hg drop (A = 0.017 cm2 ) in 0.05 M H2 SO4 ; v = 100 mV/s. (c) Adsorbate used in (b) (5 μM in solution). Points in (b) were calculated by treating this n = 2 reaction as having a 1e rate-determining first step with 𝛼 = 0.6 [Section 3.7.5(a)]. [Adapted from Laviron (26), with permission.]

If the sweep begins at a sufficiently positive potential that k fi → 0, then exp(k fi /𝛼fv) → 1, and we obtain expressions for both ΓO (t) and the i − E curve: ΓO (t) = Γ∗ e−kf ∕𝛼fv

(17.2.25a)

i = FAk f Γ∗ e−kf ∕𝛼fv

(17.2.25b)

The potential dependence of the current results from substitution for k f according to (17.2.22). The shape of the reduction peak is shown dimensionlessly in Figure 17.2.5a, and its principal characteristics are readily derived from (17.2.25b): ip =

𝛼F 2 AvΓ∗ 2.718RT

(17.2.26)

Ep =

0′ Eads

(17.2.27)

ΔEp,1∕2

( ) RT RT k 0 + ln 𝛼F 𝛼F v ( ) RT = 2.44 𝛼F

(62.7∕𝛼 mV at 25∘ C)

(17.2.28)

Again, ip is proportional to v; however, the wave is unsymmetrical and is shifted negatively from ′

0 . The peak width depends on 𝛼, but is independent the peak potential for the reversible case, Eads of both k 0 and v. An experimental example is shown in Figure 17.2.5b. Treatment of a quasireversible one-step, one-electron reaction follows the approach taken above but must include the back reaction [via (3.2.8)] and adsorption isotherms for both O and R. This case, as well as variants where coupled chemical reactions are associated with the electron-transfer reactions, has been discussed (23, 25, 26, 40, 41). Moreover, the Smith–White model [Section 17.2.2(b)] has been elaborated (38) to allow treatment of ion-pairing and double-layer effects in quasireversible systems.

765

766

17 Electroactive Layers and Modified Electrodes

17.2.4

Nernstian Processes Involving Adsorbates and Solutes

When the electrode process involves several or all of Oads , Rads , Osoln , and Rsoln , the theoretical treatment requires the full flux equation, (17.2.1), along with adsorption isotherms, the usual diffusion equations, and initial and semi-infinite conditions, as discussed in Section 17.2.1. Since the partial differential equations involving mass transfer must be employed, the mathematical treatment is more complicated. Here, we simplify by considering only nernstian systems in which either the electroreactant, O, or the electroproduct, R, is adsorbed, but not both (42). (a) Product (R) Strongly Adsorbed ∗ ≥ 100). Initially, C (x, 0) = C ∗ , C (x, 0) = 0, and For this case, 𝛽 O → 0 and 𝛽 R is large (𝛽R CO O R O ∗ ΓR = 0. To be solved are:

• • • •

Diffusion equations for O and R The flux equation, (17.2.1) The adsorption isotherm for species R, (17.2.3) Equation (17.2.9), defining the nernstian kinetics.

Adsorption equilibrium is assumed always to apply. The mathematics generally follow the process described in Section 17.2.1. The original treatment (42) accommodated the possibility that 𝛽 R varies with potential according to 𝛽R = 𝛽R0 e𝜎R nf (E−E1∕2 )

(17.2.29)

where 𝜎 R is a parameter defining the variation. When 𝜎 R = 0, 𝛽 R is independent of E. In the CV, a prewave (or prepeak) appears (Figure 17.2.6a), having the shape and properties described in Section 17.2.2 and representing the reduction of Osoln to form a layer of Rads , Osoln + ne ⇌ Rads

′

0 (Es∕ads )

(17.2.30) ′

The prewave occurs at potentials more positive than E0 for the diffusion-controlled wave because the free energy of adsorption makes reduction of Osoln to Rads easier than to Rsoln . Following the prewave in the forward scan is the response from Osoln /Rsoln , Osoln + ne ⇌ Rsoln

′

(E0 )

(17.2.31)

While the latter is largely as observed in the absence of adsorption, it is perturbed by the depletion of species O earlier in the forward scan by reduction of Osoln to Rads . A larger value of 𝛽 R implies a more negative free energy of formation of Rads and shows up as a greater positive shift of the prepeak vs. the diffusion peak (Figure 17.2.6b). The case just described resembles that seen in Figure 17.2.4b for the 2,6-AQDS/2,6-DHADS system at high bulk concentrations, but there are two important differences there. First, the electron-transfer kinetics for diffusing 2,6-AQDS/2,6-DHADS are quasireversible; hence, there is a significant separation of the corresponding forward and reverse CV waves. This effect places the reverse diffusion peak on top of the reverse adsorption peak, so that the latter can no longer be distinguished. For Figure 17.2.4a, the bulk concentration is simply too low for the diffusion wave to be visible. The second important difference in the 2,6-AQDS/2,6-DHADS system is that both O and R are adsorbed, but R is adsorbed significantly more strongly. The existence of a prepeak rests on the stronger binding of R, not on the absence of binding of O. Returning now to the fully reversible case being treated in this section, we note that the peak current of the prewave, ip,ads , increases with v [Section 17.2.2(a)], while that for the diffusion wave, ip,d , varies with v1/2 ; therefore, ip,ads /ip,d increases with increasing v (Figure 17.2.7a).

i

(n2F2/RT)ACO* DO1/2v1/2

i

(n2F2/RT)ACO* DO1/2v1/2

17.2 Cyclic Voltammetry of Adsorbed Layers

0.4

1

2

3

4

0.2

0.0 100

0

300

–100

200

100

n(E – E1/2)/mV

n(E – E1/2)/mV

(a)

(b)

0

–100

Figure 17.2.6 (a) Solid curve: CV for reduction when the product is strongly adsorbed, showing a prepeak. Scan begins at positive limit and first moves negatively. Dashed curve: In the absence of adsorption. (b) LSV for reduction when the product is strongly adsorbed. Adsorption energy increases with 𝛽R0 , expressed dimensionlessly as 4ΓR,s 𝛽R0 (nfv)1∕2 ∕(𝜋DR )1∕2 . Values of this parameter are (1) 2.5 × 106 , (2) 2.5 × 105 , (3) ∗ (𝜋D )1∕2 ∕[4Γ (nfv)1∕2 ] = 1, 𝜎 f = 0.05 mV−1 . [Reprinted with permission from 2.5 × 104 , (4) 2.5 × 103 . CO O R,s R Wopschall and Shain (42). © 1967, American Chemical Society.]

For very low bulk concentrations, only the prepeak is observed (assuming that a significant ∗ increases, Γ also increases; therefore, both the prepeak and amount of R is adsorbed). As CO R diffusion wave grow. However, ip,ads attains a limiting value as ΓR approaches ΓR,s , and ip,ads /ip,d ∗ (Figure 17.2.7b) because the diffusion peak is still then begins to decrease with increasing CO ∗ growing with CO , but the adsorption peak is not. If 𝛽 R depends on potential according to (17.2.29), the width of the prepeak, ΔEp,1/2 , varies with 𝜎 R . For 𝜎 R f = 0 mV−1 , ΔEp,1/2 = 90.6/n mV (Section 17.2.2), but for 𝜎 R f = 0.4 mV−1 , the prepeak narrows greatly, to 7.5/n mV. Details concerning the theory and treatment of data are given in reference (42). (b) Reactant (O) Strongly Adsorbed ∗ ≥ 100; Now, let us examine the opposite situation in which O is adsorbed, but R is not (𝛽O CO 𝛽 R → 0),

Oads + ne ⇌ Rsoln

′

0 (Eads∕s )

(17.2.32)

In the forward scan, the result is a postwave (or postpeak) (Figure 17.2.8) following the peak for 0′ diffusion-controlled reduction of Osoln to Rsoln (reaction 17.2.31). The negative shift of Eads∕s ′

vs. E0 reflects the stabilization of Oads compared to Osoln . In the forward scan, the diffusion wave is unperturbed by the adsorption of O if the adsorption equilibrium has been established and the solution has been homogenized before the scan is initiated. On reversal, the diffusion wave is only slightly perturbed. The reduction of dissolved O may occur either by mediation at the adsorbed layer or at the free surface. The postwave has the typical bell shape, as well as the other properties of adsorption waves discussed in Sections 17.2.2 and 17.2.4(a). We can again turn to Figure 17.2.4b for an example simply by thinking about the CV as though it began at the negative limit and first moved positively. With the long early stage of the positive scan, the system builds up a strongly reduced diffusion layer, hardly different for the purpose of

767

17 Electroactive Layers and Modified Electrodes

i

(n2F2/RT)ACO* DO1/2v1/2

0.4 1

2

0.2

3 0.0 200

100

0

–100

n(E – E1/2)/mV (a) 3 1

i

0.4 (n2F2/RT)ACO* DO1/2v1/2

768

1

0.2 2 3 0.0 200

100

0

–100

n(E – E1/2)/mV (b)

Figure 17.2.7 LSV when the product is strongly adsorbed. (a) Effect of v and ΓR, s . These variables are ∗ (𝜋D )1∕2 , which has values of (1) 1.6, (2) 0.8, (3) 0.2. With all expressed dimensionlessly as 4ΓR,s v 1∕2 (nf )1∕2 ∕CO O

∗ (D ∕D )1∕2 = 2.5 × 105 . parameters constant except v, relative scan rates are 64:16:1. 𝜎 R f = 0.05 mV−1 , 𝛽R0 CO O R ∗ , expressed as C ∗ (𝜋D )1∕2 ∕[4Γ (nfv)1∕2 ], which has values of (1) 0.5, (2) 2.0, (3) 8.0. (b) Effect of CO O R,s O

𝜎 R f = 0.05 mV−1 , 4ΓR,s 𝛽R0 (nfv)1∕2 ∕(𝜋DR )1∕2 = 1.0 × 106 . [Reprinted with permission from Wopschall and Shain (42). © 1967, American Chemical Society.]

the CV than a bulk solution of 2,6-DHADS. The adsorption waves are postpeaks relative to an initial oxidation process because the “initial reactant,” 2,6-DHADS, is more strongly adsorbed than the “initial product,” 2,6-AQDS. (c) Reactant (O) Weakly Adsorbed ∗ ≤ 2; 𝛽 → 0), the energy difference between the reducWhen adsorption of O is weak (𝛽O CO R tions of Oads and Osoln is small and a separate postwave is not observed (Figure 17.2.9a). The main effect is an increase in the height of the cathodic peak, compared to the expectation in the absence of adsorption, because both Oads and Osoln contribute to the current. The anodic

Figure 17.2.8 Solid curve: CV showing a postpeak when the reactant, O, is strongly adsorbed. Scan begins at the positive limit and first moves negatively. Dashed curve: In the absence of adsorption. [Reprinted with permission from Wopschall and Shain (42). © 1967, American Chemical Society.]

i (n2F2/RT)ACO* DO1/2v1/2

17.2 Cyclic Voltammetry of Adsorbed Layers

100

0

–100

n(E – E1/2)/mV

0.5

1 2 3

1

2 3

i

0.4

(n2F2/RT)ACO* DO1/2v1/2

i

(n2F2/RT)ACO* DO1/2v1/2

0.8

0.0

3

–0.5

0.0

2

3 2 1

–0.4 50

0

1

–1.0 –50

–100

–150

100

0

–100

n(E – E1/2)/mV

n(E – E1/2)/mV

(a)

(b)

∗ = 0.01). Scan Figure 17.2.9 Effect of v on CV in cases of weak adsorption. (a) Weakly adsorbed reactant (𝛽O CO 1/2 1/2 1/2 rate is expressed dimensionlessly as 4ΓO, s 𝛽 O v (nf ) /(𝜋DO ) , having values of (1) 5.0, (2) 1.0, (3), 0.1. ∗ = 0.01). Scan rate expressed as Relative scan rates are 2500:100:1. (b) Weakly adsorbed product (𝛽R CO 1/2 1/2 1/2 4ΓR, s 𝛽 R v (nf ) /(𝜋DR ) , having values of (1) 20, (2) 5, (3) 0.1. Relative scan rates are 4 × 104 : 2500 : 1. In each case, (3) approaches the response in the absence of adsorption. [Reprinted with permission from Wopschall and Shain (42). © 1967, American Chemical Society.]

current on reversal is also increased, but to a smaller extent, and only because there is a larger amount of R near the electrode at the time of scan reversal. As in the case of strong adsorption of O, the relative contribution of Oads increases in the forward scan at greater scan rates (Figure 17.2.10a). In the limit of very high v, ipf approaches a

769

2.0

O

* D 1/2v1/2

17 Electroactive Layers and Modified Electrodes

reactant adsorbed

O

ipf

0.446(n2F 2/RT)AC

770

1.0 product adsorbed

0.0

0.1

1

4Γj,s βj (nf)

1/2 1/2

v

10 1/2

/(πDj )

(a) 2.0 product adsorbed ipr ipf

1.0 reactant adsorbed

0.0

0.1

1 4Γj,s βj (nf)

10 1/2 1/2

v

100

1/2

/(πDj )

(b)

Figure 17.2.10 Behavior with scan rate in CV when the electroreactant or electroproduct is weakly adsorbed. ∗ = 1. For (b), (a) Forward peak current. (b) Reversal ratio. Index j applies to the adsorbed participant. 𝛽j CO reversal was at 180/n mV beyond E 1/2 . [Adapted with permission from Wopschall and Shain (42). © 1967, American Chemical Society.]

proportionality with v, while at very low v, ipf ∝ v1/2 (Problem 17.3). The reversal ratio, |ipr /ipf |, is a function of v and is smaller than unity (Figure 17.2.10b). As with strong adsorption, the ∗. relative contribution of the effect of adsorption decreases at high CO (d) Product (R) Weakly Adsorbed ∗ ≤ 2), the cathodic current on the forward scan in When R is weakly adsorbed (𝛽 O → 0; 𝛽R CO CV is only slightly perturbed, while the anodic current on reversal is enhanced (Figure 17.2.9b). The forward peak shifts slightly toward more positive potentials with increasing v because there is a decrease in Rsoln near the electrode surface resulting from uptake by adsorption. The effect resembles the positive shift observed in Epc when R is involved in a following reaction (Case Er Ci ; Section 13.3.1). The reversal ratio, |ipr /ipf |, is greater than unity and decreases with decreasing v (Figure 17.2.10b).

17.2.5

More Complex Systems

Adsorption effects in CV often involve considerations that cannot be addressed using the methods employed in Sections 17.2.2–17.2.4. Among them are

17.2 Cyclic Voltammetry of Adsorbed Layers

• Adsorption of both O and R when electroactivity for the diffusing species is also significant. • Quasireversible or totally irreversible heterogeneous kinetics for any or all electrode reactions. • Lateral interactions in the adsorbate layer, requiring more complex isotherms. • Kinetic control by the rate of adsorption. • Coupling of redox processes involving adsorbates with homogeneous chemistry or other surface chemistry. More complex problems must usually be addressed numerically. Simulation applies if the electrode processes involve diffusing species (43), and some commercial electrochemical simulators have built-in options for accommodating items on the list above. If mass transfer is not involved because only adsorbates participate in the electrode process, other numeric methods can be used to predict responses. Comparative simulation (e.g., Sections 13.2.5 and 13.3.9) is the most reliable way to evaluate parameters from real CV results; however, the reader is forewarned that proposed mechanisms involving both adsorbed and diffusing species involve more parameters than mechanisms based on diffusing species alone. The number of parameters can easily exceed a practical number for evaluation (Section 13.3.9). In that event, the investigator must simplify the model or find independent means for evaluating some parameters. 17.2.6

Electric-Field-Driven Acid–Base Chemistry in Adsorbate Layers

An interesting consequence of the electrostatics at electrode surfaces is that one can observe voltammetric peaks associated with the reversible dissociation of acid groups contained within an irreversibly adsorbed layer. This behavior was predicted (44) from the electrostatics of the double layer for an electrode bearing a monolayer of a weak acid or weak base. It was subsequently verified experimentally (45). The basic ideas are represented in Figure 17.2.11, which describes a mixed SAM created from 11-mercaptoundecanoic acid [HS(CH2 )10 COOH] and 1-decanethiol [HS(CH2 )9 CH3 ]. Neither molecule in the mixed SAM displays faradaic chemistry within a moderate potential window in aqueous solutions (−1 to 1 V). However, the film’s state of charge can be altered because the degree of –COOH dissociation depends on the electrostatic profile in the double layer. Under certain experimental conditions, this electric-field-driven acid–base chemistry results in a reversible voltammetric response that resembles the faradaic behavior of an adsorbed redox molecule. The situation of greatest interest is where the pH of the bulk solution is near pK a of the bound acid, and the electrode potential is not far from the PZC (Figure 17.2.11b). Under these conditions, the acid is partially ionized to a degree that is controlled not just by the pH, but also by the electrode potential. The latter effect occurs because of the electrostatic field, E = −𝜕𝜙∕𝜕x, in the double layer at the plane of acid dissociation (PAD). As the electrode becomes more positive, the field tends to increase the fraction of dissociation and vice versa. Accordingly, the charge density at the PAD becomes potential-dependent, implying a larger interfacial capacitance than if this effect did not operate. For Figure 17.2.11a, the behavior is different. Either of two conditions apply: The solution pH is much lower than pK a , or the potential is much more negative than the PZC. At low pH, the acid is fully protonated. If, alternatively, the potential is very negative, the field at the PAD inhibits dissociation of the acid, even for higher pH. In either case, small changes of potential do not affect the state of charge of the film, so the interfacial capacitance is low and invariant with potential.

771

772

17 Electroactive Layers and Modified Electrodes

S (a)

Ag

PAD

S S

pH < pKa or E pKa or E >> PZC

COO–

The situation in Figure 17.2.11c is opposite that in Figure 17.2.11a: The pH is much higher than pK a , or the potential is much more positive than the PZC. In either case, the acid groups are essentially fully dissociated. When CV is carried out on this system, there is only a charging current. At high or low pH relative to pK a , the interfacial capacitance near the PZC does not change much with potential because the state of protonation is invariant; thus, the CV should be flat in both scan directions. However, at a pH near pK a , the behavior should differ because the film is partially dissociated, and the degree of dissociation is potential-dependent. The interfacial capacitance has a maximum vs. potential near the PZC; accordingly, the cyclic voltammetry should show a peak charging current at approximately the same potential in both scan directions. Exactly, this behavior was borne out experimentally (45) for mixed SAMs like those of Figure 17.2.11. An illustration of these effects is provided in Figure 17.2.12a, which relates to pure SAMs of 11-mercaptoundecanoic acid (46). The treatment of this problem (44) was drawn, in effect, from the Smith–White model for CV of reversible electroactive adsorbate layers [Section 17.2.2(b)]. Assuming fast protonation/deprotonation, the fractional dissociation, f = ΓA− ∕(ΓHA + ΓA− ), is given by log

F(𝜙PAD − 𝜙S ) f = pH − pKaPAD + 1−f 2.303RT

(17.2.33)

where the subscripts HA and A− relate to the protonated and deprotonated forms, respectively, and 𝜙PAD and 𝜙S are the electric potentials at the PAD and in bulk solution, respectively. The pK a is specifically noted as applying to the PAD and may differ from the value in bulk aqueous solution. Equation 17.2.33 embodies all behavior outlined qualitatively above. This important relationship is derived by assuming equilibrium for the acid–base reaction at the PAD and equilibrium between H+ at the PAD and in the bulk solution. These two conditions,

17.2 Cyclic Voltammetry of Adsorbed Layers

(a) pH 10.0

(b) –0.20

pH 9.5

pH 9.1

Ep/V vs. SCE

–0.22 –0.24 –0.26 –0.28 –0.30 –0.32

200 nA/cm2

–0.34 pH 9.0

8.0

8.5

9.0 pH

9.5

10.0

8.5

9.0 pH

9.5

10.0

(c) 140

pH 8.5

ip/nA cm–2

120 pH 8.75

100 80 60 40

pH 7.5

20 0 8.0

–0.6 –0.4 –0.2 0.0 E/V vs. SCE

Figure 17.2.12 CV behavior of a SAM of 11-mercaptoundecanoic acid on a polycrystalline Au bead. (a) Variation of CV response vs. bulk pH in 50 mM NaF (pH adjusted with KOH or HClO4 ). More positive potentials are plotted to the right, and “anodic” currents are up. v = 20 mV/s. The vertical line corresponds to the potential where ip reaches its greatest value (near pH 9.0). (b) E p for the positive-going scan (upper in each case) vs. bulk pH. Slope is 67 mV/pH unit. (c) ip for the positive-going scan vs. bulk pH. [Adapted with permission from Burgess, Seivewright, and Lennox (46). © 2006, American Chemical Society.]

expressed in terms of electrochemical potentials (Section 2.2.4), become PAD PAD 𝜇 PAD HA = 𝜇 A− + 𝜇 H+

(17.2.34)

S 𝜇 PAD H+ = 𝜇 H+

(17.2.35)

Substitution of (17.2.35) into (17.2.34) followed by expansion of the electrochemical potentials (Section 2.2.4) gives 0PAD 0PAD 0S S S 𝜇HA + RT ln aPAD = 𝜇A + RT ln aPAD − F𝜙PAD + 𝜇H − + + RT ln aH+ + F𝜙 HA A−

(17.2.36) At the PAD, the acid dissociation constant, KaPAD , is given as 0PAD + 𝜇 0PAD − 𝜇 0PAD −RT ln KaPAD = 𝜇A − HA H+

(17.2.37)

where the right side is ΔG0 for the dissociation of HA at the PAD. By rearrangement of 0PAD − 𝜇 0PAD in (17.2.36), we obtain (17.2.37) and substitution for 𝜇A − HA −RT ln

aPAD A− aPAD HA

0PAD 0S = −RT ln KaPAD + RT ln aSH+ − F(𝜙PAD − 𝜙S ) − (𝜇H − 𝜇H + + ) (17.2.38)

773

774

17 Electroactive Layers and Modified Electrodes

With recognition that aPAD ∕aPAD ≈ f ∕(1 − f ) and conversion to common logarithms, (17.2.38) A− HA becomes log

0PAD 0S F(𝜙PAD − 𝜙S ) 𝜇H+ − 𝜇H+ f = pH − pKaPAD + + 1−f 2.303RT 2.303RT

(17.2.39)

At the PAD, the solvated environment for H+ is essentially the same as in solution; hence, we 0PAD ≅ 𝜇 0S . Thus, (17.2.39) yields (17.2.33). can assume that 𝜇H + H+ The voltammetric current associated with the acid–base chemistry results from charging of the electrode as the potential is varied. It can be expressed as (17.2.40)

i = CT v

where C T is the total interfacial capacitance. An analytical expression for C T can be obtained in terms of an equivalent circuit consisting of a Helmholtz capacitance3 in parallel with a complex impedance based on potential-dependent rates of deprotonation and protonation. The impedance model yields excellent predictions of the general shape of the acid–base voltammogram, as well as the dependence of ip on pH (46). In general, the net charge density at the PAD has its highest rate of change with potential or pH when the film is half dissociated and the ratio f /(1 − f ) is approximately unity. Consequently, C T and the voltammetric response should maximize with pH approximately when log[ f /(1 − f )] = 0, or when pH = pKaPAD −

F(𝜙PAD − 𝜙S )

(17.2.41)

2.303RT

If the second term is small [as expected near the PZC (Section 14.3)], the largest voltammetric response vs. pH should be found near pKaPAD for the acid. Figure 17.2.12c bears out the qualitative expectation of a peak pH and suggests that pKaPAD is about 8.9. For the mixed SAMs (45) of Figure 17.2.11, the peak was near pH 8.5. The agreement between these two studies is good, but the indicated pKaPAD is about 4 units higher than the values of pK a for aliphatic acids in water. The difference has been ascribed to the less polar environment encountered by the carboxylic groups at the PAD (46). Now let us rewrite (17.2.41) as 2.303RT 2.303RT pH + pKaPAD (17.2.42) F F where the subscript “p” relates to the peak response in a voltammogram. The peak potential is Ep = (𝜙M − 𝜙PAD )p + (𝜙PAD − 𝜙S )p + (𝜙S − 𝜙ref ), in which 𝜙ref is the electric potential of the reference electrode contact. The last term is invariant and needs no subscript. We can use (17.2.42) to write (𝜙PAD − 𝜙S )p = −

Ep = (𝜙M − 𝜙PAD )p + (𝜙S − 𝜙ref ) −

2.303RT 2.303RT pH + pKaPAD F F

(17.2.43)

A change in pH, ΔpH, shifts the peak potential by ΔEp . Noting that (𝜙S − 𝜙ref ) and pKaPAD are independent of pH, we find from (17.2.43) that ΔEp = Δ(𝜙M − 𝜙PAD )p −

2.303RT ΔpH F

(17.2.44)

3 In this case, the Helmholtz capacitance (Section 14.3.1) is 𝜀0 𝜀F /xPAD , where 𝜀F is the dielectric constant of the film and xPAD is the distance of the PAD from the electrode surface.

17.3 Other Useful Methods for Adsorbed Monolayers

The quantity (𝜙M − 𝜙PAD )p is a function of pH, since 𝜙PAD depends on pH in a manner that we now develop more explicitly. Let 𝛽 be the fractional part of ΔEp that occurs across the film, i.e., 𝛽 = Δ(𝜙M − 𝜙PAD )p ∕ΔEp

(17.2.45)

Since 𝜙M must track Ep exactly, Δ𝜙M /ΔEp = 1, and 𝛽 = 1 − Δ(𝜙PAD )p ∕ΔEp

(17.2.46)

Over a narrow potential range, 𝛽 may be fairly constant; thus, one can substitute (17.2.45) into (17.2.44) to obtain 2.303RT ΔEp = 𝛽ΔEp − ΔpH (17.2.47) F or 2.303RT ΔEp = − ΔpH (17.2.48) (1 − 𝛽)F This equation indicates that a plot of Ep vs. pH should be linear and have a slope of −2.303RT/(1 − 𝛽)F. The experimental results in Figure 17.2.12b bear out this expectation. The slope there is −67 mV, yielding 𝛽 = 0.12. From (17.2.46), we find that Δ(𝜙PAD )p /ΔEp = 0.88. In the experiments of Figure 17.2.12b, a pH change of 1.75 units produces ΔEp = − 120 mV. According to the model leading to (17.2.48), 𝜙PAD decreases by 105 mV over this pH change and the total potential change is partitioned across the interface as Δ(𝜙M − 𝜙PAD )p = − 15 mV and Δ(𝜙PAD − 𝜙S )p = − 105 mV. Electric-field-driven protonation/deprotonation has been proposed as a mechanism for enhancing charge storage in supercapacitors (47), as well as being a factor determining the potential at which proton-coupled electron transfer (PCET) occurs for molecules located at the electrode/electrolyte interface (48).

17.3 Other Useful Methods for Adsorbed Monolayers While CV is the most widely employed electrochemical method for examining adsorbed layers, other methods are also valuable, sometimes offering attractive simplicity or better precision in quantification. 17.3.1

Chronocoulometry

In principle, one can determine the amount of adsorbed electroreactant, e.g., ΓO , by integrating the area under the postwave in CV or LSV. However, this is workable only if the postwave is well separated from the diffusive wave, and, even then, subtraction of the baseline of the main wave and correction for double-layer charging are inevitably imprecise. Chronocoulometry (49–51) is a superior approach because it allows unambiguous determination of ΓO , regardless of the relative positions of the reduction waves for Osoln and Oads . Chronocoulometry was covered in Section 6.6, so only a few new elements are presented here. We will consider only the case where the electroproduct, R, remains unadsorbed. The potential is stepped from an initial value, Ei , to a sufficiently negative value that all Oads is reduced and C O (0, t) ≈ 0. At t = 𝜏, the potential is returned to Ei , and R created near the surface

775

776

17 Electroactive Layers and Modified Electrodes

Bf 4.0 Af 3.0

2.0

1.0 Q/μC 0.0

Blank in B Blank in A

1

t1/2/ms1/2 2

Blank in A

1.0

Figure 17.3.1 Anson plots for chronocoulometry of Cd(II) at an HMDE. Potential stepped from E i = − 0.200 to −0.900 V vs. SCE and back to E i . A = 0.032 cm2 . (Af , Ar ) 1 mM Cd(II) in 1 M NaNO3 . Slopes: Sf = Sr = 0.58 μC/ms1/2 . Intercepts: Q0f = 0.54 μC, Q0r = 0.55 μC. (Bf , Br ) 1 mM Cd(II) in 0.2 M NaSCN + 0.8 M NaNO3 . Slopes: Sf = 0.60 μC/ms1/2 . Intercepts: Q0f = 1.67 μC; Q0r = 0.86 μC. Charge values labeled “blank” refer to double-layer charging in the Cd(II)-free electrolyte solutions. [From Anson, Christie, and Osteryoung (51), with permission.]

3

4

5 θ/ms1/2

Blank in B

2.0

3.0

Ar

4.0

Br

is reoxidized. The results are analyzed using Anson plots, such as those in Figure 17.3.1. There are two branches: • The upper curve is Qf = Q(t ≤ 𝜏) vs. t 1/2 , describing the charge injected in the forward step. It is linear with an intercept of Q0f = nFAΓO + Qdl

(17.3.1)

• The lower curve is Qr (t > 𝜏) = Q(𝜏) − Q(t > 𝜏), describing the charge withdrawn during reversal. This quantity is plotted vs. 𝜃, where 𝜃 = 𝜏 1/2 + (t – 𝜏)1/2 – t 1/2 . The result is also linear and has a positive intercept. In Section 6.6.2, the intercept of the lower branch was identified as the amount of charge given in the forward step to the double layer; thus, the difference between the two intercepts was understood as nFAΓO , from which ΓO can be readily calculated. That representation is true enough for basic understanding, but it is not precise, because the treatment in Section 6.6.2 does not account for the fact that the reduction of Oads during the forward step adds to the R created in the diffusion layer by the reduction of Osoln . A full treatment (50) gives √ ) ( 2 −1 𝜏 1∕2 −1∕2 ∗ Qr (t > 𝜏) = 2nFACO DO 𝜋 𝜃 + nFAΓO 1 − sin + Qdl (17.3.2) 𝜋 t To a good approximation, the plot of Qr vs. 𝜃 is linear and follows ( ) a nFAΓO 1∕2 Qr (t > 𝜏) = 2nFAC∗O DO 𝜋 −1∕2 1 + 1 𝜃 + a0 nFAΓO + Qdl Qc

(17.3.3)

17.3 Other Useful Methods for Adsorbed Monolayers

where Qc is the total charge arising from the diffusing species during the forward step, ( ) DO 𝜏 1∕2 Qc = 2nFAC∗O (17.3.4) 𝜋 and a0 = − 0.069 and a1 = 0.97.4 Thus, the reversal branch of the Anson plot has the intercept Q0r = a0 nFAΓO + Qdl

(17.3.5)

and Qdl =

Q0r − a0 Q0f

(17.3.6)

1 − a0

Once Qdl is determined, nFAΓO can be obtained from (17.3.1). Figure 17.3.1 results from an experiment (51) involving the reduction of Cd(II) at an HMDE. In the absence of SCN− , Cd2+ is not adsorbed on Hg and the branches of the corresponding Anson plot (Af and Ar ) show equal intercepts of Q′dl . In the presence of SCN− , Cd2+ is adsorbed and the Anson plot has branches (Bf and Br ) with significantly different intercepts, allowing calculation of ΓO by approach just developed. The variation of ΓO with potential can be studied by changing Ei . Effects of varying the concentrations of O or the supporting electrolyte are readily investigated. This case provides a good example of anion-induced adsorption, in which , halide ion) also binds a metal ion from the a specifically adsorbed substance (e.g., SCN− , N− 3 solution and enables specific adsorption of that species [e.g., Cd(II), Pb(II), Zn(II)] (52, 53). Chronocoulometry can also be applied to the cases of Sections 17.2.2 and 17.2.3 where only adsorbed species are electroactive (54). In this situation, the potential step causes only double-layer charging and electrolysis of the adsorbed species. One can estimate Qdl by steps between a potential at the foot of the adsorption wave, Ei , and potentials, Ef , beyond the adsorption wave. If C d (μC/cm2 ) is not a function of E in the region of the wave, then (54): Q = Qdl + Qads = AC d (Ei − Ef ) + nFAΓO

(17.3.7)

A plot of Q vs. Ei − Ef can then be employed to determine C d and ΓO . 17.3.2

Coulometry in Thin-Layer Cells

Thin-layer methods (Section 12.6) are valuable for studying irreversibly adsorbed substances (55, 56). The cells are usually of the type shown in Figure 12.6.1b, where perhaps 40 μm of electrolyte is captured between a smooth cylindrical Pt electrode and the surrounding precision glass tubing. The area of the electrode, A, is typically about 1 cm2 ; hence, the solution volume, V , is about 4 μL. The cell is filled reproducibly by capillary action and can be flushed with pressurized inert gas. Measurement of Γ is possible for either an electroactive or an electroinactive adsorbate, but the procedure is different. Consider the case where species R is irreversibly adsorbed, but Rads is not electroactive at potentials where Rsoln shows an anodic CV wave. An example is hydroquinone (HQ; Figure 1) in 1 M HClO4 . When a solution of known concentration, C 0 , is introduced into the thin-layer cell, ΓA moles of HQ adsorb, so the concentration remaining in solution, C, is (17.3.8)

C = C0 − ΓA∕V

Anodic coulometry yields the amount of charge, Q1 , required to electrolyze the Rsoln in equilibrium with the adsorbed layer. By filling and flushing the cell several more times (without 4 a0 and a1 depend slightly on the range of 𝜃/𝜏 /2 . The given values are usually employed. 1

777

17 Electroactive Layers and Modified Electrodes

Figure 17.3.2 Surface concentration of 1,4-naphthohydroquinone vs. bulk concentration in 1 M HClO4 at 5, 25, 35, 45, and 65 ∘ C (top to bottom on the right). Measured by thin-layer coulometry at a Pt electrode in a cell like that in Figure 12.6.1b. Multiple levels of saturation suggest that the molecule adsorbs in different geometries (e.g., flat vs. edgewise) as the coverage changes. [Adapted with permission from Soriaga, White, and Hubbard (57). © 1983, American Chemical Society.]

OH

0.60 OH

1,4-naphthohydroquinone

0.50 Г/nmol cm–2

778

0.40

0.30

0.20

–5.0

–4.0 log C/M

–3.0

removing adsorbed material from the electrode), the surface of the electrode picks up enough adsorbate to be at equilibrium with the original solution concentration, so that the solution in the cell is no longer depleted by adsorption. Anodic coulometry performed on that solution gives the charge, Q0 , corresponding to C 0 . Thus, Γ = (Q0 − Q1 )∕nFA

(17.3.9)

The adsorbate layer can then be removed by oxidation at very positive potentials. If, on the other hand, the adsorbed molecule is electroactive and shows a voltammetric postpeak well separated from the voltammetric response of the dissolved species, it may be possible to measure Q1 by coulometry at a potential between the peaks for the dissolved and adsorbed species, and to measure Q0 – Q1 as the additional charge passed when the potential is changed to a value beyond the postpeak. Values of Γ (mol/cm2 ) are sometimes used to ascertain the orientation of the adsorbed molecules on the electrode surface. This is done by calculating the average area, 𝜎, occupied by the molecule, where 𝜎(nm2 ) = 1014 ∕(6.022 × 1023 Γ) = 1.66 × 10−10 ∕Γ

(17.3.10)

One can then compare that number with values obtained from molecular models assuming different orientations in close-packed, immobile structures (55) (Figure 17.3.2 and Problem 17.7). 17.3.3

Impedance Measurements

Adsorbed electroreactants are addressed in ac methods by modifying the equivalent circuit representing the electrode reaction (Section 11.3). Normally, an adsorption impedance is added in parallel with the branches bearing the Warburg impedance, ZW , and the double-layer capacitance, C d (Figure 17.3.3a) (58–62). Expressions have been suggested for reversible (59, 60) and irreversible (61, 62) systems; however, the complexity of analysis has limited the use of impedance, when Oads , Rads , Osoln , and Rsoln must all be considered. A much simpler case is where electroactivity is restricted to species adsorbed on the electrode or otherwise fixed to it [e.g., alkylthiol layers with tethered electroactive groups (Section 17.6.2)]

17.3 Other Useful Methods for Adsorbed Monolayers

Figure 17.3.3 (a) Randles equivalent circuit modified for the case where diffusing and adsorbed species are electroactive. The lowest branch describes the effect of the adsorbed layer. (b) Functional equivalent circuit for an electroactive monolayer. Ru is uncompensated solution resistance; Rct is charge-transfer resistance for heterogenous electron transfer involving solutes. Other elements defined in text.

Cd Cd Ru

Rct

ZW

Ru Rct,ads Cads

Rct,ads Cads (a)

(b)

(63–65). The Warburg impedance then becomes infinite, and the path through it is deleted from the equivalent circuit (Figure 17.3.3b). The adsorbed layer is represented by the capacitance C ads = F 2 AΓ* /4RT and the charge-transfer resistance Rct,ads = 2RT/F 2 AΓ* k f , where the electron-transfer rate constant has units of s−1 . Analysis by EIS is straightforward. Section 11.4.4 covers the related problem of EIS for a thick modification layer on an electrode bearing electroactive sites [Section 17.5.2(c)]. 17.3.4

Chronopotentiometry

For a constant-current electrolysis (66–69), the treatment depends on the order in which the adsorbed and diffusing species are electrolyzed: • If only Oads is electrolyzed, the transition time, 𝜏, is given by i𝜏 = nFAΓ∗O

(17.3.11)

A similar equation holds for a prewave where Osoln is reduced to Rads . • If both Oads and Osoln are reduced, but with Oads being reduced almost completely before reduction of Osoln , then 2

∗ n2 F 2 𝜋DO A2 CO

+ nFAΓ∗O (17.3.12) 4i • If Oads is reduced last (i.e., in a postwave), the situation becomes more complicated because the two processes are not separated in time. Some of the current must be devoted to the continuing flux of Osoln as the Oads is reduced. The overall transition time is given by 𝜏 = 𝜏 1 + 𝜏 2 , where 𝜏 1 is the transition time due only to Osoln : i𝜏 =

2

𝜏1 =

∗ n2 F 2 𝜋DO A2 CO

4i2 and 𝜏 2 is defined implicitly by 𝜋nFAΓ∗O

(17.3.13)

(𝜏 − 𝜏 ) 1 2

− 2(𝜏1 𝜏2 )1∕2 (17.3.14) i 𝜏 • For simultaneous reduction of Oads and Osoln , the behavior is still more complex and depends on the form of the adsorption isotherm, as well as the manner in which the current divides between Oads and Osoln . The problem is similar to that concerning double-layer charging in chronopotentiometry (Section 9.3.4). = 𝜏cos−1

Chronopotentiometry is not as useful as chronocoulometry for the determination of Γ∗O ; however, once Γ∗O is determined, the method can yield information about the order in which dissolved and adsorbed species are reduced.

779

780

17 Electroactive Layers and Modified Electrodes

17.4 Thick Modification Layers on Electrodes In the parlance of this chapter, a thick modification layer is one in which nearly all redox centers are remote from the electrode and cannot communicate with it by electron tunneling. Accordingly, the modification layer has a thickness, l, greater than a few monolayers, i.e., at least a few nm. Commonly, the thickness is between 10 and 500 nm, but it may even exceed 50 μm. Thick modification layers are normally created with a specific purpose, such as • Improving the selectivity of electrochemical response, e.g., by limiting access to the electrode surface. • Enabling a catalytic molecule or a catalytic particle to function in concert with the electrode, perhaps for selective analysis, electrosynthesis, or energy conversion. • Serving as a component of a layered or 3D nanostructure in which a sequence of chemical operations can take place, comprising, for example, an elaborate chemical process or a logic function. Aside from their applications, systems like these have been investigated fundamentally to improve scientific understanding of the dynamics of processes in layers on electrodes. Although the materials employed for modification are highly varied, most modifying layers are constructed using just a few strategies: 1) Polymer films. Because polymers are both versatile and processable, they are widely employed as layers on electrodes. Figure 17.4.1 illustrates the more common materials, as well as special cases of interest. a) Electroactive polymers feature oxidizable or reducible groups covalently linked to the polymer backbone. Examples are poly(vinylferrocene) (PVFc), poly(xylylviologen) (PXV), and polymerized viologen organosilane (PVOS). b) Coordinating (ligand-bearing) polymers, such as poly(4-vinylpyridine) (PVP), contain groups that can bind metal complexes at high density in the polymer matrix. Often these are redox centers capable of electron exchange, such as the Os complexes in Figure 17.8.2 (70), or catalytic centers, such as cobalt complexes active in the reduction of O2 . c) Ion-exchange polymers (polyelectrolytes) contain charged sites that can bind ions from solution via an ion-exchange process; typical examples are NafionTM (Naf ), poly(styrenesulfonate) (PSS), protonated PVP, or poly(4-vinyl-N-methylpyridinium) [often called “quaternized poly(4-vinylpyridine)” (QPVP)]. The ionically bound species are commonly redox centers (e.g., Ru(bpy)2+ ) or redox catalysts. A great range of ions, 3 and even sizable particles, can be immobilized in these matrices. d) Electronically conducting polymers, such as polypyrrole (PPy) and polythiophene (PT), can also be regarded as ion-exchange materials, since the polymer redox processes are accompanied by incorporation of ions into the polymer network. These materials are discussed in Section 20.2.3, and structures are given in Figure 20.2.5. e) Blocking polymers are insulators, such as polystyrene (PS). They form impermeable layers that block or passivate surfaces. Films can be formed on an electrode surface from a polymer solution by dip coating, spin coating, electrodeposition, or covalent attachment via functional groups. Starting with the monomer, one can produce films by thermal, electrochemical, plasma, or photochemical polymerization. 2) Inorganic layers. Solids that can be deposited on electrode surfaces frequently have well-defined nanostructures capable of capturing and hosting redox-active molecules or

17.4 Thick Modification Layers on Electrodes

)x

F2

(

F2 C

C F

(C )y

C F2

O C3F6

F2 C

C F2

(

H2 C

C H

(

C H

(

H2 C

C H

)n

SO–3Na+

SO–3

Naf

H2 C

)n

PSS

PS MeO

)n

(

OMe OMe

MeO

Si

Fe

Si + N

O

)n

+ N

PVFc

PVOS

(

(

H2 C

C H

)n N PVP

+ N

+ N PXV

)n

(

H2 C

C H

)n N

+ Me

QPVP

Figure 17.4.1 Typical polymers used for electrode modification. Chemical names are in Table 5.

catalysts in molecular-scale cavities or interlayer spaces. If the solids include transition metals, they may offer their own redox centers or catalytic capability. In addition, inorganic layers are usually stable, inexpensive, and readily available. a) Metal oxides. Films of oxides can be produced by anodization of metal electrodes (e.g., Al2 O3 on Al), with thickness controlled by the applied potential and the time of anodization. Oxide films can also be produced by chemical vapor deposition, vacuum evaporation and sputtering, and deposition from colloidal solution. b) Clays and zeolites. Aluminosilicates frequently possess well-defined structures and show capabilities for ion-exchange and catalysis. Clays are layered aluminosilicates; zeolites are networks of well-defined cages and pores. Suspended particles can be cast on substrate surfaces and will remain intact when they are used as electrodes. Electroactive cations such as Ru(bpy)2+ or MV2+ may be exchanged into them and show typical cyclic voltam3 metric responses of surface-confined species. c) Transition-metal solids. Crystalline solids based on transition metals sometimes support both electronic and ionic conductivity, allowing individual sites to undergo reversible redox conversion. In layers on electrodes, the redox processes can be faradaically driven; thus, modified electrodes bearing transition-metal solids show electrocatalytic properties (e.g., for the reduction of oxygen or for water splitting), as well as electrolytic color changes (electrochromic behavior). Of high interest are phosphates of Mn, Co, and Ni, oxides of tungsten, and transition-metal cyanide materials like Prussian Blue (a lattice of

781

782

17 Electroactive Layers and Modified Electrodes

ferric ferrocyanide). A notable aspect of the transition-metal phosphates is a capacity for self-healing (71, 72). 3) Composites. A versatile synthetic strategy is to employ one material essentially as a matrix to immobilize a different material whose properties are of primary interest. For example, metal particles, catalytic oxides, clays, zeolites, quantum dots, and nanoparticles of practically every other kind have often been cast on electrodes from suspensions of the particles in polymer-containing solutions. This approach combines the favorable film-forming properties of polymers with the chemical variety found in particles. Also common is the use of blended polymer layers, in which two or more polymers are cast simultaneously from a common solution. This approach can be useful for immobilizing a polymer that would otherwise tend to dissolve because of ready solvation or low average molecular weight. It can also help to manage the swelling of a film by solvation, which affects the dynamics inside the film (Section 17.5). 4) Biological materials. Layers based on active biological components are of high interest, especially for bioelectrocatalysis and the preparation of electrochemical sensors (Sections 17.8.2 and 17.8.3).5 The basic approach is to immobilize an enzyme, an antibody, a DNA, RNA, or polypeptide fragment, or a collection of whole cells, so that the immobilized component can recognize a target species, and in the process, produce an electrochemically detectable signal. Many modification layers based on biological components are composites in which a polymer (often a gel) provides an immobilizing matrix. A conducting component, e.g., carbon nanotubes, often helps to convey electrons between the base electrode and active redox sites. Sometimes, an enzyme or suspension is simply held against an electrode by a permeable polymer membrane. Alternative methods of immobilization include adsorption, and covalent linkage. 5) Multilayer assemblies and patterned structures. In the foregoing paragraphs, we have focused on electrodes modified by a single thick layer; however, more complicated structures have also been explored, including multiple layers of different polymers [as in Figure 17.8.4 (73)], metal films formed on a polymer layer, electrode arrays under a modifying layer, intermixed films of ionic and electronic conductor (biconductive layers). Modifying layers may also be patterned laterally in two dimensions. Systems like these often show different electrochemical properties than the simpler modified electrodes and are of interest for applications such as switches, amplifiers, and sensors.

17.5 Dynamics in Modification Layers Many measurements at modified electrodes are based on the passage of current, the magnitude of which determines the sensitivity of an analytical measurement, the speed of an electrosynthesis, or the rate of an energy conversion. Optimal performance is achieved by establishing the most favorable dynamics for conveying electrons between the electrode and the species whose oxidation or reduction is ultimately required. Selectivity is also a common goal (Section 17.8.3), and, to achieve it, one may need to incorporate a selective catalyst or to restrict access to the interior of the structure. Figure 17.5.1 provides a schematic picture wherein a primary reactant, A, in the external solution is converted to a product, B. This can occur by mass transport of A, moving through the film to the underlying electrode, or by cross-reaction with a mediator/catalyst, Q, contained 5 For example, biofuel cells are fuel cells employing enzymic catalysts.

17.5 Dynamics in Modification Layers

Figure 17.5.1 Processes at a modified electrode. Species P is a reducible mediator/catalyst in a film on the electrode surface, and A is the primary reactant in solution. (1) Heterogeneous electron transfer to P, producing the reduced form, Q. (2) Electron transfer from Q to another P in the film (electron diffusion or electron hopping). (3) Electron transfer from Q to A at the film/solution interface. (4) Penetration of A into the film (where it can react with Q or at the electrode/film interface). (5) Mass transfer of Q within the film. (6) Mass transfer of P within the film. (7) Mass transfer of A through a pinhole or channel to the electrode, where it can be reduced. [Adapted from Bard (7), with permission.]

A 1

2

P

P

e

e Q

3

4

P

A

P

e

A e

B

Q

A

A 7

Electrode

A

3

5 Q

A

B

Q A 6

1

A

Film

A

Solution

l

in the film and renewed electrochemically. Species A might react with Q in a three-dimensional zone inside the film or just at the film/solution interface.6 Several dynamic processes may occur simultaneously in a system like this, and they operate together to determine the overall behavior. In this section, we will examine the dynamic elements, both individually and in concert. The school of Savéant (74) has provided the most comprehensive treatment, and the development given here follows their approach and notation. 17.5.1

Steady State at a Rotating Disk

It is advantageous to study modified electrodes and other complex structures at steady state because one can eliminate time as a variable and simplify the treatment. Indeed, many applications of modified electrodes are based on steady-state electrolysis. Hydrodynamic voltammetry at an RDE is widely employed for the characterization of modified electrodes and will serve as the basis for our further discussion. As we saw in Section 10.2.5, a Koutecký–Levich (KL) plot, 1/il vs. 1/𝜔1/2 , is a general tool for separating the effects of a rate limitation at an electrode from those of convective diffusion in solution. When the substrate, A, is reduced, the limiting current can be written as 1∕il = 1∕iA + 1∕iF

(17.5.1)

where iF expresses the maximum rate at which A can be converted to B at the RDE (including 2∕3 any modifying film), and iA = 0.62nFAC∗A DA 𝜈 −1∕6 𝜔1∕2 is the Levich current expressing the arrival rate of species A at the outer boundary. The KL plot is a means for extrapolating behavior 6 In this book, the term substrate is used consistently in the sense of a platform, such as might be used to support the assembly of a larger structure. In much of the literature on modified electrodes, the same term is used in the sense of a consumable reactant, as often found in biochemistry. In this nomenclature, the primary reactant, A, is the substrate.

783

784

17 Electroactive Layers and Modified Electrodes

to infinite 𝜔, and it yields 1/iF as the intercept (Figure 10.2.7). Thus, iF is the limit in which there is no mass-transfer limitation on the supply of A at the outer boundary of film and the ∗. concentration there is CA The power of this approach lies in its generality (Section 10.2.5). The treatment requires no assumption about the identity of the rate limitation, and there is only one constraint on mathematical form: The overall rate of reaction must be proportional to the substrate concentration in solution at the hydrodynamic boundary surface (in this case, the outer boundary of the film). The maximum rate of reaction, iF /nFA, then occurs when the concentration of A at the outer ∗ ; thus, boundary is at its greatest value, which is CA iF nFA

∗ = k ′ CA

(17.5.2)

where k ′ is a proportionality constant describing the overall rate law. Diagnosis of the rate-controlling process can be achieved by examining variations of k ′ (or iF ) with experimental variables such as the film thickness or the concentration of mediator/catalyst within the film. 17.5.2

Principal Dynamic Processes in Modifying Films

The overall rate at which A is converted to B at a modified electrode might be determined by a single dynamic process or jointly by more than one. The main candidates are (Figure 17.5.1): • Convective diffusion in solution, which delivers A to the outer boundary of the film. • The kinetics of partitioning, which allows A to reach the interior of the film. • The diffusion of A through the film, either to the electrode surface or to redox sites in the layer. • Electron-transfer kinetics at the electrode surface or at mediator/catalyst sites. • The movement of electrons between the electrode and reaction sites in the film. For diagnosis, it is valuable to compare the rate capacities of these processes using a common basis. Our strategy is to define a set of characteristic currents expressing the conversion rates that would be observed if each individual process were entirely rate-determining. The process (or processes) responsible for the least of them would necessarily be rate-limiting. Later, we will see that these characteristic currents can also be used to express the actual current under all steady-state operating conditions. Let us begin by considering each dynamic element in turn. (a) Convective Diffusion in Solution

Let us imagine a situation in which events in the film are all very fast, so that the rate of the conversion of A is the same as the rate at which A arrives at the outer boundary of the structure. ∗ D2∕3 𝜈 −1∕6 𝜔1∕2 ; thus, the current would be i . This is the maxiThis is the Levich flux, 0.62CA A A mum conversion rate that could ever be seen in any system under any operating conditions, for it is impossible to convert A any faster than it arrives. (b) Diffusion of A within the Film

Now let us consider the situation in which convective diffusion is very fast and species A partitions into the film quickly; yet, no process converts A to B inside the film. If the heterogeneous conversion of A is fast, then the overall process (and the current) is controlled by the arrival rate of A at the electrode surface by diffusion through the film.

17.5 Dynamics in Modification Layers

Solution Film

Diffusion layer

* CA CA(l –)

CA(x)

CA(l+) δ 0

l

0

x

flux across film

current

l+δ

flux in solution equilibration at interface

Figure 17.5.2 Steady-state concentration profiles at an RDE when the current is determined by the mass 1∕3 transfer of A in the film and in solution. Film thickness is l; diffusion layer thickness is 𝛿 = 1.61DA 𝜈 1∕6 𝜔−1∕2 . Distances are not to scale; ordinarily 𝛿 ≫ l. Arrows indicate fluxes.

Figure 17.5.2 provides a schematic view. The concentration of A at the electrode surface is zero because the heterogeneous kinetics are fast. Just inside the outer boundary of the film (x = l– ), the concentration differs from that just outside the film (x = l+ ) because partitioning occurs. The concentrations are related via the partition coefficient, 𝜅=

CA (l− ) CA (l+ )

(17.5.3)

The system is at steady state; therefore, the flux of A is constant at all x. If the diffusion coefficient of A in the film, DS , is uniform everywhere, the slope of the concentration profile must be constant; hence, the profile must be linear as shown in Figure 17.5.2. Under conditions of very fast convective diffusion (𝜔 → ∞), the depletion layer outside the film would disappear, and the concentration of A exposed to the film would become the bulk ∗ and the maxivalue. The largest possible concentration just inside the outer boundary is 𝜅CA ∗ mum flux is DS 𝜅CA ∕l. This is the greatest possible rate at which A can be converted to B when the process rests entirely upon reactant diffusion in the film, and this reactant diffusion current, iS , becomes one of our conceptual descriptors.7 iS =

∗ nFADS 𝜅CA

l

(17.5.4)

At an RDE, convective transfer of A to the film surface and diffusion of A through the film occur serially, and the current follows (17.5.1), where iF = iS . Thus, the value of iS can be 7 The subscript “S” is consistently used in the literature for this conceptual current and for the diffusion coefficient of A in the film. The usage is based on identification of A as the substrate. See footnote 6.

785

17 Electroactive Layers and Modified Electrodes ∗ )−1 vs. 𝜔−1/2 for reduction of BQ at a Figure 17.5.3 (i∕CBQ Pt RDE in MeCN + 0.5 M TBABF4 . Currents are normalized ∗ . The top three data sets are for diffusion through a by CBQ ∗ = 5.82, 3.84, and PVFc film (in order from top, CBQ 1.96 mM). The bottom set is for a bare Pt RDE in 5.82 mM BQ. 𝜔 = 2𝜋f , with f in rpm. [Reprinted from Leddy and Bard (75), with permission from Elsevier Science.]

10

(i/C*BQ)–1/mol cm–3 A–1

786

8

6

4

2

0 0.00

0.01

0.02

0.03

0.04

ω–1/2/min1/2

determined from the intercept of a KL plot. An example involves the reduction of benzoquinone (BQ) on an electrode coated with a film of PVFc (Figure 17.5.3). The slopes normalized ∗ (= C ∗ ) are determined by mass transport in solution (i.e., i ) and are the same in the by CA A BQ ∗ , as predicted by (17.5.4). presence and absence of film. The intercepts depend upon l and CA Partitioning kinetics may also be rate-limiting (76). The net flux of A across the film/solution interface is 𝜒 f C A (l+ ) – 𝜒 b C A (l– ), where 𝜒 f and 𝜒 b are rate constants for transfer of A from solution into film, and from film into solution, respectively. If the serial processes of permeation and diffusion in the film are jointly rate-controlling, the KL equation becomes 1 1 1 1 = + + il iA iS iP

(17.5.5)

where the permeation current, iP , is given by ∗ iP = nFA𝜒f CA

(17.5.6)

When the maximum flux of A across the interface, expressed by iP , is much larger than iS , the last term of (17.5.5) is negligible, and the KL plot has an intercept of 1/iS . This is the case where diffusion in the film is totally rate-limiting, as discussed just above. If, instead, the partitioning kinetics are entirely rate-limiting, iP ≪ iS and the intercept of the KL plot will be 1/iP . The situations that we have just considered are sometimes said to comprise the membrane model because the film behaves as a membrane through which species A must diffuse to reach the substrate surface. (c) Diffusion of Electrons in the Film

Many potential applications of modified electrodes require the movement of electrons throughout a layer. If a given electrode process does not proceed at the electrode surface because of slow kinetics, one might still achieve it by delivery of charge through a catalyst in the film. A modified electrode designed for the oxidation of glucose is a good example (Section 17.8.2). The reaction can be selectively catalyzed by the enzyme glucose oxidase, but a supporting redox network is needed for conveyance of electrons from the reduced enzyme to the electrode.

17.5 Dynamics in Modification Layers

Figure 17.5.1 depicts two mechanisms for conveying electrons in the film by using a mediator couple, P/Q: • In the top section of the figure, substrate A does not enter the film, but electrons can move by hopping from Q to P (process 2), and, ultimately, they can be transferred to A at the film/solution interface (process 3). • The middle part of the figure depicts a different situation in which electrons can also reach the outer interface by physical diffusion of Q (process 5). In many systems, physical diffusion of mediators is insignificant because they are bound (e.g., by attachment to a polymer chain as in Figure 17.8.2). Long-range motion of P and Q is impossible or highly restricted. Even so, physical motion must still occur in the transport of charge, even by a hopping mechanism, because counterions must maintain local electroneutrality. In a constrained system, the apparent motion of Q is determined by the kinetics of electron exchange between P and Q. Theoretical work on the analogous process in homogeneous solution showed that the movement of charge continues to be a random walk, equivalent to diffusion (77–84). The apparent diffusion coefficient, DE , is the sum of two contributions—one from the physical movement of the species (governed by its translational diffusion coefficient, D) and another from the electron-exchange process. When bimolecular kinetics apply and the species can be considered as points, DE can be estimated from the Dahms–Ruff equation,8 DE = D + (1∕6)k𝛿 2 CP∗

(17.5.7)

where 𝛿 is the distance between electron-exchange sites and CP∗ is the total concentration of sites (both oxidized and reduced). While (17.5.7) is a useful approximation, it should not be taken too literally in the context of modified electrodes because the mathematics of charge diffusion in a constrained system depend significantly on the latitude for physical motion. Various limiting situations have been identified and examined (84). Now let us imagine a system in which electron diffusion is fully rate-determining for the conversion of A to B. We disallow the permeation of A into the film; hence, all electrons delivered to A must be transported entirely across the film, where a fast reaction of A takes place. The maximum current arises when the concentration of electrons in the film is at its largest possible value near the electrode surface, but approaches zero at the outer boundary, where species A arrives in high flux and quickly consumes the electrons. The largest possible concentration of electrons is the concentration of redox sites hosting them, CP∗ ; therefore, the maximum flux of electrons (formally a flux of Q outward in Figure 17.5.1) is DE CP∗ ∕l (mol cm−2 s−1 ). The corresponding current is9 iE =

FADE CP∗ l

(17.5.8)

This quantity, the electron diffusion current, is the maximum charge delivery capacity via electron diffusion in the system. (d) Cross-Reaction in the Film

A common practical situation is that the “cross-reaction” of A with Q inside the film is rate-determining for the overall conversion of A. Thus, we also need to define the largest current that cross-reaction could possibly support on its own. 8 This equation is frequently found in the literature with 𝜋/4 in place of 1/6. This substitution is due to an error in the original derivation, which was later corrected (80–82). Reference (84) provides a critical review of theory. 9 Sometimes, the amount of P in the film is given in terms of a two-dimensional concentration, Γp (mol cm−2 ), where CP∗ = ΓP ∕l.

787

788

17 Electroactive Layers and Modified Electrodes

Figure 17.5.4 Schematic concentration profiles when the rate of cross-reaction between A (solid line) and Q (dashed line) limits the current.

Solution Diffusion layer

Film

C*A

CA = κCA(l+) C*P

CQ

CA

CA(x) CA(l+) δ 0

l

0

x

l+δ

Let us imagine a system in which A partitions and permeates rapidly, so that its concentration everywhere in the film is the partitioned value in equilibrium with the concentration in solution just outside the film. Suppose also that electrons diffuse rapidly through the structure, so that their concentration is uniform. Finally, suppose that A does not react at the electrode surface. In this situation, depicted in Figure 17.5.4, A is converted to B uniformly in the film because there is uniform availability of primary reactant, electrons, and cross-reaction sites. The maximum rate of cross-reaction occurs when two conditions are satisfied: • The electrode potential must be sufficiently negative that the redox sites adjacent to the electrode are fully reduced, so that the electron concentration is CP∗ . Since electron diffusion is rapid, this electrode potential assures that this same electron concentration applies everywhere in the film. • Rotation of the electrode must be rapid enough to bring the concentration just outside the ∗ . This condition assures that film-solution boundary C A (l+ ), essentially to the bulk value, CA ∗ . Because A diffuses the concentration of A inside the film is the greatest possible value, 𝜅CA rapidly, this concentration applies everywhere. In most published treatments, the redox sites supporting electron hopping are assumed to be the same as those where A is catalytically converted to B. This assumption is often valid in real systems, and we use it here. The consequence is that the electron concentration is also the concentration of cross-reaction centers, CP∗ ; thus, the greatest possible cross-reaction rate is ∗ C ∗ (mol s−1 cm−3 ). The cross-reaction current, i , is the product of this rate, the volume k𝜅CA k P of the film, Al, and the charge passed per mole of reaction, nF, ∗ C ∗ = nFAk𝜅Γ C ∗ ik = nFAlk𝜅CA P A P

(17.5.9)

The corresponding KL expression is 1 1 1 = + il iA ik

(17.5.10)

The species involved in shuttling electrons need not be the same as that engaged in crossreaction with A; hence, the redox site concentration might differ from CP∗ in Equation 17.5.9. Adaptation of the treatment is straightforward, unless any such system also requires explicit consideration of the rate capacity for electron transfer between the shuttle and the catalyst.

17.5 Dynamics in Modification Layers

Film

Solution

C*A

C*P

Electrode

C(x)

CA A CA

CQ

χb

P+e⇌Q Q+A

k

P+B

κ = χf /χb

A χf l

δ l

0 x

l+δ

Figure 17.5.5 General concentration profiles for steady-state mediated reduction of a primary reactant, A, by electrogenerated Q. To preserve clarity, the profiles for species P and B are omitted. The electrode is held at a potential where P at the electrode surface is fully reduced, so that the concentration of Q at the electrode surface is CP∗ (= Γ∗P ∕l). In the solution (x > l), the concentration profile for A is linear near the film boundary. The rate constants for the transport of A into and out of the film are 𝜒 f and 𝜒 b , respectively. [Adapted from Leddy, Bard, Maloy, and Savéant (76), with permission from Elsevier Science.]

17.5.3

Interplay of Dynamical Elements

General steady-state concentration profiles are depicted in Figure 17.5.5 to serve as a reminder that the operation of any real modified electrode is more complex than the carefully defined limiting cases just presented. Any of the dynamic elements that we have just delineated may control or influence the rate at which an electrode reaction takes place at a modified electrode. For example, substrate A might be reduced in the film at a rate jointly determined by its diffusion in the film and its cross-reaction with mediator Q. The mathematical treatment of the general case (85) is more complicated than for the various limiting cases presented in Section 17.5.2 and requires more discussion than can be given here; however, it can be usefully summarized. In most experimental systems, only one or two of the processes are rate-limiting and the range of operating possibilities can be depicted in two-dimensional zone diagrams like those introduced in Chapter 13. As before, the zones are defined using dimensionless parameters defining the full spectrum of behavior. For the systems now before us, the most useful parameters are two current ratios: 1) i∗S ∕i∗k , comparing the availability of primary reactant to the cross-reaction rate. In this ratio, [ ] i i i∗S = iS 1 − − (17.5.11) iA iP [ ] i i i∗k = ik 1 − − (17.5.12) iA iP 2) iE ∕i∗k , comparing the availability of electrons to the cross-reaction rate. The use of these parameters is straightforward: • If i∗S ∕i∗k is small, the availability of the substrate, A, is likely to be a limiting factor because it is low relative to the capacity of the cross-reaction. This is an “S” limitation in the nomenclature of this area.

789

17 Electroactive Layers and Modified Electrodes

1 R (*2)

R + E (2)

ER (2) R + S (*2)

General case

0

log (is*/i*k)1/2

790

SR (*)

ER + S (2)

–1

S (*)

DS

S + E (*) SR + E

–2

κC*A

DE

k, l E (*)

–3 –3

–2

–1 log (iE/ik*)1/2

0

C*P 1

Figure 17.5.6 Zone diagram for mediated reduction of a primary reactant, A, at a film-covered electrode. Pattern concentration profiles are shown only inside the film and only for species A (dashed line) and species Q (solid line). Parenthesized indicators on case labels: (*) KL plot is linear for wave 1; (2) wave 2 exists. Legend at lower right shows directions of change caused by increases in the indicated parameters. [Adapted from Leddy, Bard, Maloy, and Savéant (76), with permission from Elsevier Science.]

• If iE ∕i∗k is small, the availability of electrons is likely to a limiting factor because it is low relative to the capacity of the cross-reaction. This is known as an “E” limitation. • If either i∗S ∕i∗k or iE ∕i∗k is large, the capacity of the cross-reaction is likely to be a limiting factor because it is low relative to the availability of substrate or electrons. This an “R” limitation. The full zone diagram is laid out in Figure 17.5.6, and we can explore it through a few examples. First, consider the case where the reaction rate between A and Q is slow, but A can readily penetrate the film and electron diffusion in the film is rapid (i∗S ∕i∗k and iE ∕i∗k are both large). This is Case R in the upper right corner of the diagram, where one can see pattern concentration profiles of A and Q matching those of Figure 17.5.4. Now let us think about what happens as DS decreases, so that movement of A in the film becomes slower. The operating point moves directly downward because i∗S ∕i∗k decreases without change in iE ∕i∗k . The behavior changes concomitantly, first to Case R+S and then to Case SR. In both cases, the cross-reaction rate and the diffusion of A in the film are jointly rate-controlling. The difference—illustrated by the pattern concentration profiles—is that species A is still mobile enough in Case R+S that cross-reaction cannot fully consume the supply, and the concentration profile for A extends to the electrode surface. In Case SR, which often occurs in practical systems, the cross-reaction can consume the supply of A before it can reach the electrode surface. Only for Case S, Case E, and Case R is there a single rate-controlling process. These three correspond to the limiting situations that we encountered in Sections 17.5.2(b–d), respectively.

17.6 Blocking Layers

Table 17.5.1 Information from Koutecký–Levich Plots for Limiting Currents at a Modified RDE Case

Wave(a)

Slope

Intercept

E

Only 1

0

i−1 E

ER

1

Nonlinear

Nonlinear

2+1

2∕3 [0.62nFAC∗A DA 𝜈 −1∕6 ]−1

i−1 + i−1 P S

1

Nonlinear

Nonlinear

2+1

[0.62nFAC∗A DA 𝜈 −1∕6 ]−1

ER+S R R+E

2∕3

2+1

2∕3 [0.62nFAC∗A DA 𝜈 −1∕6 ]−1 2∕3 [0.62nFAC∗A DA 𝜈 −1∕6 ]−1

1

Nonlinear

2+1

[0.62nFAC∗A DA 𝜈 −1∕6 ]−1

1

S

Only 1

S+E

Only 1

SR

Only 1

SR+E

Only 1

Nonlinear

1 2+1

i−1 + i−1 P k i−1 + i−1 P S Nonlinear

2∕3

2∕3 [0.62nFAC∗A DA 𝜈 −1∕6 ]−1 2∕3 [0.62nFAC∗A DA 𝜈 −1∕6 ]−1 2∕3 ∗ [0.62nFACA DA 𝜈 −1∕6 ]−1 2∕3 [iS ∕(iS + iE )][0.62nFAC∗A DA 𝜈 −1∕6 ]−1 2∕3 [0.62nFAC∗A DA 𝜈 −1∕6 ]−1

R+S

i−1 + i−1 P S

i−1 + i−1 P S

i−1 + [(ik iS )1∕2 tanh (ik ∕iS )1∕2 ]−1 P i−1 + (ik iS )−1∕2 tanh (ik ∕iS )1∕2 P i−1 + i−1 P S

(iP + iS )/[iP (iS + iE )] i−1 + (ik iS )−1∕2 P Nonlinear

(a) KL plot (1/il vs. 1/𝜔1/2 ) is for (Only 1) the single expected wave, (1) the first of two waves, (2+1) the sum of plateau currents for the first and second waves. Based on Leddy et al. (76). KL equations are available for all cases in the source.

For most of the labeled cases, KL plots (1/il vs. 1/𝜔1/2 ) are expected to be linear and to provide intercepts bearing information about dynamics in the modifying layer. Details are summarized in Table 17.5.1. In this table, the limiting current for wave 1 corresponds to the reduction of A by cross-reaction with Q (the catalytic reaction), while the limiting current for wave 2 arises from the direct reduction of A. The second wave occurs only in cases where A penetrates the film and reaches the metal substrate before it is fully consumed by reaction with Q (as in Case R+S). As one illustration of the use of Table 17.5.1, we can see there that Case SR produces only one wave, for which the KL plot is linear with an intercept of 1/iP + 1/(ik iS )1/2 . From the intercept, one has access to one of the fundamental parameters (𝜅, k, CP∗ , DS , or l), if the others are known independently. To interpret a KL intercept quantitatively, one must first have an accurate diagnosis of the applicable kinetic case. The diagnosis can generally be made from the behavior of the KL plots ∗ , l, and Γ . A helpful guide to diagnosis is available in the original literature with variations in CA P (76, 85), which also covers situations where the catalyst and the electron carrier are different species, as in the glucose oxidase enzyme electrode (Section 17.8.2).

17.6 Blocking Layers Layers that serve to block electron and ion transport between an electrode and a solution are also of interest, for they often have a practical role, e.g., to prevent corrosion of the surface. Such layers can also be used to study the distance dependence of electron transfer (Section 17.6.2). Since blocking may be compromised by defects, such as pinholes, the efficacy of blocking is often a primary concern. Electrochemical methods are useful for characterizing it (86):

791

17 Electroactive Layers and Modified Electrodes

8

0.4 (a)

4

0.2

(c)

0

0.0

–0.1

–4

–8 –0.6

i/μA

(b) i/μA

792

–0.4

–0.2

0.0

0.2

0.4

–0.2 0.6

E/V vs. SCE

Figure 17.6.1 Charging current in CV at 0.1 V/s for a polycrystalline gold electrode (A ≈ 1 cm2 ) in 1 M Na2 SO4 . (a) Bare electrode (left current scale). (b) With a bound layer of C18 alkylthiol (right current scale). (c) With thiol layer (left current scale). The capacitance decreases by a factor of about 50 upon binding of the thiol. [Reprinted from Finklea (86), by courtesy of Marcel Dekker, Inc.]

• To obtain the aggregate area of pinholes, one can, for example, compare the sizes of certain voltammetric peaks for the bare and filmed electrode (e.g., those for the formation and reduction of an oxide layer on Au). • To obtain the spatial distribution of pinholes, one can deposit a metal like Cu, then strip the blocking film, and perform microscopy on the resulting surface. • The formation of a blocking layer on an electrode surface decreases the capacitance compared to that of the bare electrode, since the distance of closest approach of the counterions is increased by the thickness of the layer [Equation 14.3.2 and Figure 17.6.1]. • Penetration of an electroactive species through pores to the underlying electrode surface can be measured by chronoamperometry or CV (Section 17.6.1). Usually an outer-sphere species is used as a probe under conditions where there is no electron tunneling like Ru(NH3 )3+ 6 through the film (Section 17.6.2).

17.6.1

Permeation Through Pores and Pinholes

Consider an electrode covered with a film that has continuous pores or channels from the solution to the electrode (Figure 17.5.1, process 7). One might ask how the electrolysis of a solute, A, differs at such an electrode from that at the bare electrode. The answer depends upon the characteristics of the pores and the time scale of the experiment. The situation can become complicated because pores can vary in their dimensions and tortuosity, and their distribution may not be uniform. Theoretical treatments are usually idealized. The theory for electrodes of this type is closely related to that for ultramicroelectrode arrays (Section 5.6.3) (87, 88). (a) Chronoamperometric Characterization

Perhaps, the simplest way to study such a system is to carry out a potential step causing diffusion-controlled electrolysis of a solute, then to compare the current passed at the modified electrode with the corresponding Cottrell behavior at the bare electrode. Usually simple models like those in Figure 17.6.2 are used.

17.6 Blocking Layers

2r0

2R0

(a)

2r0

2R0

(b)

Figure 17.6.2 Models for a surface with active sites (dark circles) of radius r0 , spaced 2R0 apart. (a) Hexagonal array. (b) Approximation where the inactive zones are taken as circular.

The most commonly treated situation is where the pore radius, r0 , is small and the pores are spaced far apart compared to r0 , so that the active (uncovered) fraction of the surface, 1 − 𝜃, is also small, The behavior is qualitatively like that of Figure 6.1.6, which illustrates distinct behavioral regimes (Section 6.1.5): (1) When the time scale of the experiment is small, so that (Dt)1/2 ≪ r0 , the electrode shows an electrochemical response characteristic of linear diffusion, except that the effective area is 1 − 𝜃 times that of the bare electrode. Electrolysis of a solute occurs directly at the substrate only at the active areas, without significant radial diffusion (Figure 6.1.6a). (2) At intermediate times, each site shows UME behavior and the total current is the sum of that from the individual sites (Figure 6.1.6b). (3) When the time is long enough that the diffusion layers from the individual sites merge, the electrode behavior approaches that of the unfilmed electrode and has an effective total area equal to that of the bare electrode (Figure 6.1.6c). As the time scale of the experiment is varied across these behavioral regimes, the electrochemical responses can provide information about 𝜃, r0 , and the pore distribution. For the hexagonal array of active sites shown in Figure 17.6.2a, the total current, i(t), normalized by the Cottrell current for the bare electrode, id (t), is given by (89, 90) i(𝜏) 1 = {𝜎 exp(−𝜏) − 1 + 𝜎 2 (𝜋T)1∕2 exp(T)[erf(𝜎T 1∕2 ) − erf(T 1∕2 )]} id (𝜏) 𝜎 2 − 1

(17.6.1)

in which T = 𝜏/(𝜎 2 – 1), 𝜎 = 𝜃/(1 – 𝜃), and 𝜏 = lt, where l is a function of 𝜃, D, and the size and distribution of pores (89). A thin film is assumed, so that the electroreactant makes a rapid transit through the pores. Moreover, R0 /r0 is assumed to be not very large. Plots of i(𝜏)/id (𝜏) vs. 𝜃 are given in Figure 17.6.3a. At short times (small 𝜏), the current ratio attains the limiting value of 1 − 𝜃 (case 1 above). At long times, when the diffusion layer grows to a thickness that is large compared to R0 , the ratio approaches unity (limit of case 3 above). The location of the intermediate region depends upon 𝜃 and r0 ; therefore, a plot of i(𝜏)/id (𝜏) vs. t can be used to estimate these parameters. A competing physical model with similar chronoamperometric behavior involves no array of active sites, but, instead, partitioning of the electroreactant into the film and its diffusion inside the film to the electrode surface. This is the membrane model, treated in Section 17.5.2(b). The expression for the current, normalized to that at the bare electrode, is (90) ⎧ ⎫ ∑ ( 1 − u )j ⎪ ⎪ i(𝜏) 2 = u ⎨1 + 2 exp(−j ∕𝜏)⎬ id (𝜏) 1+u ⎪ ⎪ j=1 ⎩ ⎭

(17.6.2)

793

17 Electroactive Layers and Modified Electrodes

(a)

Figure 17.6.3 Working curves for chronoamperometry at electrodes covered with a blocking film. (a) Pinhole model, curves labeled with values of 𝜃 (fraction of blocked area); (b) membrane model, curves labeled with values of u = 𝜅(DS /DA )1/2 . [Reprinted from Leddy and Bard (75), with permission from Elsevier Science.]

Pinhole model

i(τ)/id(τ)

1.0

0.6

0.2

0.5 0.6 0.7 0.8 0.9 –4

(b)

0.95 0.99 –2

0 log τ

2

4

2

4

Membrane model 1.0 0.8

i(τ)/id(τ)

794

0.6

0.2

0.6 0.4 0.3 0.2 0.1 –4

0.01 –2

0 log τ

where 𝜏 = DS t/l2 and u = 𝜅(DS /DA )1/2 . As usual, l is the film thickness, 𝜅 is the partition coefficient for species A, and DA and DS are the diffusivities for species A in solution and in the film, respectively. Plots of i(𝜏)/id (𝜏) vs. log 𝜏 are shown for the membrane model in Figure 17.6.3b. At short times, when the diffusion layer thickness is small compared to the film thickness [(DS t)1/2 ≪ l], the electrolysis occurs completely within the film and is characterized by a dif∗ . Under these conditions, i(𝜏)/i (𝜏) fusion coefficient, DS , and an initial concentration, 𝜅CA d approaches 𝜅(DS /DA )1/2 . At long times, the diffusion layer extends well into the solution phase and diffusion in the film becomes a minor aspect of the experiment. The current approaches the Cottrell current for the electrode, and i(𝜏)/id (𝜏) approaches unity. In studies of blocking layers, it is of practical interest to distinguish pinhole behavior from membrane behavior. The similarity of chronoamperometry for these two models, clearly seen in Figure 17.6.3, points to a need for careful investigation. A chronoamperometric study of this type involved poly(vinylferrocene) films (l ≈ 1 μm), where BQ or MV2+ was the electroreactant (75). For this system, a membrane model fits the experimental results better than a pinhole model and values of 𝜅 and DS were estimated. In many real systems, the simplifying assumptions identified above do not hold, and more complex models are needed. Simulation would normally be required for treatment, but the approach to interpretation would remain essentially the same. (b) Cyclic Voltammetry

When a blocked electrode is investigated by chronoamperometry, the measurements are made only at the mass-transfer controlled limit, where the electron-transfer kinetics have no effect on the results. Cyclic voltammetry is fundamentally different because the whole wave is observed. The current densities at the active sites of a blocked electrode are greater than at the bare electrode; therefore, the required overpotentials are larger, unless the kinetics are so facile that the system remains essentially reversible.

17.6 Blocking Layers

DOe

DOm

4 2

i id,c

log Λ

R0

E0′

QRm

–2

i QRe

–4

1–θ

v

E0′

0

IRm

E0′

IRe –6

k0

id,c E0′

–8 –6

–4

–2

0 2 log λ

4

6

8

Figure 17.6.4 Zone diagram for CV at a blocked electrode with disk-like active areas. Dimensionless parameters 𝜆 and Λ are defined in (17.6.4) and (17.6.5). The legend at upper right shows directions of change caused by increases in the identified experimental parameters. Zone labels are discussed in text. [Adapted from Amatore, Savéant, and Tessier (92), with permission from Elsevier Science.]

For systems in which the blocking layer is thin and the transit time in the channel is negligible, the problem is essentially the same as for CV at an array of UMEs. A general treatment is available (91), but we will consider only in an earlier, simpler approach based on the model of Figure 17.6.2b (92). In that case, 𝜋r02 ∕𝜋R20 ≈ 1 − 𝜃 and r0 = R0 (1 – 𝜃)1/2 ; therefore, 𝜃 and R0 fully describe the geometry. Apart from these variables, the CV for the one-step, one-electron reaction k 0 ,𝛼

−−−−−−− → A+e ← − B

(17.6.3)

depends only on the indicated kinetic parameters and the concentrations and diffusivities of A and B. If the latter are assumed to be equal and 𝛼 is taken as 0.5, the predicted behavior (92) can be summarized in the two-dimensional zone diagram of Figure 17.6.4, where the axes are defined by two dimensionless parameters, [D(1 − 𝜃)∕fv]1∕2 0.6R0 0 k (1 − 𝜃) Λ= (Dfv)1∕2 𝜆=

(17.6.4) (17.6.5)

For large values of 𝜆 (e.g., at low scan rates), the diffusion layer thickness is large compared to the spacing of the sites; hence, the diffusion fields of the individual sites merge as in Figure 6.1.6c. In Figure 17.6.4, the corresponding zone labels are subscripted “m” (for “merged”). The electrode behaves as though the total area—blocked and unblocked—is active, but with an 0 = k 0 (1 − 𝜃). If k 0 is sufficiently apparent decrease in the heterogeneous rate constant to kapp large that Λ remains high (zone DOm ), a nernstian CV is observed. If Λ is not so large, the CV becomes quasireversible (zone QRm ) or totally irreversible (zone IRm ).

795

796

17 Electroactive Layers and Modified Electrodes

For small values of 𝜆 (e.g., at higher scan rates), the diffusion layer thickness is smaller than the spacing of active sites and the behavior is characteristic of an ensemble of isolated disk-like UMEs as in Figure 6.1.6b (zone labels subscripted “e,” for “ensemble”). If the time scale remains long enough to establish steady-state diffusion at each active site, one observes the typical SSV for a UME, but with a limiting current scaled by the number of active sites. If the total geometric area of the electrode is A, there are p ≈ A∕𝜋R20 active “disks” and the total limiting current is ∗r p = id,c = 4FDA CA 0

∗ (1 − 𝜃)1∕2 4FADA CA

𝜋R0

(17.6.6)

In zone DOe , Λ is large enough to provide reversibility. In zones QRm and IRm , the kinetics are sluggish, so the wave broadens and shifts to a more extreme potential. The more general treatment of this problem (91) yields a more refined version of the zone diagram, but having the same nine behavioral cases. 17.6.2

Tunneling Through Blocking Films

By definition, mediated electron transfer like that discussed in Section 17.5 cannot occur in blocking films. However, for a very thin film, e.g., a SAM of alkane thiols or an oxide layer, electrons can tunnel through the barrier and cause faradaic reactions. This phenomenon is important in electronic devices, in passivation of metal surfaces, and in fundamental studies of electron-transfer kinetics. The basic concepts of electron tunneling are discussed in Section 3.5.2. The exponential decrease in the tunneling rate with distance implies that the process can be important only with blocking films thinner than about 2 nm. Studies of electron tunneling to and from redox centers have generally been of two types. One involves a blocking film and electroactive molecules in solution (Figure 17.6.5a). The other involves electroactive groups tethered on the opposite end of a linking chain from the attachment site, usually in a mixed monolayer containing similar molecules without the electroactive group (Figures 3.5.2 and 17.6.5b). In studies of either kind, the focus is on (a) how the rate constant for electron transfer depends on distance between the electroactive center and (a)

e

A

A

Au SAM

Solution (b)

e

A

Au A

Figure 17.6.5 Electron tunneling through surface layers. (a) Through a blocking layer to an electroactive solute, A and (b) to an electroactive group, A, covalently attached to the layer.

17.6 Blocking Layers

the conductive electrode surface and (b) how it is affected by potential or other experimental conditions. For meaningful measurements, the blocking film must be free of pinholes or other defects allowing direct access of electroactive solutes or tethered electroactive groups to the substrate. Moreover, the films should have a well-defined structure, so that the distance between substrate and electroactive group is constant and known. Experimental work with tethered electroactive species is less sensitive to pinholes than that with dissolved electroreactants, although heterogeneity, roughness of the substrate, and film defects can still play a role. For a tethered electroreactant, the electron-transfer rate constant, k, has units of a first-order reaction (s−1 ). Rate constants can be determined by a voltammetric method, as described earlier for electroactive monolayers (Section 17.2.3). Alternatively, potential-step chronoamperometry can be employed as delineated in Section 6.7, in which case the current follows a simple exponential decay (86, 93, 94). Results of this kind are presented in Figure 17.6.6, where one can see that the rate constant does not remain purely exponential with overpotential. The rollovers at high overpotentials are predicted by the Marcus model, but not the Butler–Volmer model [Section 3.5.4(a)]. In work with dissolved reactants, the quality of the blocking layer is very important; hence, experimental approaches have been developed for characterizing it (86, 95–97). Favorable performance has been obtained with TiO2 and Ta2 O5 (97), especially on UMEs, where defect-free areas need not be large. Because the electroreactant is freely diffusing in experiments of this kind, the rate constants are typical for heterogeneous electron transfer (e.g., k 0 or k f in cm/s) and can be extracted normally from SSV, CV, chronoamperometry, EIS, and other techniques.

14 (a)

(b)

ln k/s–1

12

10

8

Reduction of Os(III) Oxidation of Os(II) 6 –0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

η/V ′

Figure 17.6.6 Dependence of ln k on 𝜂 = E − E 0 for osmium redox centers, Os(bpy)2 Cl(pnp)3+/2+ , anchored to a Pt UME in chloroform + 0.1 M TBAP. Rate constants from chronoamperometry at potential E after a step from ′ E 0 . The ligand pnp is py(CH2 )n py, where py is 4-pyridyl. One terminal py is coordinated to Os and the other to a Pt atom on the electrode surface. (a) With p2p as the linking ligand. (b) With p3p as the linking ligand. [Adapted with permission from Forster and Faulkner (94). © 1994, American Chemical Society.]

797

798

17 Electroactive Layers and Modified Electrodes

Tunneling through a blocking layer is exploited quite differently in experiments in which a single metal particle, such as Pt with a radius of several nm, is hosted on a UME covered with a thin oxide layer (98, 99). In this case, the tunneling is between two conducting phases—the particle and the UME. If the oxide layer thickness is optimized, the UME remains fully isolated from electrochemistry with species in the electrolyte, but the particle is well supported by tunneling (98) and can serve as an active electrode. Using Pt particles of r0 = 1 − 40 nm and 3+∕2+

analysis by the KL method (Section 5.4.6), k 0 was measured as 36 cm/s for Ru(NH3 )6 in 0.1 M KNO3 (99). Experiments involving single-particle electrochemistry are discussed more fully in Chapter 19.

17.7 Other Methods for Characterizing Layers on Electrodes While electrochemical methods provide effective ways of studying thermodynamics and kinetics at modified electrodes, they provide little information about chemical or physical structure or about elemental composition. Fuller characterization requires many of the nonelectrochemical methods described in Chapter 21: • The structure of the film can be explored by scanning electron microscopy (SEM, STEM) and the different forms of scanning probe microscopy (STM, AFM, SECM, SECCM). • Elemental composition can be obtained by SEM with energy-dispersive X-ray spectroscopy (EDS) or by photoelectron spectroscopy (XPS). • Molecular identities of participants in surface layers can be determined using vibrational spectroscopy (SERS, SEIRAS, HREELS, TERS). • Mass changes during an electrochemical process can be measured by a quartz crystal microbalance (QCM). • The film thickness, l, is a key parameter in interpreting many electrochemical experiments. It is often estimated from the amount of material on the electrode by assuming a value for the film density. Ellipsometry is particularly useful for measuring l for thin films and for monitoring film growth. The thickness can also be determined by profilometry, AFM, and SECM. In-situ methods are prized because removal of a layered electrode from the electrochemical environment often causes substantial chemical and physical alteration. For example, the thickness can change appreciably if the film becomes desolvated. One often assumes that the composition and properties are uniform throughout a modifying layer, so that parameters like DS and DE are constants. However, the composition and diffusion coefficients may be functions of distance from the substrate, especially with thicker films. SECM can be used to probe inside a film and perform electrochemical experiments at a tip as a function of penetration depth (Section 18.7.3).

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification We close this chapter by recognizing four areas where electrochemical methodology has advanced by use of electrodes bearing an electroactive layer or surface modification.

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification

17.8.1

Electrocatalysis

Chapter 15 provides an extensive summary of electrocatalysis for electrode reactions of technological importance, many carried out at large scale. Electrochemical systems used for energy conversion and electrosynthesis often employ modified electrodes (100–103). The electrodes for such purposes are usually porous, so that they have enough active area to support a practical rate of electrolysis. They are commonly made of high area carbon compacted with an inert binder (often a polymer) around a wire mesh that acts as a current collector. While this structure can be robust enough for the intended application, it typically will not support the desired electrode reaction because it is not electrocatalytic. Thus, tiny amounts of electrocatalyst—often a precious active metal such as Pt or Pd, or a catalytic transition-metal compound, such as a cobalt oxide—are “decorated” onto the carbon particles before binding. Although these electrodes do not involve layered modifications, they function because they are indeed modified, and they can be understood using principles and experimental methods already presented in this chapter. Dimensionally stable anodes [DSA; Section 20.1.5(a)], widely used in the chloralkali industry, are modified electrodes of a different sort. They are structures in which a noncatalytic metal surface (often Ti) is fully covered by a layer of catalytic transition-metal oxide, such as RuO2 − TiO2 . 17.8.2

Bioelectrocatalysis Based on Enzyme-Modified Electrodes

Enzymes that catalyze electron-transfer reactions between two substrate molecules provide a means for carrying out redox reactions that are otherwise difficult to achieve electrochemically (104–106). These enzymes are called oxidoreductase enzymes10 because both oxidation and reduction occur at the same enzyme, with electrons transferred from the donor molecule to an acceptor molecule (Figure 17.8.1a). A well-known example is glucose oxidase, GOx, which catalyzes the oxidation of glucose by O2 to produce d-glucono-1,5-lactone and H2 O2 : GOx

glucose + O2 −−−−−→ gluconolactone + H2 O2

(17.8.1)

The gluconolactone then hydrolyzes into gluconic acid. Figure 17.8.1 (a) Reduction of substrate S1 and oxidation of substrate S2 by a redox enzyme. (b) Direct electron transfer between the electrode and the enzyme. (c) Mediated electron transfer between the electrode and enzyme. [Based on Milton and Minteer (107).]

(a)

S2,ox

S1,ox e

enzyme

e

S2,red

S1,red

(b) S 2,ox e

enzyme

e

electrode

S2,red (c)

M1,ox

S2,ox e S2,red 10 Also called redox enzymes in the electrochemical literature.

enzyme

e

e M1,red

electrode

799

800

17 Electroactive Layers and Modified Electrodes

The active site of GOx is the redox coenzyme, flavin adenine dinucleotide (FAD/H2 FAD), located within the enzyme interior. It serves as a redox mediator, accomplishing (17.8.1) as the net result of two distinct processes: glucose + GOx-FAD → gluconolactone + GOx-H2 FAD

(17.8.2)

GOx-H2 FAD + O2 → GOx-FAD + H2 O2

(17.8.3)

Both are complex inner-sphere reactions, involving binding of O2 and glucose to the active sites and transfers of both electrons and protons. For (17.8.2) and (17.8.3) to occur sponta′ ′ neously, the order of the redox potentials must follow E0 (O2 ∕H2 O2 ) > E0 (FAD∕H2 FAD) > ′ ′ E0 (glucose∕gluconolactone). At pH 7, the values of E0 vs. NHE decrease for these three couples in the order: 0.281, −0.080 (108), and −0.36 V (109, 110); thus, both (17.8.2) and (17.8.3) are thermodynamically downhill, as required. The catalytic activity of oxidoreductase enzymes can be used advantageously in electrochemical systems by replacing one of the substrate molecules by an electrode (Figures 17.8.1b,c). The role of the electrode is simply to supply or to remove electrons from the enzyme, allowing it to catalyze the reaction of interest. For instance, in the case of glucose oxidation, (17.8.1), the electrode can be used as an electron sink, replacing the need for the molecular oxidant, O2 . In electrochemical systems, oxidoreductase enzymes are generally employed in layers on the electrode surface, either directly in contact with the electrode, or more often as a component of a polymeric hydrogel (111). Direct electron transfer between an adsorbed enzyme and an electrode is often very sluggish because enzymes are usually relatively large (∼5 to 15 nm diameter), with the redox-active center located sufficiently far from the outer protein surface that the distance for direct electron transfer exceeds the tunneling distance of electrons (∼0.8 nm) (112). While small species like O2 and glucose can freely diffuse to the active site and react, most oxidoreductase enzymes cannot be oxidized or reduced directly at an electrode.11 Thus, a redox mediator that can access the active site is often used to facilitate electrical communication between the electrode and enzyme (Figure 17.8.1c). It may be a small redox-active solute or a redox-active polymer that is co-deposited on the electrode surface with the enzyme. Many applications of oxidoreductase enzymes in electrochemistry rely on mediated electron transfer to maximize the current that the enzyme can support. When a mediator is not used, a special pretreatment is employed to promote long-range electron transfer, e.g., modification of the electrode surface with molecules that orient the enzyme favorably for direct electron transfer (107, 113). In designing modified electrodes for sensing applications, one can take advantage of the high specificity displayed by some biological enzymes (114, 115). Figure 17.8.2a shows an example in which a redox polymer used to conduct electrons away from GOx to an electrode surface (70). This polymer is co-deposited with GOx, along with a crosslinking agent, to create a relatively thick film (∼4 μm), that, when wetted, swells to create a hydrogel structure. Redox centers (OsL2+ , where L = N,N ′ -dialkylated-2,2′ -biimidazole), tethered to the polymer backbone 3 through flexible C13 alkyl spacers, are used to transport electrons through the film via electron self-exchange [Section 17.5.2(c)]. A sufficiently positive potential is applied to the electrode to oxidize the Os(II) redox sites to Os(III), which then oxidizes H2 FAD, resulting in the oxidation of glucose and regeneration of FAD: GOx-H2 FAD + 2OsL3+ → GOx-FAD + 2OsL2+ + 2H+ 3 3

(17.8.4)

11 Reported observations of direct electron transfer between GOx and the electrode are critically discussed in reference (112).

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification 120

(a)

(b)

11

85

N+

4Cl–

N

4 N+

15

j/μA cm–2

70 20 –30 –80 COO–

O NH

–130 –0.4

–0.3

–0.2

–0.1

0.0

0.1

E/ V vs. Ag/AgCl 1200

(c) Me

2+/3+ N

Me

N

N

Os

N N

N Me

850

N

N

N

N N

Me

Me

j/μA cm–2

N

500

150

–200 –0.4

–0.2

0.0

0.2

0.4

E/ V vs. Ag/AgCl

Figure 17.8.2 (a) Structure of an Os(III)/Os(II) redox polymer designed to electrically connect the reaction centers of GOx to the electrode. (b) CV of a non-cross-linked, adsorbed film of the polymer in (a) on a 3-mm-diameter GC electrode. Under Ar in 0.1 M NaCl, 20 mM phosphate buffer (pH 7) at T = 37 ∘ C; v = 20 mV/s. Scan begins at −0.3 V and first moves positively; anodic currents are up. (c) SSV (v = 1 mV/s) at a 7-μm-diameter, 2-cm-long C fiber modified as described in the text. Under air in a phosphate buffer containing 15 mM glucose; T = 37.5 ∘ C. Scan begins at −0.4 V; anodic currents are up. [Adapted with permission from Mao, Mano, and Heller (70). © 2003, American Chemical Society.]

Effectively, OsL3+ plays the role of O2 in (17.8.3) in driving the oxidation of glucose. The 3 use of a redox polymer to transport electrons is sometimes called electrical wiring of the enzyme (116). Figure 17.8.2b shows the CV response of the OsL2+ polymer prior to mixing with GOx. There 3 is a nearly symmetrical wave with ΔEp ≈ 5 mV, characteristic of fast electron conduction within 3+∕2+

the OsL2+ film. The formal potential of OsL3 is −0.195 V vs. Ag/AgCl, slightly positive of 3 ′ 0 E (FAD∕H2 FAD) = −0.277 V vs. Ag/AgCl (1 M); thus, the oxidation of H2 FAD in (17.8.4) is thermodynamically downhill. To create a sensor for monitoring glucose, the Os2/+3 L3 polymer is mixed with GOx and cross-linker to make the hydrogel, which is then coated on the surface of a C fiber. When placed in an aqueous solution containing glucose, the modified electrode displays SSV (Figure 17.8.2c) with a limiting current proportional to the concentration of glucose over the range 0–100 mM. Enzyme-modified electrodes are often able to carry out a desired reaction at potentials very close to the thermodynamic potential for the reaction and with moderately high current densities; thus, they are also of interest as cathodes and anodes in enzyme-based fuel cells (117–119). For instance, the 4e reduction of O2 to H2 O can be carried out at physiological pH using enzyme-based cathodes of similar design to that described above, but incorporating a multi-Cu-atom oxidase as the catalytic site for O2 reduction (120). Depending on the specific

801

17 Electroactive Layers and Modified Electrodes

Poly(allylamine) backbone

CO2

NH2 NH

O

1 NH2 NH

O Co

e

Electrode

9 NH

O Co

e

Co

e

e

Electron mediator self-exchange

Formate

(a) 100

60

20 mM NaHCO3

80

50

40

Blank

20

j/μA cm–2

60 j/μA cm–2

802

40

0

20

–20

10

–40 –0.4

kM = 2.5 ± 0.2 mM jmax = 62 ± 1μA cm–2

30

0 –0.5

–0.6

–0.7

E/V vs. NHE

0

10 20 30 40 NaHCO3 concentration/mM

(b)

(c)

50

Figure 17.8.3 (a) Electroenzymatic reduction of CO2 by Mo–FDH immobilized on an electrode surface with a cobaltocene redox polymer, Co–PAA. (b) CV of a Mo–FDH/Co–PAA-modified electrode in the absence (solid curve) and presence (dashed curve) of 20 mM NaHCO3 in 1 M potassium phosphate buffer (pH 6). v = 1 mV/s. (c) Voltammetric current density at −0.66 V vs. NHE as a function of NaHCO3 concentration. [Adapted from Yuan et al. (122), with permission.]

catalyst, these electrodes can display overpotentials for the 4e reduction of O2 smaller than ′ 100 mV (vs. E0 at pH 6–7) (121), significantly below the overpotential observed at Pt or other metal electrocatalysts. Figure 17.8.3a depicts a different enzyme-modified electrode used for the reduction of CO2 to formate. Here, the enzyme is a molybdenum-dependent formate dehydrogenase (Mo–FDH), which is capable of interconverting CO2 and formate, CO2 + H+ + 2e ⇄ HCOO−

(17.8.5) ′

It is wired to a carbon electrode using a redox-active cobaltocene polymer with a E0 for ′ the Co(II/I) couple of −0.576 V vs. NHE, slightly more negative than E0 = −0.420 V for the CO2 /formate couple. Reduction of the cobaltocene centers nearest the electrode surface causes transport of electrons to the active site of Mo–FDH, where they are used to reduce CO2 to formate. Figure 17.8.3b shows the CV response of this enzymatic electrode in the absence and presence of 20 mM NaHCO3 , which at pH 6 dissociates to CO2 and H2 O. In the absence of NaHCO3 , the voltammetric response corresponds to the reversible reduction of the polymer′ bound cobaltocene centers at E0 [Co(II∕I)]. After addition of NaHCO3 , the cathodic current is enhanced and reaches a limiting plateau, while the reverse anodic wave is absent. These voltammetric features are characteristic of an electrocatalytic response (Section 13.3.4).

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification

Electrochemical measurements also provide fundamental information about the kinetics of enzymatic reactions. Figure 17.8.3c shows a plot of steady-state catalytic currents for CO2 reduction measured at −0.66 V vs. NHE as a function of the concentration of NaHCO3 . These currents follow a Michaelis–Menten model with an apparent Michaelis constant, K M , of 2.5 ± 0.2 mM NaHCO3 and a maximum enzyme velocity corresponding to the maximum current density of 62 ± 1 mA/cm2 . The faradaic efficiency for converting CO2 to formate was found to be 95%, reflecting the high efficiency of the cobaltocene polymer in transporting electrons to the electrocatalytic Mo–FDH sites. More elaborate strategies are based on enzyme-modified electrodes in which two or more enzymes are used to carry out sequential biocatalytic steps of an overall multistep reaction. This process is referred to as cascaded bioelectrocatalysis (123). 17.8.3

Electrochemical Sensors

Sensors attract broad interest for purposes such as health management, safety assurance, environmental monitoring, and process control. Their principles of operation are highly diverse. Most measure a physical or chemical variable; others simply count events. Chemical sensors, a subset, are mainly designed for an analytical role. The goal is to realize a device that responds ideally, and without interference, to a particular chemical analyte (or class of analytes) in the concentration range relevant to the intended application. Most chemical sensors are expected to operate reliably in complex situations where matrix effects may be challenging; therefore, the device must incorporate features that assure selectivity regardless of the environment. In some sensors, the key measurement or chemical transformation is electrochemical. Nearly all of these fall into four categories: 1) Amperometric or coulometric sensors, in which the primary measurement is a current (or an integrated current) arising from electrooxidation or electroreduction of the analyte. Widely used examples include Clark oxygen electrodes (Section 2.4.4), CO monitors for indoor air, and commercial electrochemical glucose sensors [Section (a) below]. 2) Potentiometric sensors, in which the primary measurement is the potential of a selective interface vs. a reference electrode. A prime example is the glass pH electrode (Section 2.4.2), which may be the oldest practical electrochemical sensor. Nearly, all other ISEs (Section 2.4) are also potentiometric sensors. 3) Voltammetric sensors, which are designed to serve as working electrodes for electroanalytical voltammetry (usually DPV, SWV, or stripping voltammetry). The primary measurement is a voltammetric peak height or wave height. An example is found in Section (b) below. 4) Electrophotonic sensors, in which an electrochemical process is detected by photons. Many examples are built on the phenomenon of electrogenerated chemiluminescence (ECL), in which an electron transfer produces an emitting excited state. ECL is covered at length in Section 20.5. It is highly sensitive and has been quite successfully exploited for analysis. Tens of thousands of publications have reported research on electrochemical sensors, and uncountable concepts have been advanced. A comprehensive summary is impractical, but it is still useful to consider major challenges and design principles, which are illustrated using two examples. (a) Glucose Sensors

The best illustration of success among commercial products is the Heller glucose sensor (73, 124), which is based on the concepts of Figure 17.8.2. The fundamentals were discussed

803

804

17 Electroactive Layers and Modified Electrodes

in Section 17.8.2. Here, we concentrate on finished products, which have been commercial in several generations since 2000.12 The point of focusing on products is to highlight ways that design challenges have been mastered. The management of diabetes rests on the measurement of glucose, ideally by the patient, either several times daily or even continuously. Historically, this task has been accomplished using blood sampled from finger-pricks. Principal goals in developing new measurement systems are to make the process more comfortable for the patient, to improve the frequency with which results can be obtained, to improve reliability and quality of performance, and to reduce the attention required from the patient. In the first commercial sensor based on the Heller concept (73), the glucose sensor was at one end of a disposable, one-time test strip requiring only 300 nL of blood, which could be taken from a tiny prick in a less sensitive area than a fingertip, such as a forearm. The opposite end of the strip held a set of electronic connectors and was inserted into a module for potential control, measurement, and readout. The strip was constructed of layered plastic with screen-printed features, including a channel of precise volume that would take up blood by capillarity. One side of the channel was a working electrode modified functionally as described for Figure 17.8.2.13 The measurement itself was coulometric. Essentially, all glucose in the captured volume was electrocatalytically converted to gluconolactone during a measurement requiring several seconds. This device and its successors remain in broad use internationally, and billions of test strips have been manufactured and used by patients. A more recent commercial product based on the Heller sensor is a continuous glucose monitor having electrodes that are implanted just under the skin. The body of the device, manifesting potential control and logging data, is not implanted, but is a small pod adhesively mounted to the skin (usually of an upper arm) at the point where the electrodes are inserted subcutaneously. The sensing cell is borne on a single element about 5 mm long and having a rectangular cross section ∼330 μm wide and ∼200 μm thick. It is roughly the same size as a fine (30 gauge) hypodermic needle. The whole element, including a modified working electrode, a reference electrode, and a counter electrode, is made by screen-printing on a polyester base. The system samples subcutaneous interstitial fluid, rather than blood, and can remain in place for up to two weeks. The carbon working electrode features two distinct modification layers (Figure 17.8.4a), both based on polymers: 1) The inner layer, in contact with the electrode surface, has essentially the character of the modification layer discussed in connection with Figure 17.8.2. It provides for mediated electrocatalytic oxidation of glucose arriving from the interstitial fluid. 2) The outer layer, facing the interstitial fluid 2, has two purposes: a) Limiting diffusion of glucose from the interstitial fluid to the inner layer. This layer is a membrane where the diffusivity of glucose is much lower than in the interstitial fluid, so that the diffusion profile supporting transport of glucose is essentially entirely contained in this membrane. While this layer lowers the flux of glucose (and, therefore, the measured current), it improves performance by providing a wider linear range and rendering the sensor insensitive to variations in transport in the interstitial fluid caused by placement of the electrodes or physical movement by the patient. This membrane also improves the selectivity of the device. 12 From Abbott Laboratories under the FreeStyle trademark. 13 But without having the Os(III/II) complexes bound to a polymer. Since the test strips were for a one-time use lasting only seconds, long-time stability of the layer after exposure to blood was not important.

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification

Carbon electrode

Subcutaneous fluid

Sensor glucose/mg dL–1

300 200 100 0

0

100

200

300

Blood glucose/mg dL–1

100

0 0

20

Insulin bolus

Layer 2: Transport-limiting and anti-fouling layer

200

Glucose infusion

Layer 1: Wired glucose oxidase layer

Glucose concentration/mg dL–1

300

40

60

80

100

120

140

160

Time/min

(a)

(b)

Figure 17.8.4 (a) Schematic illustration of the modified working electrode used in the continuous glucose monitor discussed in the text. Each layer has distinct roles. (b) Similarity of glucose concentrations in the blood and subcutaneous interstitial fluid of a rat: Triangles: In withdrawn blood. Dotted curve: In the vein, measured by the sensor. Solid curve: In the subcutaneous interstitial fluid, measured by the sensor. The rat was given a glucose infusion and a subsequent insulin dose at the indicated times. Inset in (b): Sensor measurements in interstitial fluid vs. analysis in drawn blood. [Part (b) reprinted by permission from Heller and Feldman (73). © 2010, American Chemical Society.]

b) Establishing an inert, biocompatible barrier between the functional part of the sensor and the living organism, while still admitting glucose and small ions. Because the working electrode is implanted for long periods, this is an important measure. Not only can the well-being of the patient be affected by bio-incompatibility, but the performance of the device may also degrade if the patient’s body reacts by forming deposits on the outer surface. This system provides glucose measurements by averaging the steady-state current over successive one-minute periods, and it can store several hours of minute-by-minute data. Results are sent by telemetry, either to a dedicated module or to a mobile phone, which the user can employ to examine results. Alarms can be triggered when the glucose level goes above or below acceptable bounds. An important piece of research carried out in connection with the implantable system concerned the reliability of glucose measurements in subcutaneous interstitial fluid vs. whole blood (73). Figure 17.8.4b shows results indicating that essentially the same clinical information is obtained by all approaches. The market for reliable, improved glucose sensors is large, and it drives continuing development of concepts. Extensive reviews are available and are frequently updated (124–126). (b) Sensors Based on DNA Aptamers

The second example is not a finished commercial product, but a demonstration of principle with broad potential applicability. It is well-known that biopolymers, especially polypeptides and polynucleotides, can bind target molecules, often with very high selectivity. Molecular recognition and binding take place at a segment of the biopolymer known as an aptamer. It is possible to isolate aptamers for a wide range of targets, including large and small biomolecules, organic

805

17 Electroactive Layers and Modified Electrodes

(a)

M

B

Labeled DNA aptamer Cocaine

Xe

MB

Au electrode

1.6 × 1012 cm–2

(c) 200 0.64 with 0.62 cocaine 0.60 0.58

without cocaine

0.56 0.54 –0.5 –0.4 –0.3 –0.2 –0.1 E/V vs. Ag/AgCl

e

Au electrode

Change in peak height/%

(b)

ac current/μA

806

150

2.6 × 1012 cm–2

100

4.4 × 1012 cm–2

50

0 0

200

400 600 800 1000 [Cocaine]/μM

Figure 17.8.5 (a) Schematic representation of target binding at an immobilized, labeled DNA aptamer on an Au electrode. The electroactive label, MB, is methylene blue. (b) ac voltammetry of the MB label on the immobilized aptamers. Scan begins at the positive limit and moves negatively. (c) Percentage change in ac voltammetric peak height vs. cocaine concentration in the bulk solution. Numbers by curves are the corresponding aptamer densities on the electrode. The different densities (bottom to top) were established by exposing the clean Au surface to 25-, 60-, and 500-nM concentrations of the aptamer during fabrication, respectively. The solid curves correspond (bottom to top) to dissociation constants of 327 ± 64, 101 ± 8, and 127 ± 35 μM. [Adapted with permission from White et al. (129). © 2008, American Chemical Society.]

molecules, and inorganic species (127). Here, we will see that aptamers can be exploited as modifying agents for electrodes and that they impart a high degree of selectivity in the electrochemical response at the electrode (127–130). Indeed, the very fact of molecular recognition was used to impart selectivity to the Heller glucose sensor, just discussed. In that case, the strategy was to employ an evolved enzyme in its normal biological role. Glucose oxidase was used to recognize and to oxidize glucose, just as in nature. In this section, the strategy is different. Use is made only of aptamers, which can be selected from DNA, RNA, or polypeptide samples for molecules that might be rare in nature, or not appear there at all. Figure 17.8.5a illustrates the concept (128) for a system based on a DNA sequence that selectively binds cocaine. The DNA aptamer was derivatized by addition of a thiol functionality to one end and a methylene blue (MB) label to the opposite end. The derivatized aptamer can be immobilized on a gold surface by irreversible thiol binding, just as for the SAMs discussed previously in this chapter. This is done simply by exposing a clean Au surface to a solution of the aptamer. Subsequently, the electrode is exposed to a solution of a short-chain thiol, which binds to sites uncovered by the aptamer. For Figure 17.8.5, this component was 6-mercapto-1-hexanol.

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification

The label, MB, is reducible in a 2e/1H+ heterogeneous process to leucomethylene blue (LMB), H N

N Me

N

S

Me

N+

Me + H+ + 2e

Me MB

Me

N

S

Me

Me

N Me

LMB

(17.8.6) ′ E0

For this reaction, is near −0.3 V vs. Ag/AgCl at neutral pH (131). The labeled aptamer can be interrogated voltammetrically, and Figure 17.8.5b shows that there is a well-developed peak in ac voltammetry. In the absence of cocaine, the aptamers should extend themselves, leaving the ionic MB moieties largely in the aqueous zone at the outer boundary of the SAM. Thus, one can expect that a minority of the MB in the surface layer would be close enough to the electrode surface to engage in the faradic process. Figure 17.8.5b shows that some are indeed close enough. When the modified electrode is exposed to cocaine, the ac response is significantly enhanced. This effect is attributed to folding of the aptamer upon binding of the cocaine, which is proposed to bring a larger fraction of the MB labels into faradaic engagement with the electrode. Figure 17.8.5c illustrates the percentage change in peak height from the blank value as a function of the concentration of cocaine in the pH 7 citrate buffer outside the SAM. The behavior follows expectations for a thermodynamic binding equilibrium, with saturation at high concentrations of the cocaine. These results are noteworthy because of their potential generality: • DNA aptamers can be defined for a wide range of targets. • The anchoring and labeling techniques employed here are general for DNA aptamers. • The binding of the target is thermodynamically reversible and required no additional reagent; moreover, the aptamer attachment to the electrode is lasting, so the system is reusable as fabricated. • The short-chain component of the SAM passivates areas of the underlying electrode that would otherwise be open to the analyte solution, thereby reducing the residual current and improving selectivity. There is considerable continuing activity with electrodes modified by aptamers toward sensor development (132, 133). We will encounter another example of an electrode modified with DNA in a different context in Section 20.5.5(c). (c) Sensor Research and Development

The exploration and development of electrochemical sensors is a highly active field that is covered in many specialized reviews. With the foregoing examples, we have encountered some of the principal considerations bearing on success. Continuing research is addressing all primary aspects, which we now revisit in two groups: 1) Selectivity, sensitivity, and anti-fouling. These three factors together determine a sensor’s quality of performance in a practical environment. The greatest of these three is selectivity. Electrochemical measurements are not, in themselves, highly selective; hence, a high degree of selectivity must be imparted by incorporating an additional factor prior to the

807

808

17 Electroactive Layers and Modified Electrodes

electrochemical measurement. The most effective approaches rest on selective access to the electrode. One option is to place a selective screen in front of the electrode that will admit the analyte but reject interferents. The valinomycin-based ISE for K+ (Section 2.4.3) is an example, as is the DNA-aptamer-based system just discussed. A second option is to use a selective reaction that will convert only the analyte to a species that can be detected at the electrode. Enzymatic catalysis is a practical way to achieve a selective reaction, and a good example is found in the Heller glucose sensor [Sections 17.8.2 and 17.8.3(a)]. The techniques of electrode modification discussed earlier in this chapter are widely investigated to improve selectivity in electrochemical sensors. Sensitivity is largely determined by the electrochemical method employed for measurement; however, systems that provide for preconcentration at the electrode by selective binding can enhance sensitivity. One can also gain a sensitivity improvement from an electrode modification that tends to passivate the electrode toward processes other than the desired one (as in the case of the DNA-aptamer-based system above). This is true because the passivation lowers the residual background. Anti-fouling measures are unglamorous but very important for practical sensors operating in difficult environments. Their effectiveness can govern the lifetime and economics of a sensor concept. Significant improvements can often be achieved by using an inert polymeric outer modification layer on the electrode, as in the case of the Heller glucose sensor (Figure 17.8.4a). 2) Materials, fabrication, environment, and device support. These four topics are important in the development of marketable devices. As one proceeds through the list, the concerns become more remote from the electrochemical method per se. Of course, materials have a chemical nature and may be critically involved in the recognition or measurement concept involved in a sensor. Interest continues in new materials for electrode modification, especially novel forms of carbon, including graphene and carbon nanotubes. Materials are also important to packaging, biocompatibility, anti-fouling, manufacturability, wearability, and other aspects that may not bear on the measurement carried out by the sensor. A large fraction of the sensor literature deals with such issues, but they lie outside our scope. Always of interest are fabrication options for creating a sensing electrode or a cell cavity. Among them are the immobilization of aptamers, the screen-printing of electrodes, or molecular printing for modification layers. Other techniques of fabrication are critical to the success of commercial products, as we have seen in Section (a) above; however, they may not relate to the electrochemical measurement. The word “environment” is used here to capture the setting in which a sensor would be used and its role in that setting. Often the environment defines important constraints on possible solutions. The idea of “wearability” is a good example of an environmental concept. A wearable glucose sensor, for instance, opens the door to quite different measurement concepts than apply to blood samples or subcutaneous interstitial fluid. However, an envisaged operational environment often has more to do with comprehensive device development than chemical measurement. Finally, we come to device support, which includes elements such as electric power, electronic control, telemetry, user interface, and display. These are all important for the success of a product, but they are derivative of the electrochemical measurement, not central to it. We will not dwell on them—except to mention the essentially electrochemical idea of self-powering.

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification

Many sensors are conceived for use in living systems, and there is high interest in long-term implantation. Obstacles are in the way, and one of them is a mechanism for supplying the energy needed for operation of the device. An interesting idea is to incorporate a biofuel cell driven by reagents available in vivo (134). For example, one might oxidize glucose using a modified electrode like that in Figure 17.8.2 and reduce O2 to H2 O at a cathode. Biocompatibility is a continuing concern for such systems. 17.8.4

Faradaic Electrochemical Measurements in vivo

Quantitative faradaic measurements are now carried out routinely in living organisms to monitor targeted molecules, especially vs. time after a stimulus or treatment is applied. This area is experimentally demanding because the medium, e.g., blood or intercellular fluid, is always complex; moreover, there is often a need to sample the system with spatial precision, perhaps at a particular location in a particular organ, or even inside a single cell (135, 136). It is natural to include this topic here because faradaic electrochemistry in vivo often rests on detection of adsorbed layers or the use of modified electrodes. Work like this usually falls in either of two categories: 1) Direct faradaic measurements using standard methods (e.g., CV) at a bare or modified working electrode, either truly in vivo or ex situ for examination of fresh clinical samples of fluids or tissue. 2) Ex situ faradaic measurements of clinical samples taken from a living organism and subjected to efficient separation methods, such as LC or CE, with electrochemical detection and quantification. This section is limited to the first category because the second was covered in Section 12.5.3. Faradaic electroanalysis in vivo originated with the desire to monitor neurotransmitters in functioning brain tissue (137). Since the earliest experiments (138), the field has expanded broadly (136, 139–144), and many investigators, especially in neuroscience, now make use of electrochemical measurements. (a) General Considerations

For electrodes implanted in vivo, the dual requirements of precise spatial positioning and minimal impact on the organism point immediately to the need for a UME. Indeed, the broad adoption of UMEs across electrochemistry occurred after they were introduced as working electrodes for voltammetry in living systems. Practically, all faradaic electrochemistry in vivo is now based on UMEs. Working electrodes used ex situ for fresh samples taken in vivo are not so restricted with respect to size. An extremely useful advantage of a UME is the ability to operate in a two-electrode cell configuration (Sections 1.5 and 16.8.4). For measurements in vivo, the uncompensated ohmic drop is always negligible, and electrode placement is simplified by comparison to three-electrode systems. These advantages are compelling for faradaic electrochemistry in vivo, so two-electrode cells are widely employed. If the application involves ex situ measurements on clinical samples, a three-electrode system might be required, especially with a larger working electrode. Working electrodes are often exposed carbon fibers acting as cylindrical UMEs. Useful electrodes are commercially available. Otherwise, suitable UMEs can be prepared as discussed in Sections 5.9 and 6.8.1. Because electrochemical responses often are due to adsorbate layers, the electrode material is often critical to success. Carbon is the dominant material employed in this field.

809

810

17 Electroactive Layers and Modified Electrodes

If an electrode is to be implanted in a living organism, the insulating material surrounding the electrode and its lead may have to be carefully chosen for biocompatibility, especially if implantation is for a long period. This aspect has been carefully reviewed (139). (b) Fast-Scan Cyclic Voltammetry

A widely employed method for quantification in vivo is fast-scan cyclic voltammetry (FSCV) (136, 139, 141, 145–148), some aspects of which are introduced in Section 7.6. It was invented to follow time-dependent changes in the concentrations of neurotransmitters, especially catecholamines, on timescales of several seconds. Dopamine (DA) has been a primary target (136, 139–141, 145–148) and will be used here to illustrate the method and its capabilities. The relevant electrode reaction is HO

NH2

O

NH2 + 2H+ + 2e

O

HO DA

DAQ

(17.8.7)

where DAQ is dopamine-o-quinone. In conventional CV at low v in a neutral buffer solution, this system shows an apparent quasireversible response centered near 0.2 V vs. Ag/AgCl; however, the behavior is complicated by adsorption. The concept behind FSCV is to develop a time-resolved characterization of a system as its composition changes over a timescale of seconds. At each sampled time, a whole cyclic voltammogram is recorded. To sample enough times to follow the changes of interest, one must obtain several CVs per second, and to validate the concept of sampling, each CV must be recorded quickly compared to the time interval between them. The required waveform is like that shown in Figure 17.8.6, where v = 400 V/s, so that each CV is acquired in 8.5 ms, and the hold time between the CVs is 91.5 ms. The measurement cycle is repeated at 10 Hz for the desired period. For Figure 17.8.6, one obtains a set of 151 CVs describing the system at 100 ms intervals. Because of the high scan rate, there is a large background from charging current. Faradaic sources of residual current also often contribute. In FSCV, the background is always subtracted from the CVs used for data analysis. The background curve is generally taken from the data set itself. Common approaches are simply to use the first recorded CV (i.e., CV0 in Figure 17.8.6) or to average several early CVs (e.g., CV0 − CV4 ). This practice is well adapted to the demands of in vivo measurements, where one can expect the background to be made up of many variable contributions and, therefore, subject to drift. In practice, the background remains stable over seconds, but drifts over minutes. The effectiveness of background subtraction is seen in Figure 17.8.7a, which is a CV recorded for a rat dorsal striatum in an intact animal (149). The CV shows essentially zero current except where readily assignable voltammetric peaks are observed. The CV in Figure 17.8.7a is the voltammetric signature of the DAQ/DA couple in FSCV, but the shape is quite different from the quasireversible response described above for this couple in aqueous buffer at low v. The oxidation peak near 0.6 V is displaced positively by 400 mV ′ from E0 for DAQ/DA in solution. Moreover, the peak has a symmetrical shape without a diffu′ sive tail. The reduction peak is displaced 400 mV negatively from E0 in solution. These results indicate that the FSCV arises from adsorbed DA and that the electrode kinetics have become effectively irreversible (145). The high scan rate used in FSCV promotes both aspects of the

17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification

300

(To CV150)

(Continues 15s)

Hold3

CV2

200 E (DAQ/DA, soln) 0′

t/ms

Hold2

CV1

100

Hold1

400 V/s

8.5 0

CV0

Hold0 Ei = –0.4

Eλ = 1.3 E/V vs. Ag/AgCl

Figure 17.8.6 FSCV waveform for detection and quantification of DA. Positive potentials are plotted to the right to match the data in Figure 17.8.7. The working electrode is held at E i , except during CV scans, which occur every 100 ms. Each CV is recorded at 400 V/s; hence, the cyclic sweep requires 8.5 ms. The waveform is continued for the chosen period of observation. This example shows the recording of 151 CVs, with the first, CV0 , beginning at t = 0. The experimental run would end at t = 15.0085 s, with the recording of CV150 after Hold150 .

observed results, which have been confirmed in conventional studies using simple aqueous solutions. The process of background subtraction establishes FSCV as a differential measurement. The sequence of CVs shows how the system evolves after the time when the background is measured. Living systems and many clinical samples have a pre-existing concentration of the targeted analyte (DA in the system of Figure 17.8.7). In FSCV, the voltammetric response from the pre-existing concentration is included in the background curve and is then subtracted out of all later CVs; hence, the later CVs show responses only if the analyte concentration differs from the background level. One can obtain a time profile by plotting a selected voltammetric current from each CV vs. the sampling time for the CV. An example is seen in Figure 17.8.7b, which shows the time-dependent height of the principal voltammetric peak near 0.6 V vs. Ag/AgCl. The response and relaxation arising from the stimulus is evident, and the CV of Figure 17.8.7a (recorded near the maximum in Figure 17.8.7b) confirms that the released substance is DA.

811

17 Electroactive Layers and Modified Electrodes

6 4 i/nA

812

Electrical stimulation

2 0

ΔCDA = 10 nM

–2 –4 –0.4

15 s 0.0

0.4 0.8 E/V vs. Ag/AgCl (a)

1.2

(b)

Figure 17.8.7 Use of FSCV to monitor DA at a carbon fiber UME (7-μm diameter, ∼100-μm length) implanted in the dorsal striatum of a rat. The measurement run followed parameters similar to those of Figure 17.8.6. The background was defined by early CVs in the run, and this background was subtracted from all later CVs. (a) A background-subtracted CV from the run identifying DA as the source of the response. Positive potentials are to the right and anodic currents are up. (b) Time profile of current at the peak oxidation potential of DA, as obtained from all background-corrected CVs in the run. At the noted point, an electrical stimulation of 2-ms duration was applied using a bipolar electrode situated in mid-brain (remote from the working electrode). [Reprinted with permission from Spanos et al. (149). © 2013, American Chemical Society.]

The great success of FSCV with catecholamines rests in large measure on their adsorption on carbon surfaces. This leads to • A selective preconcentration of the analyte from a complex medium. • A peak current that is linear with scan rate, so that the signal-to-background is maintained at the high sweep rates required in FSCV. • An analytical sensitivity compatible with the needs of in vivo work. Figure 17.8.7b indicates, for example, that concentration changes are being followed for DA on a nanomolar scale. The practice of FSCV, as for other forms of clinical investigation, tends to be built on wellestablished measurement protocols (145–147). When the target is DA or DAQ, for instance, the waveform of Figure 17.8.6 represents standard practice, and each detail—electrode material, v, Ei , E𝜆 , and hold time—has been defined for particular reasons based on the chemical and kinetic behavior of the DAQ/DA couple (145). Similar protocols are available for other widely targeted analytes, including adenosine, histamine, serotonin, and H2 O2 (146). This method generates large amounts of data, usually at least 100 full CVs per measurement run. Considerable effort has been expended on methods for visualizing and interpreting results (136, 139, 141, 145–147).

17.9 References 1 J. Simonet, “Electro-Catalysis at Chemically Modified Solid Surfaces,” World Scientific,

London, 2018. 2 R. C. Alkire, P. N. Bartlett, and J. Lipkowski, Eds., “Nanopatterned and

Nanoparticle-Modified Electrodes: Advances in Electrochemical Science and Engineering,” Vol. 17, Wiley-VCH, Weinheim, 2017. 3 R. Seeber, F. Terzi, and C. Zanardi, “Functional Materials in Amperometric Sensing: Polymeric, Inorganic, and Nanocomposite Materials for Modified Electrodes,” Springer, Berlin/Heidelberg, 2014.

17.9 References

4 R. C. Alkire, D. M. Kolb, J. Lipkowski, and P. N. Ross, Eds., “Chemically Modified Elec-

5

6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

trodes: Advances in Electrochemical Science and Engineering,” Vol. 11, Wiley-VCH, Weinheim, 2009. M. Fujihira, I. Rubinstein, and J. F. Rusling, Eds., “Modified Electrodes,” Vol. 10 in “Encyclopedia of Electrochemistry,” A. J. Bard and M. Stratmann, Series Eds., Wiley-VCH, Weinheim, 2007. G. A. Edwards, A. J. Bergren, and M. D. Porter, in “Handbook of Electrochemistry,” C. G. Zoski, Ed., Elsevier, Amsterdam, 2006, Chap. 8. A. J. Bard, “Integrated Chemical Systems,” Wiley, New York, 1994. G. Inzelt, Electroanal. Chem., 18, 89 (1994). R. W. Murray, Ed., “Molecular Design of Electrode Surfaces,” Vol. XXII in the series, “Techniques in Chemistry,” A. Weissberger, Founding Ed., Wiley-Interscience, New York, 1992. I. Rubinstein, in “Applied Polymer Analysis and Characterization,” Vol. II, J. Mitchell, Jr., Ed., Hanser, Munich, 1992, Part III, Chap. 1. A. J. Bard and W. E. Rudzinski, in “Preparative Chemistry Using Supported Reagents,” P. Laszlo, Ed., Academic, San Diego, CA, 1987, pp. 77–97. R. W. Murray, A. G. Ewing, and R. A. Durst, Anal. Chem., 59, 379A (1987). M. Fujihira, in “Topics in Organic Electrochemistry,” A. J. Fry and W. E. Britton, Eds., Plenum, New York, 1986. C. E. D. Chidsey and R. W. Murray, Science, 231, 25 (1986). R. W. Murray, Electroanal. Chem., 13, 1 (1983). A. J. Bard, J. Chem. Educ., 60, 302 (1983). R. W. Murray, Accts. Chem. Res., 13, 135 (1980). M. Buck, Adv. Electrochem. Sci. Engr., 11, 197, 2009. K. Uosaki, Chem. Record, 9, 199 (2009). M. M. Walczak, D. D. Popenoe, R. S. Deinhammer, B. D. Lamp, C. Chung, and M. D. Porter, Langmuir, 7, 2687 (1991). T. Nakanishi, Encycloped. Electrochem., 11, 203 (2007). C.-W. Lee and A. J. Bard, J. Electroanal. Chem., 239, 441 (1988). E. Laviron, Electroanal. Chem., 12, 53 (1982). E. Laviron, Bull. Soc. Chim. Fr., 3717 (1967). S. Srinivasan and E. Gileadi, Electrochim. Acta, 11, 321 (1966). E. Laviron, J. Electroanal. Chem., 52, 355, 395 (1974). B. E. Conway, “Theory and Principles of Electrode Processes,” Ronald, New York, 1965, Chaps. 4 and 5. A. N. Frumkin and B. B. Damaskin, Mod. Asp. Electrochem., 3, 149 (1964). P. Delahay, “Double Layer and Electrode Kinetics,” Interscience, New York, 1965. E. Laviron, J. Electroanal. Chem., 100, 263 (1979). A. P. Brown and F. C. Anson, Anal. Chem., 49, 1589 (1977). H. Matsuda, K. Aoki, and K. Tokuda, J. Electroanal. Chem., 217, 1 (1987); 217, 15 (1987). C. P. Smith and H. S. White, Anal. Chem., 64, 2398 (1992). M. Ohtani, S. Kuwabata, and H. Yoneyama, Anal. Chem., 69, 1045 (1997). T. J. Duffin, N. Nerngchamnong, D. Thompson, and C. A. Nijhuis, Electrochim. Acta, 311, 92 (2019). R. Andreu, J. J. Calvente, W. R. Fawcett, and M. Molero, Langmuir, 13, 5189 (1997). P. K. Eggers, N. Darwish, M. N. Paddon-Row, and J. J. Gooding, J. Am. Chem. Soc., 134, 7539 (2012). M. Ohtani, Electrochem. Commun., 1, 488 (1999).

813

814

17 Electroactive Layers and Modified Electrodes

39 40 41 42 43

44 45 46 47 48 49 50 51 52 53 54 55 56

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

75

P. He, R. M. Crooks, and L. R. Faulkner, J. Phys. Chem., 94, 1135 (1990). H. Angerstein-Kozlowska and B. E. Conway, J. Electroanal. Chem., 95, 1 (1979). V. Plichon and E. Laviron, J. Electroanal. Chem., 71, 143 (1976). R. H. Wopschall and I. Shain, Anal. Chem., 39, 1514 (1967). S. W. Feldberg, in “Computers in Chemistry and Instrumentation: Electrochemistry,” Vol. 2, J. S. Mattson, H. B. Mark, Jr., and H. C. MacDonald, Jr., Eds., Marcel Dekker, New York, 1972, Chap. 7. C. P. Smith and H. S. White, Langmuir, 9, 2398 (1993). H. S. White, J. D. Peterson, Q. Cui, and K. J. Stevenson, J. Phys. Chem. B, 102, 2930 (1998). I. Burgess, B. Seivewright, and R. B. Lennox, Langmuir, 22, 4420 (2006). F. Ma, X. Wang, Z. Hu, L. Hou, Y. Yang, Z. Li, Y. He, and H. Zhu, Energy Fuels, 34, 13079 (2020). M. N. Jackson, M. L. Pegis, and Y. Surendranath, ACS Cent. Sci., 5, 831 (2019). F. C. Anson, Anal. Chem., 38, 54 (1966). J. H. Christie, R. A. Osteryoung, and F. C. Anson, J. Electroanal. Chem., 13, 236 (1967). F. C. Anson, J. H. Christie, and R. A. Osteryoung, J. Electroanal. Chem., 13, 343 (1967). F. C. Anson and D. J. Barclay, Anal. Chem., 40, 1791 (1968). H. B. Herman, R. L. McNeely, P. Surana, C. M. Elliot, and R. W. Murray, Anal. Chem., 46, 1268 (1974). M. T. Stankovich and A. J. Bard, J. Electroanal. Chem., 86, 189 (1978). M. P. Soriaga and A. T. Hubbard, J. Am. Chem. Soc., 104, 2735 (1982). G. N. Salaita and A. T. Hubbard, in “Molecular Design of Electrode Surfaces,” in R. W. Murray, Ed., Vol. XXII in the series, “Techniques in Chemistry,” A. Weissberger, Founding Ed., Wiley-Interscience, New York, 1992. M. P. Soriaga, J. H. White, and A. T. Hubbard, J. Phys. Chem., 87, 3048 (1983). M. Sluyters-Rehbach and J. H. Sluyters, Electroanal. Chem., 4, 1 (1970). B. Timmer, M. Sluyters-Rehbach, and J. H. Sluyters, J. Electroanal. Chem., 18, 93 (1968). P. Delahay and K. Holub, J. Electroanal. Chem., 16, 131 (1968). P. Delahay, J. Electroanal. Chem., 19, 61 (1968). I. Epelboim, C. Gabrielli, M. Keddam, and H. Takenouti, Electrochim. Acta, 20, 913 (1975) and references therein. H. O. Finklea, M. S. Ravenscroft, and D. A. Snider, Langmuir, 9, 223 (1993). T. M. Nahir and E. F. Bowden, J. Electroanal. Chem., 410, 9 (1996). S. E. Creager and T. T. Wooster, Anal. Chem., 70, 4257 (1998). F. C. Anson, Anal. Chem., 33, 1123 (1961). W. Lorenz, Z. Elektrochem., 59, 730 (1955). W. H. Reinmuth, Anal. Chem., 33, 322 (1961). S. V. Tatwawadi and A. J. Bard, Anal. Chem., 36, 2 (1964). F. Mao, N. Mano, and A. Heller, J. Am. Chem. Soc., 125, 4951 (2003). D. A. Lutterman, Y. Surendranath, and D. G. Nocera, J. Am. Chem. Soc., 131, 3838 (2009). C. Costentin and D. G. Nocera, Proc. Natl. Acad. Sci. U.S.A., 114, 13380 (2017). A. Heller and B. Feldman, Acc. Chem. Res., 43, 963 (2010). (a) C. P. Andrieux, J. M. Dumas-Bouchiat, and J.-M. Savéant, J. Electroanal. Chem., 131, 1 (1982); (b) C. P. Andrieux and J.-M. Savéant, ibid., 134, 163 (1982); ibid., 142, 1 (1982); (c) C. P. Andrieux, J. M. Dumas-Bouchiat, and J.M. Savéant, ibid., 169, 9 (1984); (d) C. P. Andrieux and J.-M. Savéant, ibid., 171, 65 (1984); (e) F. C. Anson, J.-M. Savéant, and K. Shigehara, J. Phys. Chem., 87, 214 (1983). J. Leddy and A. J. Bard. J. Electroanal. Chem., 153, 223 (1983).

17.9 References

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

J. Leddy, A. J. Bard, J. T. Maloy, and J.-M. Savéant, J. Electroanal. Chem., 187, 205 (1985). H. Dahms, J. Phys. Chem., 72, 362 (1968). I. Ruff and V. J. Friedrich, J. Phys. Chem., 75, 3297 (1971). I. Ruff, V. J. Friedrich, K. Demeter, and K. Csaillag, J. Phys. Chem., 75, 3303 (1971). I. Ruff and L. Botár, J. Chem. Phys., 83, 1292 (1985). I. Ruff and L. Botár, Chem. Phys. Lett., 126, 348 (1986). I. Ruff and L. Botár, Chem. Phys. Lett., 149, 99 (1988). M. Majda, in “Molecular Design of Electrode Surfaces,” R. W. Murray, Ed. op. cit., p. 159. D. N. Blauch and J.-M. Savéant, J. Am. Chem. Soc., 114, 3323 (1992). C. P. Andrieux and J.-M. Savéant, in “Molecular Design of Electrode Surfaces,” R. W. Murray, Ed. op. cit., p. 207. H. O. Finklea, Electroanal. Chem., 19, 109 (1996). B. R. Scharifker, J. Electroanal. Chem., 240, 61 (1988). H. Reller, E. Kirowa-Eisner, and E. Gileadi, J. Electroanal. Chem., 138, 65 (1982). T. Gueshi, K. Tokuda, and H. Matsuda, J. Electroanal. Chem., 89, 247 (1978). P. J. Peerce and A. J. Bard, J. Electroanal. Chem., 112, 97 (1980). G. Pireddu, I. Svir, C. Amatore, and A. Oleinick, ChemElectroChem, 8, 2413 (2021). C. Amatore, J.-M. Savéant, and D. Tessier, J. Electroanal. Chem., 147, 39 (1983). (a) C. E. D. Chidsey, C. R. Bertozzi, T. M. Putvinski, and A. M. Mujsce, J. Am. Chem. Soc., 112, 4301 (1990); (b) C. E. D. Chidsey, Science, 251, 919 (1991). R. J. Forster and L. R. Faulkner, J. Am. Chem. Soc., 116, 5444 (1994). C. J. Miller, “Physical Electrochemistry. Principles, Methods, and Applications,” I. Rubinstein, Ed., Marcel Dekker, New York, 1995, Chap. 2. J. Kim, B.-K. Kim, S. K. Cho, and A. J. Bard, J. Am. Chem. Soc., 136, 8173 (2014). C. M. Hill, J. Kim, N. Bodappa, and A. J. Bard, J. Am. Chem. Soc., 139, 6114 (2017). J.-N. Chazalviel and P. Allongue, J. Am. Chem. Soc., 133, 762 (2011). J. Kim and A. J. Bard, J. Am. Chem. Soc., 138, 975 (2016). T. F. Fuller and J. N. Harb, “Electrochemical Engineering,” Wiley, Hoboken, NJ, 2018. J. Newman and K. E. Thomas-Alyea, “Electrochemical Systems,” 3rd ed., Wiley, Hoboken, NJ, 2004. M. Winter and R. J. Brodd, Chem. Rev., 104, 4245 (2004). D. Pletcher and F. C. Walsh, “Industrial Electrochemistry,” 2nd ed.; Chapman and Hall, London, 1990. H. Chen, O. Simoska, K. Lim, M. Grattieri, M. Yuan, F. Dong, Y. S. Lee, K. Beaver, S. Weliwatte, E. M. Gaffney, and S. D. Minteer, Chem. Rev., 120, 12903 (2020). J. A. Cracknell, K. A. Vincent, and F. A. Armstrong, Chem. Rev., 108, 2439 (2008). H. Chen, F. Dong, and S. D. Minteer, Nat. Catal., 3, 225 (2020). R. D. Milton and S. D. Minteer, J. R. Soc. Interface, 14, 20170253 (2017). S. Vogt, M. Schneider, H. Schäfer-Eberwein, and G. Nöl, Anal. Chem., 86, 7530 (2014). K. Burton, Ergeb. Physiol, 49, 275 (1957). P. A. Loach, in “Handbook of Biochemistry: Selected Data for Molecular Biology,” H. A. Sober, Ed., CRC Press, Cleveland, OH, 1968, p. J-27. A. Heller, Curr. Opin. Chem. Biol., 10, 664 (2006). P. N. Bartlett and F. A. Al-Lolage, J. Electroanal. Chem., 819, 26 (2018). I. Mazurenko, V. P. Hitaishi, and E. Lojou, Curr. Opin. Electrochem., 19, 113 (2020). T. Adachi, Y. Kitazumi, O. Shirai and K. Kano, Sensors, 20, 4826 (2020). P. Bollella and E. Katz, Sensors, 20, 3517 (2020). A. Heller, Acc. Chem. Res., 23, 128 (1990).

815

816

17 Electroactive Layers and Modified Electrodes

117 X. Xiao, H. Xia, R. Wu, L. Bai, L. Yan, E. Magner, S. Cosnier, E. Lojou, Z. Zhu, and A. Liu, 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

Chem. Rev., 119, 9509 (2019). A. Ruff, F. Conzuelo, and W. Schuhmann, Nat. Catal., 3, 214 (2020). J. C. Ruth and A. M. Spormann, ACS Catal., 11, 5951 (2021). N. Mano and A. de Poulpiquet, Chem. Rev., 118, 2392 (2018). V. Soukharev, N. Mano, and A. Heller, J. Am. Chem. Soc., 126, 8369 (2004). M. Yuan, S. Sahin, R. Cai, S. Abdellaoui, D. P. Hickey, S. D. Minteer, and R. D. Milton, Angew. Chem. Int. Ed., 57, 6582 (2018). Y. S. Lee, K. Lim, and S. D. Minteer, Annu. Rev. Phys. Chem., 72, 467 (2021). A. Heller and B. Feldman, Chem. Rev., 108, 2842 (2008). M. Wei, Y. Qiao, H. Zhao, J. Liang, T. Li, Y. Luo, S. Lu, X. Shi, W. Lu, and X. Sun, Chem. Commun., 56, 14553 (2020). H. Teymourian, A. Barfidokht, and J. Wang, Chem. Soc. Rev., 49, 7671 (2020). I. Willner and M. Zayats, Angew. Chem. Int. Ed., 46, 6408 (2007). Y. Xiao, A. A. Lubin, A. J. Heeger, and K. W. Plaxco, Angew. Chem. Int. Ed., 44, 5456 (2005). R. J. White, N. Phares, A. A. Lubin, Y. Xiao, and K. W. Plaxco, Langmuir, 24, 10513 (2008). Y. Xiao, B. D. Piorek, K. W. Plaxco, and A. J. Heeger, J. Am. Chem. Soc., 127, 17990 (2005). S. O. Kelley, J. K. Barton, N. M. Jackson, and M. G. Hill, Bioconjugate Chem., 8, 31 (1997). M. A. Pellitero, A. Shaver, and N. Arroyo-Currás, J. Electrochem. Soc., 167, 037529 (2020). J. K. Barton, P.L. Bartels, Y. Deng, and E. O’Brien, Methods Enzymol., 591, 355 (2017). M. Rasmussen, R. E. Ritzmann, I. Lee, A. J. Pollack, and D. Scherson, J. Am. Chem. Soc., 134, 1458 (2012). D. J. Eves and A. G. Ewing, Chem. Anal., 172, 215 (2007). E. S. Bucher and R. M. Wightman, Annu. Rev. Anal. Chem., 8, 239 (2015). P. T. Kissinger, J. B. Hart, and R. N. Adams, Brain Res., 55, 20 (1973). R. N. Adams, Progr. Neurobiol., 35, 297 (1990). N. T. Rodeberg, S. G. Sandberg, J. A. Johnson, P. E. M. Phillips, and R. M. Wightman, ACS Chem. Neurosci., 8, 221 (2017). E. E. Ferapontova, Electrochim. Acta, 245, 664 (2017). G. S. Wilson and A. C. Michael, Eds., “Compendium of In Vivo Real-Time Monitoring in Molecular Neuroscience,” 3 vols., World Scientific, Singapore, 2015. D. L. Robinson, A. Hermans, A.T. Seipel, and R.M. Wightman, Chem. Rev., 108, 2554 (2008). A. C. Michael and L. M. Borland, Eds., “Electrochemical Methods for Neuroscience,” CRC Press, Boca Raton, FL, 2007. A. A. Boulton, G. B. Baker, and R. N. Adams, Eds., “Voltammetric Methods in Brain Systems,” Humana Press, Totowa, NJ, 1995. B. J. Venton and Q. Cao, Analyst, 145, 1158 (2020). P. Puthongkham and B. J. Venton, Analyst, 145, 1087 (2020). J. G. Roberts and L. A. Sombers, Anal. Chem., 90, 490 (2018). D. L. Robinson, B. J. Venton, M. L. A. V. Heien, and R. M. Wightman, Clin. Chem., 49, 1763 (2003). M. Spanos, J. Gras-Najjar, J. M. Letchworth, A. L. Sanford, J. V. Toups, and L. A. Sombers, ACS Chem. Neurosci., 4, 782 (2013). C. M. Elliott and R. W. Murray, J. Am. Chem. Soc., 96, 3321 (1974).

17.10 Problems

17.10 Problems ′

17.1

0 and Consider the system of (17.2.5) when the initial potential is well negative of Eads * the initial surface concentrations are ΓR (0) = Γ and ΓO (0) = 0. Beginning at t = 0, the potential is swept positively at rate v. (a) Derive the i − E response and show that (17.2.13) applies to the positive-going, as well as the negative-going, scan. (b) Derive the equation for the peak potential. (c) Derive the equation for the peak current.

17.2

From the curves in Figure 17.2.5b, estimate the amount of trans-4,4′ -dipyridyl-1,2ethylene adsorbed per cm2 . Assume n = 2.

17.3

∗ = 1) at 25 ∘ C, estimate from Figure 17.2.10 the For weak adsorption of a reactant (𝛽O CO ranges of v where (a) ip is proportional to v and (b) ip is proportional to v1/2 , expressed in terms of 𝛽 O , ΓO,s , and DO .

17.4

Using the data in Figure 17.3.1, calculate DO and ΓO [where O is Cd(II)]. Also calculate Qdl and C d in the absence and presence of SCN− .

17.5

Chronocoulometric experiments were used to measure the surface excess of Tl+ at a mercury/electrolyte interface (Figure 17.10.1). Of interest was the influence of bromide on the adsorption. (a) Explain how such measurements would be carried out. (b) Explain the results in terms of chemical processes.

17.6

The amount of adsorbed O, ΓO , can also be determined in a chronocoulometric experiment from the ratio of the slopes of the curves for the forward (Sf ) and reverse (Sr ) phases. Explain how.

ΓTl+ × 1010/mol cm–2

16 14 (a) (b)

12

(c)

(d)

(e)

10 8 6 4 2 0

0

5

10

15

20 25 * –/mM CBr

30

35

–0.1 –0.2 –0.3 Ei/V vs. SCE

Figure 17.10.1 Surface excesses of Tl+ at Hg in the presence of Br− . Step potential = −0.70 V vs. SCE. Values of C ∗ + and initial potential, E i , vs. SCE: (a) 1 mM, −0.30 V; (b) 1 mM, −0.20 V; (c) 0.5 mM, −0.30 V; Tl (d) 0.5 mM, −0.20 V. For (e), E i was varied with 1 mM Tl+ , 14 mM Br− . Arrows mark precipitation of TlBr from bulk solution. [Reprinted with permission from Elliott and Murray (150). © 1974, American Chemical Society.]

817

818

17 Electroactive Layers and Modified Electrodes

17.7 A thin-layer cell with a Pt electrode (A = 1.2 cm2 ) and a layer thickness of 40 μm is used to determine the amount of hydroquinone (HQ) adsorbed on Pt. The cell is first filled with a 0.100 mM solution of HQ. An irreversible adsorption is allowed to occur, then a potential step chronocoulometric experiment is carried out to oxidize the dissolved HQ (adsorbed HQ is not electroactive). Oxidation of HQ (n = 2) required 32 μC. This solution is flushed from the cell, which is rinsed and purged several times with fresh solution. A fresh aliquot is introduced. Another chronocoulometric experiment shows that 96 μC is now required. (a) Calculate the amount of HQ adsorbed, Γ (mol/cm2 ), and the area each molecule occupies, 𝜎 (nm2 /molecule). (b) From the structure of the HQ molecule (Figure 1), what orientation of the molecule on the electrode seems most likely?

819

18 Scanning Electrochemical Microscopy Electrochemists have become increasingly interested in observing electrochemical processes microscopically because they are often affected strongly by local structure, which varies even down to the atomic level. That theme has developed progressively since Chapter 14 and will continue through the remainder of the book. This chapter describes scanning electrochemical microscopy (SECM), an entirely electrochemical methodology, in which the key element is a tiny working electrode (called a tip) that can be moved three dimensionally to address different locations across a surface (called a substrate) (1–4). This method allows spatially resolved observations of electrochemical processes on the micro- and nanometer scales.

18.1 Principles The apparatus for SECM (Figure 18.1.1) consists of piezoelectric controllers for moving the tip, plus a bipotentiostat for controlling the potentials of the tip and substrate. The tip is usually a Pt wire or C fiber of radius between 1 and 25 μm, sealed in glass and then polished to form a disk UME. The glass surrounding the disk is normally beveled, so that the thickness of insulator surrounding the conductive disk (the shroud) is small. This makes it easier to position the tip very close to a substrate without the shroud touching the surface. Smaller SECM tips, 10–500 nm in radius, can be prepared by sealing a Pt wire in a pulled glass pipet (Section 5.9.1) or by etching a wire to a sharp point and then insulating it with wax or another coating. In the SECM experiment, the tip and substrate are immersed in an electrolyte solution containing an electroactive ∗ and with diffusivity D ). The cell also contains species (e.g., substance O at concentration CO O reference and counter electrodes. The principal measurables are the current at the tip, iT , and the current at the substrate, iS , as functions of the spatial position of the tip. When the tip is far from the substrate and a potential is applied, the steady-state current, iT, ∞ , is ∗a iT,∞ = GnFDO CO

(18.1.1)

where a is the radius of the tip electrode (Figure 18.1.2a),1 and G is a geometric factor that depends on the shape of the electrode and the radius of the insulating shroud (6). For a disk electrode embedded in an infinite plane, G = 4 (Section 5.2.2; equation 5.2.18b). When the tip is brought very near the substrate surface (within a few tip radii), the tip current is perturbed. The surface blocks diffusion of O to the tip, tending to decrease the current 1 Although the radius of a disk electrode is symbolized by r0 elsewhere in this book, we use a for the tip electrode radius in SECM, following standard practice in the literature. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

820

18 Scanning Electrochemical Microscopy

Tip detail Glass Electrode Tip Reference Piezo positioner

Counter

Piezo controller

Waveform generator

Bipotentiostat

Substrate

Computer and user interface

Figure 18.1.1 Schematic diagram of SECM apparatus. [Reprinted with permission from Bard et al. (5). © 1990, American Chemical Society.] (a) Hemispherical diffusion

(b) Hindered diffusion

(c) Feedback diffusion

R Insulating substrate

O

Conductive substrate

Figure 18.1.2 Basic principles of SECM: (a) Tip far from the substrate leads to a steady-state current, iT, ∞ . (b) Tip near an insulating substrate, resulting in hindered diffusion and iT < iT, ∞ . (c) Tip near a conductive substrate, resulting in diffusional feedback and iT > iT, ∞ . [Adapted from Bard et al. (5). © 1990, American Chemical Society.]

(Figure 18.1.2b). However, if the substrate is an electrode that can reoxidize the product of the tip reaction, R, the result is a larger flux of O to the tip, which causes the current to increase (Figure 18.1.2c). Thus, iT is a function of both the distance, d, between the tip and the substrate and the rates of reactions at the substrate generating species that are electroactive at the tip. This form of operation of the SECM is called the feedback mode. It is also possible to work in

18.2 Approach Curves

the substrate generation/tip collection mode, where the tip is held close to the substrate at a potential where electroactive products produced at the substrate are detected by the tip.

18.2 Approach Curves A plot of the tip current, iT , as a function of tip–substrate separation, d, is called an approach curve, because it is often recorded as the tip is moved toward the substrate from a distance several tip diameters away. As suggested in Section 18.1, this curve provides information about the nature of the substrate. Examples of approach curves for a disk-shaped tip in a thin insulating planar shroud are shown in Figure 18.2.1, both for a perfectly insulating substrate (where the tip-generated species, R, does not react) and for an active substrate (where oxidation of R back to O occurs at a diffusion-controlled rate). The shapes of approach curves are sensitive to the tip geometry. As noted above, disk-shaped SECM tips are generally fabricated with a thin shroud to allow close approach of the tip to the substrate. Thus, the diffusion field feeding the tip is not confined to the solution over the face of the tip, but also includes a part of the solution situated behind the plane of the electrode face (6–8). That is, diffusion of the electroreactant occurs in some measure from the “back side” of the electrode, resulting in a larger value of iT than would occur with an infinite shroud. When this effect is significant, the geometric factor, G, used to define iT, ∞ in (18.1.1) becomes larger than 4. Thus, SECM approach curves depend on both the radius of the shroud, rg , and the radius of the electrode, a (Figure 18.2.2). These two parameters are generally grouped together as a dimensionless geometric factor, RG = rg /a. Table 18.2.1 lists representative values of G as a function of RG as d → ∞. For RG < 10, diffusion from the back side of the tip significantly influences the value of iT, ∞ . Accordingly, SECM approach curves depend on RG when RG < 10 (9). For an approach to a conductive substrate, the effect of RG on iT is modest, since iT is dominated in that case by the feedback mechanism (Figure 18.1.2c). However, when approaching an insulating substrate, small values of RG Figure 18.2.1 SECM approach curves for steady-state currents at (a) a conductive substrate and (b) an insulating substrate. IT is the normalized tip current, iT /iT, ∞ ; L is the normalized distance, d/a. RG = 10. Solid curves correspond to (a) (18.2.2) and (b) (18.2.1). Filled squares correspond to values computed by finite-element simulations. [Courtesy of G. T. Solymosi, University of Utah, 2021, unpublished.]

8.0 6.0 IT 4.0 2.0 0.0

0

1

2

3

4

5

3

4

5

(a) 1.0 IT

0.8 0.6 0.4 0.2 0.0

0

1

2

L (b)

821

822

18 Scanning Electrochemical Microscopy

Figure 18.2.2 Geometry of a disk-shaped SECM tip. [Adapted with permission from Shao and Mirkin (9). © 1998, American Chemical Society.]

Electrode Shroud a rg

Table 18.2.1 Values of G as a Function of RG(a) RG

∞

G

4

10 4.07

5

2

1.5

4.16

4.44

4.65

(a) Values calculated from equation 19 of reference (8).

lead to significantly greater currents than for RG > 10, as diffusion from the back side of the tip significantly increases the total flux into the gap between the tip and substrate (Figure 18.1.2b). Approach curves are presented in the dimensionless form of I T (L) = iT /iT, ∞ vs. L = d/a, so that they become independent of disk radius, diffusion coefficient, and solute concentration. However, they remain dependent upon RG as indicated above. For RG ≥ 10, the following approximate analytical expression has been proposed for approach to an insulating substrate: [ ]−1 ITi (L) = 0.4572 + 1.4604∕L + 0.4313 exp(−2.3507∕L) [ ] (18.2.1) + −0.14544L∕(5.5769 + L) For approach to a conductive substrate, the corresponding expression is ITc (L) = 0.68 + 0.78377∕L + 0.3315 exp (−1.0672∕L)

(18.2.2)

Equations 18.2.1 and 18.2.2, both based on RG ≥ 10, have been shown to be accurate to within 1% of simulated approach curves (Figure 18.2.1). In both cases, the substrate is assumed to be much larger than the tip radius, a, and the rate of electron-transfer at the tip is assumed to be mass-transfer limited. Expressions analogous to (18.2.1) and (18.2.2) have been developed for RG values less than 10 and are required for more accurate analysis when using tips with a thin insulating layer (9–11). They are particularly important when using small SECM tips prepared using pulled pipet methods, where RG is often less than 5. The approach curve over a conductor can also indicate when the conductive part of the tip is recessed inside the insulating sheath, which often happens with very small tips. In this case, only a small amount of positive feedback is observed before the insulating shroud contacts the substrate, and iT levels off. Since direct observation of small tips, e.g., by electron microscopy, is difficult, SECM is a useful way of learning about the tip size and configuration (12, 13). Spherical or conical tips display different approach curves than disk-shaped ones and are reviewed in reference (10). In addition to the limiting cases described earlier (i.e., either no reaction at the insulating substrate or rapid conversion of R to O at a conductive substrate), one can calculate approach

18.2 Approach Curves

curves as a function of the rate of conversion of R to O at the substrate (14, 15). Let us begin by assuming that only O is initially present in solution and that its reduction at the tip is reversible, O + ne ⇌ R

(at tip, reversible)

(18.2.3)

while the oxidation of R occurs at a large planar substrate electrode and is described by the Butler–Volmer kinetic model. k 0 ,𝛼

R −−−−→ O + ne

(at substrate, irreversible)

(18.2.4)

Under these conditions, the tip approach current as a function of the substrate potential, ES , expressed in dimensionless form, I T (ES , L), is given by (2): ( ) ITi IT (ES , L) = IS 1 − c + ITi (18.2.5) IT where ITi and ITc are given by (18.2.1) and (18.2.2), and I S is the kinetically controlled substrate current 0.68 + 0.3315 exp (−1.0672∕L) 0.78377 IS = + (18.2.6) L(1 + 1∕Λ) 1 + F(L, Λ) The dimensionless functions in (18.2.6) are defined as 11∕Λ + 7.3 F(L, Λ) = 110 − 40L Λ = KS d∕a

(18.2.7) (18.2.8)

where K S is the dimensionless rate constant at the substrate, KS = kS a∕DR In (18.2.9), k S is the BV rate constant for (18.2.4), [ ] ′ kS = k 0 exp (1 − 𝛼)f (ES − E0 )

(18.2.9)

(18.2.10) ′

where k 0 and 𝛼 apply to (18.2.4), occurring at the substrate as indicated above, and E0 is for the O/R redox couple. By comparison to numerical simulations, (18.2.5) has been shown to be accurate to within ∼1–2% for 0.1 < L < 1.5 and −2 < K S < 3, when RG = 10 (16). A closed form solution more suitable for smaller RG has been reported (17). Figure 18.2.3 shows a family of I T (ES , L) vs. log K S curves over a range of separation distances. Each curve correponds to a different value of log L = log(d/a), where we recall that d is the separation distance between the substrate and tip, and a is the SECM tip radius. Among these curves, the tip is positioned closest to the electrode in (a) and furthest away in (p). Although complex, the dimensionless working curves in Figure 18.2.3 provide a wealth of information on how the tip current varies with d, ES , and k 0 . One can readily identify several limiting cases and applications from this figure: • Tracing along any individual curve from right to left is equivalent to ES becoming more negative at fixed d, corresponding to a voltammogram of the tip current vs. substrate potential, i.e., I T (ES , L) for a specified standard rate constant, k 0 at the substrate. • A vertical slice through curves (p) to (a) corresponds to an approach curve at constant k S (or, equivalently, at constant ES and k 0 ). For example, consider the situation where the rate of ′ electron transfer at the substrate is fast, i.e., ES ≫ E0 and K S ≫ 1. This situation corresponds to the right-hand side of Figure 18.2.3 (e.g., at log K S = 3). A vertical slice through the curves

823

18 Scanning Electrochemical Microscopy

14.0 (a) 12.0 (b)

10.0 IT(ES, L) = iT/iT,∞

824

(c)

8.0

(d) 6.0

(e)

4.0 2.0 0.0 –2.0

}

(p) –1.0

(a) 0.0 1.0 log KS = log (kSa/D)

2.0

(f) (g) (h) (i) (j) – (p)

3.0

Figure 18.2.3 SECM tip voltammograms for different values of the dimensionless heterogeneous rate constant, K S , for the conversion of R to O at the substrate electrode. Curves (a) to (p) correspond to log L = log(d/a) = − 1.2, −1.1, −1.0, … 0.3. Curve (p) is a horizontal line at IT (E S , L) = 1. It is the bottom of the envelope on the right, but the top of the envelope on the left. Curve (a) is the bottom of the envelope on the left. [Reprinted with permission from Bard, Mirkin, Unwin, and Wipf (15). © 1992, American Chemical Society.]

at this point corresponds to the approach curve for pure diffusion-controlled positive feedback, (18.2.2). • With increasing d, which corresponds from moving through the curves from (a) to (p), I T (ES , L) monotonically converges on unity, regardless of the chosen value of ES or k 0 . This limit corresponds to a tip current of iT, ∞ at large separation distance, as given by (18.1.1). ′ • The situation of ES ≤ E0 and small K S corresponds to the approach to a perfect insulator (18.2.1) and is represented by a vertical slice on the left-hand side of Figure 18.2.3. However, ′ at ES ≈ E0 and large K S , significant feedback occurs at sufficiently small d. A method for analyzing tip voltammograms for either reversible or quasireversible substrate kinetics under nonsteady-state conditions has been presented (18). SECM is generally useful in probing heterogeneous kinetics at electrode surfaces, as well as at other surfaces, like enzyme-containing membranes. Section 18.4.1 covers the use of SECM for measuring k 0 when quasireversible kinetics apply to the tip reaction (O + e ⇄ R). The largest values of k 0 that can be measured are of the order of D/d. This limit depends experimentally upon d, which is the characteristic dimension in SECM. Analogous measurements at a UME in bulk solution are similarly limited and provide a maximum k 0 on the order of D/a (Section 5.4.3). Pure diffusional positive feedback can also occur when a conductive substrate is not electrically connected to the potentiostatic circuit. This phenomenon can be useful in applications where applying a potential might cause undesirable reactions (e.g., metal dissolution or oxide formation), or when imaging an electrically isolated particle or region of the surface. The effect results from the fact that most of the conducting surface is located far from the tip and is bathed in a solution containing only O, thus maintaining the entire substrate at a potential, ES , posi′ tive of E0 for the O/R couple, in accordance with the Nernst equation. Figure 18.2.4 illustrates

18.3 Imaging Surface Topography and Reactivity

Figure 18.2.4 SECM feedback at an unbiased electrode. No net faradaic current passes across the substrate/ electrolyte interface. The tip cathodic current is balanced by an anodic current at the counter electrode (not shown). [Adapted with permission from Xiong, Guo, and Amemiya (19). © 2007, American Chemical Society.]

Tip e

O

R

O

R

O

R

e

O

R

e Substrate

the resulting dynamics. As R is generated at the tip, it diffuses to the substrate, where it is oxidized back to O, creating feedback. Since no net current can flow across the interface of an unbiased substrate, charge neutrality is maintained by the reduction of an equivalent amount of O somewhere else on the substrate. If the substrate is much larger than the tip, this event is likely to occur remotely from the region near the tip. In essence, the substrate acts a bipolar electrode, where the driving force for the parallel cathodic and anodic reactions arises from the difference of solution composition directly beneath the tip compared to that far away from the tip–substrate gap. When the dimensions of a planar and unbiased substrate are similar to the radius of the tip, approach curves show purely negative feedback because no part of the substrate is bathed in solution containing only O; thus, there is no driving force for the reduction of the R on the substrate beneath the tip. This limiting situation is of importance when SECM is carried out in surface interrogation mode (Section 18.5). Quantitative treatments provide details of how the feedback depends of the relative dimensions of the tip and substrate, the tip–substrate separation distance, and k 0 (19–21). The application of (18.2.5) for measuring interfacial redox kinetics is not limited to the study of solid electrodes. The rates of electron transfer between two redox molecules at the interface between immiscible electrolyte solutions have been determined by measuring the tip current as the tip approaches the liquid/liquid interface and applying theory similar to that presented above (16, 22, 23).

18.3 Imaging Surface Topography and Reactivity 18.3.1

Imaging Based on Conductivity of the Substrate

If the tip is systematically scanned over a uniformly conductive substrate in the x − y plane at constant z,2 the surface topography can be imaged by recording iT , because the tip current reflects variations in d as the tip position changes laterally. With a substrate that has both conductive and insulating regions, the tip current at a given z differs over the different regions, as implied by Figure 18.2.3. Over a conducting region, iT > iT, ∞ , while over the insulating portions, iT < iT, ∞ . One can also differentiate between these regions by modulating the tip in the z-direction with a sinusoidal voltage applied to the z-piezo and noting the phase of the modulated tip current with respect to the modulated distance (24). Over a conductive zone, as the tip approaches the sample, the current increases, while over an insulating zone the current decreases. Thus, the ac component of the current over a conductor and that over an insulator are 180∘ out of phase. 2 Scanning systematically in two dimensions is called rastering.

825

826

18 Scanning Electrochemical Microscopy

18.3.2

Imaging Based on Heterogeneous Electron-Transfer Reactivity

The SECM can also be used to map zones of gradually varying reactivity on an electrode surface, because the amount by which iT differs from iT, ∞ is a function of the electron-transfer rate constant at the substrate (Section 18.2). For species R generated at a tip scanned over the substrate at constant d, the feedback current is a measure of the rate at which R is oxidized on different parts of the substrate (14). The same strategy can, of course, be applied to study the rate of reduction of O at the substrate. In that case, R is initially present in solution and is reduced to O at the tip. The expressions developed in Section 18.2 involve assumptions that the substrate is large and displays uniform electrochemical activity. In SECM imaging, one often wishes to measure spatial variations in activity; thus, questions arise regarding the resolution achievable by SECM. In general, it is limited by the radially outward diffusion of R generated at the tip, resulting in positive or negative feedback from an area on the substrate larger than the tip area. This problem has been treated quantitatively (15), but we present a brief qualitative description here to provide the reader with a general idea of the factors that control resolution. When the tip is positioned above the substrate, only the region of substrate beneath the tip participates in the feedback process. The area of this region defines the spatial resolution that can be achieved by SECM and is dependent on the tip radius, a, and the separation distance between the tip and substrate, d. For example, consider the problem of imaging a circular conductive disk of radius aS embedded in a large insulating plane. If we assume for the moment that the tip radius, a, is smaller than aS , there are two easily identified limiting cases. • If the tip is positioned close to the conductive disk, such that aS ≫ d, then the conducting disk effectively acts an infinitely large conductive substrate, resulting in positive feedback, and iT > iT, ∞ , as given by (18.2.2). • Conversely, if d ≫ aS , species R generated at the tip diffuses radially outward from the tip and interacts with a surrounding insulating region of the substrate that is much larger than the area of the conductive disk. In this limit, negative feedback results, and iT < iT, ∞ , approximately as given by (18.2.1). Clearly, the tip radius, a, is a parameter determining the resolution that can be achieved in feedback SECM. From digital simulations of I T as a function of aS , a, and L, an empirical approximation has been proposed (15), a∞ ≈ a(1 + 1.5L) S

(18.3.1)

in which a∞ corresponds to the radius of a conductive disk at which iT can be approximated S by pure positive diffusional feedback from a large conducting surface, i.e., (18.2.2.). For instance, consider using SECM in feedback mode to image a circular conductive site of radius aS = 50 nm, surrounded by an infinite insulating plane. If a = 10 nm and L = 1.5 (corresponding to d = 15 nm), then a∞ = 32.5 nm, which is significantly smaller than the disk S to be imaged, having aS = 50 nm. Thus, an SECM tip of 10-nm radius, rastered at 15 nm above the surface, provides sufficient resolution to image the conductive site. If instead a 50-nm tip were employed and rastered at d = 100 nm, then a∞ becomes 400 nm, much larger than the S feature to be imaged. In this latter case, iT would always contain a large negative feedback contribution from the insulating region. Clearly, high-resolution SECM imaging requires attention to the preparation of very small tips (25). Figure 18.3.1 shows an example of high-resolution imaging in feedback mode to study 20-nm-diameter Au nanoparticles attached to an insulating polymer film that had been deposited on HOPG (26). SECM images were obtained in a solution containing 1 mM

18.3 Imaging Surface Topography and Reactivity

8 10 iT/pA

6 5 0

4 0

25 x/nm

50 0

50 25 m y/n

2

1.0

2

I = iT/iT,∞

I = iT/iT,∞

(a)

1

0

0

2

L = d/a (b)

4

6

0.5

0.0

0

5 L = d/a (c)

10

Figure 18.3.1 (a) High-resolution SECM image in feedback mode of an Au nanoparticle on an insulating HOPG/polyphenylene substrate obtained using a 14-nm radius Pt tip electrode. The solution contained 1 mM FcMeOH and 0.1 M KCl. The tip was held at a potential of 400 mV vs. Ag/AgCl to oxidize FcMeOH. The substrate was unbiased. (b) Approach curve over the Au nanoparticle showing positive feedback. (c) Approach curve over the HOPG/polyphenylene substrate away from the nanoparticle showing negative feedback. [Adapted from Sun, Yu, Zacher, and Mirkin (26), with permission.]

FcMeOH using a 14-nm-radius Pt nanodisk tip. To acquire an image, FcMeOH is oxidized at the SECM tip at the diffusion-controlled rate, and FcMeOH+ is reduced at the substrate. The image contrast in Figure 18.3.1a corresponds to the much higher rate of reduction of FcMeOH+ on the Au nanoparticle than on the HOPG/polyphenylene substrate. Approach curves recorded above the Au nanoparticle (Figure 18.3.1b) and the HOPG/polyphenylene substrate (Figure 18.3.1c) are well described by (18.2.2) and (18.2.1), respectively, indicating very fast reduction of FcMeOH+ on Au and essentially no redox activity on the HOPG/polyphenylene support. Similar measurements using nanometer-scale tips, but operating in substrate generation/tip collection mode (Section 18.4.2) allows the evaluation of kinetic parameters associated with the HER on Au nanoparticles (26) and the oxygen evolution reaction (OER) on nickel oxide nanosheets (27). Approximately 15-nm spatial resolution of kinetic activity was obtained using SECM tips with radii on the order of 10 nm.

18.3.3

Simultaneous Imaging of Topography and Reactivity

In SECM imaging experiments, the SECM tip is generally rastered at a constant height above the surface (i.e., constant z). If the surface is relatively smooth and parallel to the plane in which tip is being rastered, then the tip current reflects only the rate of electron-transfer reactivity. However, many surfaces display features such as steps and pits, or are modified by layers and particles (e.g., metal or proteins) that result in the tip–substrate distance, d, varying as a function of the lateral (x − y) tip position. In this situation, the tip current is a function of both the reactivity and the topography of the surface.

827

828

18 Scanning Electrochemical Microscopy

Combining atomic force microscopy (AFM; Section 21.1.2) and SECM provides a means to maintain the tip at a constant d based on one of the force-based imaging mechanisms of AFM (e.g., noncontact mode, Section 21.1.2), while simultaneously measuring the electrochemical current at the tip (28). This requires an AFM/SECM tip that serves two roles: • Sensing the forces between the tip and substrate in order to maintain a constant d. • Measuring the electrochemical current, as discussed in all preceding sections. This dual functionality allows SECM images to be obtained at constant d, allowing the effect of surface topography to be separated from the electrochemical surface reactivity. A byproduct is that AFM topographical images are simultaneously acquired with the SECM image. Thus, one can correlate structural and reactivity information obtained in a single experiment. Much effort has been invested in designing AFM probes that incorporate capabilities for measuring local electrochemical reactivity. The key requirement for simultaneous imaging of topography and reactivity is having a geometrically well-defined SECM tip that is comparable in size to the force-sensing tip. Additionally, the SECM tip should be located as close as possible to the AFM tip, so that the electrochemical and topographic information are in spatial registry. All parts of the AFM-SECM probe immersed in the electrolyte must be insulating or coated by an insulator, except for the electroactive region. Probe designs include those in which the apex of a conducting AFM tip serves to measure both force and reactivity. Some of these are based on a metal wire that is insulated except at the very end to expose a sharp tip; others involve coating a conventional insulating AFM tip with a metal layer (29). Alternatively, AFM-SECM probes prepared by microlithographic methods allow the SECM electrode to be separated from, but adjacent to, a nonconductive apex (30–33). These probes are generally designed with the exposed electroactive area located just behind the nonconductive AFM tip in order to prevent an electrical short circuit between the probe and substrate; thus, they allow measurements in contact-mode AFM. The geometries of AFM-SECM probes are very different from that of a conventional SECM tip (i.e., a disk shrouded in an insulating plane); thus, the electrochemical interaction between the AFM-SECM probe and substrate, which is still largely based on diffusion of a redox species, is different. However, expressions for the current at geometrically complex AFM-SECM probes can be readily obtained by finite-element simulations (34, 35). AFM-SECM probes have been applied with great success to image surface topography and electrochemical reactivity simultaneously in a broad range of applications, including studies of enzyme activity, electrocatalytic activity at particles, and localized corrosion. Frequently, the electrochemical image is obtained by generating a redox species at the substrate and detecting it at the AFM-SECM probe (i.e., substrate generation/tip collection mode). An extensive review of AFM-SECM technology and applications is available (36).

18.4 Measurements of Kinetics 18.4.1

Heterogeneous Electron-Transfer Reactions

In Section 18.2, we considered the case where the tip was held at a potential to generate R at the diffusion-controlled rate, while the reaction at the substrate was considered to be irreversible. The mathematical expression for the tip current, (18.2.5), and Figure (18.2.3) then provide means for understanding variations in iT when imaging surfaces with conducting and insulating regions, or when the substrate rate constant varies as a function of potential.

18.4 Measurements of Kinetics

We now consider the case where a quasireversible conversion, O + e → R, occurs at the tip, while the substrate potential is held at a potential where the reverse reaction, R → O + e, occurs at a diffusion-controlled rate. Under these conditions, the tip current is given for RG = 10 by (37). IT (ET , L) =

where

0.68 + 0.3315 exp(−1.0672∕L) 0.78377 + [ )] ( )( L(𝜃 + 1∕𝜅) 2𝜅𝜃 + 3𝜋 𝜋 𝜃 1+ 𝜅𝜃 4𝜅𝜃 + 3𝜋 2

[ ] ′ 𝜃 = 1 + (DO ∕DR ) exp nf (ET − E0 ) [ ] ′ k 0 exp −𝛼f (ET − E0 ) 𝜅= mO 4DO [ iT ] mO = 0.68 + 7877∕L + 0.3315 exp(−1.0672∕L) = 2 𝜋a 𝜋a nFC ∗O

(18.4.1)

(18.4.2)

(18.4.3) (18.4.4)

In (18.4.3), the standard rate constant, k 0 , corresponds to the value at the tip electrode, and mO is the mass-transfer coefficient for O. Like (18.2.5), (18.4.1) can be used to compute the voltammetric response at the tip (at constant d) or to compute approach curves (at constant ET ). ′ When the reaction at the tip becomes very fast, i.e., at conditions corresponding to ET ≪ E0 and large k 0 , then 𝜃 ≈ 1 and 𝜅 → ∞. In this limit, (18.4.1) reduces to the expression for pure diffusional feedback, (18.2.2), where both the tip and substrate reactions are fast. Figure 18.4.1 shows an experimental approach curve using a 46-nm Pt tip in a solution containing 1 mM FcMeOH and 0.2 M NaCl. A smooth Au electrode was used as the substrate. The potential of the tip was held at a value corresponding to the diffusion-limited oxidation of FcMeOH (FcMeOH → FcMeOH+ + e), while the Au electrode was unbiased. These conditions correspond to pure diffusional feedback; thus, (18.2.2) describes the approach curve. 8.0

6.0 IT 4.0

2.0

0.0 0.0

1.0

2.0

L

3.0

4.0

5.0

Figure 18.4.1 SECM approach curve recorded in 1 mM FcMeOH + 0.2 M NaCl at a 46-nm-radius Pt tip. The substrate was an unbiased Au film substrate, and the tip potential was held at a value where FcMeOH is oxidized at the diffusion-controlled rate. Thus, the approach curve corresponds to pure diffusional feedback. Symbols: Experimental data. Solid curve: Equation (18.2.2) using parameters noted in text. IT (L) is computed for data points by normalizing the tip current by iT, ∞ . [Adapted from Sun and Mirkin (37). © 2006, American Chemical Society.]

829

830

18 Scanning Electrochemical Microscopy

0

–10

(a) (b)

iT/pA –20

(c) (d)

–30 –100

0

100

200

300

ET/mV vs. Ag/AgCl

Figure 18.4.2 SECM tip voltammograms recorded in 1 mM FcMeOH + 0.2 M NaCl with a 36-nm-radius Pt tip. Tip–substrate separation distances, d: (a) ∞, (b) 54 nm, (c) 29 nm, and (d) 18 nm. v = 50 mV/s. Substrate was an unbiased Au film. Solid curve through each voltammogram corresponds to the best fit of (18.4.1) to the data using k0 and 𝛼 as adjustable parameters. [Adapted from Sun and Mirkin (37). © 2006, American Chemical Society.]

The solid line in Figure 18.4.1 is a plot of (18.2.2), demonstrating excellent agreement with the ∗ = 1 mM, and a = 46 nm as the only experimental data, using DO = DR = 7.8 × 10−6 cm2 /s, CO input parameters. SECM tip voltammograms for the same chemical system are shown in Figure 18.4.2. The curves are recorded as a function of the potential applied to a 36-nm radius tip. Included in Figure 18.4.2 are fits of (18.4.1) using k 0 and 𝛼 as adjustable parameters. Mean values of the kinetic parameters determined from the voltammograms, and additional experiments using a range of tip sizes (a = 25 − 290 nm) yielded k 0 = 6.8 ± 0.7 cm/s and 𝛼 = 0.42 ± 0.03. The uncertainties represent 95% confidence levels. No correlations of k 0 and 𝛼 with tip size were observed. SECM tip voltammetry experiments provide extremely reliable and precise estimates of kinetic parameters for the reduction and oxidation of outer-sphere redox species (37, 38). Values of k 0 > 10 cm/s listed in Table 18.4.1 are among the largest heterogenous rate constants reported in the literature. Table 18.4.1 Electrochemical Kinetic Parameters Determined by SECM Tip Voltammetry(a) , (b) k0 /cm s−1

Mediator

Electrode

Electrolyte

𝜶

Ru(NH3 )3+ 6

Pt(c)

H2 O + 0.5 M KCl

0.45 ± 0.03

17 ± 0.9

Au(d)

H2 O + 0.5 M KCl

0.45 ± 0.09

13.5 ± 2

Pt(d)

H2 O + 1 M KF

0.40 ± 0.05

11.9 ± 0.5

Au(d)

H2 O + 1 M KF

0.42 ± 0.03

9.3 ± 0.4

Pt(c)

MeCN

0.47 ± 0.02

8.4 ± 0.2

Au(d)

MeCN

0.44 ± 0.03

8.0 ± 0.5

Pt(c)

H2 O

0.42 ± 0.03

6.8 ± 0.7

Au(d)

H2 O

0.42 ± 0.06

8±1

TCNQ

Pt(c)

MeCN

0.42 ± 0.02

1.1 ± 0.4

TTF

Pt(d)

DCE

0.375 ± 0.02

8.8 ± 0.4

Au(d)

DCE

0.395 ± 0.02

9.0 ± 0.4

Fc FcMeOH

(a) (b) (c) (d)

Adapted with permission from Amemiya (39). Chemical abbreviations identified in Table 5 (front of book). Data from reference (37). Data from reference (38).

18.4 Measurements of Kinetics

In the methods described to this point, the tip is held at a potential, ET , to carry out the reaction O + e → R, while the potential of the substrate, ES , is held a value where the opposite reaction, R → O + e, occurs. Alternatively, one can operate SECM in shielding mode, where the same redox process, e.g., O + e → R, occurs at both the substrate and tip (40, 41). In this situation, the substrate actively shields the tip, resulting in a tip current smaller than observed when the substrate is an insulator. This SECM experiment is analogous to shielding experiments at an RRDE [Section 10.3.2(b)]. The magnitude of the shielding effect on iT depends on rate constant, k 0 , of the reaction at the substrate, as well as the value of ES . Thus, SECM operated in shielding mode can be used obtain information about the kinetic reversibility of the reaction at the substrate. Shielding has also been developed for imaging localized electrocatalysis. In this mode (referred to as redox competition SECM, RC-SECM) (42), both the tip and the substrate compete for the same electroreactant. Thus, a decrease in the tip current over a location on the substrate signals greater electroactivity at that location, perhaps because an electrocatalytic particle or surface feature is present there. An interesting application of RC-SECM involves monitoring of the O2 concentration at a Pt tip placed above an insulating surface on which glucose oxidase is deposited. Addition of glucose to the solution causes partial consumption of the available O2 due to the enzymatic oxidation of glucose, resulting in a decreased tip current. Thus, iT may be used to quantify the rate of the enzymatic reaction (43).

18.4.2

Homogeneous Reactions

SECM can also be used in feedback and generation-collection modes to study homogeneous reactions of products generated at either the tip or the substrate (44, 45). Measurements of this kind are analogous to experiments that we have already encountered for other dual-electrode systems, such as an RRDE (Section 10.3.2) or an interdigitated band electrode (Section 5.6.3). In feedback experiments, the effect of the coupled reaction on the tip current is studied, while in generation-collection experiments one electrode, such as the tip, behaves as the generator electrode, producing the product of interest (R), while the other, such as the substrate, is the collector, operating at a potential where R is oxidized back to O. The case just described is the widely used tip generation/substrate collection (or TG/SC) mode of SECM. When species R is stable, and the tip is close to the substrate (a/d < 2), essentially all electrogenerated R is collected (i.e., oxidized) by the substrate. Under these conditions, the steady-state substrate current, iS , is equal in magnitude to the tip current, iT ; thus, the collection efficiency, |iS /iT |, is 1. This behavior contrasts with that at the RRDE, where even with close spacing between disk and ring and with a large ring, the collection efficiency is usually much less than 1, because much of the disk-generated product diffuses outward before being collected by the ring. Now suppose the tip-generated species, R, is not stable, but decomposes to an electroinactive species (the Er Ci case; Section 13.3.1). If R reacts appreciably before it can diffuse across the tip–substrate gap, the collection efficiency will be smaller than unity, approaching zero for a very rapidly decomposing R. One can illuminate the kinetics by determining |iS /iT | as functions ∗ , and the concentrations of coreactants. The decomposition also decreases the amount of d, CO of positive feedback of O to the tip, so that iT becomes smaller than in the absence of any kinetic complication. Accordingly, a plot of iT vs. d represents an alternative means for determining the rate constant for R decomposition, k. For both collection and feedback, k is determined from working curves in the form of dimensionless current vs. distance (e.g., iT /iT, ∞ vs. d/a)

831

832

18 Scanning Electrochemical Microscopy

for different values of the dimensionless kinetic parameter, K = kd2 /D (first-order reaction) or ∗ ∕D (second-order reaction). K ′ = k ′ d2 CO As an illustration, consider the electroreductive hydrodimerization of acrylonitrile (AN), which is used commercially to produce adiponitrile, (ADN), a precursor in Nylon production (46). The proposed mechanism in dry DMF (47) is an Er C2 process [Section 13.1.1(a)] involving dimerization of the acrylonitrile anion radical, AN ∙. N

N

+e AN

N

•

–

+

N

•

(18.4.5) –

N –

– –

N

• – AN –•

–

N

+ 2H+

N

(18.4.6)

N ADN

N

(18.4.7)

In Figure 18.4.3b, the tip voltammogram for the reduction of AN in DMF + 0.1 M TBAPF6 shows a reduction wave at −2.0 V vs. a QRE. The tip-generated species, AN ∙, is so unstable that in most experiments, even fast cyclic voltammetry, one does not observe its oxidation in reversal. However, when the SECM tip is close to a gold substrate held at −1.75 V vs. QRE, one sees a collection wave for oxidation of AN ∙ at the substrate as the tip is scanned through the reduction wave. By studying the dependences of the collection efficiency on d and AN concentration, the rate constant of reaction (18.4.6) was found to be 6 × 107 M−1 s−1 . These results apply to dry DMF; however, the mechanism of AN reduction and product distribution depends strongly upon the choice of solvent and solution conditions (48). Figure 18.4.3 SECM TG/SC voltammograms for 1.5 mM AN in DMF + 0.1 M TBAPF6 . The tip (a = 2.5 μm) was spaced 1.36 μm from a 60-μm diameter gold electrode held at E S = − 1.75 V vs. a silver QRE. E T was scanned at 100 mV/s to produce a voltammogram of iT for reduction of AN (dashed curve). The substrate current, iS (solid curve), shows the oxidative collection of AN ∙ generated at the tip. [Reprinted with permission from Zhou and Bard (47). © 1994, American Chemical Society.]

2a Tip

d

AN

iT

e

AN–•

Products

AN

AN Substrate

AN–• e iS (a)

1.4 1.0 iT

0.6 i/nA 0.2 –0.2 –0.6

iS 0

–1 –2 E/V vs. Ag QRE (b)

–3

18.4 Measurements of Kinetics

SECM operated in TG/SC mode is especially useful for the detection of very short-lived species generated in solution by an electrochemical reaction. The observational time scale associated with SECM is approximately 𝜏 obs = d2 /D (Table 13.4.1), where D is the diffusion coefficient of the species being studied. Thus, very short tip–substrate distances, d, are generally required for an unstable species to diffuse across the gap to be detected before undergoing chemical reaction. Using very smooth tip and substrate electrodes, d values as small as 10 nm are possible, allowing detection of intermediates with lifetimes on the order of 100 ns. Comparable observational timescales are provided by CV at ultra-high scan rates (∼106 V/s; Section 7.6); however, SECM measurements are conducted at steady-state and are less prone to measurement interference arising from large double-layer charging and adsorption, which make detection of short-lived species very difficult by fast scan CV. The capabilities of SECM are illustrated by the detection of transient CO2 ∙, generated during the electrochemical reduction of CO2 (49). There is broad interest in pathways for CO2 reduction because they bear on the potential conversion of CO2 into fuels or other products. In the absence of adsorption (50), the first step is the 1e reduction of CO2 to yield dissolved CO2 ∙, CO2 + e ⇌ CO2 ∙

(18.4.8)

Depending on the electrode material and the solution pH and composition, the electrogenerated CO2 ∙ rapidly undergoes homogeneous reactions to produce a number of products, including oxalate, C2 O2− , which results from rapid dimerization of two CO2 ∙ radical ions, 4 kc

2CO2 ∙ −−−→ C2 O2− 4

(18.4.9)

Figure 18.4.4 shows a schematic of the SECM configuration used to detect CO2 ∙ and C2 O2− 4 produced in dry DMF containing 0.1 M TBAPF6 . The experiment is initiated by reducing CO2 at a 5-μm-radius Hg/Pt UME tip prepared by electrodeposition of mercurous ions onto the Pt surface. The hemispherical shape of the Hg electrode and its atomically smooth surface allow the tip to closely approach a 12.5-μm radius Au UME, which is employed as the substrate. The tip is held at a constant potential, ET , to generate CO2 ∙ continuously at steady-state rate. Detection of the CO2 ∙ is performed by holding the potential of the Au substrate, ES , at a value where CO2 ∙ is oxidized. Due to the hemispherical tip shape and the rapid dimerization rate, only CO2 ∙ generated at the Hg surface in close proximity to the Au substrate is detected. Figure 18.4.5a shows the SSV response at the Hg/Pt UME in the bulk DMF solution (i.e., far from the substrate) in the absence (black curve) and presence (gray curve) of 20 mM CO2 . Tip Hg/Pt UME

d

CO2 +e CO− • 2 −e

CO2

+e

CO2−•

CO− 2•

2−

C2O4

Substrate Au UME

Figure 18.4.4 Schematic diagram of the collection of the CO2 ∙ radical in TG/SC mode of SECM. The separation between the tip and substrate, d, is much smaller than indicated in the figure. [Adapted with permission from Kai et al. (49). © 2017, American Chemical Society.]

833

18 Scanning Electrochemical Microscopy

150 120.8 100

120.4

iT/nA

i/nA

50

ET = –2.8 V d = 1 μm

iT

0.0 iS –0.4

0 –2.0

Collection efficiency, |iS/iT|/%

834

–2.2

–2.4 –2.6 ET/V vs. Pt/PPy (a)

–2.8

–0.8

–1.2 –1.6 ES/V vs. Pt/PPy (b)

–2.0

iT

1.2 100 0.8 i/nA 0 ES = 0.5 V d = 1 μm

0.4 –100

iS

0.0 0

2

4 6 d/μm (c)

8

10

–2.2

–2.4 –2.6 ET/V vs. Pt/PPy (d)

–2.8

Figure 18.4.5 (a) CV at an Hg/Pt UME tip (a = 5 μm) in DMF + 0.1 M TBAPF6 in the absence (black curve) and presence (gray curve) of 20 mM CO2 . v = 100 mV/s. (b) Comparison of iT and iS for generation and collection of CO2 ∙ , recorded simultaneously as E S is scanned from −2.0 to −0.8 V at 100 mV/s, while E T is held at −2.8 V. Substrate is a 12.5-μm-radius Au UME. (c) Collection efficiency of CO2 ∙ , |iS /iT |, vs. the tip-substrate distance. Solid points: Experimental values. Solid curve: Simulated for (18.4.8) and (18.4.9) with kc = 6 × 108 M−1 s−1 , taking into account the tip and substrate geometries. (d) Comparison of iT and iS for the generation of CO2 ∙ and collection of C2 O2− , recorded by holding E S at 0.5 V, while scanning E T from −2.2 to −2.8 V. Collection 4 efficiency, |iS /iT |, is nearly unity. [Adapted from Kai et al. (49). © 2017, American Chemical Society.]

Reduction of CO2 in this system begins at about −2.45 V vs. a Pt/PPy QRE (−0.19 V vs. SCE; Section 2.5.2). The tip is then positioned above the Au substrate, and ET = − 2.8 V vs. Pt/PPy is applied to generate CO2 ∙ in the TG/SC experiment. Figure 18.4.5b shows SSVs recorded at both the tip and substrate, while scanning ES from −2.0 to −0.9 V and maintaining ET at −2.8 V. When the tip is positioned sufficiently close to the substrate (d < 2 μm), the collection of CO2 ∙ is indicated by an increase in substrate current, iS , beginning at −1.6 V. A small increase in the tip current, iT , is also observed, due to feedback of CO2 produced at the substrate. At d = 1 μm, the collection efficiency, |iS /iT |, is 0.005, indicating that only 0.5% of the CO2 ∙ generated at the tip is collected at the substrate, mainly a consequence of the rapid dimerization to produce C2 O2− . The low collection efficiency also reflects the hemispherical tip shape that results in 4 most CO2 ∙ being generated far from the narrowest gap between the tip and substrate. Larger collection efficiencies for CO2 ∙ are obtained as d is decreased toward 200 nm (Figure 18.4.5c). Numerical simulation based on the actual SECM configuration and the EC reaction mechanism, (18.4.8) and (18.4.9), allows prediction of |iS /iT | as a function of d and determination of the rate constant for the dimerization reaction, k c . A value of k c = 6 × 108 M−1 s−1 was obtained by fitting simulated and experimental results (Figure 18.4.5c). This rate constant approaches the

18.5 Surface Interrogation

diffusion-controlled limit for a bimolecular reaction and is consistent with estimates obtained by fast CV (51). As demonstrated in Figure 15.4.5d, the product of dimerization, C2 O2− , can also be directly 4 detected in the same experiment by holding ES at a positive value (0.5 V vs. Pt/PPy) where C2 O2− is rapidly oxidized to CO2 , 4 C2 O2− → CO2 + 2e 4

(18.4.10)

At the same time, ET is scanned across the wave for reduction of CO2 to CO2 ∙ (18.4.8). The high collection efficiency for capture of C2 O2− (|iS /iT | ≈ 0.9 at d = 1 μm) as CO2 ∙ is generated 4 2− confirms that C2 O4 is the predominant species produced during CO2 reduction in DMF. TG/SC experiments have been used to detect other interesting intermediates, including • An Sn(III) species generated during the 2e reduction of Sn(IV)Br2− (52). 6 • Superoxide generated in aqueous solution during O2 reduction (53). It is also possible to carry out substrate generation/tip collection (SG/TC) experiments, where the tip probes the products of a reaction at the substrate. However, this approach to studies of homogeneous kinetics is less straightforward, since the larger substrate electrode does not attain a steady-state condition, and the collection efficiency for this case, |iT /iS |, is much less than unity, even in the absence of a homogeneous kinetic complication. Even so, this mode has been used to examine concentration profiles above a substrate (54).

18.5 Surface Interrogation Section 18.4 focused on the use of SECM for determining mechanisms and kinetics of reactions involving dissolved redox reactants and products. Such experiments generally involve steady-state measurements, e.g., approach curves and steady-state tip voltammetry. Modes of SECM based on measurements of transient behavior allow investigation of many different types of interfacial processes (55) such as • Adsorption/desorption of protons at oxide electrodes (56). • Lateral proton diffusion in Langmuir monolayers (57, 58). • The rate of molecule transfer across an air/water interface (59). Especially useful for investigating the chemical state of a surface is the surface interrogation mode (SI-SECM) (60, 61), in which the tip is used to generate a redox titrant that chemically reacts with an adsorbed species on the substrate. Analysis of SI-SECM tip voltammograms allows one to measure the surface coverage of an adsorbate directly as a function of the substrate potential. Figure 18.5.1a–d shows the steps involved in an SI-SECM experiment, which is usually performed with tip and substrate electrodes of comparable size (60, 61). Both are frequently UMEs with radii in the 10–50 μm range and RG values of ∼1.5. The interrogation experiment is performed in a solution containing one redox form (either O or R) of a mediator couple ′ with a well-defined E0 . In Figure 18.5.1 and the discussion that follows, it is assumed that only O is present in the bulk solution. Because of the microscopic size of the substrate, the first step (Figure 18.5.1a) is to align the tip and substrate electrodes with respect to each other, using positive feedback to guide the experimentalist; maximum positive feedback current is obtained when the tip is positioned directly over the substrate. The tip-to-substrate separation is then reduced to L ∼ 0.1 in order to maximize the tip current during the interrogation step.

835

836

18 Scanning Electrochemical Microscopy

ET

ET

OC

ET

Tip O

R

O

R

O

R

O

R

P +O R AAA

AAA

Substrate

ES

ES

(a)

(b)

OC (c)

OC (d)

25 20

Positive feedback

15 iT/nA 10

Negative feedback

10 nC 5 nC

5

2.5 nC 0 0.15

0.10

0.05

0.00 –0.05 –0.10 –0.15

(ET – E0′)/V (e)

Figure 18.5.1 Sequence of steps in SI-SECM. (a) Both tip and substrate are under active potential control (at potentials E T and E S , respectively) to generate diffusional feedback for alignment purposes; (b) a reactive species, A, is chemically or electrochemically adsorbed on the substrate under active potential control, while the tip is at open circuit. (c) The substrate is then taken to open circuit, while the tip is held at a potential to generate species R (the “titrant”), which reacts with species A to produce transient positive feedback. (d) After complete consumption of A, the tip response corresponds to negative feedback. The tip-to-substrate distance is greatly exaggerated for the purpose of showing the reactive species; typically, L = d/a is on the order of ∼0.1. (e) Digital simulations of voltametric tip responses corresponding to diffusional positive feedback (dashed curve), transient titration of adsorbed A (solid curves), and negative feedback (dotted curve). Each peak-shaped voltammogram provides an electrical charge from the integral under the peak. This charge is proportional to ∗ = 1 mM, a = 1.25 μm, d = 1.25 μm, the amount of adsorbed A. Simulation parameters: v = 50 mV/s, CO 4 −1 −1. RG = 1.5, and k = 5.0 × 10 M s [Part (e) adapted with permission from Rodríguez-López, Alpuche-Avilés, and Bard (60). © 2008, American Chemical Society.]

During the second step (Figure 18.5.1b), the tip is disconnected from the potentiostatic circuit and remains at its open-circuit potential (OCP). At the same time, the substrate electrode surface is chemically or electrochemically modified by adsorption of a redox-active species, A. For instance, this step might involve holding the substrate potential for a predetermined time at a value where oxygen is chemisorbed on the electrode surface. In the third step (Figure 18.5.1c), the substrate is disconnected from the potentiostatic circuit, while the tip is reconnected and ′ held at a constant ET > E0 , so that no reduction of O occurs at the tip. The tip potential is then slowly scanned in the negative direction to reduce O to R, which diffuses across the gap and reacts by reducing adsorbed A, regenerating O and resulting in positive feedback.

18.5 Surface Interrogation

A secondary product, P, is derived from the reduction of A. Positive feedback occurs only as long as species A is present to react with R; thus, during the interrogation voltammogram, the tip current decreases as A becomes exhausted. Since a stoichiometric number of O molecules is produced by the reaction of one molecule of A, the time integral of the positive feedback current corresponds precisely to the initial amount of adsorbed A. Once A is consumed, only O that diffuses into the gap from the bulk is reduced at the tip, corresponding to the case of pure negative feedback (Figure 18.5.1d). Tip and substrate electrodes of comparable size are required to obtain negative feedback when all of A has reacted and the substrate is at open circuit. (See Section 18.2 for discussion of imaging unbiased substrates.) Figure 18.5.1e shows digital simulations of voltammetric tip responses expected in a typical SI-SECM experiment, corresponding to pure diffusion feedback during the alignment procedure (dashed curve), transient interrogation feedback (solid curves), and pure negative feedback (dotted curve). The three voltammograms for transient interrogation feedback correspond to three different amounts of adsorbed A, expressed using electrical units: 10, 5, and 2.5 nC. It is clear from the different responses that the interrogation voltammogram provides quantitative evaluation of the amount of adsorbate on the substrate. The SI-SECM experiment is a form of coulometric titration, in which a titrant (R in this case) is generated at the tip and used to titrate the substrate surface to determine the amount of A. The decrease in voltammetric current when all of A has reacted marks the end point of the titration (Figure 18.5.1e). Figure 18.5.2 shows an example of SI-SECM transients obtained in an investigation of oxide formation on Ir (62). As discussed in Chapter 15, the adsorption of oxygen or hydroxyl and the formation of oxide layers play a role in many electrocatalytic reactions, such as for the oxidation and reduction of H2 O2 (Section 15.3.7) and the oxidation of methanol (Section 15.3.3). SI-SECM is particularly useful for evaluating the presence and amounts of different oxygen-containing species on metal and semiconductor surfaces, as well as other adsorbed

9

8

300

7 iT/nA 6 200

5 4 3 2

100

1

0 –0.9

–1.0

–1.1

–1.2

–1.3

ET /V vs. Ag/AgCl

Figure 18.5.2 SI-SECM transients of tip current vs. tip potential (solid curves) used to measure the amount of oxide deposited on an Ir disk electrode (a = 62.5 μm) as a function of the electrode potential, E S . The tip was a 50-μm-radius GC electrode. Oxide was deposited on the Ir electrode at (1) E S = − 1.00 V, (2) −0.90 V, (3) −0.80 V, (4) −0.70 V, (5) −0.60 V, (6) −0.50 V, (7) −0.40 V, (8) −0.30 V, and (9) −0.20 V. v = 50 mV/s. The solution contained 10 mM Fe(TEA)(OH) and 2.0 M NaOH. Digital simulations of the SI-SECM transients (open circles) were fit to the experimental curves assuming kchem = 4.0 ± 0.5 × 104 M−1 s−1 . Other simulation details are presented in reference (62). [Adapted from Arroyo-Currás and Bard (62). © 2015, American Chemical Society.]

837

838

18 Scanning Electrochemical Microscopy

species, e.g., bromine. Iridium oxide films are specifically important because of their use for the OER on dimensionally stable anodes [DSA; Section 20.1.5(a)], as well as their applications to other purposes, e.g., for pH sensors. It is generally understood that the electrocatalytic activity of Ir for the OER is due to the formation of an oxide film; thus, there is interest in knowing the thickness, structure, and stability of the oxide as a function of the applied potential. The SI-SECM interrogation voltammograms shown in Figure 18.5.2 were recorded employing a 50-μm-radius GC tip, a 62-μm-radius Ir substrate, and the redox mediator Fe(III)-TEA, where TEA is triethanolamine. This mediator undergoes a reversible 1e reduction in 2.0 M NaOH (62). Fe(III)-TEA + e ⇌ Fe(II)-TEA

′

E0 = −1.05 vs. Ag∕AgCl

(18.5.1)

As described earlier, the tip and substrate are first aligned, and the tip is positioned close to the substrate (1–3 μm) using positive feedback at a potential where no oxide is present on the Ir electrode. In the second step, the tip is disconnected from the circuit, and a layer of hydrous oxide layer is deposited on the Ir by holding the substrate potential at a constant value for a specified time. The Ir electrode is then taken to open circuit. The GC tip is reconnected, and its potential is swept at 10 mV/s from −0.9 to −1.3 V, generating the titrant, Fe(II)-TEA, which diffuses to the substrate and reacts with the Ir hydrous oxide: kchem

Fe(II)-TEA + Ir-(OH)ads −−−−−→ Fe(III)-TEA + Ir + OH−

(18.5.2)

The chemical reduction of Ir - (OH)ads by Fe(II)-TEA produces a transient feedback current corresponding to the reduction of Fe(III)-TEA at the GC tip. When the Ir - (OH)ads has been fully consumed, the tip current falls to a constant negative feedback level determined by L and the electrode geometries. Figure 18.5.2 shows the results of experiments in which the Ir potential, ES , was held at potentials between −1.0 and −0.2 V for 70 s in the 2 M NaOH solution. At ES = − 1.0 V, the Ir surface is free of oxides or adsorbed oxygen; thus, after taking the Ir potential to open circuit, the interrogation voltammogram corresponds to a pure negative feedback response, displaying a sigmoidal shape (curve 1 in Figure 18.5.2). When the experiment is repeated, holding the Ir surface at progressively more positive ES values, the interrogation voltammogram develops a peak-shaped response, the height of which increases for more positive ES (curves 2–9). The area underneath each interrogation voltammogram, after subtraction of the pure negative feedback response (curve 1) and division by the scan rate, corresponds to the quantity of electrical charge, Q, used in titration during the experiment. Therefore, a plot of Q vs. ES provides a description of oxide formation on the Ir surface as a function of its potential. For the experiment shown in Figure 18.5.2, titration of the oxide layer at −0.2 V corresponds to a total charge density of ∼300 μC/cm2 , after normalizing Q to the area of the Ir surface. Similarly, using a different redox mediator, adsorbed H can be titrated to determine its potential-dependent surface coverage on Ir (62). The open circles overlaid on the interrogation voltammograms are computed values of iT based on digital simulations of the processes leading to the feedback current. These simulations are sensitive to the rate constant, k chem , for reduction of the Ir hydrous oxide by the titrant, Fe(II)-TEA in (18.5.2). A value of k chem = 4.0 ± 0.5 × 104 M−1 s−1 gave the best fit of simulated values to the experimental voltammograms in Figure 18.5.2. Related studies have shown that ′ the value of k chem depends upon the E0 of the mediator used to titrate the oxide film (63). SI-SECM experiments have been used to measure oxide coverages on Pt and Au electrodes in acidic media (60), as well as the amount of H adsorption on MoS2 (64). They have also been used to titrate photogenerated OH ∙ on the surface of TiO2 (65), hematite (66), and W/Mo-BiVO4 (67) semiconductor electrodes.

18.7 Other Applications

18.6 Potentiometric Tips The tips discussed so far have been amperometric probes, typically made of a metal or carbon, that produce faradaic currents reflecting redox processes. However, it is also possible to carry out SECM using potentiometric tips, such as ISEs based on micropipets (68–70). These produce potentials vs. a reference electrode that depend logarithmically on the solution activity of a specific ion (Section 2.4). Tips of this type have been described for measurements of H+ , Zn2+ , Cl− , Ca2+ , NH+ , and K+ with spatial resolution of a few μm. They are often prepared from a 4 pulled glass pipet that has a small tip opening at one end. Capillary action is used to draw a liquid ion-exchanger cocktail, containing an ionophore selective for the desired ion (Section 2.4.3), into the pipet through the small opening (68). An internal filling solution containing an electrolyte and an Ag/AgCl wire is then introduced through the large opening of the pipet to complete the SECM tip. Tips prepared by packing the end of the pipet with IrO2 particles (71), or using an Sb microelectrode (72), have been employed for localized pH measurements. Potentiometric tips are especially useful for measurements of electroinactive ions that cannot be detected by amperometric SECM tips. However, potentiometric tips are passive, in that they detect the local activity of a given species, but do not sense the presence of the substrate. They cannot be used to determine d, so they must be positioned with respect to a substrate by visual observation with a microscope, by resistance measurements, or by using a double-barrel tip that contains both an amperometric element and a potentiometric one. Once a reference height relative to the surface is established, the potential of the tip can be measured as a function of d. This relationship can then be converted to a plot of concentration or pH vs. distance when the tip response with respect to concentration (or pH) has been calibrated. Potentiometric and amperometric SECM tips have been used extensively for mapping pH and metal ion concentrations in localized corrosion (73–75). In these systems, the corrosion reaction frequently produces a metal ion in the vicinity of an active site, which is accompanied by a decrease in the local solution pH due to metal hydrolysis. Consequently, tips that are selective for pH or metal-ion concentrations can detect and track corrosion events vs. time with micrometer spatial resolution. pH-sensitive SECM tips have also found applications in biological systems, including measurements of extracellular pH changes under electrical stimulation (76) and real-time imaging of pH changes above an acid-producing Streptococcus mutans biofilm (77).

18.7 Other Applications SECM is versatile and has been employed ingeniously for diverse purposes, as detailed in reviews (78–80) and a monograph (81). This section is meant to illustrate the versatility through three areas of application. Section 19.6 describes a fourth, which is the use of SECM to detect individual molecules based on redox cycling between the tip and substrate. 18.7.1

Detection of Species Released from Surfaces, Films, or Pores

SECM can be employed to study the flux of species produced at a modified electrode surface, such as one with a film of polymer (Section 17.4). In one type of experiment, the tip is held at a potential where it can detect an electroactive ion released from the polymer film during a redox process (82–84). For example, SECM was used to measure the release of Br− during the reduction of an oxidized polypyrrole (PPy) film [expressed in the form, PPy+ Br− ] (84). During a reductive CV scan, Br− was found to be released only in a later part of the scan, after an

839

840

18 Scanning Electrochemical Microscopy

1 4

3

2

5

3

1

(d)

1

2

(e)

(c) (b)

1 iT

1

300 × 300 μm

(f)

2 iS 2 μA

(a)

(g)

KI + K2SO4 Only K2SO4 0.0

0.5

1.0 1.5 ES/V vs. Ag/AgCl

2.0

2.5

Figure 18.7.1 (a)–(g) Images of iT vs. x − y position for a 300 μm × 300 μm area of a Ta electrode covered by a Ta2 O5 film in 10 mM NaI + 0.1 M K2 SO4 as the Ta potential is scanned to more positive values. Pointers show the potentials corresponding to the images. The tip potential was 0.0 V vs. Ag/AgCl, where iodine formed at the substrate is reduced. The vertical scale in (b) is about 0–2 nA and applies to all images. The LSV curves show iS vs. E S in the presence and absence of iodide, as indicated. [Adapted from Basame and White (87). © 1999, American Chemical Society.]

appreciable fraction of cathodic charge had passed. This result suggested that during the early phase of PPy+ reduction, the uptake of cations, rather than the release of anions, maintained charge balance in the film. By positioning the tip close to the surface of a porous membrane (e.g., skin or a microporous polymer), SECM can be used to measure the flux of ions driven through the pores by an electric current (iontophoresis) (85). Similarly, the rates of electron and ion transfer across a liquid/liquid interface (ITIES) can be studied (16, 86). SECM can be particularly useful in imaging electrode surfaces and probing films on surfaces. For example, the nature of the structural defects in a blocking film of Ta2 O5 on a Ta electrode was studied as a function of potential by mapping the locations where iodide could be oxidized on the substrate (with detection of iodine at the tip) (Figure 18.7.1) (87). Activity for I− oxidation at these sites increases as the potential is scanned positive and then disappears as the thickness of the oxide layer on the Ta surface increases. 18.7.2

Biological Systems

SECM has been widely employed in studies of biological systems and for diverse biotechnical purposes, including monitoring of enzyme activity, small spot immunoassays, readout of DNA arrays, and imaging, in addition to applications using potentiometric tips (Section 18.6). Reviews detail these applications (88–90). SECM is especially well suited for noninvasive imaging and monitoring of living cells (91). Figure 18.7.2a shows an example in which SECM is used to assay the viability of tissues grown in an array of microwells in a Petri dish (92). The experiment is based on measuring the concentration of O2 in the solution above the tissues using RC-SECM (Section 18.4.1). Living tissue with high respiratory activity consumes O2 in the solution above a microwell, resulting in a decrease in the SECM tip current. As shown in Figure 18.7.2b, the local depletion of O2 by viable tissue is clearly reflected by a ∼40% decrease in tip current above each microwell. After exposing the tissues to a 50% ethanol solution, the measured tip current remains constant as

Normalized current

18.8 Scanning Electrochemical Cell Microscopy

SECM tip

O2

H2O

(a)

1.0

Dead Live

0.8 0.6 0.4

0

1000 2000 3000 Distance/μm (b)

4000

Figure 18.7.2 Live/dead assay of tissues using SECM with a 10 μm Hg-coated Pt tip. (a) Schematic of SECM measurement of O2 uptake by cells, performed by scanning the tip at a constant height of 15 μm above the surface of a microwell array containing tissue. The circular microwells were 200 μm deep, 400 μm in diameter, and spaced apart by 800 μm. (b) Tip current for O2 reduction recorded during a line scan (10 μm/s) over an array of viable tissues (gray curve) and after exposure to ethanol (black curve). Currents are normalized to the tip current for reduction of O2 far from the surface (>250 μm). Measurements performed in HEPES buffer at a tip potential of −0.6 V vs. Ag/AgCl. [From Sridhar, de Boer, van den Berg, and Le Gac (92), with permission.]

the tip is scanned across the microwells, indicating that the cells are no longer alive. In contrast to traditional fluorescence and colorimetric assays, SECM assays do not require any extensive sample preparation. 18.7.3

Probing the Interior of a Layer on a Substrate

An example of probing a modifying film in the z-direction involves a Nafion layer containing Os(bpy)2+ (93). The tip current for oxidation of Os(bpy)2+ was measured as a conical tip moved 3 3 from the solution phase into the film (Figure 18.7.3). The point where the tip just enters the film is signaled by the onset of the anodic current. It increases until the conical section is completely in the film, after which a steady-state plateau current is recorded. When the tip approaches the ITO substrate, positive feedback and, finally, tunneling occur. These measurements allowed a precise determination of the film thickness in situ (l = 220 nm), as well as estimation of electrochemical parameters within the film. For example, by obtaining a voltammogram when the 3+∕2+ electrode reaction tip was inside the film (Figure 18.7.3c), the value of k 0 for the Os(bpy)3 was estimated as 1.6 × 10−4 cm/s (93).

18.8 Scanning Electrochemical Cell Microscopy Scanning electrochemical cell microscopy (SECCM) is a distinct form of scanned-probe microscopy in which the tip of a micro- or nanopipette, filled with electrolyte, is placed in contact with the substrate surface, which is the working electrode (94, 95) (Figure 18.8.1a). The electrochemical cell is formed by placing a reference-counter electrode inside the pipet, and the current is measured in a two-electrode configuration. In effect, the electrolyte in the cell becomes part of the scanning probe, and the interface between the liquid meniscus and electrode surface defines the area of the substrate being interrogated. The SECCM response corresponds to redox processes occurring within that localized area of the substrate. A pulled glass pipet is used to fabricate the probe, which can be made readily with a tip opening of 100 nm radius. Different types of electrochemical measurements, including voltammetry and chronoamperometry, can be used to probe the electrochemical behavior of the area in contact with the tip. Interpretation of the i − E or i − t responses is largely as for macroscopic

841

842

18 Scanning Electrochemical Microscopy

UME tip Nafion film

Os(bpy) 32+

Os(bpy) 32+

ITO (a)

–e

Os(bpy)33+

Os(bpy) 32+

–e

Os(bpy) 33+

(c)

(b)

3+ Os(bpy) 32+ –e +e Os(bpy)3

Tunneling

(d)

(e) 3

1.0

(f) 2

iT/nA

iT/pA 1 0.5

e 0 d

c

b

a

0.0 1000

1100 Tip displacement/nm

1200

Figure 18.7.3 Five stages of an SECM experiment in which a tip penetrates from solution into a Nafion film containing Os(bpy)2+ : (a) Tip in solution above film. (b) Tip just penetrates the film. (c) Conical electroactive 3 portion of tip completely in film. (d) Tip near the substrate, where positive feedback occurs. (e) Tip in the tunneling region. (f ) Observed tip current vs. displacement at two different sensitivities: (squares) left current scale, (crosses) right current scale. Labels a–e correspond to stages (a)–(e). E T = 0.80 V vs. SCE; E S = 0.20 V vs. SCE. The tip moved in the z-direction at 3 nm/s. [Reprinted with permission from Mirkin, Fan, and Bard (93). © 1992, American Association for the Advancement of Science.]

electrodes, although correction for the effect of the tip and meniscus geometries on mass transport is required in some situations. In the simplest mode of operation, SECCM imaging is performed by rastering a single-barrel probe (Figure 18.8.1a) across the surface using piezoelectric drivers, while continuously measuring the current at a fixed potential. Using a dual-barrel SECCM probe (Figure 18.8.1b), either a dc or ac current passing between the two barrels in response to an applied V2 can be measured during scanning to control the tip height above the surface. In hopping-mode imaging, the SECCM probe is retracted to detach the meniscus from the surface after the initial measurement, and the probe is then moved to a new position above the electrode surface and lowered until the electrolyte meniscus again makes contact with the surface. This process is repeated hundreds or thousands of times to record the electrochemical response at different positions on the electrode surface. The individual responses are then used to construct the SECCM image. Image resolution is determined by the meniscus contact radius, which scales

18.8 Scanning Electrochemical Cell Microscopy

V2 V1

iS

iB

V1

O

R

Electrode (a)

iS

O

R

Electrode (b)

Figure 18.8.1 Schematics of SECCM probes and circuitry: (a) Single-barrel. (b) Dual-barrel. Each barrel is a micropipet or nanopipet filled with electrolyte and includes a quasireference-counter electrode. Single-barrel probes can be used for SECCM scans of conductive surfaces, while dual-barrel probes are also applicable to insulating surfaces. With dual-barrel probes, the ion current, iB across the meniscus generated by the potential bias between two barrels, V 2 , is monitored for positioning the tip above the surface. Meniscus contact with the surface is indicated by a change in iB . The dual-barrel probe can also be modulated vertically at a set frequency using a sinusoidal signal, and the resulting ac signal used for positioning. In either (a) or (b), the voltage V 1 sets the driving force for electrochemical reactions, and iS is measured using a current follower, which keeps the substrate at instrument ground (Section 16.2.1). In the simpler single-barrel case, the potential of the electrode in the tip is controlled vs. instrument ground at V 1 ; therefore, E S = − V 1 vs. that reference (Section 16.4). [Adapted from Wahab, Kang, and Unwin (96), with permission.]

with the tip radius, but also depends on the wetting properties of the electrode surface. The meniscus contact area can often be precisely determined following the electrochemical experiment using optical or electron microscopy to image electrolyte residue left on the surface. Figure 18.8.2 illustrates the measurement of local variations in electrochemical behavior on an electrode using SECCM (97). As previously discussed in Section 15.2.2(e), the edge planes of MoS2 layered nanoparticles display enhanced electrocatalytic activity for hydrogen evolution. A hopping-mode SECCM image (Figure 18.8.2a) of MoS2, obtained at constant potential using an elliptically shaped3 dual-barrel pipet in aqueous 0.1 M HClO4 , displays significantly higher HER activity at step edges relative to the basal plane. The location and heights of steps determined by SEM and AFM topographical imaging (Figure 18.8.2b) are correlated with the spatial HER activity mapped by SECCM. In the example in Figure 18.8.2, increased rates of HER are identified with 2-nm and 40-nm high steps. When the SECCM tip is brought down on a location containing a step defect, the contact area defined by the meniscus was estimated as being predominantly basal plane (90%), plus a smaller area associated with the step. Thus, the measured current arises from both the higher activity at the step and the lower activity on the basal plane on each side of the step. LSVs recorded on the basal plane far from the steps, and over the small and larger steps are shown in Figure 18.8.2c. Consistent with prior understanding, the SECCM studies show that the kinetics of the HER on the edge planes of MoS2 are enhanced relative to the basal plane. Semi-quantitative Tafel-plot analysis of the SECCM i − E responses indicates that the standard exchange current density at the edge plane of MoS2 (j0 ∼ 10−4 A/cm2 ) is approximately 3 A dual-barrel pipet has a thin glass wall running down the middle of the circular tubing. When pulled, there is an asymmetry in the force due to the glass wall, and an elliptical overall shape develops. Each barrel has about half of the open area at the end.

843

18 Scanning Electrochemical Microscopy

p1

Ste

ep St 2

0 nA

–4 nA

10 μm (a)

0 nm

–1 iS/nA –2 –3

75 nm

3 μm (b)

40

0

Basal plane Step 1 Step 2

–1.0 –0.8 –0.6 –0.4 –0.2 ES/V vs. RHE (c)

Height/nm

844

30 20

Step 2

10 0

Step 1 0.0

0.4 0.8 1.2 Distance/μm (d)

Figure 18.8.2 (a) SECCM image of HER activity on the basal plane of MoS2 obtained with E S = − 1.05 V vs. RHE. The image was recorded in 100 mM HClO4 using an elliptically shaped dual-barrel pipet (220- and 100-nmradius major and minor axes, respectively), with V 2 = 0.05 V. (b) AFM topographical image of the area in (a) enclosed within the dashed square. (c) LSVs obtained at pixel locations on the basal plane, on Step 1, and on Step 2. The LSVs are averages of multiple sweeps recorded at 0.25 V/s (top to bottom: 1500, 33, and 14 sweeps, respectively). (d) AFM line profile corresponding to the arrow in (b). [Adapted from Bentley et al. (97), with permission.]

40 times larger than on the basal plane (j0 ≈ 2.5 × 10−6 A/cm2 ). The value of j0 on the edge plane is approximate due to the uncertainty in determining the step edge area, but is ∼40× smaller than the value observed at polycrystalline Pt. These relative values are consistent with the j − E response and Tafel plots shown in Figure 15.2.2 for MoS2 layered nanoparticles and Pt. In Figure 18.8.3, we turn to a quite different application of SECCM. Depicted there is an experiment in which SECCM, operated in carrier generation-tip collection mode (CG-TC mode), is employed to visualize and to quantify photogenerated carrier transport distances and recombination rates in a semiconductor electrode (Section 20.3) (98). In this measurement, a narrow gaussian-shaped beam of light illuminates a thin layer of the n-type semiconductor WSe2 , supported in air on an ITO electrode (Figure 18.8.3a). Illumination from the backside (through the ITO) produces majority and minority carriers (electrons and holes, respectively) that recombine within the WSe2 layer when the photoanode is not connected to a cathode. To measure lateral diffusional transport of photogenerated holes, a 150-nm-radius SECCM probe containing 0.1 M NaI and 10 mM I2 is brought in contact with the top side of the WSe2 layer (Figure 18.8.3b). When an appropriate potential is applied between the ITO substrate and an Ag/AgI counter-reference electrode located inside the SECCM probe, photogenerated holes that have diffused laterally from the site of illumination can be collected at the WSe2 surface in contact with the SECCM tip, where they can be used to oxidize I− . An equal number of electrons is collected at the ITO substrate. The reaction, 3I− → I− + 2e, results in 3

18.8 Scanning Electrochemical Cell Microscopy

i Electrolyte-filled pipet

3I– h

I3–

e ITO

h

n-WSe2 Microscope objective

rp (a)

(b)

0.2 V 0.3 V 0.4 V 0.6 V

1.0 i/nA 0.5

0.0 –10

–5

0 rp/μm (c)

5

10

Figure 18.8.3 Carrier photogeneration-tip collection SECCM. (a) Minority carriers (holes) are generated within a layer of n - WSe2 nanosheets by local illumination through the ITO support and diffuse outward. (b) As carriers reach the SECCM probe, they are used to drive the oxidation of iodide (3I− → I− + 2e) on the n - WSe2 basal 3 plane. (c) Cross-section of SECCM currents resulting from illumination of a 90-nm thick n - WSe2 sample with a light beam of 𝜆 = 633 nm having a gaussian profile (𝜎 = 0.75 μm). Current measured with a 150-nm-radius SECCM probe filled with 0.1 M NaI and 10 mM I2 . [From Hill and Hill (98), with permission.]

a photocurrent that passes through the external circuit and is recorded. However, the holes must survive long enough to diffuse to the site of the SECCM probe. As the probe is positioned further from the site of photogeneration, a larger fraction of the holes will recombine with electrons inside the semiconductor prior to reaching the probe, thereby reducing the measured photocurrent. Thus, measurement of the photocurrent at the SECCM probe as a function of radial distance, rp , allows one to determine the diffusion length of the minority carriers. Figure 18.8.3c shows a plot of the SECCM current as a function of the distance from the illumination spot, recorded at different applied potentials. The spatial resolution is about 1 μm. The data indicate that holes diffuse radially from the site of generation and can be collected at distances of up to 10 μm from the photogeneration site. Analyses of these and additional CG-TC SECCM imaging experiments reveal highly anisotropic hole transport, with in-plane and out-of-plane hole diffusion lengths of 2.8 μm and 5.8 nm, respectively. Experiments in which photoexcitation and SECCM carrier detection were performed at locations separated by well-defined step edges resulted in lower photocurrents, providing direct evidence of a dramatic increase in recombination rates at these defects. SECCM is particularly well suited for imaging and quantifying electrochemical phenomena associated with individual particles (99), nanowires (100), bubbles (101, 102), grain

845

846

18 Scanning Electrochemical Microscopy

boundaries (103), or any other feature that can be isolated within the diameter of the probe tip. SECCM has also been used to map the PZC on individual grains of polycrystalline Au (104). Whereas the current recorded at an SECM tip reflects the electrochemical activity at a substrate indirectly via feedback or collection of products, an SECCM current is a direct measurement of processes at the substrate. As such, SECCM currents do not require analysis of diffusional transport of electroactive species between the tip and substrate and are, thus, simpler to analyze. SECCM measurements of systems that require rigorously dry or O2 -free solutions are challenging because gas transport across the air/solution interface at the nanoscopic meniscus is extremely rapid. Such measurements require performing the measurement in an inert environment. Reviews describing applications of SECCM can be found in the literature (95, 96, 105). An extensive review of electrochemical imaging methods, including SECM and SECCM, is presented in reference (106).

18.9 References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

A. J. Bard, F.-R. F. Fan, J. Kwak, and O. Lev, Anal. Chem., 61, 132 (1989). A. J. Bard, F.-R. F. Fan, and M. V. Mirkin, Electroanal. Chem., 18, 243 (1994). M. V. Mirkin, Anal. Chem., 68, 177A (1996). A. J. Bard, F.-R. F. Fan, and M. V. Mirkin, in “Physical Electrochemistry: Principles, Methods and Applications,” I. Rubinstein, Ed., Marcel Dekker, New York, NY, 1995, p. 209. A. J. Bard, G. Denuault, C. Lee, D. Mandler, and D. O. Wipf, Accts. Chem. Res., 23, 357 (1990). D. Shoup and A. Szabo, J. Electroanal. Chem., 160, 27 (1984). Y. Fang and J. Leddy, Anal. Chem., 67, 1259 (1995). C. G. Zoski and M. V. Mirkin, Anal. Chem., 74, 1986 (2002). Y. Shao and M. V. Mirkin, J. Phys. Chem. B, 102, 9915 (1998). M. V. Mirkin, in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 5. J. L. Amphlett and G. Denuault, J. Phys. Chem. B, 102, 9946 (1998). M. V. Mirkin, F.-R. F. Fan, and A. J. Bard, J. Electroanal. Chem., 328, 47 (1992). P. Sun and M. V. Mirkin, Anal. Chem., 79, 5809 (2007). D. O. Wipf and A. J. Bard, J. Electrochem. Soc., 138, 469 (1991). A. J. Bard, M. V. Mirkin, P. R. Unwin, and D. O. Wipf, J. Phys. Chem., 96, 1861 (1992). C. Wei, A. J. Bard, and M. V. Mirkin, J. Phys. Chem., 99, 16033 (1995). R. Cornut and C. Lefrou, J. Electroanal. Chem., 621, 178 (2008). N. Nioradze, J. Kim, and S. Amemiya, Anal. Chem., 83, 828 (2011). H. Xiong, J. Guo, and S. Amemiya, Anal. Chem., 79, 2735 (2007). A. I. Oleinick, D. Battistel, S. Daniele, I. Svir, and C. Amatore, Anal. Chem., 83, 4887 (2011). C. G. Zoski, N. Simjee, O. Guenat, and M. Koudelka-Hep, Anal. Chem., 76, 62 (2004). M. V. Mirkin and M. Tsionsky, in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 8. J. Zhang and P. R. Unwin, Phys. Chem. Chem. Phys., 4, 3820 (2002). D. O. Wipf and A. J. Bard, Anal. Chem., 64, 1362 (1992). F.-R. F. Fan and C. Demaille, in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 3. T. Sun, Y. Yu, B. J. Zacher, and M. V. Mirkin, Angew. Chem. Int. Ed., 53, 14120 (2014).

18.9 References

27 T. Sun, D. Wang, M. V. Mirkin, H. Cheng, J.-C. Zheng, R. M. Richards, F. Lin, and H. L.

Xin, Proc. Natl. Acad. Sci. U.S.A., 116, 11618 (2019). 28 J. V. Macpherson and C. Demaille, in “Scanning Electrochemical Microscopy,” M. V.

Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 17. 29 J. V. Macpherson and P. R. Unwin, Anal. Chem., 7, 276 (2000). 30 C. Kranz, G. Friedbacher, B. Mizaikoff, A. Lugstein, J. Smoliner, and E. Bertagnolli, Anal.

Chem., 73, 2491 (2001). 31 M. A. Derylo, K. C. Morton, and L. A. Baker, Langmuir, 27, 13925 (2011). 32 A. Eifert, B. Mizaikoff, and C. Kranz, Micron, 68, 27 (2015). 33 P. S. Dobson, J. M. R. Weaver, M. N. Holder, P. R. Unwin, and J. V. Macpherson, Anal.

Chem., 77, 424 (2005). 34 K. Leonhardt, A. Avdic, A. Lugstein, I. Pobelov, T. Wandlowski, M. Wu, B. Gollas, and

G. Denuault, Anal. Chem. 83, 2971 (2011). 35 O. Sklyar, A. Kueng, C. Kranz, B. Mizaikoff, A. Lugstein, E. Bertagnolli, and G. Wittstock, 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Anal. Chem., 77, 764 (2005). X. Shi, W. Qing, T. Marhaba, and W. Zhang, Electrochim. Acta, 332, 135472 (2020). P. Sun and M. V. Mirkin, Anal. Chem., 78, 6526 (2006). J. Velmurugan, P. Sun, and M.V. Mirkin, J. Phys. Chem. C, 113, 459 (2008). S. Amemiya in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 6, Table 6.1. C. G. Zoski, C. R. Luman, J. L. Fernández, and A. J. Bard, Anal. Chem., 79, 4957 (2007). C. B. Ekanayake, M. B. Wijesinghe, and C. G. Zoski, Anal. Chem., 85, 4022 (2013). K. Eckhard, X. Chen, F. Turcu, and W. Schuhmann, Phys. Chem. Chem. Phys., 8, 5359 (2006). I. Morkvenaite-Vilkonciene, A. Ramanaviciene, and A. Ramanavicius, RSC Adv., 4, 50064 (2014) P. R. Unwin and A. J. Bard, J. Phys. Chem., 95, 7814 (1991). F. Zhou, P. R. Unwin, and A. J. Bard, J. Phys. Chem., 96, 4917 (1992). D. E. Danly, J. Electrochem. Soc., 131, 435C, (1984). F. Zhou and A. J. Bard, J. Am. Chem. Soc., 116, 393 (1994). D. E. Blanco, B. Lee, and M. A. Modestino, Proc. Natl. Acad. Sci. U.S.A., 116, 17683 (2019). T. Kai, M. Zhou, Z. Duan, G. A. Henkelman, and A. J. Bard, J. Am. Chem. Soc., 139, 18552 (2017). A. R. Paris and A. B. Bocarsly, ACS Catal., 9, 2324 (2019). A. Gennaro, A. A. Isse, J.-M. Savéant, M.-G. Severin, and E. Vianello, J. Am. Chem. Soc., 118, 7190 (1996). J. H Chang and A. J. Bard, J. Am. Chem. Soc., 136, 311 (2014). M. Zhou, Y. Yu, K. K. Hu, and M. V. Mirkin, J. Am. Chem. Soc., 137, 6517 (2015). R. C. Engstrom, T. Meany, R. Tople, and R. M. Wightman, Anal. Chem., 59, 2005 (1987). P. R. Unwin and J. V. Macpherson, in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 13. P. R. Unwin and A. J. Bard, J. Phys. Chem., 96, 5035 (1992). J. Zhang and P. R. Unwin, J. Am. Chem. Soc., 122, 2597 (2002). J. Zhang and P. R. Unwin, J. Am. Chem. Soc., 124, 2379 (2002). J. Zhang, and P. R. Unwin, Langmuir, 18, 1218 (2002). J. Rodríguez-López, M. A. Alpuche-Avilés, and A. J. Bard, J. Am. Chem. Soc., 130, 16985 (2008).

847

848

18 Scanning Electrochemical Microscopy

61 J. Rodríguez-López, C. G. Zoski, and A. J. Bard, in “Scanning Electrochemical Microscopy,”

62 63 64 65 66 67 68

69 70 71 72 73

74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 16. N. Arroyo-Currás and A. J. Bard., J. Phys. Chem. C, 119, 8147 (2015). J. Rodríguez-López, A. Minguzzi, and A. J. Bard, J. Phys. Chem. C, 114, 18645 (2010). H. S. Ahn and A. J. Bard, J. Phys. Chem. Lett., 7, 2748 (2016). D. Zigah, J. Rodríguez-López, and A. J. Bard, Phys. Chem. Chem. Phys., 14, 12764 (2012). Y. Ma, P. S. Shinde, X. Li, and S. Pan, ACS Omega, 4, 17257 (2019). H. S. Park, K. C. Leonard, and A. J. Bard, J. Phys. Chem. C, 117, 12093 (2013). G. Denuault, G. Nagy, and K. Toth, in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 10. B. R. Horrocks, M. V. Mirkin, D. T. Pierce, A. J. Bard, G. Nagy, and K. Toth, Anal. Chem., 65, 1213 (1993). C. Wei, A. J. Bard, G. Nagy, and K. Toth, Anal. Chem., 67, 34 (1995). E. El-Giar and D. O. Wipf, J. Electroanal. Chem., 609, 147 (2007). D. Filotás, B.M. Fernández-Pérez, J. Izquierdo, A. Kiss, L. Nagy, G. Nagy, and R. M. Souto, Corros. Sci., 129, 136 (2017). D. E. Tallman, M. B. Jensen, and K. Toth, in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 14. R. M. P. da Silva, J. Izquierdo, M. X. Milagre, A. M. Betancor-Abreu, I. Costa, and R. M. Souto, Sensors, 21, 1132 (2021). N. Payne, L. I. Stephens, and J. Mauzeroll, Corrosion, 73, 759 (2017). Q. Xiong, R. Song, T. Wu, F. Zhang, and P. He, J. Electroanal. Chem., 887, 115169 (2021). V. S. Joshi, P. S. Sheet, N. Cullin, J. Kreth, and D. Koley, Anal. Chem., 89, 11044 (2017). D. Polcari, P. Dauphin-Ducharme, and J. Mauzeroll, Chem. Rev., 116, 13234 (2016). C. G. Zoski, J. Electrochem. Soc., 163, H3088 (2016). A. Preet and T.-E. Lin, Catalysts, 11, 594 (2021). M. V. Mirkin and A. J. Bard, Eds., “Scanning Electrochemical Microscopy,” Taylor and Francis Group, Boca Raton, FL, 2012. J. Kwak and F. C. Anson, Anal. Chem., 64, 250 (1992). C. Lee and F. C. Anson, Anal. Chem., 64, 528 (1992). M. Arca, M. V. Mirkin, and A. J. Bard, J. Phys. Chem., 99, 5040 (1995). E. R. Scott, H. S. White, and J. B. Phillips, J. Membr. Sci., 58, 71 (1991). M. Tsionsky, A. J. Bard, and M. V. Mirkin, J. Am. Chem. Soc., 119, 10785 (1997). S. B. Basame and H. S. White, Anal. Chem., 71, 3166 (1999). F. Conzuelo, A. Schulte, and W. Schuhmann, Proc. R. Soc. A, 474, 20180409 (2018). I. Beaulieu, S. Kuss, and J. Mauzeroll, Anal. Chem., 83, 1485 (2011). B. R. Horrocks and G. Wittstock, in “Scanning Electrochemical Microscopy,” M. V. Mirkin and A. J. Bard, Eds., Taylor and Francis Group, Boca Raton, FL, 2012, Chap. 11. S. Bergner, P. Vatsyayan, F. M. Matysik, Anal. Chim. Acta, 775, 1 (2013). A. Sridhar, H. L. de Boer, A. van den Berg, and S. Le Gac, PLoS One, 9, e93618 (2014). M. V. Mirkin, F.-R. F. Fan, and A. J. Bard, Science, 257, 364 (1992). N. Ebejer, M. Schnippering, A. W. Colburn, M. A. Edwards, and P. R. Unwin, Anal. Chem., 82, 9141 (2010). N. Ebejer, A. G. Güell, S. C. S. Lai, K. McKelvey, M. E. Snowden, and P. R. Unwin, Annu. Rev. Anal. Chem., 6, 329 (2013).

18.10 Problems

96 O. J. Wahab, M. Kang, and P. R. Unwin, Curr. Opin. Electrochem., 22, 120 (2020). 97 C. L. Bentley, M. Kang, F. M. Maddar, F. Li, M. Walker, J. Zhang, and P. R. Unwin, Chem.

Sci.., 8, 6583 (2017). 98 J. W. Hill and C. M. Hill, Chem. Sci., 12, 5102 (2021). 99 M. Choi, N. P. Siepser, S. Jeong, Y. Wang, G. Jagdale, X. Ye, and L. A. Baker, Nano Lett.,

20, 1233 (2020). 100 A. G. Güell, N. Ebejer, M. E. Snowden, K. McKelvey, J. V. Macpherson, and P. R. Unwin,

Proc. Natl. Acad. Sci. U.S.A., 109, 11487 (2012). 101 Y. Wang, E. Gordon, and H. Ren, J. Phys. Chem. Lett., 10, 3887 (2019). 102 Y. Liu, C. Jin, Y. Liu, K. H. Ruiz, H. Ren, Y. Fan, H. S. White, and Q. Chen, ACS Sens., 6, 103 104 105 106

355 (2021). R.G. Mariano, K. McKelvey, H.S. White, and M.W. Kanan, Science, 358, 1187 (2017). Y. Wang, E. Gordon, and H. Ren, Anal. Chem., 92, 2859 (2020). C. L. Bentley, Electrochem. Sci. Adv., e2100081 (2021). C. L. Bentley, J. Edmondson, G. N. Meloni, D. Perry, V. Shkirskiy, and P. R. Unwin, Anal. Chem., 91, 84 (2019).

18.10 Problems 18.1

SECM is carried out with a disk-shaped Pt tip of 10-μm diameter embedded in a glass insulating sheath, RG = 10, over a Pt substrate. An experiment was performed with a ∗ = 5.0 mM and D = 5 × 10−6 cm2 /s. When the tip, at solution of species O having CO O a potential where O is reduced to R at a diffusion-controlled rate, was held near the Pt substrate surface, with the substrate at a potential where R is totally oxidized to O, the ratio iT /iT, ∞ was 2.5. (a) At what distance, d, was the tip from the surface? (b) What is iT, ∞ ? (c) If this tip is held over a glass substrate at the same d value, what would iT /iT, ∞ be?

18.2

(a) Qualitatively sketch the flux lines for diffusion of species O to a disk-shaped SECM tip as the tip approaches an insulating substrate. Assume RG = 10 and L = 100, 10, and 1. (b) Repeat for RG = 1.5.

18.3

Use a spreadsheet and expressions (18.2.5)–(18.2.10) to calculate tip voltammograms (iT vs. ES ) resulting from feedback for rate constants at the substrate, k 0 , of 10−2 , 10−4 , and 10−6 cm/s. Assume O + e → R at the tip, and take L = 0.1, a = 10 μm, RG = 10, DO = DR = 10−5 cm2 /s, 𝛼 = 0.5, and T = 25 ∘ C. Explain how decreasing L would affect the shape of the voltammograms.

18.4

SECM is used to study an Er C i reaction (O + e ⇌ R; R → Z). A 10-μm tip is used to reduce O while being positioned over a Pt electrode where R is oxidized back to O at a diffusion-controlled rate. When the tip is 0.2 μm from the surface, the approach curve shows the same feedback current as for a mediator where the product is stable. However, when the tip is 4.0 μm away, the response is close to that for an insulating substrate.

849

850

18 Scanning Electrochemical Microscopy

(a) Estimate the rate constant for the decomposition of R to Z. (b) If this reaction were studied by CV, approximately what scan rate would be needed to find a nernstian response? (c) What are the advantages of SECM in studying this kind of reaction compared to CV? 18.5 Consider the same system as in Problem 18.1. Assume that the tip is biased about 0.5 V with respect to the Pt substrate. At about what distance, d, will the current attributable to direct tunneling between the tip and substrate become larger than the SECM feedback current? Do you think that a tip like that described in Problem 18.1 could attain such a d value? Why?

851

19 Single-Particle Electrochemistry In previous chapters, we have seen that electrochemical experiments are typically performed in solutions containing 1 μM to 10 mM of a redox species, corresponding to ∼1018 to 1022 molecules per liter of solution. We refer to such a large collection of molecules as an ensemble. Similarly, the i − E or i − t responses considered to this point have been understood on the basis of deterministic equations (e.g., Fick’s laws) describing transport and reaction kinetics for large ensembles of molecules. We now consider situations and experiments that represent the opposite extreme, in which the electrochemical response is associated with an individual particle and displays stochastic characteristics. We use the term particle to describe a wide range of entities, including atoms, molecules, metal nanoparticles, emulsion droplets, and biological cells, with sizes ranging from a fraction of a nanometer to tens of micrometers. At sufficiently low particle concentrations, the electrochemical i − E or i − t response associated with an individual particle can often be observed. The strategies employed in observing and interpreting different types of electrochemical responses associated with particles are presented in this chapter. These include amperometric detection of collisions of nanoparticles with an electrode surface, detection of single molecules using steady-state redox cycling between two closely spaced electrodes, and voltammetry at single metal atoms and clusters.

19.1 General Considerations in Single-Particle Electrochemistry Interest in single-particle electrochemistry is driven by the desire to observe behavior that cannot be seen in ensembles, as well as to develop new methods of ultra-sensitive electroanalysis. In general, the descriptions of electrochemical systems developed in Chapters 1–18, based on averages of behavior over large ensembles, mask the true complexity of transport and reaction in electrochemistry. For instance, we have seen in previous chapters that an oxidation or reduction reaction at the electrode surface, coupled to the random thermal motion of a large ensemble of molecules, results in a net flux to the electrode well described by Fick’s laws. However, one cannot observe the individual random motions of molecules in a large ensemble, nor can one observe the electron-transfer reaction of a single molecule as it arrives at the electrode. Conversely, single-particle measurements provide a means to study the motion and reactivity on a microscopic scale. They also allow the heterogeneity of a sample of particles to be explored, e.g., the dispersion in size and catalytic activity of metal nanoparticles. Throughout this chapter, we will consider special situations and methodologies in which individual particles are detected and their properties and behavior are characterized by electrochemical methods. In general, some form of physicochemical amplification of the current Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

852

19 Single-Particle Electrochemistry

resulting from the interaction of the electrode and particle is required if one is to successfully detect or quantify electrochemical behavior associated with an individual particle. We have seen in Section 16.8.1 that the smallest currents measurable using special low-noise electronics are on the order of 10 fA, or about 60,000 electrons per second. Thus, to detect a single particle, a very large current amplification—on the order 105 or more—is essential. A variety of electrochemical amplification methods have been developed over the past two decades for the detection of single particles (e.g., including atoms and molecules). For instance, the repeated oxidation and reduction of an individual redox molecule at two electrodes separated by a distance on the scale of 10–100 nm result in a current on the order of 10 fA to 1 pA, which is readily measured (Section 19.6). Similarly, a sudden increase in current on the order of 10 pA to 1 nA resulting from the adsorption of an electrocatalytic particle at an electrochemically inert electrode allows that particle to be readily detected. The detection of insulating particles is different: A sudden decrease in current from an electroactive solute is observed when such a particle collides with the surface, blocking transport of a large number of redox molecules to the surface.

19.2 Particle Collision Experiments Electrochemical measurements used to detect the collision of a particle with the electrode are applicable to metal, metal oxide, and polymer nanoparticles, emulsion droplets, carbon nanotubes, and biological particles, including enzymes and single cells. Literature reviews describe methods and applications of nanoparticle collision in electrochemistry (1–4). Collisions result from transport of particles from the bulk solution to the electrode, usually by diffusion and migration in an unstirred solution. They may result in particle adsorption (“sticky” collisions) or transient interactions with the electrode (“reflective” collisions). Most particle collision experiments are based on recording the chronoamperometric response at the electrode following the introduction of the particles into the solution. The electrode potential is held at a constant value based on the redox chemistry used in detecting a collision event. Figure 19.2.1 schematically depicts an experiment that is often performed using an UME in the presence of a supporting electrolyte. Both two- and three-electrode cells can be employed because the currents associated with particle collisions at the UME are typically very small. For two reasons, UMEs with radii between 1 and 10 μm are generally employed to detect particle collisions. First, one generally wishes to minimize simultaneous particle collisions to simplify the analysis of the i − t response. Collision rates of 100 to 0.01 s−1 allow one to conveniently resolve individual particle collisions, while providing an adequate number of collisions for a meaningful statistical analysis. This range of collision frequency can typically be obtained by decreasing the particle concentration to sub-nM levels while using a 1- to 10-μm-radius UME. Second, the current associated with an individual particle collision is generally in the 0.01–100 pA range and must be measured above the background current at the electrode. The latter arises from transient capacitive currents and extraneous faradaic reactions associated with impurities, e.g., oxygen reduction or electrode-surface redox chemistry. Decreasing the electrode size reduces these background interferences, allowing easier detection and quantification of behavior associated with a single-particle collision. However, the advantage gained by decreasing the electrode size to reduce background currents is eventually offset by an inconveniently low particle collision rate; thus, 1- to 10-μm-radius UMEs have generally been found to provide sufficiently low background currents while maintaining a reasonable collision rate.

19.2 Particle Collision Experiments

Figure 19.2.1 (a) UME in an electrolyte containing a uniform dispersion of nanoparticles (NPs). (b) Idealized i − t response on a long timescale (e.g., 50 s) resulting from stochastic collisions of individual NPs with the electrode surface. Blips in the i − t response are observed following introduction of the NPs into the electrolyte. (c) The i − t response on an expanded time axis (e.g., 50 ms) showing a single particle collision. The behavior depicted in (b) and (c) is typical of collisions resulting in the oxidation or reduction of the particle, generating a transient current spike. Differences in the amplitude and width of the spikes reflect nonuniform particle size and/or reactivity.

Electrolyte containing NPs

UME (a)

10 s

Introduce NPs to solution ↑ i 0

t→ (b)

10 ms i 0 t→ (c)

A general approach for determining the concentration of particles in solution is based on measuring the frequency of particle collisions at the electrode, which is practical for particle concentrations in the picomolar (10−12 M) to femtomolar (10−15 M) range. However, not all particle collisions necessarily result in an electrochemical signature. In order for the particle to be detected, it must make electrical contact with the electrode surface for a time period sufficient to carry out an electron transfer reaction. This criterion generally requires that the particle be adsorbed or located within tunneling distance from the electrode surface. Accordingly, significant attention must be given to remove any contamination or insulating film from the electrode surface that interferes with electrical contact between the electrode and particle. Particle collision rate measurements often provide fundamental insights into particle size and particle transport dynamics. Nanoparticle aggregation can be problematic in solutions of high ionic strength but can be alleviated by working in low electrolyte concentrations, or in the case of metal nanoparticles, by using ionic surface-capping agents. Often, particle collision experiments are useful to probe aggregation and the polydispersity of particle aggregate sizes. Electrochemical methods used to detect particles fall in three categories, depending on their basis: 1) Blockage of the flux of a redox species in solution by an insulating particle (blocking collisions). 2) Amplification of the electrochemical reaction (electrocatalytic amplification collisions). 3) Electrolysis of the particle (direct electrochemical detection). Figure 19.2.2 shows representative examples of each of these types of particle collision experiments. Section 19.4 provides a deeper look into the type of information that can be obtained in each type of measurement.

853

19 Single-Particle Electrochemistry

–60 –70

–90

150 20

25

t/s (a)

30

35

–100 36 38 40 42 44 46 t/s (b)

10 ms

800

–80 i/pA

i/pA

160 i/pA

854

Q = 0.77 pC

400 0 0.51

0.52 t/s (c)

0.53

Figure 19.2.2 Typical chronoamperometric responses resulting from particle collisions: (a) Blocking response following adsorption of 0.5-μm-radius insulating polystyrene particles at the surface of a 2.5-μm-radius Pt electrode in 2 mM FcMeOH. Each sudden drop in current corresponds to adsorption of one particle, partially blocking the diffusion of FcMeOH to the electrode [Reprinted with permission from Quinn, van’t Hof, and Lemay (5). © 2004, American Chemical Society.]. (b) Electrocatalytic amplification collisions corresponding to the transient adsorption of IrOx nanoparticles at a 5-μm-radius Au electrode, resulting in electrocatalytic oxidation of H2 O [From Kwon et al. (3), with permission.]. (c) Collision and oxidation of an individual 35-nm radius Ag nanoparticle at a 6.25-μm-diameter Au microelectrode using a 10 kHz low-pass filter and a 50-kHz sampling rate [Reprinted with permission from Robinson et al. (6). © 2017, American Chemical Society.]

19.3 Particle Collision Rate at a Disk-Shaped UME The motion and electron-transfer processes of individual particles in time and space are stochastic in nature. The term stochastic indicates that the random nature of diffusion or electrode reaction of a single molecule precludes use of deterministic continuum equations. Thus, the mathematical approaches we encounter in this section to treat single-particle electrochemistry differ fundamentally from those presented in the preceding chapters. Essentially all single-particle electrochemical responses contain some element of stochastic behavior. One cannot precisely predict the time when a particle will collide with an electrode surface, but one can compute the expected number of collisions in an experimental time interval. We will see that descriptions and analyses of these type experiments can be made based on statistical analyses or probability theory (7). 19.3.1

Collision Frequency

The frequency at which particles in a solution collide with an electrode surface is a key parameter in many of the experiments described below. Experimentally, the collision frequency, f (s−1 ), is determined by simply counting the number of current steps or spikes during a period of time, assuming that all collisions result in an observable electrochemical response. Often, f is found to be proportional to the particle concentration. We will develop expressions for (a) the average collision frequency, (b) the expected variance in the number of collisions during an observation period, and (c) the expected mean time for the first particle collision at the beginning of the experiment. These three quantities fully describe the time dependence of collision behavior and can be used to extract information regarding the concentration, activity, and size of the particles. For simplicity, we assume that particles of identical size are randomly distributed throughout the solution at the beginning of the experiment with a concentration C * . In real solutions, the particles display a polydispersity in radius, which can be reflected by a dispersity in their behavior, e.g., collision peak heights. Particle concentrations typically range between pM and

19.3 Particle Collision Rate at a Disk-Shaped UME

fM, corresponding to interparticle distances ranging from 10 to 100 μm, respectively. These interparticle distances are generally equal to or larger than the radius of the UME used to observe particle collisions. The large interparticle distances, relative to the electrode size, is the underlying physical basis for the stochastic nature of particle collisions.1 In most particle collision experiments (but not all), the particle sticks to the surface upon collision and may be consumed electrolytically by a redox reaction; thus, the electrode can be considered as a sink or absorber of the particles. The methods used in Chapters 5 and 6 for obtaining the i − t response of a UME following a potential step can be directly used in such cases to compute the flux of particles to a UME. If all arriving particles stick to the electrode or are eliminated by electrolysis, then the flux of nanoparticles, J NP , becomes diffusion-limited. For a disk-shaped UME with radius r0 , the flux magnitude is then given at steady state by |JNP | = 4DC ∗ ∕𝜋r0

(19.3.1)

Multiplying both sides by the Avogadro constant, N A , to express the flux in particles (s−1 cm−2 ), and then by the area of the disk UME, 𝜋r02 , one obtains the expected collision frequency in units of s−1 . f = 4DC ∗ r0 NA

(19.3.2)

Equation 19.3.2 represents the average particle collision frequency during a diffusion-limited steady-state measurement. The expected number of collisions during any time interval, Δt, is given by N(Δt) = 4DC ∗ r0 NA Δt 19.3.2

(19.3.3)

Variance in the Number of Particle Collisions

Due to the stochastic nature of the particle distribution and the random motion of individual particles, the actual number of particle collisions observed in an experiment during a period Δt will fluctuate about the expected value of N(Δt) given by (19.3.3). The statistical fluctuation in 2 numbers of collisions is described by a Poisson distribution, where the variance, 𝜎N(Δt) , is equal to the number of collisions, N(Δt), 2 𝜎N(Δt) = N(Δt)

(19.3.4)

Thus, the fluctuation around the expected value of N(Δt) for a disk UME is expressed by the standard deviation 𝜎N(Δt) = (4DC ∗ r0 NA Δt)1∕2

(19.3.5)

For example, for a particle collision experiment in which the average collision frequency, f , is 1 s−1 , the expected number of collisions in a 100 s period would be 100 ± 10, i.e., N(Δt) ± 𝜎 N(Δt) . An important consequence of the Poisson distribution is that the relative fluctuation in number of collisions, 𝜎 N(Δt) /N(Δt), increases dramatically with a decrease in either f or Δt. For example, in the previous example, if Δt is decreased to 10 s, the expected number of collisions would be 10 ± 3.3. Thus, the relative error in C * increases by a factor of 3 (from 10% 1 The appropriateness of using of continuum differential equations to describe the particle concentration in very dilute solutions of particles, where the interparticle distance is comparable or exceeds the dimensions of the electrode, is not obvious. However, the time-dependent local concentration, C(x,t), in the differential equations can be interpreted as the probability of finding a particle at a point, x, in the solution at time t (7).

855

856

19 Single-Particle Electrochemistry

to 33%) for the shorter measurement. For ultralow concentrations of particles, where collisions are very infrequent, it is often difficult to obtain a precise or accurate measure of f or N(Δt) because the number of counts is so small. One often wishes to know the probability of observing a particular number of collisions in a time Δt. This is readily computed from the Poisson probability distribution, P(k, 𝜆) = 𝜆k exp(−𝜆)∕k

(19.3.6)

which describes the probability of observing k particle collisions in a time period Δt, given that 𝜆 is the average or most probable value of the number of collision events in that period. It should be apparent that 𝜆 is equivalent to N(Δt), as given by (19.3.3). As an example, if f = 0.1 s−1 , then for a 20 s observation time, 𝜆 = 2, and the probabilities of observing 0, 1, 2, or 3 collisions are 0.14, 0.27, 0.27, and 0.18, respectively. 19.3.3

Time of First Arrival

A method for estimating C * in ultralow concentration ( E 0 (O/R)

O

e

d

Eb < E 0 (O/R)

Figure 19.6.1 Schematic diagram of a single molecule undergoing redox cycling between two electrodes with independent potential control.

875

876

19 Single-Particle Electrochemistry

molecule to diffuse (by Brownian motion) between two electrodes separated by a distance d is given by td = d2 ∕2D

(19.6.1)

where D is the diffusion coefficient of the molecule. Recognizing that both O and R must diffuse the distance d to complete one redox cycle, the total time for the cycle is t2d =

d2 d2 d2 + =2 2DO 2DR 2De

(19.6.2)

where De is the effective diffusion coefficient of the O/R couple. This equation can be readily rearranged to De =

2DO DR DO + DR

(19.6.3)

For reversible redox couples undergoing a simple electron-transfer reaction, it is often true that DO ≈ DR = D. In this situation, (19.6.3) reduces to De = D. For d = 10 nm and assuming DO = DR = 1 × 10−5 cm2 /s, t d ∼ 100 ns. In one second, the molecule will make 107 round trips between the two electrodes, generating a current of 1.6 pA. Currents in this range are readily measured using standard amplifiers found in electrochemical laboratories. 0′ , then redox cycling If Et and Eb are held at values sufficiently positive and negative of EO∕R occurs at the maximum diffusion-controlled rate. The current for this condition can be expressed as i = nNe∕2t2d

(19.6.4)

where N is the number of trapped molecules, n is the number of electrons for the reaction O + ne ⇌ R, and e is the electronic charge. By substitution from (19.6.1), one obtains i = nNeDe ∕d2

(19.6.5)

The diffusion-limited redox cycling current can also be expressed in terms of the bulk concentration of the trapped species, C * = N/AdN A , using the equation for a macroscopic dual-electrode thin-layer cell (53, 54): i = nFADe C ∗ ∕d

(19.6.6)

Since (19.6.5)–(19.6.6) can be derived from each other, they are equivalent (Problem 19.6). In addition to measuring redox cycling currents at the diffusion-controlled rate with both electrodes held at fixed potential, it is also possible to hold one electrode at a constant potential, while scanning the potential of the other to generate voltammetric waves corresponding to a single molecule (50, 55). Two strategies based on these principles have been used in designing experiments for single-molecule detection (SMD). In “closed” systems, one traps a single molecule between a small metal electrode and a much larger conducting substrate (Figure 19.6.2a,b). The first example of SMD (50) involved the use of a scanning electrochemical microscope (SECM) to position a Pt–Ir tip with a radius of ∼5 nm within ∼10 nm of an ITO or TiO2 conductive substrate (Figure 19.6.2a). An insulating wax shroud around the metal tip provided lateral confinement of the molecule. For a 2 mM solution of an electroactive solution, it was most likely that only one molecule would be trapped in the 10−18 cm3 volume between the tip and substrate when the wax insulator surrounding the SECM tip was pushed up against

19.6 Single-Molecule Electrochemistry

Pt–lr tip

Pt

15 nm Wax insulator

O

e

Fc+

Fc

2+

10 nm

+e –e

R

Hg ITO substrate

e (a)

(b) Quasi-reference counter electrodes

Top working electrode

Probe working electrode

Cl Ag / Ag

Solution e Solution

R

O e

d

Confined solution volume

e R O

Bottom working electrode (c)

Substrate working electrode e

(d)

Figure 19.6.2 Experimental approaches to trapping single molecules for detection by redox cycling. (a) SECM tip positioned above a conductive substrate [From Fan and Bard (50), with permission.] (b) Recessed Pt nanoelectrode immersed in a Hg pool [Adapted with permission from Sun and Mirkin (55). © 2008, American Chemical Society.]; (c) Lithographically fabricated nanogap cell containing two Pt electrodes [Adapted with permission from Mampallil, Mathwig, Kang, and Lemay (56). © 2014, American Chemical Society.] (d) Multi-barrel SECCM probe positioned above a conductive substrate [Adapted with permission from Byers et al. (57). © 2015, American Chemical Society.] Both (a) and (b) represent “closed” systems in which one or a few redox molecules are trapped between two electrodes at the start of the experiment, while (c) and (d) are “open” systems where the redox molecules may enter and exit the volume of solution between the electrodes over time. In each system, the two electrodes are typically separated by 10–100 nm, allowing sufficiently fast redox cycling to detect a single molecule.

the surface. A similar strategy (Figure 19.6.2b) is based on pushing a Pt nanodisk electrode (r0 = 10 − 100 nm), recessed by etching a few nanometers below the surface of its glass shroud, into a pool of Hg (55). In this example, redox cycling occurs between the Pt nanodisk and Hg electrodes. The distance d corresponds to the depth of the recess (∼5–10 nm), while the area is defined by that of the Pt nanodisk. Both the SECM and Pt/Hg pool systems are “closed” because, in principle, the molecule(s) are trapped into a sealed volume at the beginning of the experiment, and no molecules can subsequently enter or leave. While the average number of molecules trapped is based on the bulk solution concentration and confined volume, the actual number in any trial is dictated by Poisson statistics (55). In contrast, “open” systems for SMD allow molecules to diffuse into and out of the detection volume during the period of measurement. Figure 19.6.2c shows a lithographically prepared thin-layer cell, comprising two Pt electrodes of relatively large area (50 μm length × 1.5 μm width) separated by 70 nm. Openings connect the solution between the Pt electrodes to the bulk solution (58); thus, redox molecules can enter or leave the confined sensing volume,

877

19 Single-Particle Electrochemistry

where they are detected by redox cycling. A similar principle is applied in using scanning electrochemical cell microscopy (SECCM) with a multi-barrel pipette (Figure 19.6.2d). In this system, one barrel is filled with the electrolyte and redox molecule at the appropriate concentration, while additional barrels are filled with pyrolytically deposited carbon that serve as electrodes for redox cycling. The pipette, with the electrolyte meniscus extending from the pipette, is carefully driven toward a conducting substrate, creating a volume of solution defined by the pipette radius and the distance between the pipette (diameter ∼3 μm) and the conducting substrate (∼20 nm). The SECCM approach has been demonstrated to have exceptionally low baseline noise for SMD (∼1 fA), an advantage mainly due to the small contact area between the substrate and solution (57). Each of the four systems represented in Figure 19.6.2 is intended for SMD of freely diffusing redox molecules, and each has its own advantages and disadvantages, as discussed in the original reports. Figure 19.6.3a shows an example of data from the SECM-based system, in which an ∼10−18 -cm3 volume of solution containing 2.0 mM FcTMA+ (Figure 1) and 2.0 M NaNO3 is confined between the Pt tip and the conductive ITO substrate. Based on the estimated solution volume and bulk concentration of FcTMA+ , approximately 1 molecule of FcTMA2 + /1+ is predicted to be trapped between the Pt tip and the ITO. The SECM tip was held at 0.55 V vs. SCE to oxidize FcTMA+ , while the ITO was held at −0.3 V to reduce FcTMA2+ , both at diffusion-controlled rates. When the tip was moved to within ∼10 nm of the ITO substrate, fluctuations began to be observed in the i − t response. Although the signal is noisy, the i − t trace shows clear peaks of ∼0.5 and 1.0 pA, as well as periods of essentially zero average current. These fluctuations in the i − t response are interpreted as corresponding to one or two FcTMA+ molecules trapped in the ∼10 nm gap between the tip and substrate but also able to diffuse into and out of the sensing region directly below the tip, where redox cycling occurs. The probability density function (PDF) plot (Figure 19.6.3b) corresponding to these data displays two Gaussian peaks centered at 0.5 and 1.0 pA, again suggesting that a discrete number of molecules (0, 1, or 2) are undergoing redox cycling at any moment (51). Figure 19.6.4 shows a redox cycling experiment (58, 59) performed in the lithographically prepared thin-layer cell, having a 70-nm gap between the two Pt electrodes (Figure 19.6.2c). While similar to the preceding experiment, the thin-layer cell also allows recording of the redox cycling current at both electrodes. Figure 19.6.4 shows an example of using this cell to detect 5

2

4 PDF/PA–1

1 iT/pA

878

0

3 2 1

–1 0

50

100

t/s (a)

150

200

0 0.0

0.5

1.0

1.5

iT/pA (b)

Figure 19.6.3 (a) i − t trace observed at a SECM Pt tip (r0 = 7 nm) positioned ∼10 nm above an ITO electrode (Figure 19.6.2a). The aqueous solution contained 2.0 mM FcTMA+ and 2.0 M NaNO3 . The SECM tip was held at 0.55 V vs. SCE to oxidize FcTMA+ , while the ITO electrode was held at −0.3 V to reduce FcTMA2+ . (b) Probability density function (PDF) analysis for the data in (a). The most probable tip currents are spaced 0.5 pA apart, with a standard deviation of 0.1 pA. [Reprinted from Fan, Kwak, and Bard (51). © 1996, American Chemical Society.]

19.7 References

40 20 0

0 –50

No Fc

i/fA

i/fA

50

(a)

0 –20 –40

i/fA

50 0 –50

0

120 pM Fc 0

10

20

t/s (b)

30

40

50

5

10 t/s (c)

15

Figure 19.6.4 Amperometric detection of single molecules using a nanogap cell (Figure 19.6.2c) that allows molecules to enter and exit by diffusion. Simultaneously recorded i − t curves at the top (gray) and bottom (black) Pt electrodes for a nanogap cell containing MeCN/0.1 M TBAPF6 in the (a) absence and (b) presence of 120 pM ferrocene. The two overlapping electrodes (active areas of 50 μm × 1.5 μm) were separated by a distance of d = 70 nm. The i − t traces were recorded while holding the potentials of the top and bottom electrodes at E t = 0.35 V and E b = 0.1 V, to oxidize Fc and reduce Fc+ , respectively. Discrete, anticorrelated fluctuations in the i − t traces with amplitude ∼20 fA are observed at the two Pt electrodes, corresponding to one (or more) Fc molecules entering the device, undergoing redox cycling, and subsequently exiting. (c) An i − t trace exhibiting current plateaus corresponding to 0, 1, and 2 molecules inside the nanogap. [Reprinted with permission from Lemay, Kang, Mathwig, and Singh (58). © 2013, American Chemical Society.]

single molecules of Fc in which the potentials applied to the two electrodes were set to oxidize Fc and reduce Fc+ at the diffusion-limited rates. Figure 19.6.4a shows the background current in the absence of Fc, giving an indication of the background noise, while Figure 19.6.4b shows the observed fluctuations in electrode currents when Fc is added to the solution and a discrete number of Fc molecules (N = 0, 1, 2, 3…) diffuse into the volume between the two electrodes. Because molecules are freely diffusing between the thin-layer cell and the bulk solution, the probability of finding 0, 1, 2, …, N molecules between the electrodes at any moment in time is given by the Poisson distribution, (19.3.6). Measuring the currents separately at the top and bottom Pt electrodes allows the experimenter to examine the correlation between oxidation and reduction, which is confirmatory in identifying redox cycling events. Any such event requires that correlated equal and opposite currents appear at the two electrodes. Inspection of the i − t traces in Figure 19.6.4b,c clearly demonstrates that discrete, anticorrelated fluctuations in the anodic and cathodic i − t traces with amplitude of ∼20 fA are observed at the two Pt electrodes when Fc is present in the solution.

19.7 References 1 W. Cheng and R. G. Compton, Trends Anal. Chem., 58, 79 (2014). 2 S. V. Sokolov, S. Eloul, E. Kätelhön, C. Batchelor-McAuley, and R. G. Compton, Phys. Chem.

Chem. Phys., 19, 28 (2017). 3 S. J. Kwon, H. Zhou, F.-R. F. Fan, V. Vorobyev, B. Zhang, and A. J. Bard, Phys. Chem. Chem.

Phys., 13, 5394 (2011). 4 A. J. Bard, H. Zhou, and S. J. Kwon, Isr. J. Chem., 50, 267, 2010. 5 B. M. Quinn, P. G. van’t Hof, and S. G. Lemay, J. Am. Chem. Soc., 126, 8360 (2004). 6 D. Robinson, Y. Liu, M. A. Edwards, N. J. Vitti, S. M. Oja, B. Zhang, and H. S. White, J. Am.

Chem. Soc., 139, 16923 (2017).

879

880

19 Single-Particle Electrochemistry

7 S. Eloul, E. Kätelhön, C. Batchelor-McAuley, K. Tschulik, and R. G. Compton, J. Phys. 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

Chem. C, 119, 14400 (2015). A. Boika and A. J. Bard, Anal. Chem., 87, 4341 (2015). A. Boika, S. N. Thorgaard, and A. J. Bard, J. Phys. Chem. B, 117, 4371 (2013). J. E. Dick, C. Renault, and A. J. Bard, J. Am. Chem. Soc., 137, 8376 (2015). M. R. Newton, K. A. Morey, Y. Zhang, R. J. Snow, M. Diwekar, J. Shi, and H. S. White, Nano Lett., 4, 875 (2004). X. Xiao, F.-R. F. Fan, J. Zhou, and A. J. Bard, J. Am. Chem. Soc., 130, 16669 (2008). X. Xiao and A. J. Bard, J. Am. Chem. Soc., 129, 9610 (2007). P. A. Defnet, C. Han, and B. Zhang, Anal. Chem., 91, 4023 (2019). Z.-P. Xiang, H.-Q. Deng, P. Peljo, Z.-Y. Fu, S. L. Wang, D. Mandler, G.-Q. Sun, and Z.-X. Liang, Angew. Chem. Int. Ed., 57, 3464 (2018). S. E. F. Kleijn, B. Serrano-Bou, A. I. Yanson, and M. T. M. Koper, Langmuir, 29, 2054 (2013). J. H. Bae, R. F. Brocenschi, K. Kisslinger, H. L. Xin, and M. V. Mirkin, Anal. Chem., 89, 12618 (2017). S. J. Kwon and A. J. Bard, J. Am. Chem. Soc., 134, 7102 (2012). S. J. Kwon, F.-R. F. Fan, and A. J. Bard, J. Am. Chem. Soc. 132, 13165 (2010). H. Zhou, F.-R. F. Fan, and A. J. Bard, J. Phys. Chem. Lett., 1, 2671 (2010). S. E. F. Kleijn, S. C. S. Lai, T. S. Miller, A. I. Yanson, M. T. M. Koper, and P. R. Unwin, J. Am. Chem. Soc., 134, 18558 (2012). Y. Zhang, D. A. Robinson, K. McKelvey, H. Ren, H. S. White, and M. A. Edwards, J. Electrochem. Soc., 167, 166507 (2020). P. Li, Q. He, H.-X. Liu, Y. Liu, J.-J. Su, N. Tian, and D. Zhan, ChemElectroChem, 5, 3068 (2018). Z.-P. Xiang, A.-D. Tan, Z.-Y. Fu, J.-H. Piao, and Z.-X. Liang, J. Energy Chem., 49, 323, (2020). P. A. Bobbert, M. M. Wind, and M. Vlieger, J. Physica, 141A, 58 (1987). X. Li, J. Dunevall, and A.G. Ewing, Acc. Chem. Res., 49, 2347 (2016). Y.-G. Zhou, N. V. Rees, and R. G. Compton, Angew. Chem. Int. Ed., 50, 4219 (2011). C. A. Little, R. Xie, C. Batchelor-McAuley, E. Kätelhön, X. Li, N. P. Young, and R. G. Compton, Phys. Chem. Chem. Phys., 20, 13537 (2018). C. Batchelor-McAuley, J. Ellison, K. Tschulik, P. L. Hurst, R. Boldt, and R. G. Compton, Analyst, 140, 5048 (2015). J. Ustarroz, M. Kang, E. Bullions, and P.R. Unwin, Chem. Sci., 8, 1841 (2017). W. Ma, H. Ma, J.-F. Chen, Y.-Y. Peng, Z.-Y. Yang, H.-F. Wang, Y.-L. Ying, H. Tian, and Y.-T. Long, Chem. Sci., 8, 1854 (2017). S. M. Oja, D. A. Robinson, N. J. Vitti, M. A. Edwards, D. A. Robinson, H. S. White, and B. Zhang, J. Am. Chem. Soc., 139, 708 (2017). D. Robinson, M. A. Edwards, H. Ren, and H. S. White, ChemElectroChem, 5, 3059 (2018). S. Gutierrez-Portocarrero, K. Sauer, N. Karunathilake, P. Subedi, and M. A. Alpuche-Aviles, Anal. Chem., 92, 8704 (2020). B.-K. Kim, J. Kim, and A. J. Bard, J. Am. Chem. Soc., 137, 2343 (2015). H. Deng, J. E. Dick, S. Kummer, U. Kragl, S. H. Strauss, and A. J. Bard, Anal. Chem., 88, 7754 (2016). M. W. Glasscott, A. D. Pendergast, S. Goines, A. R. Bishop, A. T. Hoang, C. Renault, and J. E. Dick, Nat. Commun., 10, 2650 (2019). A. von Weber, E. T. Baxter, H. S. White, and S. L. Anderson, J. Phys. Chem. C, 119, 11160 (2015).

19.8 Problems

39 N. Cheng, S. Stambula, D. Wang, M. Norouzi Banis, J. Liu, A. Riese, B. Xiao, R. Li, T.-K.

Sham, L.-M. Liu, G. A. Botton, and X. Sun, Nat. Commun., 7, 13638 (2016). 40 S. Yang, J. Kim, Y. J. Tak, A. Soon, and H. Lee, Angew. Chem. Int. Ed., 55, 2058 (2016). 41 J. Deng, H. Li, J. Xiao, Y. Tu, D. Deng, H. Yang, H. Tian, J. Li, P. Ren, and X. Bao, Energy

Environ. Sci., 8, 1594 (2015). 42 B. Qiao, A. Wang, X. Yang, L. F. Allard, Z. Jiang, Y. Cui, J. Liu, J. Li, and T. Zhang, Nat.

Chem., 3, 634 (2011). 43 C. Zhu, S. Fu, Q. Shi, D. Du, and Y. Lin, Angew. Chem. Int. Ed., 56, 13944 (2017). 44 R. M. Ara´ n-Ais, Y. Yu, R. Hovden, J. Solla-Gullo´n, E. Herrero, J. M. Feliu, and H. D. Abruña,

J. Am. Chem. Soc., 137, 14992 (2015). J. Huang, J. Zhang, and M. H. Eikerling, J. Phys. Chem. C, 121, 4806 (2017). M. Gara, K. R. Ward, and R. G. Compton, Nanoscale, 5, 7304 (2013). M. Zhou, J. E. Dick, and A. J. Bard, J. Am. Chem. Soc., 139, 17677 (2017). M. Zhou, S. Bao, and A. J. Bard, J. Am. Chem. Soc., 141, 7327 (2019). Z. Jin and A. J. Bard, Proc. Natl. Acad. Sci. U.S.A., 117, 12651 (2020). F.-R. F. Fan and A. J. Bard, Science, 267, 871 (1995). F.-R. F. Fan, J. Kwak, and A. J. Bard, J. Am. Chem. Soc., 118, 9669 (1996). A. J. Bard and F.-R. F. Fan, Acc. Chem. Res., 29, 572 (1996). L. B. Anderson and C.N. Reilley, J. Electroanal. Chem., 10, 538 (1965). L. B. Anderson and C.N. Reilley, J. Electroanal. Chem., 10, 295 (1965). P. Sun and M.V. Mirkin, J. Am. Chem. Soc., 130, 8241 (2008). D. Mampallil, K. Mathwig, S. Kang, and S. G. Lemay, J. Phys. Chem. Lett., 5, 636 (2014). J. C. Byers, B. P. Nadappuram, D. Perry, K. McKelvey, A. W. Colburn, and P. R. Unwin, Anal. Chem., 87, 10450 (2015). 58 S. G. Lemay, S. Kang, K. Mathwig, and P. S. Singh, Acc. Chem. Res., 46, 369 (2013). 59 M. A. G. Zevenbergen, P. S. Singh, E. D. Goluch, B. L. Wolfrum, and S. G. Lemay, Nano Lett., 11, 2881 (2011). 45 46 47 48 49 50 51 52 53 54 55 56 57

19.8 Problems 19.1

Consider the electrochemical oxidation of an electroactive molecular film following a potential step (for instance the oxidation of Fc tethered to Au by an alkanethiol; Chapter 17). Fc → Fc+ + e The rate of Fc oxidation is characterized by a first-order rate constant, k (s−1 ), which is dependent on the electrode potential, and describes how fast, on average, Fc is converted to Fc+ . (a) Assuming that one starts with a large number of Fc (N 0 ∼ 1012 ), sketch the relative number of Fc molecules, N(t)/N 0 , as a function of time, t, following a potential step. (b) On the same plot, show how N(t)/N 0 might vary for the same experiment but starting with five Fc molecules, i.e., N 0 = 5. How does the plot vary if the experiment with N 0 = 5 is repeated?

19.2

Consider an experiment in which a 1-μm-radius Pt UME is used to oxidize ferrocene (Fc → Fc+ + e) in a cyclic SSV experiment (Section 5.1.4). If Fc is present in solution at 1 mM, and the scan rate is 10 mV/s, then ∼1011 molecules of Fc are oxidized during the measurement, assuming that the electrode potential is scanned in both directions over a

881

882

19 Single-Particle Electrochemistry

0.5 V range, given D = 10−5 cm2 /s. Calculate the number of Fc molecules oxidized during a very narrow potential range (e.g., a 1 mV interval) corresponding to the limiting current of the SSV response. Comment on whether this current is expected to be governed by deterministic or stochastic approaches. 19.3 Zhou, Bao, and Bard (48) reported the electrodeposition of individual Ptn clusters on a 2− − 120-nm-radius Bi UME via reduction of PtCl2− 6 in water (PtCl6 + 4e → Pt + 6Cl ). In 23 independent trials, the authors observed the following distribution of clusters containing n Pt atoms after electrodeposition for 20 s in a 300 fM PtCl2− 6 solution. n

0

1

2

3

4

5

Occurrences

5

5

7

3

2

1

(a) Assuming that PtCl2− 6 is reduced to a Pt atom at the diffusion-controlled rate with D = 1.2 × 10−5 cm2 /s, compute the expected rate of Pt atom deposition (s−1 ) at the Bi UME. (b) Compute the expected distribution of cluster sizes based on Poisson statistics. (c) Plot the experimental and theoretical distributions of Ptn clusters on the same graph. (d) Does a Poisson distribution adequately describe the experimental results? 19.4 Use (4.4.3) to demonstrate that (19.6.1) defines the average time of transit, t d , between two planar electrodes during redox cycling. 19.5 Equation 19.6.6 corresponds to the diffusion-limited, steady-state current for a dual-electrode thin-layer electrochemical cell where the potentials of the two electrodes are independently controlled relative to a reference electrode. Derive this equation assuming that the electrode surfaces (of area A) are parallel and opposite each other (Figure 19.6.1) and are separated by a distance, d. In addition, assume that species O and R are linked by a simple redox reaction, O + e ⇌ R, that only species O is present in solution at the start of the experiment, and that DO = DR . One electrode is held at a potential where O is reduced at the diffusion-limited rate, and the other electrode is held at a potential where R is oxidized at the diffusion-limited rate. (Hint: The solution involves integration of Fick’s second law, assuming steady-state conditions (equation 4.5.11), and application of the appropriate boundary conditions corresponding to this problem statement.) 19.6 Consider an experiment for detecting single redox molecules using the twin-electrode “nanogap” thin-layer cell of Figure 19.6.2c. Assume that the two parallel Au electrodes are separated by 70 nm and that the volume of the solution between the overlapping regions of the electrodes is defined by a cell width of 1.5 μm and a length of 50 μm. (a) Based on the Poisson probability distribution function, calculate the probability of finding 0, 1, 2, …, N molecules of ferrocene (Fc) between the two electrodes when the cell is in equilibrium with an external acetonitrile solution containing 120 pM Fc and 0.1 M TBAPF6 . What is the corresponding concentration of Fc between the electrodes when N = 1, 2, 3…? (b) Using the results in (a), show that, while the instantaneous concentration of Fc within the cell will fluctuate with time as molecules diffuse into and out of the detection

19.8 Problems

region between the two electrodes, the time-averaged concentration of Fc inside the cell is equal to the bulk value in the external solution (120 pM). (c) Using (19.6.6) (derived in Problem 19.5), calculate the time-averaged current for this cell as Fc molecules diffuse between the cell and external solution and the current when one Fc molecule is present in the cell. Assume DFc = DFc+ = 1.7 × 10−5 cm2 /s. (d) Equation 19.6.4 provides a separate expression for computing the current based on the transit time, t 2d , required for an O/R molecule to complete one redox cycle. Use this equation to find the current when one Fc molecule is in the cell, assuming the same cell dimensions as given above. Show that your result is the same as the value of the steady-state current calculated from (19.6.6). (e) The answer in (c) for the current corresponding to a single Fc molecule was computed from (19.6.6), which is derived from Fick’s second law, a continuum-based differential equation (Problem 19.5). This differential equation assumes that the concentration distribution of Fc is continuous between the two electrodes, so that the steady-state concentration profiles C Fc (x) and CFc+ (x) can be computed with high accuracy from Fick’s second law. However, in the real experiment, the single Fc molecule is undergoing Brownian motion and its position within the cell cannot be predicted at any instant. Given these different descriptions, explain why the continuum-based expression, (19.6.6), yields a correct prediction of the current corresponding to a single redox molecule.

883

885

20 Photoelectrochemistry and Electrogenerated Chemiluminescence In this chapter, we examine experiments in which photons participate in electrode processes. Photoelectrochemistry (Sections 20.3 and 20.4) is the domain, in which photons are, in effect, reactants, contributing their energy to drive an electrochemical process. In electrogenerated chemiluminescence (Section 20.5), photons are products of an electrode reaction, carrying away released energy. To prepare for these topics, we first consider the properties of solid materials (Section 20.1) and electrochemical behavior at semiconductors in the dark (Section 20.2).

20.1 Solid Materials Solids have always been important in electrochemistry. Sections 14.4–14.7 and Chapters 15 and 17 have already provided extensive coverage of the properties and modification of solid electrode surfaces. In this section, we focus on bulk properties of solids, especially on electronic conduction. The goal is simply to define and to distinguish categories of solids that are used in electrochemical systems. Solid materials are much more diverse and complex than we are able to convey here. For the interested reader, an elaborate literature is available (1–4). 20.1.1

The Band Model

The electronic properties of solids are usually described in terms of the band model, which involves the behavior of electrons in the combined fields of atomic nuclei and other electrons in a regular array (5–8). Consider the formation of a crystalline solid (e.g., Au, Si, or TiO2 ). As isolated atoms are assembled into a lattice containing N atoms, each type of valence orbital splits into N different energy levels. Because so many atoms are involved (5 × 1022 atoms/cm3 ), the orbitals of a common character (arising from the same atomic orbitals) are closely spaced in terms of energy (ΔE ∼ 10−22 eV) and exist in a “band” of states covering an essentially continuous distribution (Figure 20.1.1). • Bonding orbitals form the valence band (VB). • Antibonding orbitals form the conduction band (CB). In most crystalline solids, the bottom of the conduction band is separated from the top of the valence band by an energy range, Eg , called the band gap, sometimes several eV in height, as at spacing d′ in Figure 20.1.1b. There are almost no orbitals with energies in a band gap, which is sometimes also called the forbidden region. In these solids, the states of the VB are filled with electrons, just as for bonding orbitals in molecules, and the states of the CB are empty of electrons, as in the case of antibonding orbitals in molecules. In some solids the bands overlap, leaving no gap between them, as one sees at spacing d′′ in Figure 20.1.1b. In these materials, electrons fill the lower portion of the states, whether in the Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

Semiconductors and insulators Isolated atoms

Metals Electron energy, E

886

CB CB EF

Vacant

VB

EF

Eg VB

Bands and state Occupancy at dʺ (a)

d″

Occupied

d′ Interatomic spacing, d (b)

Figure 20.1.1 Formation of electronic energy bands in solids by assembly of isolated atoms into a lattice. At far right in (b), energy levels correspond to the highest occupied and lowest unoccupied states of isolated atoms. As interatomic spacing decreases, bonding and antibonding create bands of states, with the conduction band (CB) derived from the lowest unoccupied atomic orbitals, and the valence band (VB) derived from the highest occupied. Gray curves define the top and bottom of the CB, with many closely spaced states between. The VB is defined likewise by the black curves. At an intermediate spacing, d′ , the CB and VB are separated in energy by a band gap, Eg . At closer spacing, e.g., d′′ , the gap disappears and the CB and VB overlap. The band structure and electronic occupancy at spacing d′′ are depicted in (a). There is a continuum of electronic states from the bottom of the VB to the top of the CB, and the electrons corresponding to the highest occupied atomic orbitals fill them in order of increasing energy (shaded zone). The Fermi energy, EF , marks the division between filled and unfilled states. The existence and magnitude of the band gap is not just a matter of spacing, but also reflects the nature of the bonding and bond strength. Depending on the character of bonding, a material with an interatomic spacing at or to the left of d′ could be a metal, a semiconductor, or an insulator.

VB or the CB, until the electrons are accommodated. Since the states form a continuum, there are unfilled states immediately above the highest filled states (Figure 20.1.1a.). 20.1.2

Categories of Pure Crystalline Solids

The electrical and optical properties of any crystalline solid are strongly influenced by the band structure, which varies among solids depending on • the interatomic spacing in the lattice, as indicated in Figure 20.1.1b; • the nature and strength of chemical bonding leading to the formation of the lattice; and • the number of atoms in the lattice. All three factors determine the size of Eg ; the last determines the spacing of states on the energy scale. Based on electrical and optical properties, one can classify pure materials into several categories (Table 20.1.1). When the valence and conduction bands strongly overlap, a material displays metallic conduction of electricity (e.g., Cu and Ag). Filled and vacant electronic energy levels coexist in a narrow range near the top of the filled states (i.e., near the Fermi energy; Figure 20.1.1a and Section 20.1.4). An electron can move from one level to another with only a tiny energy of activation, and this feature provides electrical mobility for electrons in the solid, allowing them to respond to an electric field. In contrast, electrons in a completely full band, with no empty levels nearby in energy, have no means for redistributing themselves spatially in response to a field, so they cannot support electrical conduction.

20.1 Solid Materials

Table 20.1.1 Principal Categories of Pure Crystalline Materials Category

Eg

Examples

Metal

None

Au, Ag, Cu, Fe, and Pt

Semimetal

< kT

Graphite, Sn, Sb, and Bi

Insulator

>2 eV

SiO2 and Si3 N4

Intrinsic semiconductor

∼0.3 to 4 eV

Si, Ge, GaAs, and CdSe

Quantum dot

Variable(a)

Si, GaAs, and CdSe

(a) The band model does not strictly apply for a QD, but the energy gap between the highest bonding and lowest antibonding states is optically determinable. It is near the band gap of the intrinsic semiconductor when the particle has as many as 105 atoms, but shifts progressively toward the blue for a QD with fewer atoms.

If the band gap is small compared to kT (25.7 meV at 25 ∘ C), mobile electrons remain available, but there are relatively few states near the band edges, so the carrier density is small and electronic conduction is less effective than in a metal. A material of this kind is called a semimetal. If a pure solid has Eg ≫ kT (e.g., for Si, where Eg = 1.1 eV), its VB is almost filled and its CB is almost vacant. Conduction becomes possible because of thermal excitation of electrons from the VB into the CB (Figure 20.1.2). This process, called intrinsic semiconduction, produces electrons in the CB that have electrical mobility because they can transfer freely among vacant levels in the CB. It also leaves “holes” in the VB that have mobility because VB electrons can rearrange themselves to shift the spatial location and energy of the vacancy. The electrons and holes are charge carriers, existing in a dynamic thermal equilibrium. They are created by dissociation and eliminated by recombination;1 and their densities—ni for CB electrons and pi Conduction band

EC

EC EF

Eg Forbidden gap EV

+

Valence band

Si

Si Si

Si (a)

Si Si

Si

Si

Si

EV

Si

Si

Si

+

Conduction band electrons

Si Si

+

Si

Si Valence band holes

Si

Si (b)

Figure 20.1.2 Energy bands and two-dimensional representation of an intrinsic semiconductor lattice. (a) At absolute zero, the lattice is perfect and no holes or mobile electrons exist. (b) At a higher temperature, some lattice bonds are broken, yielding electrons in the conduction band and holes in the valence band. EF represents the Fermi energy. 1 Recombination is the annihilation of an electron and a hole. It is often promoted by surface states at interfaces [Section 20.2.1(c)].

887

888

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

for VB holes—adhere to an equilibrium constant defined by Eg . In an intrinsic semiconductor, the electron and hole densities are equal and are given approximately by (8) ( ) −Eg 19 cm−3 ni = pi ≈ 2.5 × 10 exp (near 25 ∘ C) (20.1.1) 2kT For silicon, ni = pi ≈ 1.4 × 1010 cm−3 . The mobile carriers move in the semiconductor in a manner analogous to the movement of ions in solution (Sections 2.3.3 and 2.4.2), but with mobilities, un and up , that are orders of magnitude larger than for solvated ions (Table 2.3.2). For example, in silicon, un = 1350 cm2 V−1 s−1 and up = 480 cm2 V−1 s−1 . If a pure material has Eg > 2 eV, so few carriers are produced by thermal excitation at room temperature that the solid will be an electrical insulator. Examples are SiO2 (Eg = 8.9 eV), C (diamond) (Eg = 5.5 eV), and TiO2 (Eg = 3.0 eV). For all categories just discussed, the band model is assumed to apply, and electronic states are described by continuous distributions over ranges of energy. The validity of the continuous model requires that the crystalline solid includes enough atoms that the energy spacing between adjacent states within a band is negligible. Quite small samples can fulfill this condition. For example, a cubic grain of Au 100 nm × 100 nm × 100 nm would contain 4 × 107 atoms, which are sufficient to manifest the characteristic band structure and bulk properties of crystalline Au. However, if the particle size becomes significantly smaller, the antibonding and bonding states become too few to be treated as a continuous distribution and the discreteness of states on the energy scale starts to become evident (9–13). Crystalline materials in this range are often called nanocrystals. This effect appears electrochemically in Figure 20.1.3a, which shows DPV for particles of only 25 atoms of Au (14). One can readily see the sequential addition or subtraction of electrons; thus, the continuity of states clearly no longer applies. This method allows measurement of the Eg

2.0 1.5 Eg /eV

Rest potential –1e

1.5

1.0

–2e 0e 0.5

1.0 0e

+1e

0.0

–1e 0.0

–0.5 –1.0 E/V vs. QRE (a)

0.5

–1.5

–2.0

0

20

40 60 80 100 120 140 160 Number of Au atoms (b)

Figure 20.1.3 (a) DPV for particles containing 25 atoms of Au in CH2 Cl2 + 0.1 M TBAPF6 . The particles were stabilized by the adsorbed surfactant phenylethanethiol (PhC2S) and are characterized by the formula Au25 (PhC2S)18 . The voltammetric responses are charging currents for the sequential placement or removal of single electrons on the particles, as indicated. The particles are uncharged at the rest potential and gain electrons toward negative potentials or lose them toward positive potentials. Eg in eV is numerically equal to the potential difference in V between the peaks corresponding to the addition and subtraction of the first electron at an uncharged particle. (b) Measured Eg vs. particle size. [Adapted from Sardar, Funston, Mulvaney, and Murray (14)/American Chemical Society. Original data in (a) from Lee et al. (15).]

20.1 Solid Materials

energy gap between the HOMO and LUMO (not really a “band” gap, but still symbolized by Eg ), and Figure 20.1.3b shows how the measured values of Eg depend on particle size. The gap is quite large for tiny particles, but shrinks with the number of Au atoms in the particle and approaches zero for more than 100 Au atoms (although many more atoms per particle would be required to fully develop the characteristics of metallic Au). Semiconductor particles with diameters in the range of 2–30 nm, hosting 102 − 106 atoms per particle, are often called quantum dots (QDs) (9–11). In some respects, they behave more like molecules than crystalline solids. For our purposes, they are of interest because they are both electroactive and photoluminescent. We will encounter them in Sections 20.3.4 and 20.5. In electrochemistry, one occasionally works with sizable single crystals as electrodes or other cell components; however, polycrystalline materials are much more common. In a typical solid, the average grain size is larger than the cubic grain of Au imagined above, with an edge length of 100 nm; thus, one can usually expect that the polycrystalline material will have the bulk electronic properties of the crystalline material. However, one must be alert to the effects of grain boundaries, which represent internal discontinuities in a polycrystalline sample. As we will see more fully below, they can substantially affect some aspects of behavior. Amorphous solids [e.g., conducting polymers (Section 20.2.3), amorphous Si, glassy carbon, or even glassy SiO2 ] are also of practical importance in electrochemistry. The concepts of this section apply qualitatively to them; however, models for the behavior of amorphous solids are typically more complex and less predictive than those for crystalline solids, which benefit from simplifications based on a geometrically regular array. 20.1.3

Doped Semiconductors

In the range of band gaps where pure solids show intrinsic semiconductivity or even insulating behavior (perhaps 0.5–6 eV), it is possible to increase the conductivity of a solid by deliberately substituting acceptor and donor atoms (called dopants) into the lattice (1–5). The resulting materials are called extrinsic semiconductors. A doped semiconductor can be orders of magnitude more conductive than the intrinsic material from which it is derived. There are two varieties, determined by whether the dopant is an electron donor or an electron acceptor. For example, an As atom (a Group V element) behaves as an electron donor when substituted into crystalline Si (a Group IV element), introducing an energy level at ED , just below the bottom of the CB (within ∼0.05 eV). At room temperature, most of the donor atoms are ionized, each yielding a CB electron and leaving behind an isolated positive site at the donor atom (Figure 20.1.4a). If the amount of dopant is about 1 ppm, the donor density, N D , is ∼5 × 1016 cm−3 and this will essentially be the CB electron density, n. The hole density, p, given by the electron–hole equilibrium relationship, p=

n2i ND

(20.1.2)

is much smaller. For this example of As-doped Si, p ≈ 4000 cm−3 at 25 ∘ C. In such a material, practically all of the electrical conductivity arises from the CB electrons, which are the majority carriers. The holes, as minority carriers, make only a small contribution. A material doped with donor atoms is called an n-type semiconductor.2 Alternatively, one might introduce an acceptor atom (e.g., Ga, a Group III element) into intrinsic Si, creating an energy level, EA , just above the top of the VB (Figure 20.1.4b). In this 2 Delocalized electrons and holes can also be introduced by atomic vacancies in the lattice. For example, n-type conductivity in the intrinsic insulator TiO2 can be produced by oxygen vacancies in the lattice.

889

890

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

EF

+

+

+

EC ED

EV

+

Si Si

EF

+

–

+

+

+

Si Si

+ As Si

–

Positive site at ionized donor atom

(a)

EA EV Valence band hole

Si

Si

Si

–

Conduction band electron

Si

Si

EC

Si Si

Si

Si

– Ga

Si Si

Negative site at ionized acceptor atom

(b)

Figure 20.1.4 Energy bands and two-dimensional representation of extrinsic semiconductor lattices. (a) n-type. (b) p-type.

case, electrons are thermally excited from the VB into these acceptor sites, leaving mobile holes in the VB and fixed, negatively charged acceptor sites. Thus, the acceptor density, N A , is essentially the same as the hole density, p, and the CB electron density, n, is given by n=

n2i NA

(20.1.3)

With N A = p = 5 × 1016 cm−3 in Si, for example, n ≈ 4000 cm−3 . The holes are the majority carriers, the electrons are the minority carriers, and the material is called a p-type semiconductor. 20.1.4

Fermi Energy

An important concept in the description of semiconductor behavior is the Fermi energy, EF , defined as the energy where the probability is exactly 0.5 that a level is occupied by an electron (i.e., the energy where it is equally probable that the level is occupied or vacant; Section 3.5.5). • In a metal, both occupied and vacant states are present at energies near EF because there is an unbroken continuum of states from filled to unfilled (Figure 20.1.1a). • For an insulator or intrinsic semiconductor at room temperature, EF lies within the forbidden gap, essentially midway between the CB and VB. Neither filled nor unfilled levels exist near EF (Figure 20.1.1b at spacing d′ ). • For a doped material, the location of EF depends on the identity and quantity of the dopant. For moderately doped n-type solids, EF lies slightly below the CB edge (Figure 20.1.4a). Similarly, for moderately doped p-type materials, EF lies just above the VB edge (Figure 20.1.4b).3 In Section 2.2.5, the Fermi energy of a phase 𝛼, E𝛼F , is shown to be the electrochemical potential of electrons in 𝛼, 𝜇𝛼e , expressed on a per-electron basis. This identification is valuable because it allows the electronic properties of a semiconductor electrode to be correlated with those of an adjacent electrolyte. 3 The edge of a band is the boundary with the band gap; thus, the VB edge is the state of highest energy in the VB and the CB edge is the state of lowest energy in CB.

20.1 Solid Materials

Vacuum, E=0

Vacuum, E=0 EA Φ

Φ

EC EF

EF

EV (a) Metal

(b) Semiconductor

Figure 20.1.5 Relationships between energy levels, work function, Φ, and electron affinity, EA, for (a) metal and (b) semiconductor.

The scale for EF is based on a defined zero energy for a free electron in vacuo, and values of EF for metals and semiconductors can be determined from measurements of work functions, Φ, or electron affinities, EA (Figure 20.1.5 and Section 2.2.5). Electrons in real materials are at a lower energy than in vacuum; therefore, values of EF are negative (e.g., about −5.1 eV for Au or −4.8 eV for intrinsic Si). 20.1.5

Highly Conducting Oxides

Conductive metal oxide ceramics have a long history of application in electrochemistry. Although many have been exploited, we will use just three categories to illustrate characteristics and possibilities. (a) RuO2 –TiO2

Dimensionally stable anodes (DSAs), usually involving an RuO2 − TiO2 coating on a metallic support, such as Ti (16), transformed the chloralkali industry by greatly improving the energy efficiency and economics of electrolysis. Such electrodes are called “dimensionally stable” because they do not lose mass through corrosion over time. They replaced the large carbon anodes historically employed in chloralkali cells. During operation, these electrodes did lose mass, causing the electrolyte gap between anode and cathode to widen gradually, progressively increasing power consumption by resistive heating of the solution. The corrosion also created a need for periodic replacement of the anodes, the cost of which could be avoided with DSAs. Moreover, the Ru-based surface proved catalytic for the evolution of Cl2 , offering an economic advantage of different kind. This concept has become successful well beyond the chloralkali industry because (a) the metallic conductivity of RuO2 − TiO2 makes these systems electrically efficient, (b) catalysis by Ru applies to many important electrode reactions, and (c) the oxides are robust in difficult environments. Dimensionally stable electrodes (cathodes, as well as anodes) have now been devised for many practical purposes. Often, they continue to be based on RuO2 − TiO2 ; however, they may also, or alternatively, involve IrO2 or PtO2 as a catalytic component. A generic name for this kind of system is mixed metal oxide (MMO) electrode. (b) Ti4 O7

When TiO2 , a crystalline insulator, is heated in the presence of a reductant (often H2 ), it can lose oxygen from the lattice and convert to any of several sub-stoichiometric oxides, Tin O2n − 1 , known as Magnéli phases. The crystal structures, band structures, electronic properties, and optical properties vary extensively across the Magnéli series (17). Most prominent among these materials is Ti4 O7 , which has been commercially developed (18) and utilized in electrochemical systems.

891

892

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

At room temperature and above, Ti4 O7 is semimetallic, with an electronic conductivity comparable to that of graphite. This character is ascribed to a crystal structure allowing delocalization of d-band electrons (17). Although TiO2 is bright white, Ti4 O7 is almost black and is sometimes called Ebonex , one of its trade names (18). This material is available as powders or as monolithic ceramic objects, including disks or rods, which may be compact or porous. In powder form, it can be coated onto substrates, such as Ti or Al. In electrochemical cells, Ti4 O7 can act as either a cathode or anode, and it can host an electrocatalyst, such as Pt. Because Ti4 O7 resists corrosion and is dimensionally stable in difficult environments, it has been explored for application in molten-salt cells (19, 20). It was successfully employed as an anode for evolution of O2 from CaCl2-CaO-SiO2 eutectic, supporting the deposition of Si at a cathode (20). This cell allowed the CO2 -free production of Si.

®

(c) Indium Tin Oxide and Fluorine-Doped Tin Oxide

Indium tin oxide (ITO) and fluorine-doped tin oxide (FTO) form transparent coatings on glass or plastic substrates. Coated materials are readily available commercially and are commonly used as transparent electrodes for spectroelectrochemistry (Section 21.3) or photoelectrochemistry (Section 20.3). Both materials are degeneratively doped n-type semiconductors. Operationally, this term means that the doping level is so high that the material acts electrochemically more like a metal electrode than a semiconductor electrode (Section 20.2.2). Electrodes made from ITO or FTO are effective for outer-sphere electrode reactions over a useful working range, and the films typically offer 80% transparency over the visible spectrum.

20.2 Semiconductor Electrodes Let us now consider the formation of an interface between a doped semiconductor and a solution containing an O/R couple (5, 8, 21). There are distinctive aspects to this kind of interface, and they lead to faradaic behavior that differs significantly from what we have seen previously for metallic or carbon electrodes. An important distinction is that semiconductors can utilize the energy from absorbed photons to drive an electrode process. We will address that phenomenon in Section 20.3; however, our immediate goal is to understand the basics of these interfaces without illumination. Many detailed reviews of this area have appeared (5, 8, 21–25). 20.2.1

Interface at a Semiconducting Electrode in the Dark

Suppose an n-type semiconductor is brought into contact with an O/R solution, as in Figure 20.2.1. When electrostatic equilibrium is attained by charge transfer between the phases [Section 2.2.5(d)], 𝜇 e (or, equivalently, the Fermi energy) becomes equalized across the interface. In Figure 20.2.1, where EF of the semiconductor initially lies above that in solution, electrons will flow from the semiconductor (which becomes positively charged) to species O in the solution phase (which becomes negatively charged).4 (a) The Space Charge Region

The excess charge in the semiconductor does not reside at the surface, as it would in a metal, but instead is distributed in a three-dimensional space charge region. This distribution is a 4 Although this description is frequently given of the semiconductor–solution junction, the behavior is often more complicated, especially with aqueous solutions. Interference to equilibration can arise from corrosion of the semiconductor, from the formation of a surface film (e.g., an oxide), or from inherently slow electron transfer across the interface. Sometimes, the behavior of the semiconductor electrode even approaches ideal polarizability (Section 1.6.1).

20.2 Semiconductor Electrodes

E/eV

E0/V vs. NHE

Vacuum 0

0 EC EF

EF(Au) –5.1

E/eV

Cr3+/Cr2+ –0.41

–4.0

H+/H2

–4.4

0.00 O 0 EO/R R Fe3+/Fe2+ 0.77

EV n-type semiconductor

EO/R –5.2

Solution (a)

Δϕ EF

O R

Eg n-type Solution semiconductor (b)

Figure 20.2.1 Formation of a junction between an n-type semiconductor and a solution containing a redox couple, O/R. (a) Before contact. Typical energy levels are shown vs. NHE (E 0 ) and vs. vacuum (E). (b) After contact and electrostatic equilibration. The space charge region is the zone in the semiconductor where the bands are curved. [Adapted from Bard (26)/with permission of Elsevier.]

consequence of the much lower carrier concentration in a semiconductor vs. a metal. It is analogous to the diffuse double layer in solution at a metal electrode (Section 14.3), but is usually much thicker.5 In a space charge region at equilibrium, the charge distribution creates an electric field that affects the local energies of all electrons. Thus, the band energies in this region differ from those in the bulk semiconductor, where there is no field. In the case of Figure 20.2.1, the energy levels at the surface of the semiconductor remain substantially unchanged as the space charge is established because the relationship between energy states on the two sides of the interface is unaffected by charging deeper in the semiconductor. However, the positive charge in the space charge region causes the band energies to become steadily lower with increasing distance from the interface and then to remain flat in the field-free bulk (Figure 20.2.1b). The result is called band bending. When the space charge is positive, as in this case, the bands are often said to be bent “upward” (relative to the bulk semiconductor). When an n-type semiconductor is placed in solution of O/R, as in the example of Figure 20.2.1, a positive space charge develops by depletion of the majority carriers (electrons) in the cathodic electrode reaction. The net charge exchange at the interface leaves behind an excess positive charge made up mostly of fixed, ionized donor sites. The minority carriers (holes, in this case) would also contribute to the positive space charge, but very modestly, because there are so few holes in the material. 5 A moderately doped semiconductor typically has a majority carrier concentration of 1015 − 1017 cm−3 , 5–7 orders of magnitude smaller than for a typical metal (1022 cm−3 ). The carrier density in the semiconductor is also 2–4 orders of magnitude smaller than for 0.1 M aqueous ionic solution (1019 cm−3 ). The Debye length, 𝜅 −1 , for the diffuse layer in 0.1 M electrolyte at a metal electrode is about 1 nm (Table 14.3.1). Given the lower carrier density in a doped semiconductor and the lower dielectric constant, one can expect the space charge depth to be larger by orders of magnitude. Typical figures are tens to thousands of nm for moderate doping (Problem 20.2).

893

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

There is a force on any carrier in the space charge region, consistent with the electric field: • Electrons tend to flow to lower energy down the CB edge. • Holes tend to flow to lower energy up the VB edge. In Figure 20.2.1b, the force on an electron is toward the bulk semiconductor, while that on a hole is toward the interface. These forces exist on any mobile carrier, but thermal randomization is also a factor, and the space charge distribution is an equilibrium outcome, just as in the case of a diffuse layer on the solution side of a double layer. (b) Polarization of Semiconductor Electrodes

We have just seen how the space charge can develop at a semiconductor–electrolyte interface from interfacial electron exchange. However, the existence of the space charge does not depend on having a redox couple in solution. One can alter the space charge simply by altering the potential of the semiconductor electrode with a potentiostat. For any electrode, a change in potential amounts to a change of Fermi energy. If the electrode is shifted by −0.5 V vs. a reference electrode, its Fermi energy is elevated by 0.5 eV. For a semiconductor electrode, this means that the whole band structure in the bulk semiconductor rides up or down the energy scale as the potential is made more negative or more positive. If the band structure at the interface remains invariant with potential, the bands must bend and a corresponding space charge is produced. At the potential of zero charge, Ez , no excess charge exists in the semiconductor. There is no electric field and no space charge region, and the bands are not bent (Figure 20.2.2a,d). This flat-band potential, Efb , is an important point of reference.

Electron energy

E = Efb

E > Efb EF pushed down vs. flat-band

E > Efb EF pushed up vs. flat-band e

EC EF

EC EF

e

EC EF

Depleted majority carriers

Accumulated majority carriers

EV

EV Semiconductor

Electrolyte

EV

Semiconductor

Semiconductor

Electrolyte

(b)

(a)

Electrolyte

(c)

n-type EC Electron energy

894

EC EF EV

EC EF EV Semiconductor

(d)

Electrolyte

EF EV

h Semiconductor

(e)

p-type

Accumulated majority carriers Electrolyte

h

Semiconductor

Depleted majority carriers

Electrolyte

(f)

Figure 20.2.2 Polarization of a semiconductor–electrolyte interface. (a–c): n-type electrode. (d–f ): p-type electrode. (a, d): At E fb . (b, e): At a positive potential vs. E fb . (c, f ): At a negative potential vs. E fb . Bold vertical arrows show energy shifts of the Fermi energy from the flat-band position. When a depletion layer forms (b, f ), band edges at the surface remain essentially fixed at the flat-band values. As an accumulation layer forms (c, e), the band edges at the surface become released to move with the Fermi energy.

20.2 Semiconductor Electrodes

There are four different space charge conditions that we need to recognize, depending on the type of semiconductor and the potential, E, vs. Efb . When the potential of the semiconductor electrode is on the positive side of Efb (Figure 20.2.2b,e), the bands are bent upward, reflecting a net positive space charge. If E is more negative than Efb (Figure 20.2.2c, f ), the bands are bent downward, reflecting a net negative space charge. This behavior is independent of the type of semiconductor. However, the character of the space charge and the behavior of the interface do depend on the type of semiconductor. Let us take n-type first: • When E > Efb (Figure 20.2.2b), the band bending drains majority carriers (electrons) toward the bulk. While there is a tendency to collect holes at the interface, very few exist in an n-type material; thus, the net positive space charge is made up mostly of immobile, ionized donor sites. The density of carriers is lower (often very much lower) than in the bulk, and the region is called a depletion layer. • When E < Efb (Figure 20.2.2c), the band bending collects the majority carriers and their density is higher at the interface than in the bulk. The region is known as an accumulation layer. For a p-type electrode, the same conditions exist, but in the opposite potential ranges: • When E > Efb (Figure 20.2.2e), the band bending collects the majority carriers (holes), so that their density at the interface exceeds that of the bulk. An accumulation layer exists. • When E < Efb (Figure 20.2.2f ), the band bending drains majority carriers (holes) toward the bulk. The net negative space charge is made up mostly of immobile, ionized acceptor sites. The density of carriers is lower than in the bulk, and a depletion layer exists. One can see in the figure that the extent of band bending and the depth of the space charge region are significantly smaller for an accumulation layer vs. a depletion layer. These are consequences of the small energy difference in the bulk semiconductor between the Fermi energy and the adjacent band edge (CB for n-type or VB for p-type). If the potential is changed so that the Fermi energy moves in the direction of that adjacent edge, it creates an accumulation layer. In the space charge region, the Fermi energy can cross the band edge at the surface and actually enter the band (Figure 20.2.2c,e). This condition, called degeneracy, is of great consequence because it creates a situation where there is a high density of partially occupied states near the Fermi energy, as in a metal. A high majority carrier density is established near the surface, and the distribution of states at the surface becomes released from the energies at Efb , so that they move with the Fermi energy (and the electrode potential). We will see in Sections 20.2.2 and 20.3 that the electrochemical behavior at a semiconductor electrode is dominated by the various conditions we have just identified. (c) Surface States

A complication in the model just presented arises from surface states (5, 22, 23), which are energy levels corresponding to orbitals localized near a surface. It is easy to see, for example, that silicon atoms in a surface plane cannot be surrounded with the tetrahedral symmetry found in the bulk solid. Thus, the electronic properties of these atoms must differ, and they yield localized orbitals. When surface states have energies in the band gap, they can have big effects on the electronic properties of any junction made with the surface, because they tend to trap carriers and can mediate electron exchange with redox species in solution. For a single crystal semiconductor electrode, the only surface states of concern are at the electrode/solution interface. However, a polycrystalline electrode can feature surface states at every grain boundary, and they can complicate charge transport in the interior of the electrode.

895

896

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

(d) The Mott–Schottky Relationship

Using an approach similar to the Gouy–Chapman treatment of differential capacitance at a metal electrode (Section 14.3), one can derive relationships for the excess charge in the semiconductor, the distribution of electric potential in the space charge region, and the differential capacitance (5, 8). For an intrinsic semiconductor, the relationship between the space charge density, 𝜎 SC , and the potential of the surface with respect to the bulk of the semiconductor, Δ𝜙, is given by (14.3.20), with the following replacements: 𝜎 M by 𝜎 SC , 𝜙0 by Δ𝜙, and n0 by ni , with z = 1, and 𝜀 now referring to the dielectric constant of the semiconductor. Similarly, the space charge capacitance, C SC , is given by (14.3.22) with analogous replacements. For a doped semiconductor, the situation is more complicated because the countercharge for the majority carriers is made up principally of the ionized dopant sites, which are not mobile. A case of interest for the semiconductor/liquid interface is where a depletion layer forms (Figure 20.2.2b, f ). For an n-type electrode, the space charge capacitance can be shown to be (5, 8) ( )1∕2 ( )−1∕2 e2 𝜀𝜀0 ND eΔ𝜙 − −1 (20.2.1) CSC = 2kT kT Rearrangement yields the Mott–Schottky equation (27, 28) (first derived for the metal/semiconductor junction): ( )( ) 1 2 kT = −Δ𝜙 − (20.2.2) e𝜀𝜀0 ND e C2 SC

Recognizing that –Δ𝜙 = E – Efb and evaluating constants for T = 298 K, N D in cm−3 , C SC in μF cm−2 , and potentials in V, one obtains 1 1.41 × 1020 = (E − Efb − 0.0257) 2 𝜀ND CSC

(20.2.3)

2 vs. E should be linear. The potential where the line Thus, a Mott–Schottky plot of 1∕CSC intersects the potential axis yields the value of Efb , and the slope can be used to obtain the doping level N D . Examples (29) are shown in Figure 20.2.3. In Problem 20.4, the reader is invited to interpret the data in that figure. An equation analogous to (20.2.3) applies for a depletion layer in a p-type electrode and allows for determination of the acceptor density, N A , and Efb . While Mott–Schottky plots have been useful for characterizing semiconductor/solution interfaces, they must be used with caution because perturbing effects, such as those attributable to surface states can cause deviations from the predicted behavior (30). One should verify that the parameters obtained from such plots are independent of the frequency employed in the capacitance measurements.

20.2.2

Current–Potential Curves at Semiconductor Electrodes

Electron transfer processes at a semiconductor/electrolyte interface are strongly affected by the density of available carriers (electrons and holes) at the interface. The observed i − E behavior differs from that at metals and carbon, where there is always a large density of carriers in the conductor. The Marcus–Gerischer kinetic model (Section 3.5.5) is widely used to treat electrochemical dynamics at semiconductor electrodes, with rate constants described by (3.5.62) and (3.5.63). In this section, we use a descriptive approach rooted in that treatment.

5

100

4

80

3

60

2

40

1

20

0 –0.5

0.0

0.5 E/V vs. NHE

1.0

–2 2 4 104CSC /F m

–2 2 4 104CSC /F m

20.2 Semiconductor Electrodes

0 1.5

Figure 20.2.3 Mott–Schottky plots for n- and p-type InP, (111) face, in contact with 1 M KCl + 0.01 M HCl. Right axis for n-type: (O) 200, (◽) 2500, (Δ) 20,000 Hz. Left axis for p-type: (•) 200, (◾) 2500, (▴) 20,000 Hz. [From Van Wezemael, Laflère, Cardon, and Gomes (29), with permission.]

An electroactive solute has energy levels flanking E0 for the couple, with the empty O states above E0 and the filled R states below (Section 3.5.5). If E0 lies in the band gap of the semiconductor at the surface (Figure 20.2.1b), the majority carrier usually dominates the electrochemical behavior. Thus, a moderately doped n-type material can carry out reductions, but not oxidations, because there are electrons available in the conduction band to transfer to an oxidized solution species, but few holes to accept an electron from a reduced species. Figure 20.2.4a illustrates this behavior for a solution of the stable cation radical of thianthrene ′ + (TH; Figure 1) (31). At Pt, the TH ∙∕TH couple is reversible with E0 = 1.23 V vs. SCE. At + n-TiO2 one can see the reduction of TH ∙ to TH, but the reoxidation simply does not occur in the reverse scan. Since Efb in this system is near −0.7 V vs. SCE, a depletion layer exists in the n-TiO2 over the entire potential range in Figure 20.2.4a. A considerable overpotential exists + for the reduction of TH ∙ at n-TiO2 . In the potential range between 1.2 V and about 0.7 V, there are just too few electrons at the surface to allow the reduction to proceed, but the density of majority carriers rises at the interface as the potential becomes more negative and eventually enables the reduction. For an n-type electrode at E > Efb , the current for a reduction of species O is given by (5, 21, 23) i = nFAkf′ ns CO (x = 0)

(20.2.4)

where ns (cm−3 ) is the concentration of electrons at the interface and kf′ (cm4 /s) is the heterogeneous rate constant. The units of kf′ differ from those used with a metal electrode, where the carrier density in the metal is high and is included in k f [cm/s; (3.5.62)]. The behavior of a p-type material is analogous, but opposite to that of an n-type electrode. The excess holes allow oxidations, but not reductions, of electroactive solutes with E0 in the band gap. When E < Efb , a depletion layer exists and the current for an oxidation of species R

897

898

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

1

TiO2

Pt

20 μA

2

i 0.5 μA

1.6

1.2

0.8 0.4 E/V vs. SCE (a)

0.0

–0.4

0.5 μA Pt 20 μA TiO2

i

–0.7

TiO2

–1.1

Pt

–1.5 –1.9 E/V vs. SCE (b)

–2.3

Figure 20.2.4 CV at an unilluminated n-TiO2 single crystal (A = 0.5 cm2 ) or a Pt disk (A = 7.5 × 10−3 cm2 ) in + MeCN + TBAP. All scans begin near 1.6 V vs. SCE and first move negatively. v = 200 mV/s. (a) For 1.2 mM TH ∙ . + Curve 1 is the initial scan at n-TiO2 . The second scan (curve 2) is lower because TH ∙ reduced near the electrode in the first scan is not restored in the reversal. (b) For 0.4 mM Ru(bpy)3+ . By the time the illustrated potential 3 range is reached, the first 1e reduction has occurred and the species near the electrode is largely Ru(bpy)2+ . 3 The three waves on the forward scan are for further 1e reductions. [Adapted from Frank and Bard (31)/ American Chemical Society.]

at a p-type semiconductor is given by i = nFAkb′ ps CR (x = 0)

(20.2.5)

where kb′ is the heterogeneous rate constant for oxidation and ps (cm−3 ) is the concentration of holes at the surface. In these equations, both the rate constant (kf′ or kb′ ) and the majority carrier concentration at the interface (ns or ps ) can be affected by the applied potential. The rate constants kf′ or kb′ follow

20.2 Semiconductor Electrodes ′

expressions like (3.3.7a,b) with E − E0 replaced by the potential drop across the Helmholtz layer at the semiconductor–solution interface. The surface concentrations of the carriers are given by ns = ND e−f (E−Efb )

ps = NA ef (E−Efb )

(20.2.6a,b)

where E − Efb represents the amount of band bending. For an n-type semiconductor at E > Efb , ns < N D , while for a p-type semiconductor at E < Efb , ps < N A . When the potential is altered under these conditions of depletion, most of any change will drop over the semiconductor space charge region, rather than across the Helmholtz layer. This effect can be understood in another way by comparing the capacitance of the space charge region, C SC , given in (20.2.1), with the capacitance of the Helmholtz and diffuse layers, C GCS (14.3.30), and noting that generally C SC ≪ C GCS . In tracing a current–potential curve with a semiconductor electrode, it is the variation of ns or ps , rather than kf′ or kb′ , that dominates the observed behavior in the depletion region. When the potential of an n-type material is made more negative than Efb , it establishes an accumulation layer and then becomes degenerate. It behaves much like a metal electrode toward redox species with E0 values lying in the CB. Changes in potential no longer affect ns significantly but mainly appear across the Helmholtz layer and primarily affect kf′ or kb′ . Similarly, p-type materials become degenerate at potentials positive of Efb , where they begin showing metallic electrode behavior.6 Figure 20.2.4b illustrates behavior of this kind in the n-TiO2 system previously discussed. Over the whole potential range in that frame, the n-TiO2 electrode has E < Efb (31); hence, an accumulation layer exists. The electrode provides reversible voltammograms for the three sequential 1e reductions of Ru(bpy)2+ , just as at a Pt electrode. 3 The shapes of i–E curves at semiconductors are determined by an interplay of kinetics and mass transfer, just as a metal electrode; however, complications may arise from • processes in parallel with the desired electron-transfer reaction, such as corrosion of the semiconductor material, • effects of the resistance of the electrode material, and • charge transfer reactions that occur via surface states.7 Careful studies of electrode kinetics have been carried out at semiconductor electrodes (32, 33). The details exceed our scope, but some results are cited and discussed in Section 3.5.4(f ). 20.2.3

Conducting Polymer Electrodes

Electronically conducting polymers are of broad interest for application in electrochemical systems (34–37), particularly for modification of electrode surfaces, for current collection in sensors (Section 17.8.3), and for energy-related devices such as supercapacitors and batteries. They are also of interest in photoelectrochemical systems (Section 20.3.3). These materials are usually linear π-conjugated polymers (Figure 20.2.5), in which the band structure relating to conductivity arises from the large number of p-orbitals making 6 Some materials are doped at such high levels that the Fermi energy in the bulk semiconductor lies below the VB edge (p-doping) or above the CB edge (n-doping). This condition is called degenerative doping. A common example is an ITO coating on glass [Section 20.1.5(c)]. A degeneratively doped electrode has such a large carrier concentration that it forms no appreciable space charge and behaves electrochemically essentially like a metal. + 7 The case is made in reference (31) that the reduction of TH ∙ in Figure 20.2.4a is facilitated by surface states.

899

900

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

CH2(CH2)4CH3

S

n P3HT

N H PPy

O

O

S

n

PEDOT

n

S

Figure 20.2.5 Common conducting polymers: P3HT, poly(3-n-hexylthiophene); PEDOT, poly(3,4-ethylenedioxythiophene); PPy, poly(pyrrole); PT, poly(thiophene). All of these materials are doped by partial oxidation, which may be accomplished electrochemically or chemically.

n

PT

up a conjugated chain. They are semiconductors with Eg typically in the range of 1.5–3 eV. The solid-state physics and electronic properties have been reviewed in detail (38). Conductivity is normally established by partial oxidation or reduction of the polymer (35–39). This process is called “doping,” just as for a crystalline semiconductor; however, it is not accomplished by lattice substitution, but by a redox action that requires charge neutralization with a counterion introduced into the material during the doping process. Most electronically conducting polymers in practical use, including all shown in Figure 20.2.5, are “p-doped” by oxidation and are often called “hole transport materials” (HTMs). Materials doped by reduction [n-doping, producing electron transport materials (ETMs)] are often inconvenient to work with because they are not chemically stable in air in a conducting state. Many conducting polymers are synthesized by oxidative polymerization of the corresponding monomer, sometimes in homogeneous solution, but often directly on an electrode by repeated cycling to positive potentials (35, 37). Many are rather insoluble and difficult to process unless they are blended with another polymer, such as poly(styrenesulfonate) (PSS). By comparison with moderately doped crystalline semiconductors, the doping levels in conducting polymers are very high, usually in the range of several percent. Accordingly, these materials are essentially degeneratively doped (Section 20.2.2). In electrochemical systems where the conducting polymer is in contact with an electrolyte, the polymer is typically much more solvated in the conducting state. In some respects, one can regard each polymer chain as an individual electrode operating in parallel with the others; thus the “electrode phase” becomes three-dimensional. Because of both the degenerative doping and the lack of a sharp electrochemical interface, one cannot think about electrochemistry at a conducting polymer using the semiconductor model developed in Sections 20.2.1 and 20.2.2. When conducting polymers are used in solid-state systems, the band model can be applied, but high doping prevents the formation of any appreciable space charge region (35, 37, 38). As at many working electrodes, one can think of electrochemistry at a conducting polymer electrode as being either “of the electrode” or “at the electrode:” • In the former case, one observes only an electrochemical response of the electrode material itself. As the potential is altered, currents from faradaic changes in doping level and nonfaradaic double-layer charging are superimposed. At a conducting polymer, these phenomena are difficult to distinguish and may not be meaningfully distinct (40). In CV, the overall result is a voltammetric response proportional to the scan rate, v (Figure 20.2.6). The current is small in the potential region where the polymer is undoped (more negative than about −0.2 V for PXDOT in Figure 20.2.6). As the potential moves into the doped regime, the doping level increases as the polymer is oxidized, giving rise to a greatly increased voltammetric current, reflecting both faradaic and capacitive components. Reversal of the scan changes the sign of the current and eventually leads to de-doping.

20.3 Photoelectrochemistry at Semiconductors

Figure 20.2.6 Inset: CV for v = 50 − 500 mV/s at a poly(3,4-ortho-xylenedioxythiophene) (PXDOT) electrode in MeCN + 0.1 M TBAP. More positive potentials are to the right, and anodic currents are up. Scans begin at −0.5 V and first move positively. Curves for progressively greater v are in sequence from inside to outside. Points show ipa vs. v. [Adapted from Ibanez et al. (35)/American Chemical Society.]

100 90 60

80

30 i/μA 0 –30

60

–60

i/μA

–0.5

0.0 0.5 1.0 E/V vs. Ag/AgCl

40 O

20

O

S n PXDOT

0

20

40

60

80

100

v/mV s

–1

• If the system contains an electroactive species other than the electrode material itself, that species may undergo a faradaic reaction at the conducting polymer when it is in a doped state. It is often possible to observe the electrode reaction using a conventional method such as CV. Because background currents are relatively high at a conducting polymer electrode, methods that tend to reject background signals, such as DPV or SWV, are often favored, especially for analytical applications.

20.3 Photoelectrochemistry at Semiconductors By comparison with metal or semimetal electrodes, semiconductor electrodes are distinctive in their ability to support photoelectrochemistry. Under the right conditions, one can irradiate an electrode with light absorbed by the semiconductor and it will respond by producing a photocurrent. The variations of this current with wavelength, electrode potential, and solution composition provide information about the nature, energetics, and kinetics of the photoprocess. Photoelectrochemical studies are frequently carried out to explore the fundamentals of electrode–solution interfaces (5, 32, 33, 41, 42); however, they are rather more broadly investigated for potential practical applications because photoelectrochemistry entails the conversion of light energy to electrical and chemical energy (43–48). 20.3.1

Photoeffects at Semiconductor Electrodes

Let us return to the n-type semiconductor in contact with a solution containing the couple O/R (Section 20.2.1; Figure 20.2.1). The semiconductor develops a depletion layer typically 50–1000 nm thick, depending on the doping level and Δ𝜙 [Section 20.2.1(a)]. The direction of the electric field is such that any excess holes created in the space charge region would move toward the surface and any excess electrons would move toward the bulk semiconductor. When the interface is irradiated with light of energy greater than the band gap, Eg , an electron–hole pair is created by each photon absorbed (Figure 20.3.1). Some pairs, especially any formed in the bulk semiconductor, recombine with the evolution of heat or the emission of light. However, those created in the space charge region can become separated by the space charge field and avoid immediate recombination. The separated holes migrate to the surface, where they become

901

902

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

–

– O

EF hν > Eg

O hν > Eg

R e +

n-type semiconductor (a)

Solution

e

R

EF + p-type semiconductor (b)

Solution

Figure 20.3.1 Separation of electron–hole pairs in a depletion layer. (a) At an n-type electrode, driving a photo-oxidation of species R in solution. (b) At a p-type electrode, driving a photo-reduction of species O in solution. [Adapted from Bard (26)/with permission of Elsevier.]

available at an effective potential corresponding to the VB edge at the surface. There they can be used for oxidation of R to O. The separated electrons migrate into the bulk semiconductor, where they can give rise to current flow in the external circuit. Thus, we see that irradiation of an n-type semiconductor electrode can promote a photooxidation (or cause a photoanodic current). However, this is only possible when a depletion layer exists (E > Efb for n-type). If, instead, there is an accumulation layer, as in Figure 20.2.2c, the space charge field may still separate the photoproduced electron–hole pair; however, the hole will migrate toward the bulk semiconductor, where it will inevitably encounter an electron and recombine. In that case, the absorption event produces no net change in the available carriers, and the light energy becomes thermalized. The absorbed photon cannot contribute to a photocurrent. In an analogous manner, a p-type semiconductor can produce a photocurrent when it has a depletion layer at the interface (E < Efb ; Figure 20.2.2f ). In this case, the space charge field separates an electron–hole pair by sending the electron to the interface and the hole toward the bulk. Thus, a p-type semiconductor can support a photoreduction, providing a photocathodic current. We can summarize both cases by saying that photogenerated minority carriers are collected at the interface, where they produce the electrochemistry. Photogenerated majority carriers are sent into the bulk semiconductor, ultimately supporting current flow in the external circuit. Photoelectrochemical effects are illustrated voltammetrically in Figure 20.3.2. We take the n-type case first (Figure 20.3.2a): • In the dark (curve 1), essentially no current flows when the potential of the semiconductor electrode is made more positive than Efb because there are few holes in the semiconductor to accept electrons from the reduced form of a redox couple located at potentials within the gap.8 • Under irradiation (curve 2), a photoanodic current, iph , flows, while the potential of the electrode remains more positive than Efb , so that electron-hole pair separation can occur. Thus, the onset of the photocurrent is near Efb (unless surface recombination processes move the onset potential toward more positive values). The photo-oxidation of R to O occurs at less positive applied potentials than those required to carry out this process at an inert metal electrode (curve 3). This is possible because the light energy helps to drive the oxidation process. Such processes are photoassisted electrode reactions. 8 At very positive potentials, a “dark” anodic current can flow from breakdown phenomena.

20.3 Photoelectrochemistry at Semiconductors

Figure 20.3.2 Current–potential curves: (a) n-type semiconductor in a solution of species R in the dark (curve 1) and under irradiation (curve 2). (b) p-type semiconductor in a solution of species O in the dark (curve 1) and under irradiation (curve 2). For both (a) and (b), curve 3 is the i − E curve at a platinum electrode in a solution of both O and R.

O→R Eeq

i

(a) n-type

Pt Efb(n) E iph

1 3

2

R→O O→R 2 Efb(p)

Eeq

iph

i

(b) p-type

Pt 1 E

3 R→O

The behavior of a p-type semiconductor with a couple having a redox potential in the gap region (Figure 20.3.2b) is analogous to that of the n-type material, but opposite in character: • In the dark (curve 1), there is essentially no current when E < Efb because there are few electrons in the semiconductor to transfer to the oxidized form of a couple with E0 in the gap. • Under irradiation (curve 2), a photocathodic current flows, while the potential is negative vs. Efb . The photoreduction of O to R is at less negative applied potentials than are needed an inert metal electrode (curve 3); thus, the reduction is photoassisted. The photocurrent, iph , depends upon the light absorption rate and the availability of electroreactants at the semiconductor electrolyte interface. In technological systems, one generally seeks to operate under conditions where the current is limited by the available light, rather than by mass-transfer or kinetic effects. However, a real photoelectrochemical system may exhibit performance limitations based on nonideal electrode kinetics. It may also exhibit a current efficiency below 100% because a second photoprocess, such as oxidation of the semiconductor, operates in parallel with the intended reaction. 20.3.2

Photoelectrochemical Systems

A wide variety of photoelectrochemical cells based on semiconductor electrodes has been investigated because of their possible use in the conversion of radiant energy to electrical or chemical energy. Three categories can be identified (26). (a) Photovoltaic Cells

If the cell is designed so that the reaction at the counterelectrode simply reverses the photoassisted process at the semiconductor, then the cell operates to convert light to electricity, ideally without change in the solution composition or the electrode materials. This is a photovoltaic cell (Figure 20.3.3). The operating characteristics of such a cell can be deduced from i–E curves

903

904

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

e

e e

e O R

hν > Eg

O R

hν > Eg

e

e

n-type semiconductor

Solution Metal electrode (a)

p-type semiconductor

Solution Metal electrode (b)

Figure 20.3.3 Photovoltaic cells based on (a) an n-type or (b) a p-type semiconductor electrode. The light irradiates the semiconductor/solution interface. Only photons absorbed in the space charge layer of the semiconductor can drive the cell. e

e

– O′ R′ hν > Eg

–

e hν > Eg

O R e

e O′ R′

e O R

+ n-type Solutions Metal semiconductor electrode (a)

+ p-type Solutions Metal semiconductor electrode (b)

Figure 20.3.4 Photoelectrosynthetic cells based on (a) an n-type and (b) a p-type semiconductor electrode. Initially present in the cell are R and O′ . As in Figure 20.3.3, the light irradiates the semiconductor/solution interface. The vertical double-dashed line is a separator, which might be required to keep the products, O and R′ , apart.

such as those in Figure 20.3.2. Examples include (49–54): n-TiO2 ∕NaOH, O2 ∕Pt

(20.3.1)

2− n-CdSe∕Se− 2 , Se2 ∕Pt

(20.3.2)

3+

2+

p-MoS2 ∕Fe , Fe ∕Pt

(20.3.3)

(b) Photoelectrosynthetic Cells

One can sometimes use light absorbed at a semiconductor electrode to drive an overall cell reaction in the nonspontaneous direction. In this case, the radiant energy is stored as chemical energy and the system is a photoelectrosynthetic cell (Figure 20.3.4). For either cell in Figure 20.3.4, the net reaction is h𝜈

R + O′ −−−→ O + R′

(ΔG > 0 in the dark)

(20.3.4)

With an n-type semiconductor, the necessary condition for driving the reaction is that the 0 energy corresponding to EO∕R lies above the valence band edge, while that corresponding to

0 0 EO ′ ∕R′ lies below the conduction band edge (such that EO′ ∕R′ > Efb ). If this condition does not

20.3 Photoelectrochemistry at Semiconductors

exist, it may still be possible to drive the reaction in the desired direction by applying an external bias to the cell. An analogous condition applies for an electrosynthetic cell based on a p-type semiconductor. The reader can easily work it out from Figure 20.3.4b. Examples of photoelectrosynthetic cells include (55–58): n-SrTiO3 ∕H2 O∕Pt

(driving water splitting )

(20.3.5)

p-GaP∕CO2 (pH = 6.8)∕C

(driving reduction of CO2 )

(20.3.6)

(c) Photocatalytic Cells

Sometimes one desires to use the energy of an absorbed photon to drive a reaction in the spontaneous direction (ΔG < 0) because the reaction is slow in the dark. In this case, the light energy is used to overcome the activation energy of the process. Such a system, called a photocatalytic cell (Figure 20.3.5), is similar to a photoelectrosynthetic cell, except that the relative locations of the potentials of the O/R and O′ /R′ couples are reversed. For either cell in Figure 20.3.5, the net reaction is h𝜈

R + O′ −−−→ O + R′

(ΔG < 0 in the dark)

(20.3.7)

n-TiO2 ∕CH3 COOH∕Pt

(CH3 COOH → C2 H6 + CO2 + H2 )

(20.3.8)

p-GaP∕DME, AlCl3 , N2 ∕Al

(reduction of N2 by Al)

(20.3.9)

Examples include (59, 60):

While some applications are interesting, the efficiencies of many of the processes carried out in photoelectrosynthetic and photocatalytic cells are often rather low. 20.3.3

Dye Sensitization

In the systems that we have so far discussed, the light employed must have a photon energy greater than that of the band gap Eg because photons of lower energy are not absorbed by the semiconductor. Indeed, a plot of photocurrent vs. the wavelength of irradiating light can be employed to determine Eg . For n-TiO2 , for example, only light of photon energy above 3.0 eV is useful. Since more than 95% of the total energy in the solar spectrum at the Earth’s surface lies below this energy, sunlight is not utilized very effectively by a TiO2 electrode. e

e –

– O R hν > Eg

e

e O′ R′

e + n-type Solutions Metal semiconductor electrode (a)

O R

hν > Eg

e

O′ R′ + p-type Solutions Metal semiconductor electrode (b)

Figure 20.3.5 Photocatalytic cells based on (a) an n-type and (b) a p-type semiconductor electrode. Initially present in the cell are R and O′ . As in Figure 20.3.3, the light irradiates the semiconductor/solution interface. The vertical double dashed line is a separator, which might be desired to maintain purity on the two sides.

905

906

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

2

Figure 20.3.6 Dye sensitization of a photoprocess at an n-type semiconductor electrode, for example, n-ZnO/rose bengal/I– /Pt (65).

– hν

Conduction band e

O

1

3

R

+

Valence band

Semiconductor

Dye (D)

Solution

Dye sensitization of a semiconductor (61–64) allows utilization of longer wavelength light and also enables interesting experiments in their own right. The principles are illustrated in Figure 20.3.6 for an n-type electrode. A thin layer of dye, D, is coated on the semiconductor, and when D is excited by photon absorption (step 1), it injects an electron into the conduction band (step 2), becoming oxidized to D+ . In the absence of a suitable couple in solution, this photoprocess would cease when the D becomes fully consumed; however, if species R in solution is capable of reducing D+ (step 3), then D is regenerated. In this manner, dye sensitization allows longer-wavelength light to be employed in photoelectrochemistry, although the hole generated by the light is at a less positive potential than one produced in the valence band of the semiconductor without dye sensitization. This concept is applicable to any of the systems discussed in the preceding section. With a p-type semiconductor, one uses a dye whose energy levels bracket the valence band edge, and hole injection into the semiconductor follows light absorption by the dye. The depiction of the energy-level diagram for this case and a description of the processes involved are left as an exercise for the reader (Problem 20.3). Dye sensitization has been used successfully to create photoelectrochemical cells based on conducting polymers (35). Photoeffects are also observed at metal electrodes: • The direct irradiation of a metal electrode can cause the photoemission of an electron into the solvent. If this electron is scavenged by some reactant in solution, a net cathodic photocurrent results (66, 67) (Section 20.4.1). • Excitation of dyes adsorbed on metals can also lead to photocurrents, but they are usually much smaller than the photocurrents obtainable at semiconductor electrodes under comparable conditions (62). This low efficiency of net conversion of photons to external photocurrent is attributed to the ability of a metal to act as a quencher of excited states at or very near the surface by either electron or energy transfer (68, 69). 20.3.4

Surface Photocatalytic Processes at Semiconductor Particles

The principles governing electrochemistry at semiconductor electrodes can also be applied to redox processes in particle-based systems. In this case, one considers the rates of the oxidation

20.3 Photoelectrochemistry at Semiconductors

and reduction half-reactions that occur on a particle, usually in terms of a current, as a function of particle potential. As we saw in Section 3.6, one can use i − E curves to estimate the nature and rates of mixed heterogeneous reactions on surfaces. This approach applies not only to semiconductor particles, but also to metal particles that behave as catalysts and to surfaces undergoing corrosion. Let us consider first a heterogeneous catalytic reaction that occurs on a metal particle. Many thermodynamically favorable reactions are slow in homogeneous solution because the mechanisms are complex. A good example is +

MV ∙ + H+ → MV2+ + 1/2H2

(20.3.10) +

where MV2+ is methyl viologen (Figure 1) and MV ∙ is its stable cation radical. The evolution of H2 requires two electrons, while the reactants are 1e reductants. Moreover, the 1e interme∙ , requires a very negative potential (Section 15.2). diate from proton reduction, Haq In the presence of a Pt particle, the overall process is facilitated, with the particle acting as an electrode (and electron reservoir) at which both anodic and cathodic half-reactions take place (Figure 20.3.7a). The reaction rate can be obtained by considering the current–potential curves for both half-reactions (Figure 20.3.7b) (70, 71). A cathodic current, ic , represents the rate of proton reduction at the particle surface, while an anodic current, ia , represents the oxidation + of the reducing agent, such as MV ∙. At steady state, the rates of these reactions must be equal, so the particle would establish a potential [sometimes called a mixed potential (Section 3.6)], where ic = − ia (Figure 20.3.7b).9 At irradiated semiconductor particles, the rate of the photoreaction is balanced by that of a dark reaction. For example, consider a TiO2 particle immersed in a solution containing acetic acid and oxygen. Under irradiation with UV light absorbed by TiO2 , the particle behaves as a short-circuited version of the cell shown in Figure 20.3.4a (72). At its surface, the oxidation of acetic acid by photogenerated holes (to CO2 and CH4 ) is balanced by the reduction of oxygen (Figure 20.3.7c). In a more chemical view, one can describe the photoexcitation process as a ligand-to-metal transition, such that the photogenerated hole is manifested as oxidized hydroxide ion at the

+

MV •

H2

Pt e

1.0 0.8 H+ + e → ½H2 i/il 0.6 0.4 υcath 0.2 0 υ –0.2 anod –0.4 –0.6 MV+• – e → MV2+ –0.8

e

MV2+

H+

0.5

(a)

0.0

–0.5

–1.0

E/V vs. NHE (b)

H2O2

hν

TiO2

e

CH3COOH

O2 e

h

CH4 + CO2

–1.5

(c)

Figure 20.3.7 (a) Schematic representation of the two half-reactions occurring at a Pt particle, resulting in + catalysis of the overall reaction MV ∙ + H+ → MV2+ + 1/2H2 . [MV2+ = methyl viologen; Figure 1.] (b) Current–potential curves for the two half-reactions. Arrows show predicted currents and the mixed potential where vcath = vanod (or ic = − ia ). (c) Half-reactions occurring during irradiation of a TiO2 particle in an oxygen-saturated acetic acid solution. 9 This same principle applies to corrosion reactions on metal surfaces. On metallic iron, for example, the oxidation of Fe to Fe2 O3 occurs at one site and is balanced by the reduction of O2 or protons at another.

907

908

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

surface of TiO2 (hydroxyl radical), which in turn reacts with acetic acid. The electron reduces a Ti(IV) center to Ti(III), which then reacts with oxygen. The generation of hydroxyl radicals at the surface of irradiated TiO2 has been established by electron spin resonance and spin trapping techniques (73). Electrochemical methods can be used to characterize photoprocesses in slurries of semiconductor particles, where irradiation either generates dissolved species that can be detected at an electrode or produces excess charge in the particles that can be collected at an electrode. Consider a deaerated suspension of TiO2 particles in an electrochemical cell (74, 75). Irradiation in the presence of an irreversible electron donor (such as acetic acid or EDTA) will result in reaction of the photogenerated holes with the donor, leaving the electrons trapped at the surface of the particle. These can be collected at an anode held at an appropriate potential, resulting in an anodic photocurrent during irradiation. Alternatively, irradiation in the presence of an irreversible electron acceptor, such as oxygen, can produce cathodic photocurrents. If the experiment with the irreversible donor is carried out in the presence of a mediator that can be reversibly reduced at the semiconductor particles, such as methyl viologen (MV2+ ) or Fe3+ , greatly enhanced photocurrents can be measured. In this case, the photoprocess produces + reduced forms (MV ∙ or Fe2+ ), which are then oxidized at an electrode held at a potential where the required anodic process can occur (76). The large enhancement of the current occurs because the mediator is much more mobile than the particles and establishes better communication with the anode. While it might seem surprising that rather large (μm-scale) particles can be characterized as slurries in solution by electrochemical techniques, there have been many papers concerned with the dark electrochemistry of suspended solids. For example, voltammetric studies of suspensions of AgBr are possible (77). In addition, there have been numerous studies of solid particles mounted on electrode surfaces (78). Electrodes can be prepared with films of semiconductor particles. A straightforward approach involves suitable chemical treatment of a substrate metal, e.g., chemical or anodic oxidation of Ti to form a film of TiO2 , which is polycrystalline, and, therefore, made up of separate particles. An alternative is to spread a film of semiconductor particles on an electrode surface. Films of nm-size semiconductor particles (quantum dots, QD, Section 20.1.2) have been of special interest (9) because of their distinctive electronic and optical properties. Moreover, such particles have very large surface area/volume ratios, and electrodes formed from them tend to have a high porosity and large roughness factors (48). Suspensions of particles can be prepared by a variety of techniques used in colloid formation. For example, TiO2 nanocrystals can be formed by hydrolysis of TiCl4 or alkoxides of Ti(IV) (79–81). Electrodes prepared by coating ITO/glass with such nanocrystalline films have been of interest in dye-sensitized photoelectrochemical cells and other devices (48, 80, 81).

20.4 Radiolytic Products in Solution Redox-active solutes are often generated under irradiation of a system by photons or electrons. These species can be detected or studied through electrode reactions.10 20.4.1

Photoemission of Electrons from an Electrode

A metal surface exposed to intense light can eject electrons that travel 2 to 10 nm into the electrolyte and then become solvated (67, 82–86). These electrons are reactive and produce 10 This topic was covered at greater length in the second edition, Section 18.3.

20.4 Radiolytic Products in Solution

chemistry if scavengers are available to interact with them. In the absence of such species, the electrons return to the electrode by diffusion and no net loss of charge is detected. The time for recollection of the electrons is 100 ns to 1 ms. If a scavenger exists, e.g., N2 O in water, some react and fail to return, and, therefore, the faradaic charge transfer can be detected: eaq + N2 O + H2 O → N2 + OH− + OH∙

(20.4.1)

The OH∙ diffuses to the electrode and, depending on the potential, may withdraw additional charge. Usually, the stimulus is a pulsed laser. Since the quantum yield for photoemission is low, an intense source is needed (∼10–100 kW/cm2 ). The detection method resembles that of the coulostatic technique (Section 9.7), in that one observes the potential shift caused by charge ejection, followed by relaxation back to the original state as charge is re-collected. Figure 9.9.2 offers some data for the N2 O system, and Problem 9.9 deals with its interpretation. This technique has been applied to the electrochemistry of free radicals (e.g., CH3 ∙, ∙CH2 OH, or phenyl) that have lifetimes too short for convenient study by purely electrochem∙ , the chemistry of which bears on electrolytic ical techniques (87). Problem 9.10 concerns Haq hydrogen evolution (Section 15.2). 20.4.2

Detection and Use of Radiolytic Products in Solution

Pulse radiolysis involves irradiation of a solution by a pulsed (∼20 ns) beam of high-energy electrons. Their passage creates solvated electrons, radicals, and ions, whose distribution can often be controlled by known chemistry. Significant concentrations of such species are produced, and they can sometimes be detected faradaically (88–90). If the radiolytic product of interest has a fairly long lifetime, then a potentiostatic experiment will yield a Cottrell-like response after the pulse, because the faradaic current is controlled by diffusion or, perhaps in part, by electrode kinetics. If the species is short-lived, the current transient is also influenced by the homogeneous decay. Similar in concept is the use of electrochemistry to monitor the products of flash photolysis, in which excitation is by pulsed UV or visible photons (82, 91–96). A more recent development is the deliberate use of radiolytic products generated by a TEM beam to carry out surface chemistry of electrochemical interest (97, 98). This is done inside a TEM cell that can hold a liquid sample for a prolonged period in vacuo between ∼50-nm-thick windows of Si3 N4 or graphene layers (99–101). The primary electron beam generates reactive species like those discussed above, and they are used to generate or to modify a surface that is observed continuously by TEM. An example is the in situ radiolytic reduction of Pt(II) to produce Pt nanocrystals, whose facet development could be observed in detail (97). 20.4.3

Photogalvanic Cells

It is possible to drive homogeneous redox systems in a nonspontaneous direction by using light can absorb light to produce an excited state that energy. For example, the complex Ru(bpy)2+ 3 is a fairly good reductant. Thus, one observes the reaction: 2+∗

Ru(bpy)3

+ Fe3+ → Fe2+ + Ru(bpy)3+ 3

(20.4.2)

In general, the products, Fe2+ and Ru(bpy)3+ , can be expected to restore the starting materials 3 by spontaneous reversal of (20.4.2). The net effect would be only a quenching of the excited complex, by which the absorbed photon becomes thermalized. However, the reverse of (20.4.2) to

909

910

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

produce ground-state products is not very fast (102), and one can build up appreciable concentrations of Ru(bpy)3+ and Fe2+ in the system. With the addition of electrodes, one can harvest 3 energy electrically by allowing spontaneous reduction of Ru(bpy)3+ at a cathode and oxidation 3 of ferrous iron at an anode. Usually, one of these electrodes is chosen to be reversible toward only one of the half-reactions, to enforce specific behavior at each electrode. These systems, called photogalvanic cells, have been explored for solar energy conversion (103, 104). The efficiency of conversion depends strongly on the degree to which the kinetics can be optimized (103–107).

20.5 Electrogenerated Chemiluminescence A striking aspect of certain very exothermic electron-transfer reactions is an accompanying chemiluminescence. Even though the emitted light almost always comes from reactions in solution, it is frequently studied by experiments involving the electrolytic production of participants; hence, it is called electrogenerated chemiluminescence (or electrochemiluminescence, ECL). The phenomenon was initially of interest because it offers insight into the way in which reaction products accommodate released energy in electron transfer (108–113); however, it later led to analytical applications (108, 114–123), which have become, by far, the principal basis of new research. 20.5.1

Chemical Fundamentals

ECL was discovered and initially examined in systems based on redox reactions of radical ions, typically in MeCN or DMF. The following cases, involving radical ions of rubrene (R), TMPD, and p-benzoquinone (BQ) (Figure 1), are representative: +

R−∙ + R ∙ → 1 R∗ + R +

R−∙ + TMPD ∙ → 1 R∗ + TMPD +

R ∙ + BQ−∙ → 1 R∗ + BQ

(20.5.1) (20.5.2) (20.5.3)

In all three cases, the emission is the yellow fluorescence of R, arising from the first excited singlet, 1 R* . 1 ∗

R → R + h𝜈

(20.5.4)

The formation of an electronically excited state as a result of electron transfer is a kinetic manifestation of the Franck–Condon principle (108, 110–112). The reactions are very exothermic (typically 2–4 eV) and very fast (probably on the time scale of molecular vibration for the actual transfer). Since it is difficult for the molecular frames to accept a large amount of released energy in a mechanical form (e.g., vibrationally) on a short time scale, there is a significant probability that an electronically excited product will be produced instead. Research in this area has addressed the fundamentals of energy disposition in fast, energetic reactions (108, 110, 111), and it can serve as a test for theories of electron transfer. The free energy released in a redox process producing ground-state products, e.g., +

R−∙ + R ∙ → 2R

(20.5.5)

20.5 Electrogenerated Chemiluminescence 1

TMPD* + R

3

1 3 –

+

R• + R•

TMPD* + R

1 *

BQ* + R

3

BQ* + R

R +R

2 –

+

R • + TMPD • +

ΔG0/eV

–

R • + BQ •

3 *

R +R

1

Emission

Ground states of ion precursors

0

Figure 20.5.1 Energetics for chemiluminescent reactions of rubrene radical ions. All energies are measured with respect to ground-state neutral species. Dashed arrow shows S route. Dotted arrows show T route. Promotion from 3 R* + R to 1 R* + R requires another rubrene triplet. [Adapted from Faulkner (111)/with permission of Elsevier.]

approximates the energy available for exciting a product.11 Since ΔG0 is readily computed from reversible standard potentials for ion/precursor couples, it can easily be compared with excited state energies obtained spectroscopically. Excited states lower than the available energy may be populated in the reaction. Higher states are energetically inaccessible. Figure 20.5.1 shows the energetics of reactions (20.5.1)–(20.5.3). Since all excited states of BQ and TMPD are inaccessible, the three electron transfers can produce only the rubrene singlet, 1 R* , the rubrene triplet, 3 R* , and products in their ground states. The emitting state, 1 R* , is directly accessible in (20.5.1); hence, that reaction can be an elementary process as written. This path to ECL is called the S (for singlet) route, and the system is sometimes termed energy-sufficient. In contrast, (20.5.2) and (20.5.3) are energy-deficient because the emitter is not accessible to the electron-transfer process. Accordingly, (20.5.2) and (20.5.3) cannot be elementary processes and must involve more than one step. The usual explanation for energy-deficient ECL involves triplet intermediates, for example, +

R−∙ + TMPD ∙ → 3 R∗ + TMPD

(20.5.6)

3 ∗

(20.5.7)

R +3 R∗ → 1 R∗ + R

The second step, called triplet–triplet annihilation, allows the energy from two electron transfers to be pooled into the production of 1 R* . There is much evidence for this mechanism 11 Actually, the energy available for exciting a product is the standard internal energy change, ΔE0 . Because the reaction is in a condensed phase, ΔE0 ≈ ΔH 0 , which is ΔG0 + TΔS0 . Since TΔS0 ∼ 0.1 eV for this kind of reaction, ΔE0 ≈ ΔG0 .

911

912

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

(108, 110–112), which is called the T (for triplet) route. It may operate in both energy-sufficient and energy-deficient systems. The Marcus theory (Sections 3.5.3 and 3.5.4) can help us to understand the formation of an excited state in an electron-transfer reaction (108, 124). With all other factors remaining equal (especially the reorganization energy, 𝜆), the relative rate constants of electron-transfer reactions become determined by the overall free-energy change, ΔG0 . For modest ΔG0 , the rate constant increases as ΔG0 becomes more negative. However, for large negative ΔG0 values—in the “inverted region” [Sections 3.5.4(e,g)]—the rate constant becomes progressively much smaller. In ECL reactions, the ΔG0 for direct production of ground state species is large, while that for producing an excited state is very much smaller. Thus, the Marcus model predicts that an energetic electron-transfer reaction can have a rate constant for excited state formation exceeding that for formation of ground-state products. Indeed, ECL was the first experimental evidence for an inverted region in electron-transfer reactions (125). Hundreds of ECL reactions have been reported, and many are spectroscopically simple enough to be understood in these terms. Others offer emission bands due to excimers [excited dimers such as (DMA)∗2 , where DMA is 9,10-dimethylanthracene], exciplexes [excited-state + complexes, such as (TPTA ∙BP−∙), where TPTA is tri-p-tolylamine and BP is benzophenone; Figure 1], or decay products of the radical ions. More complicated mechanisms are needed to describe such situations. Many studies involve radical ions of aromatic compounds, but others have dealt with metal complexes [especially Ru(bpy)2+ (bpy = 2,2′ -bipyridine)], superoxide, 3 solvated electrons, and classical chemiluminescent reagents, such as lucigenin (108, 110–113). 20.5.2

Fundamental Studies of Radical-Ion Annihilation

ECL experiments focused on radical-ion annihilation [e.g., (20.5.1)–(20.5.3)] are carried out in conventional electrochemical apparatus (112), but procedures must be modified to allow the electrogeneration of two reactants. In addition, one must pay scrupulous attention to the purity of the solvent/supporting electrolyte system. Water and oxygen are particularly harmful; thus, apparatus is constructed to allow transfer of solvent and degassing on a high-vacuum line or operation in an inert-atmosphere box. Other constraints may be imposed by optical equipment used to monitor the light. The primary goals are often to define the nature of the emitting state, the mechanism by which it is produced, and the efficiency of excited-state production. (a) Dual-Potential Generation of ECL

Many ECL experiments require that reactants be electrogenerated at two different (often widely separated) potentials. To observe light from (20.5.1), for example, one usually generates R −∙ + and R ∙ by reducing rubrene beyond −1.5 V vs. SCE and by oxidizing it beyond 1.0 V. We will use the term dual-potential generation for this form of experimentation, whether the reactants are produced at a single electrode or at separate electrodes. Much research in ECL has been carried out by generating the reactants sequentially at a single electrode. For example, one might start with a solution of rubrene and TMPD in DMF. A Pt disk stepped to −1.6 V vs. SCE produces R−∙ in the diffusion layer. After a forward generation + time, t f , which might be 10 μs to 10 s, the potential is changed to +0.35 V to produce TMPD ∙, − which diffuses outward. Since R ∙ is oxidized at this potential, its surface concentration drops + effectively to zero and R−∙ in the bulk begins to diffuse back toward the electrode. Thus, TMPD ∙ and R−∙ move together and react in the diffusion layer. If the reaction rate constant is very large, their concentration profiles do not overlap and the reaction occurs in the plane where they meet, as shown in Figure 20.5.2. As the experiment proceeds, the R−∙ gradually is used up and

20.5 Electrogenerated Chemiluminescence

1.0

Concentration/mM

Figure 20.5.2 Concentration profiles near an electrode during an ECL step experiment. Data apply to 1 mM R and 1 mM TMPD. (a) Profiles of R and R−∙ at the end of the forward + step. (b) Concentrations of TMPD ∙ and R−∙ during the second step. Reaction boundary is shown by the dotted line. Curves apply for a time 0.4tf into the second step, where tf is the forward step duration. Diffusion coefficients are taken as equal.

R–•

R

(a)

0.0 1.0

+

TMPD •

(b)

–

R• 0.0 0.0

2.0 x/(Dtf)1/2

4.0

the reaction plane moves farther from the electrode. The light appears as a pulse that decays with time because of the depletion of R−∙ . Experiments like these may be carried out with just one step in each direction, producing a single pulse of light, or a train of alternating steps, yielding a sequence of light pulses. Other useful approaches to ion-annihilation ECL involve the dual-potential generation of the reactants at two different electrodes in close proximity. For example, an RRDE (Section 10.3.2) can be employed, with one reactant generated at the disk and the other at the ring. These are swept together by diffusion and convection,12 resulting in reaction and a ring of light on the inner edge of the ring electrode (126, 127). Other experiments (128, 129) employ dual-working-electrode systems with thin-layer geometry, interdigitated electrodes, or flow streams to move the reactants together. (b) Modes of Exploration

ECL experiments often involve recording the spectrum of emitted light, which is essential for the identification of emitting species. Figure 20.5.3 offers an example based on the reaction + between TMPD ∙ and the radical anion of pyrene, Py−∙. The fluorescence of a solution of pyrene (Py) and TMPD excited at 350 nm shows a sharp band at 400 nm, ascribed to 1 Py* , and a minor shoulder at 450 nm, due in part to the excimer 1 Py∗2 , which emits in the dissociative process: 1

Py∗2 → 2Py + h𝜈

(20.5.8)

In contrast, the ECL produced by electrolysis at an RRDE (Py reduced at the ring and TMPD oxidized at the disk) shows strong emission from the excimer. The chemiluminescent system evidently has a specific path for efficient production of the excimer. It is believed to be the triplet–triplet annihilation involving 3 Py* . Figure 20.5.4 shows light intensity from this same system vs. disk potential. In the upper frame, Py−∙ was generated at the ring and one sees that light results from the oxidation (at the + disk) of TMPD to either TMPD ∙ (first wave) or TMPD2+ (second wave). Oxidation products + generated at the most positive potentials (perhaps Py ∙ or its decay products) quench the ECL entirely. Interpretation of Figure 20.5.4b is left to the reader as Problem 20.8. 12 More detail in the first edition, Section 14.4.2.

913

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

Figure 20.5.3 (a) Steady-state ECL intensity vs. wavelength, 𝜆, from the reaction between Py−∙ + and TMPD ∙ in DMF. Ion radicals were electrogenerated at an RRDE in a solution of 1 mM TMPD and 5 mM Py. (b) Fluorescence spectrum of the same solution under excitation at 350 nm. [Adapted from Maloy and Bard (126)/American Chemical Society.]

Relative intensity

(a)

(b)

300

400

λ/nm

500

600

700

ECL intensity

200

1.5

1.0

ED /V vs. SCE (a)

0.5

0.0

ECL intensity

914

–1.5

–2.0 ED /V vs. SCE (b)

–2.5

Figure 20.5.4 Steady-state ECL intensity at an RRDE vs. disk potential, E D , in the pyrene–TMPD system. (a) At 393 nm, with Py−∙ generated at the ring [E R in the plateau region of (b)]. The disk potential was swept toward + positive potentials, first to generate TMPD ∙ , and then to produce TMPD2+ . (b) At 393 nm (upper) and 470 nm + (lower), with TMPD ∙ generated at the ring [E R in the first plateau region of (a)]. [Maloy and Bard (126)/American Chemical Society.]

20.5 Electrogenerated Chemiluminescence

Figure 20.5.5 (a) Chemiluminescence from FA−∙ + and 10-MP ∙ in DMF. Reactants were generated from solutions containing 1 mM FA and 10-MP. (b) Spectrum upon addition of anthracene. Inset shows fluorescence from anthracene at 10 μM in DMF. Shortest wavelength anthracene peak is not seen in ECL because of self-absorption. [Original data from Freed and Faulkner (133). © 1971, American Chemical Society. Figure from Faulkner (134). Reproduced with permission of the American Chemical Society.]

Relative intensity

The mechanism of light production is always of interest, and many experiments have been devised to probe it. One approach is based on shapes of single pulses of light produced in a step sequence like that described above (110–112). The basic idea is to find the dependence of light intensity on the rate of redox reaction between the oxidant and reductant. For example, the S route calls for a linear dependence, whereas the T route generally would yield a relationship of higher order. The diffusion-kinetic problem for step generation of ECL has been solved (130), and the time decay of the redox reaction rate can be calculated for a given system. It can then be compared with the observed intensity transient. Problem 20.9 provides an example. By using a UME, it is possible to shorten the time scale. In one set of experiments (131), a continuous symmetric square wave with step times down to 5 μs was applied to Pt disks with r0 = 1 − 5 μm, and the emission was measured by single-photon counting. By comparing the shape of the relative intensity vs. time to theoretical behavior obtained by digital simulation, it was possible to find the rate constant for the ion-annihilation reaction. The + annihilation rate constants for DPA−∙ + DPA ∙ and for Ru(bpy)3+ + Ru(bpy)+ were both at the 3 3 10 −1 −1 diffusion-controlled value (2 × 10 M s ) in MeCN. When the temporal resolution was extended to the nanosecond regime (132), individual annihilation reaction events could be observed. Experiments have been devised to intercept intermediates, such as triplets or singlet oxygen. Results for such a case are given in Figure 20.5.5, relating to the energy-deficient reaction between the cation radical of 10-methylphenothiazine (10-MP) and the anion radical of fluoranthene (FA) (133). This system is believed to produce light by the triplet–triplet annihilation of 3 FA* , producing 1 FA* . Addition of anthracene (An) did not disturb the electrochemistry, but did transform the emission spectrum from that of 1 FA* to that of 1 An* . Apparently, this result

(b)

400

500

(a)

300

400

λ/nm

500

600

915

916

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

was caused by the energy transfer: An + 3 FA∗ → FA + 3 An∗

(20.5.9)

followed by mutual annihilation of 3 An* , producing 1 An* . Sometimes magnetic fields enhance ECL intensities, and studies along this line have been used for mechanistic diagnosis (112). The effects seem to arise from field-dependent rate constants for certain reactions involving triplets; hence, they are associated with the T route. 20.5.3

Single-Potential Generation Based on a Coreactant

An important development was the demonstration that ECL can be produced at a single potential when a suitable coreactant is present (135–137). Here, we use the term single-potential generation to label such cases. In some instances, the potential may be alternated between two values, but only one gives rise to reactants for the ECL process. The single-potential strategy permits ECL to be observed in aqueous solutions, which is tremendously important for analytical application (108, 114–122). , which can be oxidized at Pt to Consider, for example, an aqueous solution of Ru(bpy)2+ 3 2+

Ru(bpy)3+ at about +1 V vs. SCE. To form the excited state Ru(bpy)3 ∗ (2.04 eV above the 3 ground state), one needs a reductant produced at a potential more negative than −1 V. This potential is not attainable at a Pt electrode in aqueous solution before the cathodic background limit. However, there is an alternative: an electrode reaction with coupled homogeneous chemistry (Chapter 13) can sometimes generate a strong reductant as an intermediate, and it might , which be captured for the generation of ECL. An example is the oxidation of oxalate, C2 O2− 4 produces the strong reducing agent, CO2 −∙. Thus, one can carry out the oxidations of C2 O2− 4 3+ and CO − and Ru(bpy)2+ simultaneously at ∼1 V vs. SCE, so that Ru(bpy) ∙ are generated 2 3 3 together in the same space near the electrode. Their annihilation produces the ECL. The reaction sequence is (136): → Ru(bpy)3+ +e Ru(bpy)2+ 3 3

(20.5.10)

2+ − Ru(bpy)3+ + C2 O2− 4 → Ru(bpy)3 + CO2 + CO2 ∙ 3

(20.5.11)

2+∗

Ru(bpy)3+ + CO2 −∙ → Ru(bpy)3 3

+ CO2

(20.5.12)

The direct production of CO2 −∙ at the electrode surface by oxidation of C2 O2− also occurs: 4 − C2 O2− 4 → C2 O4 ∙ + e

(20.5.13)

C2 O4 −∙ → CO2 + CO2 −∙

(20.5.14)

This process can add to the production of the excited state by increasing the rate of (20.5.12). Still another possibility is an indirect light-producing path: CO2 −∙ + Ru(bpy)2+ → CO2 + Ru(bpy)+ 3 3

(20.5.15) 2+∗

Ru(bpy)3+ + Ru(bpy)+ → Ru(bpy)2+ + Ru(bpy)3 3 3 3

(20.5.16)

In this system, the CO2 −∙ becomes available because C2 O4 −∙ decomposes with the formation of very strong bonds in CO2 . The intermediates C2 O4 −∙ and CO2 −∙ have been detected, and their lifetimes have been estimated using SECM (138) (Section 18.4.2). Both the direct and indirect mechanisms involve the oxidation of Ru(bpy)2+ at the electrode 3 surface, consistent with observations that ECL is observed only at potentials where Ru(bpy)3+ 3

20.5 Electrogenerated Chemiluminescence

is produced (136). In either mechanism, emission of one ECL photon requires at least two electrons passed by anodic reactions. The C2 O2− is called a “coreactant” because it is oxidized in parallel with the emitting reagent, 4 2+ Ru(bpy)3 . Other coreactants, such as tertiary amines, are also practical for use with Ru(bpy)2+ . 3 The mechanisms can become complex, and the details depend upon the lifetimes of intermediates generated from decomposition of the coreactants. The mechanism for the important Ru(bpy)2+ /tripropylamine system is discussed in Section 20.5.5(b). 3 Alternatively, it is possible to use a reductive mode for single-potential generation of ECL from Ru(bpy)2+ , in which case the coreactant must be reduced in a process ultimately yielding 3 a strong oxidant (137). Usually, peroxydisulfate, S2 O2− , is employed. At a potential where both 8 2+ 2− Ru(bpy)3 and S2 O8 can be coreduced, the powerful oxidant, SO4 −∙, gives ECL by direct or indirect oxidation of the electrogenerated Ru(bpy)+ . The proposed direct mechanism is (137) 3 Ru(bpy)2+ + e → Ru(bpy)+ 3 3

(20.5.17)

2+ 2− − Ru(bpy)+ + S2 O2− 8 → Ru(bpy)3 + SO4 + SO4 ∙ 3

(20.5.18)

2+∗

SO4 −∙ + Ru(bpy)+ → SO2− 4 + Ru(bpy)3 3

(20.5.19)

The oxidant SO4 −∙ is also produced directly at the electrode surface by reduction of S2 O2− : 8 3− S2 O2− 8 + e → S 2 O8 ∙

(20.5.20)

S2 O38 −∙ → SO4 −∙ + SO2− 4

(20.5.21) 2+

This sequence may enhance the generation of Ru(bpy)3 ∗ through (20.5.19). In this system, the indirect path for production of ECL is analogous to (20.5.15)–(20.5.16): 3+ SO4 −∙ + Ru(bpy)2+ → SO2− 4 + Ru(bpy)3 3

(20.5.22) 2+∗

Ru(bpy)3+ + Ru(bpy)+ → Ru(bpy)2+ + Ru(bpy)3 3 3 3 20.5.4

(20.5.23)

ECL Based on Quantum Dots

ECL is not limited to small molecules but can also be observed with nanocrystalline semiconductors known as quantum dots (QD; Section 20.1.2). Results were obtained initially with Si QD (139) but have since been extended broadly to Cds, CdSe, PbS, and many other materials. The area has been reviewed (9, 114, 140). QDs can often be faradaically reduced or oxidized in 1e steps in an aprotic solvent to produce QD+ and QD− , which can annihilate with generation of an excited species QD* , QD+ + QD− → QD∗ + QD

(20.5.24)

Light can arise from the immediate product, QD* , or from a successor state after energy transfer. In the initial work with Si QD (2–4 nm in diameter), dual-potential generation of QD+ and QD− was carried out with alternating steps. The ECL spectrum (Figure 20.5.6a) is notable because the emission maximum at 640 nm is far to the red of the photoluminescence maximum at 420 nm from the same QD. Clearly, the emitter in ECL differs from that in photoluminescence. A luminescent surface state was proposed to explain the ECL. Since this state is not visible in photoluminescence, it appears not to be a trap for the excited state giving rise to that phenomenon. Consequently, the emitting state in ECL seems to be generated directly as QD* in (20.5.24).

917

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

Figure 20.5.6 ECL from Si quantum dots. (a) Dual-potential generation at a single working electrode using 10-Hz alternating steps between the first reduction and first oxidation of the QD in MeCN with 0.1 M tetrahexylammonium perchlorate. (b) Single-potential generation involving simultaneous oxidation of the QD and C2 O2− as coreactant. (c) 4 Single-potential generation involving simultaneous reduction of the QD and S2 O2− as coreactant. [From 8 Deng et al. (139), with permission.]

(a) ECL intensity

918

(b)

200

400

600 800 Wavelength/nm (c)

1000

QD can be used more effectively by single-potential generation with a coreactant, as illustrated for the Si system in Figure 20.5.6b,c. The effectiveness of the reductive process involving S2 O2− is evident. This is a common finding with QD and has led to broad interest in 8 single-potential generation with S2 O2− . 8 20.5.5

Analytical Applications of ECL

Because the light intensity is usually proportional to the concentration of the emitting species, ECL can be exploited analytically for either the emitter (which often serves as a label on a molecule of interest) or a coreactant (108, 114–122). Very low light levels can be measured (e.g., by single-photon counting); thus, high sensitivity can be achieved. A light source is not used; therefore, scattered light and interferences from emission by luminescent impurities are not problems. In effect, ECL provides the darkest background of all analytical methods based on luminescence. (a) Assays Based on Labeled Reagents

The most widely used ECL-active label is Ru(bpy)2+ because its ECL can be generated in 3 aqueous solutions with a suitable coreactant (e.g., C2 O2− for oxidations and S2 O2− for 4 8 reductions). Moreover, the emission is intense and fairly stable. The ECL is proportional to the concentration over a wide range (e.g., 10−7 to 10−13 M) (141). By attachment of a suitable group to the bipyridine moieties, Ru(bpy)2+ can be linked to biologically interesting molecules, 3

20.5 Electrogenerated Chemiluminescence

such as antibodies or DNA, where it serves as a label for analysis in a manner analogous to the use of radioactive or fluorescent labels (142). Commercial instruments are available for ECL assays of antibodies, antigens, and DNA (108, 114–122, 143–145). These are usually based on the use of magnetic bead technology (142–144). The principles of a typical sandwich assay of an antigen are outlined in Figure 20.5.7. The surfaces of commercially available magnetic beads are modified by attaching an antibody to a particular antigen of interest (e.g., prostate-specific antigen, PSA). These beads, the sample of interest, and Ru(bpy)2+ -labeled antibodies are mixed. If antigen is present, as shown in 3 Figure 20.5.7, the labeled antibody becomes attached to the magnetic bead because the antigen behaves as a bridge to form a “sandwich” structure. If no antigen is present, the labeled antibody does not attach to the bead. The magnetic beads are then flushed into an ECL cell, where they are captured on the working electrode by applying a magnetic field (Figure 20.5.8). The beads are washed, and a solution of the appropriate composition containing a coreactant (usually tri-n-propylamine, TPrA; Figure 1) is pumped into the cell. Upon application of a sweep or step to positive potentials, oxidation of the coreactant occurs with emission of light from the bead-bound Ru(bpy)2+ , which is detected with the photomultiplier. Following the measure3 ment, the beads are washed from the cell, which is then cleaned and made ready for the next sample. Automated instruments for clinical diagnostics, capable of handling multiple samples without operator intervention, are available.

Ru Magnetic bead

Antibody

Antigen

Labeled antibody

Ru (a) Sample without antigen (labeled antibody unattached to bead)

Ru (b) Sample with antigen (labeled antibody attaches to bead)

Figure 20.5.7 Representation of ECL-based immunoassay. Black circle with “Ru” indicates labeling with Ru(bpy)2+ . Sample unbound to antigen in (a); bound in (b). 3

919

920

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

Magnetic beads

Cell window

Photomultiplier O-ring

Cell body Working electrode

Solution inflow

Magnet

Solution outflow

Figure 20.5.8 Flow cell used in a commercial instrument for ECL-based immunoassay employing magnetic beads. [Based on Bard and Whitesides (142), Yang et al. (143), and Blackburn et al. (144)].

(b) Mechanism of ECL with TPrA as a Coreactant

The chemistry of the Ru(bpy)2+ ∕TPrA system has received careful attention (146–148) and 3 now appears well-understood. The central concept is the same as in the case of oxalate as a coreactant; viz. that the oxidation of the coreactant is followed by homogeneous chemistry leading to a powerful reductant, which can drive the light-producing process. However, the details in this case are more intricate. Figure 20.5.9 lays out the principal path to ECL. + Key participants are TPrA ∙, which is the product of 1e oxidation of TPrA at the electrode, + and TPrA ∙, which is formed from TPrA ∙ by loss of H+ with bond reorganization (not just a + simple deprotonation). In subsequent chemistry, both TPrA ∙ and TPrA ∙ are redox agents, but + of greatly different character, with TPrA ∙ being a strong oxidant, and TPrA ∙, a strong reductant. + Light production arises from the annihilation of TPrA ∙ and Ru(bpy)+ , giving the emitting 3 2+

state, Ru(bpy)3 ∗ . The role of the strong reductant TPrA∙ is to generate the Ru(bpy)+ for that 3 + reaction. In the sandwich assay ECL system described above, the lifetimes of TPrA ∙ and TPrA ∙ Electrode

TPrA

–H+

+H+

TPrAH+

e +

TPrA

TPrA• –H+ TPrA•

Ru(bpy)+3

P1

Ru(bpy)3

2+

* Ru(bpy)2+ 3

2+

hν

Ru(bpy)3

Figure 20.5.9 Principal mechanism for light production in the Ru(bpy)2+ ∕TPrA system, normally in pH 8–9 3 aqueous phosphate buffer. For the TPrAH+ /TPrA acid–base system, pK a = 10.4; hence, TPrA is predominantly protonated. The gray arrows mark the light-producing reaction. P1 is assumed to be an inert product. [Adapted from Miao, Choi, and Bard (148)/American Chemical Society.]

20.5 Electrogenerated Chemiluminescence

in phosphate buffer are such that both exist in concentrations that support these reactions with the immobile Ru(bpy)2+ -labeled antibodies on the magnetic beads. 3 Evidence of great variety supports the mechanism of Figure 20.5.9 (148), which explains many details of experimental behavior, including the fact that light can be generated without using a potential quite as positive as required for electrogeneration of Ru(bpy)3+ . In systems where 3 3+ Ru(bpy)3 is mobile and can be produced electrolytically, then annihilation of Ru(bpy)3+ and 3 Ru(bpy)+ becomes a parallel path to light production. 3 (c) Elaboration of Analytical ECL

The concepts and tools for analytical use of ECL extend well beyond the chemistry, assay methods, and instrumentation described above: • A great range of luminescent labels (luminophores) has been explored (108, 114–122). Most are complexes of Ru(II) related to Ru(bpy)2+ ; however, high interest exists in complexes of 3 Ir(III), which are efficient emitters. Most of these involve pyridine-based ligands or acetylacetonates. Considerable effort has also been invested in luminophores based on aromatic organic species like derivatives of DPA and on nanomaterials, especially QD of CdS, CdSe, CdTe, and other semiconductors (9, 140). • Microfluidic ECL cells have been developed for detection in FI, LC, and CE (108, 114–122, 149, 150). These typically utilize single-potential generation of ECL from Ru(II)-based luminophores, where the detected species, such as amines, NADH, and amino acids, often serve as the coreactant. Observation of ECL with flowing streams can also provide information about the hydrodynamics in the detector cell (151). • Spatial imaging of ECL has been developed (114–116, 152). Light collection has been through an optical microscope, providing spatial resolution on the order of the wavelength of the emission. • The success of the magnetic-bead-based systems for assay by ECL made it clear that the luminophore can be immobilized, as long as it can be reached by the essential electrolytic products of the coreactant (preceding subsection). This recognition has led to the investigation of a large range of ECL-based sensors in which molecular recognition is achieved through antigen–antibody interactions or hybridization of DNA segments (108, 114–122). An interesting example in the latter category (153) is represented in Figure 20.5.10. All of the electrochemistry driving the ECL takes place on a tiny metallic strip that has no external electrical connections. Instead, the system is driven by placing the strip lengthwise along an ohmic drop in a surrounding electrolyte. The ohmic drop is established by two “driver” electrodes passing a sizable current between them (Figure 20.5.10a). If the ohmic field along the metallic strip is large enough, then the difference of electric potential between metal and solution, 𝜙M − 𝜙S , at one end of the strip can correspond to a very different value of E than at the other end. As a consequence, different electrode reactions can occur at the two ends. However, there can be no buildup of charge on the metal strip, so one of the reactions must be anodic and the other must be cathodic, and they must occur at the same rate, i.e., ic = − ia . The metal strip is called a bipolar electrode. Such electrodes can operate in large arrays without need of individual connections to any element (154). In the particular case of Figure 20.5.10, the electrolyte contains Ru(bpy)2+ and TPrA, and 3 the right end of the metal strip is modified with a segment of probe DNA (Figure 20.5.10b). If a target DNA segment functionalized with Pt nanoparticles is introduced into the solution, and if it can hybridize the probe DNA (being chemically recognized), then the reduction of O2 on the Pt nanoparticle becomes enabled on the right end, and that reaction can drive the single-potential generation of ECL on the left end. Because ic = − ia , the rate of light production

921

922

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

(a)

(b)

2+

Ru(bpy)3 + TPrA

hν

ECL reporting site

H2O

O2

Pt DNA sensing site

Figure 20.5.10 (a) Top view of cell with three parallel bipolar electrodes of 1 mm length. Driver electrodes are on the right and the left. (b) Chemistry on the opposite ends of a bipolar electrode. (c) Micrograph of three parallel electrodes under illumination. (d) Micrograph of ECL with the same optical conditions as for (c), except in the dark. Only the top two electrodes were exposed to the labeled target. [Adapted from Mavré et al. (153)/American Chemical Society.]

1 mm (c)

(d)

on the left end (Figure 20.5.10d) is proportional to the rate of O2 reduction, which is, in turn, proportional to the amount of labeled DNA bound at the cathodic end. 20.5.6

ECL Beyond the Solution Phase

In its classic form, ECL is regarded as a solution-phase process, on the basis of both direct evidence (Problem 20.9) and the expectation that metal electrodes quench excited states (68, 69). However, exceptions to that idea are widespread (108, 114–122). • As we have seen above, the single-potential generation of ECL can operate beneficially with an immobilized luminophore. There are other examples. Surface films, such as monolayer assemblies and polymer-modified electrodes (Chapter 17), have been reported to produce ECL. For example, a Langmuir–Blodgett monolayer of a long chain hydrocarbon with an attached Ru(bpy)2+ group or a similar self-assembled monolayer will show ECL emission 3 on oxidation in the presence of a coreactant (155, 156). In fact, monolayers at the air/water interface that contain ECL-active groups will emit when contacted by touching the horizontal film with an ultramicroelectrode tip from the air side. In this experiment, the counter and reference electrodes and the coreactant are all contained in the aqueous medium in the Langmuir trough (157). • The band structure of semiconductor electrodes (Section 20.1) sometimes removes the tendency toward quenching of excited states, and emission from excited states produced directly in heterogeneous charge transfer at semiconductors can occur (158–160). • Polymer films on electrodes, such as poly(vinyl-DPA) (161), a polymerized film of in Nafion (163), will also tris(4-vinyl-4′ -methyl-2,2′ -bipyridyl)Ru(II) (162), or Ru(bpy)2+ 3 produce ECL. ECL processes that produce light in polymer layers, sometimes in the absence of solvent (electroluminescent polymers), are of interest in connection with display applications (108, 119, 164–167).

20.6 References 1 2 3 4

S. H. Simon, “The Oxford Solid State Basics,” Oxford, 2013. C. Kittel, “Introduction to Solid State Physics,” 8th ed., Wiley, Hoboken, NJ, 2005. N. W. Ashcroft and N. D. Mermin, “Solid State Physics,” Saunders, New York, 1976. N. F. Mott and R. W. Gurney, “Electronic Processes in Ionic Crystals,” 2nd ed., Clarendon Press, Oxford, 1948.

20.6 References

5 Yu. V. Pleskov and Yu. Ya. Gurevich, “Semiconductor Photoelectrochemistry,” P. N.

Bartlett, Transl. Ed., Consultants Bureau, New York, 1986. 6 D. Madelung, in “Physical Chemistry—An Advanced Treatise,” Vol. X, W. Jost, Ed., Aca-

demic Press, New York, 1970, Chap. 6. 7 G. Ertl and H. Gerischer, ibid., Chap. 7. 8 V. A. Myamlin and Yu. V. Pleskov, “Electrochemistry of Semiconductors,” Plenum, New

York, 1967. 9 W. W. Zhao, J. Wang, Y.-C. Zhu, J.-J. Xu, and H.-Y. Chen, Anal. Chem., 87, 9520 (2015). 10 A. M. Smith and S. Nie, Acc. Chem. Res., 43, 190 (2010). 11 D. J. Norris, M. G. Bawendi, and L. E. Brus, in “Molecular Electronics,” J. Jortner and M. 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

A. Ratner, Eds., Blackwell Science, Malden, MA, 1997, pp. 281–323. M. L. Steigerwald and L. Brus, Acc. Chem. Res., 23, 183 (1990). A. Henglein, Chem. Rev., 89, 1861 (1989). R. Sardar, A. M. Funston, P. Mulvaney, and R. W. Murray, Langmuir, 25, 13480 (2009). D. Lee, R. L. Donkers, G. Wang, A. S. Harper, and R. W. Murray, J. Am. Chem. Soc., 126, 6193 (2004). S. Trasatti, Electrochim. Acta, 45, 2377 (2000). L. Liborio, G. Mallia, and N. Harrison, Phys. Rev. B, 79, 245133 (2009). P. C. S. Hayfield, “Development of a New Material—Monolithic Ti4 O7 Ebonex Ceramic,” Royal Society of Chemistry, Cambridge, 2002. K. McGregor, E. J. Frazer, A. J. Urban, M. I. Pownceby, and R. L. Deutscher, ECS Trans. 2, 369 (2006). J. Ge, X. Zou, S. Almassi, L. Ji, B. P. Chaplin, and A. J. Bard, Angew. Chem. Int. Ed., 58, 16223 (2019). H. Gerischer, Electrochim. Acta, 35, 1677 (1990). A. J. Nozik and R. Memming, J. Phys. Chem., 100, 13061 (1996). C. A. Koval and J. N. Howard, Chem. Rev., 92, 411 (1992). H. Gerischer, in “Physical Chemistry—An Advanced Treatise,” Vol. IXA, H. Eyring, D. Henderson, and W. Jost, Eds., Academic Press, New York, 1970, p. 463. H. Gerischer, Adv. Electrochem. Electrochem. Eng., 1, 139 (1961). A. J. Bard, J. Photochem., 10, 59 (1979). W. Schottky, Z. Phys., 113, 367 (1939); 118, 539 (1942). N. F. Mott, Proc. R. Soc. (London), A171, 27 (1939). A. M. Van Wezemael, W. H. Laflère, F. Cardon, and W. P. Gomes, J. Electroanal. Chem., 87, 105 (1978). E. C. Dutoit, F. Cardon, and W. P. Gomes, Ber. Bunsenges. Phys. Chem., 80, 1285 (1976). S. N. Frank and A. J. Bard, J. Am. Chem. Soc., 97, 7427 (1975). N. S. Lewis, J. Phys. Chem. B, 102, 4843 (1998). N. S. Lewis, Annu. Rev. Phys. Chem., 42, 543 (1991). A. F. Diaz, K. K. Kanazawa, and G. P. Gardini, J. Chem. Soc., Chem. Commun., 635 (1979). J. G. Ibanez, M. E. Rincón, S. Gutierrez-Granados, M. Chahma, O. A. Jaramillo-Quintero, and B. A. Frontana-Uribe, Chem. Rev., 118, 4731 (2018). P. G. Pickup, Mod. Asp. Electrochem., 33, 549 (1999). A. F. Diaz, M. T. Nguyen, and M. Leclerc, in “Physical Electrochemistry,” I. Rubinstein, Ed., Marcel Dekker, New York, 1995, Chap. 12. A. J. Heeger, Chem. Soc. Rev., 39, 2354 (2010). T. A. Skotheim and J. Reynolds, Eds., “Handbook of Conducting Polymers,” 3rd ed.; CRC Press, Boca Raton, FL, 2007. S. W. Feldberg, J. Am. Chem. Soc., 106, 4671 (1984).

®

923

924

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

41 N. S. Lewis, Acc. Chem. Res., 23, 176 (1990). 42 L. M. Peter and D. Vanmaekelbergh, Adv. Electrochem. Sci. Eng., 6, 77 (1999). 43 M. R. Nellist, F. A. L. Laskowski, F. Lin, T. J. Mills, and S. W. Boettcher, Acc. Chem. Res.,

49, 733 (2016). 44 L. M. Peter and K. G. U. Wijayantha, Chem. Phys. Chem., 15, 1983 (2014). 45 M. G. Walter, E. L. Warren, J. R. McKone, S. W. Boettcher, Q. Mi, E. A. Santori, and N. S. 46 47 48 49 50 51 52 53

54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

Lewis, Chem. Rev., 110, 6446 (2010). P. Dale and L. Peter, Adv. Electrochem. Sci. Eng., 12, 1 (2010). H. J. Leverenz, Adv. Electrochem. Sci. Eng., 12, 61 (2010). M. Grätzel, Philos. Trans. R. Soc. A, 365, 993 (2007). A. Fujishima and K. Honda, Bull. Chem. Soc. Jpn., 44, 1148 (1971); Nature, 238, 37 (1972). D. Laser and A. J. Bard, J. Electrochem. Soc., 123, 1027 (1976). B. Miller and A. Heller, Nature, 262, 680 (1976). G. Hodes, D. Cahen, and J. Manassen, Nature, 260, 312 (1976). M. S. Wrighton, A. B. Bocarsly, J. M. Bolts, A. B. Elllis, and K. D. Legg, in “Semiconductor Liquid-Junction Solar Cells,” A. Heller, Ed., The Electrochemical Society, Princeton, NJ, Proc. Vol. 77-3, 1977, p. 138. H. Tributsch, Ber. Bunsenges. Phys. Chem., 81, 361 (1977). M. S. Wrighton, A. B. Ellis, P. T. Wolczanski, D. L. Morse, H. B. Abrahamson, and D. S. Ginley, J. Am. Chem. Soc., 98, 2774 (1976). J. G. Mavroides, J. A. Kafalas, and D. F. Kolesar, Appl. Phys. Lett., 28, 241 (1976). T. Watanabe, A. Fujishima, and K. Honda, Bull. Chem. Soc. Jpn., 49, 355 (1976). M. Halman, Nature, 275, 115 (1978). B. Kraeutler and A. J. Bard, J. Am. Chem. Soc., 99, 7729 (1977). C. R. Dickson and A. J. Nozik, J. Am. Chem. Soc., 100, 8007 (1978). H. Gerischer and F. Willig, “Topics in Current Chemistry,” Vol. 61, Springer-Verlag, Berlin, 1976, p. 31. H. Gerischer, J. Electrochem. Soc., 125, 218C (1978). M. T. Spitler and B. A. Parkinson, Acc. Chem. Res., 42, 2017 (2009). F. Bella, C. Gerbaldi, C. Barolo, and M. Grätzel, Chem. Soc. Rev., 44, 3431 (2015). H. Tsubomura, M. Matsumura, Y. Nomura, and T. Amamiya, Nature, 261, 402 (1976). G. A. Kenney and D. C. Walker, Electroanal. Chem., 5, 1 (1971). G. C. Barker, Ber. Bunsenges. Phys. Chem., 75, 728 (1971). E. A. Chandross and R. E. Visco, J. Phys. Chem., 72, 378 (1968). H. Kuhn, J. Chem. Phys., 53, 101 (1970). M. Spiro, J. Chem. Soc., Faraday Trans. 1, 75, 1507 (1979). D. S. Miller, A. J. Bard, G. McLendon, and J. Ferguson, J. Am. Chem. Soc., 103, 5336 (1981). D. Meissner and R. Memming, Electrochim. Acta, 37, 799 (1992). C. D. Jaeger and A. J. Bard, J. Phys. Chem., 83, 3146 (1979). W. W. Dunn, Y. Aikawa, and A. J. Bard, J. Am. Chem. Soc., 103, 3456 (1981). M. D. Ward and A. J. Bard, J. Phys. Chem., 86, 3599 (1982). M. D. Ward, J. R. White, and A. J. Bard, J. Am. Chem. Soc., 105, 27 (1983). I. M. Kolthoff and J. T. Stock, Analyst, 80, 860 (1955). F. Scholz and B. Meyer, Chem. Soc. Rev., 23 341 (1994); Electroanal. Chem., 20, 1 (1998). N. Vlachopoulos, P. Liska, J. Augustynski, and M. Grätzel, J. Am. Chem. Soc., 110, 1216 (1988). B. O’Regan and M. Grätzel, Nature, 353, 737 (1991). T. Gerfin, M. Grätzel, and L. Walder, Prog. Inorg. Chem., 44, 345 (1997).

20.6 References

82 A. B. Bocarsly, H. Tachikawa, and L. R. Faulkner, in “Laboratory Techniques in Electroana-

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114

115 116

lytical Chemistry,” 2nd ed., P. T. Kissinger and W. R. Heineman, Eds., Marcel Dekker, New York, 1996, Chap. 28. Yu. V. Pleskov and Z. A. Rotenberg, Adv. Electrochem. Electrochem. Eng., 11, 1 (1978). G. C. Barker, D. McKeown, M. J. Williams, G. Bottura, and V. Concialini, Faraday Discuss. Chem. Soc., 56, 41 (1974). Yu. V. Pleskov, Z. A. Rotenberg, V. V. Eletsky, and V. I. Lakomov, Faraday Discuss. Chem. Soc., 56, 52 (1974). A. Brodsky and Yu. V. Pleskov, in “Progress in Surface Sciences,” Vol. 2, Part 1, S. G. Davidson, Ed., Pergamon, Oxford, 1972. P. Hapiot, V. V. Konovalov, and J.-M. Savéant, J. Am. Chem. Soc., 117, 1428 (1995). A. Henglein, Electroanal. Chem., 9 163 (1976). M. S. Alam, E. Maisonhaute, D. Rose, A Demarque, J.-P. Larbre, J.-L. Marignier, and M. Mostafavi, Electrochem. Commun., 35, 149 (2013). A. Latus, M. S. Alam, M. Mostafavi, J.-L. Marignier, and E. Maisonhaute, Chem. Commun., 51, 9089 (2015). S. P. Perone and J. R. Birk, Anal. Chem., 38, 1589 (1966). G. L. Kirschner and S. P. Perone, Anal. Chem., 44, 443 (1972). R. A. Jamieson and S. P. Perone, J. Phys. Chem., 76, 830 (1972). J. I. H. Patterson and S. P. Perone, J. Phys. Chem., 77, 2437 (1973). D. D. M. Wayner, D. J. McPhee and D. Griller, J. Am. Chem. Soc., 110, 132 (1988). B. A. Sim, P. H. Milne, D. Griller, and D. D. M. Wayner, J. Am. Chem. Soc., 112, 6635 (1990). H.-G. Liao, D. Zherebetskyy, H. Xin, C. Czarnik, P. Ercius, H. Elmlund, M. Pan, L.-W. Wang, and H. Zheng, Science, 345, 916 (2014). N. Hodnik, G. Dehm, and K. J. J. Mayrhofer, Acc. Chem. Res., 49, 2015 (2016). H. Rasool, G. Dunn, A. Fathalizadeh, and A. Zettl, Phys. Status Solidi B, 253, 2351 (2016). Y. Sasaki, R. Kitaura, J. M. Yuk, A. Zettl, and H. Shinohara, Chem. Phys. Lett., 650, 107 (2016). W. Xin, I. M. De Rosa, P. Ye, L. Zheng, Y. Cao, C. Cao, L. Carlson, and J.-M. Yang, J. Phys. Chem. C, 123, 4523 (2019). C. T. Lin and N. Sutin, J. Phys. Chem., 80, 97 (1976). W. J. Albery, Acc. Chem. Res., 15, 142 (1982). M. D. Archer, J. Appl. Electrochem., 5, 17 (1975). W. J. Albery and M. D. Archer, Electrochim. Acta, 21, 1155 (1976). W. J. Albery and M. D. Archer, J. Electrochem. Soc., 124, 688 (1977). W. J. Albery and M. D. Archer, J. Electroanal. Chem., 86, 1, 19 (1978). A. J. Bard, Ed., “Electrogenerated Chemiluminescence,” Marcel Dekker, New York, 2004. A. Kapturkiewicz, Adv. Electrochem. Sci. Eng., 5, 1 (1997). L. R. Faulkner and R. S. Glass, in “Chemical and Biological Generation of Excited States,” W. Adam and G. Cilento, Eds., Academic Press, New York, 1982, pp. 191–227. L. R. Faulkner, Methods Enzymol., 57, 494 (1978). L. R. Faulkner and A. J. Bard, Electroanal. Chem., 10, 1 (1977). T. Kuwana, Electroanal. Chem., 1, 197 (1966). N. Sojic, Z. Ding, A. Kapturkiewicz, W. Miao, G. Xu, I. Svir, F. Paolucci, J. Li, C. Hogan, and R. Forster, “Analytical Electrogenerated Chemiluminescence,” Royal Society of Chemistry, London, 2020. C. Ma, Y. Cao, X. Gou, and J. J. Zhu, Anal. Chem., 92, 431 (2020). H. Qi and C. Zhang, Anal. Chem., 92, 524 (2020).

925

926

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

117 K. Hiramoto, E. Villani, T. Iwama, K. Komatsu, S. Inagi, K. Y. Inoue, Y. Nashimoto, K. Ino,

and H. Shiku, Micromachines, 11, 530 (2020). 118 L. Li, Y. Chen, and J.-J. Zhu, Anal. Chem., 89, 358 (2017). 119 Z. Liu, W. Qi, and G. Xu, Chem. Soc. Rev., 44, 3117 (2015). 120 S. Parveen, M. S. Aslam, L. Hu, and G. Xu, “Electrogenerated Chemiluminescence: Proto121 122 123 124 125 126 127

128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144

145 146 147 148 149 150 151 152 153 154

cols and Applications,” Springer, Dordrecht, 2013. R. J. Forster, P. Bertoncello, and T. E. Keyes, Annu. Rev. Anal. Chem., 2, 359 (2009). W. Miao, Chem. Rev., 108, 2506 (2008). A. W. Knight and G. M. Greenway, Analyst, 119, 879 (1994). R. A. Marcus, J. Chem. Phys., 43, 2654 (1965). R. A. Marcus (Nobel Lecture), Angew. Chem. Int. Ed. Engl., 32, 1111 (1993). J. T. Maloy and A. J. Bard. J. Am. Chem. Soc., 93, 5968 (1971). J. T. Maloy, in “Computers in Chemistry and Instrumentation,” Vol. 2, “Electrochemistry,” J. S. Mattson, H. B. Mark, Jr., and H. C. MacDonald, Jr., Eds., Marcel Dekker, New York, 1972, Chap. 9. G. H. Brilmyer and A. J. Bard, J. Electrochem. Soc., 127, 104 (1980). J. E. Bartelt, S. M. Drew, and R. M. Wightman, J. Electrochem. Soc., 139, 70 (1992). L. R. Faulkner, J. Electrochem. Soc., 124, 1725 (1977). M. M. Collinson, R. M. Wightman, and P. Pastore, J. Phys. Chem., 98, 11942 (1994). M. M. Collinson and R. M. Wightman, Science, 268, 1883 (1995). D. J. Freed and L. R. Faulkner, J. Am. Chem. Soc., 93, 2097 (1971). L. R. Faulkner, Int. Rev. Sci.: Phys. Chem. Ser. Two., 9, 213 (1975). M. Chang, T. Saji, and A. J. Bard, J. Am. Chem. Soc., 99, 5399 (1977). I. Rubinstein and A. J. Bard, J. Am. Chem. Soc., 103, 512 (1981). H. S. White and A. J. Bard, J. Am. Chem. Soc., 104, 6891 (1982). T. Kai, M. Zhou, S. Johnson, H. S. Ahn, and A. J. Bard, J. Am. Chem. Soc., 140, 16178 (2018). Z. Deng, B. M. Quinn, S. K. Haram, L. E. Pell, B. A. Korgel, and A. J. Bard, Science, 296, 1293 (2002). Q. Zhai, J. Li, and E. Wang, ChemElectroChem, 4, 1639 (2017). D. Ege, W. G. Becker, and A. J. Bard, Anal. Chem., 56, 2413 (1984). A. J. Bard and G. M. Whitesides, U. S. Patent 5,221,605 (June 22, 1993). H. Yang, J. K. Leland, D. Yost, R. J. Massey, Nat. Biotechnol., 12, 193 (1994). G. F. Blackburn, H. P. Shah, J. H. Kenten, J. Leland, R. A. Kamin, J. Link, J. Peterman, M. J. Powell, A. Shah, D. B. Talley, S. K. Tyagi, E. Wilkins, T.-G. Wu, and R. J. Massey, Clin. Chem., 37, 1534 (1991). N. R. Hoyle, J. Biolumin. Chemilumin., 9, 289 (1994). J. B. Noffsinger and N. D. Danielson, Anal. Chem., 59, 865 (1987). J. Leland and M. J. Powell, J. Electrochem. Soc., 137, 3127 (1990). W. Miao, J.-P. Choi, and A. J. Bard, J. Am. Chem. Soc., 124, 14478 (2002). J. A. Holeman and N. D. Danielson, J. Chromatogr., 679, 277 (1994). T. M. Downey and T. A. Nieman, Anal. Chem., 64, 261 (1992). L. L. Shultz, J. S. Stoyanoff, and T. A. Nieman, Anal. Chem., 68, 349 (1996). M. Sentic, M. Milutinovic, F. Kanoufi, D. Manojlovic, S. Arbault, and N. Sojic, Chem. Sci., 5, 2568 (2014). F. Mavré, R. K. Anand, D. R. Laws, K.-F. Chow, B.-Y Chang, J. A. Crooks, and R. M. Crooks, Anal. Chem., 82, 8766 (2010). K.-F. Chow, F. Mavré, J. A. Crooks, B.-Y. Chang, and R. M. Crooks, J. Am. Chem. Soc. 131, 8364 (2009).

20.7 Problems

155 156 157 158 159 160 161 162 163 164 165 166 167 168

X. Zhang and A. J. Bard, J. Phys. Chem., 92, 5566 (1988). Y. S. Obeng and A. J. Bard, Langmuir, 7, 195 (1991). C. J. Miller, P. McCord, and A. J. Bard, Langmuir, 7, 2781 (1991). M. Gleria and R. Memming, Z. Phys. Chem., 101, 171 (1976). L. S. R. Yeh and A. J. Bard, Chem. Phys. Lett., 44, 339 (1976). J. D. Luttmer and A. J. Bard, J. Electrochem. Soc., 125, 1423 (1978). F.-R. F. Fan, A. Mau, and A. J. Bard, Chem. Phys. Lett., 116, 400 (1985). H. D. Abruña and A. J. Bard, J. Am. Chem. Soc., 104, 2641 (1982). I. Rubinstein and A. J. Bard, J. Am. Chem. Soc., 102, 6641 (1980). K. M. Maness, R. H. Terrill, T. J. Meyer, R. W. Murray, and R. M. Wightman, J. Am. Chem. Soc., 118, 10609 (1996). Q. Pei, Y. Yang, G. Yu, C. Zhang, and A. J. Heeger, J. Am. Chem. Soc., 118, 3922 (1996). H. C. Moon, T. P. Lodge, and C. D. Frisbie, J. Am. Chem. Soc., 136, 3705 (2014). H. K. Seok, J. I. Lee, S. Kim, and S. K. Moon, ACS Photonics, 5, 267 (2018). P. R. Michael and L. R. Faulkner, J. Am. Chem. Soc., 99, 7754 (1977).

20.7 Problems 20.1 The properties of a semiconductor photovoltaic cell can be deduced (neglecting the effect of internal resistance) from i − E curves such as those in Figure 20.3.2a. Assume that couple O/R is Fe(CN)3− (0.1 M)∕Fe(CN)4− (0.1 M), that Efb = − 0.20 V vs. SCE, and 6 6 that the limiting photocurrent is governed by the light flux, 6.2 × 1015 photons/s, which is completely absorbed and converted to separated electron–hole pairs. (a) For a cell comprising the n-type semiconductor and a Pt electrode in the electrolyte, what are the maximum open-circuit voltage and the short-circuit current under illumination? (b) Sketch the expected output current vs. output voltage for this cell. (c) What is the maximum output power for the cell? 20.2 An expression frequently used for the “thickness of the space-charge region,” L1 , is (5, 8, 24, 25) ( )1∕2 ( )1∕2 2𝜀𝜀0 Δ𝜙 L1 = Δ𝜙 ≈ 1.1 × 106 𝜀 cm (ND in cm−3 , Δ𝜙 in V) eN D ND (20.7.1) (a) Sketch the variation of L1 with Δ𝜙 for N D values of 5 × 1017 , 5 × 1016 , 5 × 1015 , and 5 × 1014 cm−3 for a semiconductor with 𝜀 = 10. (b) For efficient utilization of light, most of the radiation should be absorbed within the space charge region. The absorption of light follows a Beer’s law relationship with an absorption coefficient, 𝛼 (cm−1 ); hence, a “penetration depth” of ∼1/𝛼 can be estimated. If 𝛼 = 105 cm−1 and a band bending of 0.5 V is to be used, what is the recommended doping level, N D ? (c) Why would much lower doping in a semiconductor photoelectrochemical cell be undesirable? 20.3 Consider the dye sensitization of a p-type semiconductor electrode with energy levels as depicted in Figure 20.7.1. Explain how this system operates under illumination.

927

20 Photoelectrochemistry and Electrogenerated Chemiluminescence

Electron energy →

928

O R

Semiconductor

Dye

Solution

Figure 20.7.1 Energy levels in separated components of a dye-sensitized electrode.

20.4 Consider the Mott–Schottky plots in Figure 20.2.3. (a) Estimate the doping levels of the two InP electrodes and their flat-band potentials in 1 M KCl, 0.01 M HCl. (b) How does the difference in Efb for n- and p-type InP compare to Eg for this material (1.3 eV)? 20.5 The following cell is proposed as a photoelectrochemical storage battery: n-TiO2 ∕0.2M Br− (pH = 1)∕0.1 M I− 3 (pH = 1)∕Pt The Efb for n-TiO2 under these conditions is −0.30 V vs. SCE. Under irradiation, a photo-oxidation producing Br2 occurs at TiO2 . (a) Write the half-cell reactions that occur at both electrodes during irradiation. What is the maximum open-circuit voltage under illumination (assuming no liquid junction potential)? (b) During the “photo-charge cycle” Br2 and I− accumulate in the cell. If half of the I− is 3 converted during charging, and a platinum electrode is used in the Br2 /Br− cell for dark discharge, what is the cell voltage of the charged cell? Write the half reactions that take place during dark discharge. 20.6 It is possible to measure the efficiency of triplet production in some redox processes by using trans-stilbene as an interceptor. For example, the fluoranthene (FA) triplet undergoes the reaction 3

FA∗ + trans-stilbene → 3 stilbene∗ + FA

The stilbene triplet decays to cis and trans ground states with a known partition ratio. Devise a bulk electrolysis experiment to measure the efficiency of triplet formation + (triplets per redox event) in the reaction between 10-MP ∙ and FA−∙. Derive an equation relating the measured quantities to this efficiency. Why is bulk electrolysis needed? 20.7 Use the half-wave potentials for the waves of ECL intensity in Figure 20.5.4 to esti′ + mate E0 for TMPD ∙ + e ⇌ TMPD and Py + e ⇌ Py−∙. Then, estimate the free energy

20.7 Problems

released in the reaction +

TMPD ∙ + Py−∙ → TMPD + Py From the fluorescence spectrum in Figure 20.5.3, estimate the energy of 1 Py* relative to + Py. Comment on the probability of forming 1 Py* in the reaction between TMPD ∙ and − Py ∙ and account for the light. 20.8 Interpret Figure 20.5.4b. 20.9 Figure 20.7.2 provides an ECL transient for the energy-sufficient reaction between the cation radical of thianthrene (TH) and the anion radical of 2,5-diphenyl-1,3,4-oxadiazole (PPD; Figure 1). The ECL is from 1 TH* . The transient was generated by a sequence of four potential steps. Step 1 generated PPD−∙ for t f = 500 ms, then Step 2 produced + TH ∙ for 50 ms. For the next 10 ms (Step 3, 0.10 ≤ t r /t f ≤ 0.12), the electrode was + returned to an intermediate potential where TH ∙ was reduced heterogeneously. + Finally, the electrogeneration of TH ∙ was resumed in Step 4 at the same potential used in Step 2. (a) The dip in light output lagged the application of the Step 3. This point was cited as evidence that the light-producing reaction took place in solution at a distance from the electrode. Argue this point. Can you estimate the distance of the reaction zone from the surface? + (b) The redox reaction rate between PPD−∙ and TH ∙ can be predicted by simulation, given the diffusion coefficients for the system. Figure 20.7.2 shows that the observed light intensity followed the square of the reaction rate. Rationalize this observation mechanistically.

Step 3

Step 4

ECL intensity or (reaction rate)2

Step 2

0.05

0.10

tr/tf

0.15

0.20

Figure 20.7.2 ECL intensity (open circles) and square of redox reaction rate (filled circles) vs. time for + reaction of TH ∙ and PPD−∙ . Time, tr , is measured beyond Step 1 of a four-step transient experiment. Step 1 duration, tf , was 500 ms. ECL intensity is in arbitrary units, so a vertical scaling factor is needed to fit the light intensity and the square of reaction rate. No other parameters are required. [Adapted from Michael and Faulkner (168)/American Chemical Society.]

929

931

21 In situ Characterization of Electrochemical Systems Electrochemical investigators often employ nonelectrochemical methods (e.g., forms of spectroscopy or microscopy), to gather additional information about their systems (1–7). In this chapter, we explore both the opportunities and the challenges involved in coupling an electrochemical cell to an apparatus for nonelectrochemical characterization. Three modes are often identified: • In-situ approaches involve the nonelectrochemical examination of a cell component (e.g., the electrode surface) without removal, and perhaps even while the cell remains under potential control with passage of current. • Operando approaches involve the nonelectrochemical characterization of an operating system while it is providing electrochemical performance data. In electrochemistry, this term is used to label nonelectrochemical measurements on operating practical devices, such as batteries or fuel cells.1 • Ex situ approaches involve removal of a component from a cell for subsequent examination in a different environment, e.g., in air or high vacuum. Many types of ex-situ measurements are possible; however, one must be on guard for alteration of the electrochemical component upon removal from the cell. In Sections 21.1–21.7, we will focus on widely used methods having in-situ or operando capabilities. Most can also be employed ex situ; however, we will not separately discuss ex-situ usage. In Section 21.8, we turn briefly to important tools that function exclusively ex situ (or almost so). For all methods, references are mainly to principal reviews.2

21.1 Microscopy The invention of scanning tunneling microscopy (STM) by Binnig and Rohrer (8) provided a completely new concept for imaging surfaces and was quickly recognized by the Nobel Prize. Their innovative use of piezoelectric micropositioning to probe a surface on an atomic distance scale inspired the development of other methods, including SECM (Chapter 18), which is fundamentally electrochemical. Here, we address several nonelectrochemical forms of microscopy that are effective for characterizing electrochemical systems in situ.

1 In original intent, the word operando related to a subgroup of in situ methods, but the two terms are increasingly used as synonyms. 2 Chapter 17 of the second edition covers fundamental principles of a wide range of nonelectrochemical characterization methods, as well as more detail about implementation than is included here. Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

932

21 In situ Characterization of Electrochemical Systems

21.1.1

Scanning Tunneling Microscopy

STM is useful for studying well-defined, atomically smooth surfaces. The method rests on the measurement of the tunneling current between the electrode surface and a sharp metal tip (W or Pt) as the tip is brought close to the surface and scanned across it (Figure 21.1.1). The movement of the tip is controlled by piezoelectric elements (piezos), essentially exactly as employed in SECM. Tunneling occurs when the tip is so close to the surface that the wave functions of the tip atoms overlap those of the substrate atoms (Section 3.5.2). This current is nonfaradaic and produces no chemical change. A simplified expression for the tunneling current, itun , is itun = (constant)V exp (−2𝛽d) = V ∕Rtun

(21.1.1)

where V is the tip–substrate bias, d is the distance between tip and surface, and Rtun is the effective resistance of the tunneling gap, typically 109 to 1011 Ω. The pre-exponential constant is related to the overlap of density-of-states distributions (filled and vacant), and 𝛽 depends on the energy barrier between tip and sample, which is related to the work function of the sample (9–12). Usually, 𝛽 ≈ 1 Å−1 . Appreciable currents (pA to nA) flow only when the tip is closer than a few nm from the surface and under a bias of mV to a few V. STM is generally carried out in the constant current mode, in which one obtains an image by moving the tip toward the surface until a tunneling current flows and then scanning the tip over the surface in the x − y plane. The tunneling current is kept constant by moving the tip up and down (i.e., by varying z), and the image is z vs. (x, y). One can also obtain an image by measuring itun with the z-piezo left fixed (the constant height mode). Then, the image is itun vs. (x, y). The observed STM image basically consists of a topographic trace of the surface, modified by local variations in work function. A typical image is presented as a color or grayscale plot, where different heights are depicted as different shades or different colors, as in Figure 21.1.2a (13). To obtain high-resolution STM images, the tip must be very sharp; its movement must be controlled within fractions of an Å; and thermal changes and vibrational movements must be avoided. In electrochemical STM (EC-STM) (1, 14–23), the working electrode is mounted horizontally at the bottom of a small cell that also contains auxiliary and reference electrodes. The scanning tip is held above the working electrode. The potentials of the substrate and the tip, ES and ET , are controlled independently with a bipotentiostat (Section 16.4.4) so that ES produces z piezo Tip

y piezo x piezo i Tip d ≤ 1 nm

Tunneling current

(a)

(b)

Figure 21.1.1 (a) Tunneling between tip and sample atoms. (b) Tip attached to three piezo elements used to position the tip and scan it across a surface.

21.1 Microscopy

10 nm 0.2

0.1 5 nm 0.0

nm 0.12

2

0.06

1 nm

0.00 0 1 nm 0

5 nm

(a)

10

2

0

0

(b)

Figure 21.1.2 STM images of HOPG in air. (a) Grayscale image at low resolution. (b) Higher resolution topographic plot of the area in the square in (a). [Liu, Chang, and Bard (13). Reprinted with permission from American Chemical Society.]

the reaction of interest and ET gives the desired bias. Because only the tunneling current is of interest in STM, electrochemical reactions that occur at the tip are undesirable. Thus, in EC-STM (as opposed to ex situ STM in air or vacuum) the tip is coated with glass or polymer, with only a very small area at the very bottom left exposed. Moreover, the tip potential is chosen to be in a region where electrochemical reactions do not occur. The thickness of the electrolyte layer over the working electrode must be small, so that only the tip, and not the tip holder or a piezo, contacts the solution. Experimental details are available (14–23). EC-STM is usually used to examine HOPG or single crystals of metals or semiconductors. In many cases, one can resolve the atomic structure of the electrode and see structural features. Examples are found in Figure 21.1.2, showing STM images of an HOPG basal plane surface (13), and Figure 14.5.1a, providing an image of Au(100) surface with adsorbed iodide (24). Such images are maps of the electron density distribution across the surface. The observed corrugations, reflecting the periodicity of this distribution, are much larger for HOPG, where the electron density is less delocalized through the material, than for Au. The variations in intensity can sometimes be identified with the actual atomic structure of the surface, but other aspects of the sample can also influence the image. With HOPG, for example, only half of the surface carbon atoms are imaged. Alternate surface atoms are situated directly over carbon atoms in the second layer (Figure 14.4.3), and, for those surface atoms, the electron density is directed downward and is less available for overlap with the tip than for the other surface atoms (13). When good atomic resolution is obtained, it is generally possible to obtain a full view of atomic arrangements on a surface [e.g., on Au(100)] and to determine the interatomic spacings (24). Sometimes, one can distinguish atoms of one type from neighbors of a different type, because the difference in electronic structure between elements leads to different tunneling behavior, which provides contrast in the images. This effect is readily visible in Figure 14.5.1a. For the most part, chemical information is sparse in STM. The investigator mainly gains structural insights. However, EC-STM adds a chemical dimension, because the behavior in STM often varies as the substrate potential is altered. Figure 14.5.1a provides an example in which the substrate potential was stepped to a new value during the image acquisition, so that the structural change caused by the potential step was fully registered in a single image. Time-resolved STM now allows the real-time observation of atomic-level structural changes on surfaces (1, 25). Many are chemically significant. Figure 21.1.3 provides an example in

933

934

21 In situ Characterization of Electrochemical Systems

Figure 21.1.3 (a–f ) STM images taken sequentially as a Bi island was electrodeposited on Au(111) at −0.40 V vs. SCE from a solution of Bi3+ in 0.1 M HClO4 . The imaged area is 25 Å × 38 Å. [Matsushima, Lin, Morin, and Magnussen (26) with permission from Royal Society of Chemistry.]

(a) 0.0 s

(b) 0.1 s

(c) 0.2 s

(d) 0.3 s

(e) 0.4 s

(f) 0.5 s

which Bi is being electrodeposited on an Au(111) substrate (26). A well-ordered island of Bi has developed, and in the successive frames of Figure 21.1.3, one can see the continued growth of the island over a period of 0.5 s by addition of atoms along the edge. The process involves kink propagation, in which two rows of Bi are jointly developed in a zig-zag pattern. Time-resolved EC-STM has also allowed direct observation of phase changes in adsorbed CO layers on Pt(111) during a voltammetric scan (27). Another interesting chemical example involves the EC-STM of mixed monolayers on HOPG of protoporphyrin IX (PP; Figure 1) and iron protoporphyrin IX [FePP; Figure 1 (MPP, M = Fe)] (28). Results are presented in Figure 21.1.4. PP is not electroactive in the studied potential region, but FePP shows a reduction wave characteristic of a surface species (Section 17.2.2) with Ep at −0.48 V (Figure 21.1.4b). With a tip bias of −0.1 V and a substrate potential of −0.41 V, the tunneling current is greater when the tip is above FePP compared to PP, so FePP molecules appear much brighter (Figure 21.1.4a). The maximum difference between FePP and PP and the largest tunneling current (i.e., apparent height) are both observed near Ep for the FePP (Figure 21.1.4c,d). The increase in current for FePP near Ep was ascribed to resonant tunneling through the molecules. Although STM can be wonderfully illuminating for certain questions, it is a demanding method, mainly practiced using custom-designed and -constructed apparatus. Commercial sources of equipment are limited. 21.1.2

Atomic Force Microscopy

Atomic force microscopy (AFM) is based on varying vertical deflections of a tiny cantilever holding a sharp tip, frequently of Si3 N4 or SiO2 , as the tip is scanned over a surface (Figure 21.1.5) (9, 12, 29–32). These deflections are caused by short-range forces between tip and surface. They are measured by observing the position of a laser beam reflected from the cantilever. In an atomic force microscope, the cantilever and tip are stationary, and the sample is mounted on a piezo scanner, which moves it in the z-direction (toward and away from the tip) as well as in the x- and y-directions. The cantilever and integral tip are fabricated together, usually by photolithography of Si. The incident beam is reflected to an array of photocells

21.1 Microscopy

1 1′

10

5

0 5′

–10

1Å

5 nm 6 2

2′

10 6′

0 –10

7

j/mA cm–2 3′

3

7′

10 0 –10

4′

8

8′

10

4

0 –10

9′

9

–0.8 –0.6 –0.4 –0.2 E/V vs. SCE (a)

(b)

0 (c)

5 10 15 20 Distance/Å (d)

Figure 21.1.4 (a) STM images of mixed adsorbed layers of PP and FePP on HOPG. Wax-coated Pt tip in 0.05 M Na2 B4 O7 with FePP:PP ratios of (1) 0:1, (2) 1:4, (3) 4:1, (4) 1:0. HOPG at −0.41 V vs. SCE; tip-substrate bias, −0.1 V; tunneling current, 30 pA. (b) CV at 200 mV/s, where (1’−4’) correspond to (1–4). Scans begin near −0.1 V vs. SCE and first move negatively. Cathodic currents are down. (c) STM images of FePP embedded in an ordered array of PP for different values of substrate potential: (5) −0.15, (6) −0.30, (7) −0.42, (8) −0.55, and (9) −0.65 V vs. SCE, with other conditions as in (a). (d) Cross-sections along the white line in (5) covering three molecules: PP/FePP/PP, where (5′ −9′ ) correspond to (5–9). Imaging was done at a constant current of 30 pA; response is an apparent height. [Adapted from Tao (20) with permission of the American Physical Society.]

by a metal coating on the top surface of the cantilever. Movement of the cantilever due to a force between tip and surface causes a change in the amount of light on each cell, creating a differential electrical signal which is recorded. Further details about the construction and operation of force microscopes are available (9, 12, 29–32). The imaging process depends upon surface forces (33, 34), which have various origins. Consider the plots of cantilever deflection against z-piezo displacement in Figure 21.1.6a. When the tip and the surface are immersed in a solution and are separated by 20 nm or more, there is essentially no force between them, so the cantilever is straight. As the z-piezo moves upward, eventually shortening the distance between sample and tip below 3 nm, attractive van der Waals forces come into play, and the tip moves toward the surface (i.e., the cantilever deflects downward). If the rate of change in force per unit of displacement becomes greater than the spring constant of the cantilever, the tip will “jump to contact” with the substrate. At “contact” the forces are strongly repulsive, and as the z-piezo moves further upward, the cantilever moves up by essentially the same amount (assuming that the tip does not significantly deform the surface). This region allows one to calibrate cantilever deflection in terms of z-piezo travel.

935

21 In situ Characterization of Electrochemical Systems Diode laser Positionsensitive detector

Incident beam

Mirror Incident beam

Reflected beam Reference

Reflected beam

AFM tip AFM tip mount and cell top (stationary)

Counter

Reflective cantilever spring

Tip

Electrolyte

O-ring seal

Cell base with working electrode

Working Piezoelectric scanner (mobile)

z

y x

Surface

(a)

(b)

Figure 21.1.5 (a) A system for AFM in an electrochemical cell. (b) AFM tip near the surface of a sample, such as a working electrode. This diagram is not to scale. The sharp end of the tip is almost on the atomic scale, but the opposite end and the cantilever must be on scale of micromachined parts. Changes in the force on the tip cause the cantilever spring to respond, altering the angle at which the incident beam is reflected. 1.5

2.4

0.6

2 Extend 0

–2 –10

0.3

Retract 10 30 50 70 90 Sample displacement/nm

110

0.0

–0.3 –0.6

8

2.1 (Force/radius)/(mN/m)

Tip deflection/V

0.9

Contact region

Tip deflection/V

4

1.2 (Force/radius)/(mN/m)

936

1.8 1.5 1.2

Contact region

6 4 2

Extend

0

–2 –10

0.9

Retract 10

30 50 70 90 Sample displacement/nm

110

0.6 0.3 0.0

0

20 40 60 Tip–sample separation/nm

(a)

80

–0.3

0

20 40 60 Tip–sample separation/nm

80

(b)

Figure 21.1.6 Effects in AFM of sample movement in the z direction for (a) an attractive interaction (largely van der Waals forces) and (b) a repulsive interaction caused by excess like charge on tip and substrate. The inset shows the raw data (cantilever deflection), which is converted to the force data based on the radius of the tip and the force constant of the cantilever. In this study, the tip is an SiO2 sphere ∼10 μm in diameter. In (a), the lower curve is for retraction after contact and shows the extra force required to break adhesion. [Reproduced with permission from Hillier, Kim, and Bard (35). © 1996, American Chemical Society.]

Figure 21.1.6b relates to a different situation in which there is a repulsive force on the tip at distances greater than 10 nm from the surface. At such large distances, the only forces are electrostatic, which may be either attractive or repulsive. In illustrated case, they are repulsive because there are excess like charges on both the tip and the substrate. They cause the cantilever to deflect away from the sample surface. From the amount of deflection and the known spring constant of the cantilever, typically of the order of 0.01–0.4 N/m, the repulsive force can be estimated.

21.1 Microscopy

AFM has several operating modes, the principal ones being (29) • Contact mode, in which the tip is held in the strongly repulsive zone of the force curve. In effect, the tip is “in contact” with the surface. This mode is prone to damage the surface as the tip is scanned laterally. • Non-contact mode, in which the tip is held away from the surface by 5–15 nm, usually in the flat or attractive portion of the force curve. The tip is operated in vertical oscillation. As the force changes during scanning, the oscillation frequency or phase tends to change, but a feedback loop counters any change by advancing or retracting the tip. This mode is especially useful for samples of softer materials. • Tapping mode, in which the tip is made to oscillate vertically over a range of ∼20 nm, with closest approach made to the strongly repulsive zone of the force curve. This mode avoids most of the surface damage caused in contact mode, because the tip is not in contact during its lateral motion. AFM is much more widely employed than STM, because it is easier to use. Equipment, including tips, are commercially available, and the method is readily accessible to occasional users. AFM is a regular tool in many laboratories, while STM is not. AFM is also more versatile. Substrates need not support a tunneling current, so they can be insulating. Moreover, the examined distance scales can cover a wide range in AFM, from fractions of a nanometer to micrometers. The work represented in Figure 15.6.8, for example, involved AFM measurements of particle sizes in the range of 5–50 nm in a 1 μm × 1 μm field. In electrochemistry, AFM has mainly been used to examine changes in electrode surfaces resulting from underpotential deposition (UPD), corrosion, etching, adsorption, or other forms of electrode modification (1, 14, 21–23). Figure 15.6.8 provides an example. AFM is also used effectively in combined operation with SECM (Section 18.3.3). AFM has found use for quantitative measurements of surface forces (36), which can then be related to surface charge density or other parameters (33–36). For example, a small silica sphere attached to the cantilever attains a charge in aqueous solutions. At pH values above 4, this surface charge is negative due to deprotonation of silanol (SiOH) surface groups. As the tip is moved through the diffuse double layer, the force between the tip and an electrode is a measure of the surface charge (35, 37). 21.1.3

Optical Microscopy

Various forms of optical microscopy have long been used to examine electrochemical systems, both in situ and ex situ (38, 39). The basic principles and practice of optical microscopy are amply described in general sources (40, 41) and will be generally familiar to the reader. Conventional microscopy is relatively easy to use with an operating electrochemical cell, because one can easily include an observation window in the cell. In some cases, the microscope objective can simply be immersed in the electrolyte, perhaps just above the face of a working electrode. For optical microscopy based on light of wavelength 𝜆, the theoretical spatial resolution is the Abbé diffraction limit, in the range of 𝜆/2. Visible photons have wavelengths of 400–800 nm, so the practical resolution is in the hundreds of nanometers. Conventional optical microscopy is well-suited to examining structural changes in electrodeposition, corrosion, or battery technology, which involve features on the micrometer scale or greater (42, 43).

937

938

21 In situ Characterization of Electrochemical Systems

There are many variations of optical microscopy (39, 40), several of which are applied in electrochemistry. We mention three: • Fluorescence microscopy, in which the primary optical beam serves to excite luminescence from the sample. The imaging system is allowed to receive only the luminescence. In electrochemical systems, this methodology is used, for example, to follow fluorescent probes in modification layers on electrodes (44) or native luminescence from nanoparticles (45, 46). • Dark-field microscopy, in which the image is formed only from light scattered from the primary beam. Thus, scatterers appear bright, and a non-scattering background is dark. This can be a favorable mode for imaging particles (38). • Surface-enhanced Raman microscopy, in which the sample is irradiated with monochromatic light from a laser and a Raman-shifted wavelength of light is detected. This becomes practical for microscopy because surfaces can strongly enhance the Raman emission probability, so the method is useful for characterizing specific surface interactions (38). Raman spectroscopy, including the surface enhancement effect, is covered in Section 21.4.2. An entirely different approach is near-field scanning optical microscopy (NSOM) (41, 47). The primary beam does not impinge directly on the sample, but is obstructed by a tiny aperture just a few nanometers in diameter—much smaller than the wavelength range of the light. The beam cannot exit and is reflected. Reflection creates an evanescent wave, which is an extension of the electromagnetic field of the light beyond the plane of reflection. This evanescent wave enables light absorption for a fraction of a wavelength beyond the aperture, and the absorption can be detected in the reflected wave. By rastering the aperture over the sample and measuring the degree of absorption vs. position, one obtains a near-field micrograph. Thus, NSOM is a type of scanning-probe method. The resolution is not limited by diffraction, but by the size of the aperture, and can be much higher than in conventional microscopy. NSOM is capable of imaging on the low nanometer scale and has been employed in situ to study electrode reactions at nanoparticles (38). 21.1.4

Transmission Electron Microscopy

In an electron microscope, electrons are used to “illuminate” the sample (48), and all elements of the instrument, including the sample, are in vacuum. To facilitate management of the beam, especially focusing, the electrons are accelerated to a uniform energy; thus, they are monochromatic, having a single de Broglie wavelength.3 Typically, they are at 10–300 keV, corresponding to wavelengths of 12–2 pm. Since these wavelengths are 4–5 orders of magnitude smaller than those of visible light, electron microscopy is capable of imaging features far smaller than those accessible to optical microscopy. The principal varieties are scanning electron microscopy (SEM) (48, 49), transmission electron microscopy (TEM) (48, 50), and scanning transmission electron microscopy (STEM) (48, 50). All are employed very largely ex situ and are treated in Section 21.8.1; however, TEM and STEM have now been developed for in-situ electrochemical studies. In TEM and STEM, images are created by collecting electrons that have passed through extremely thin samples; hence, they can display the interior structure of the sample. With the thinnest samples, these methods can resolve individual atoms, but resolution degrades with sample thickness. In-situ characterization of electrochemical cells by TEM or STEM is extremely demanding, because the entire cell, including the walls, must be thin enough 3 A moving particle of mass m has a de Broglie wavelength, 𝜆 = h/(2Em)1/2 , where h is the Planck constant, and E is the kinetic energy. Relativistic corrections apply when the velocity approaches c.

21.1 Microscopy

to support electron transmission. Moreover, one must be concerned with the possibility of damage by the electron beam. Experimental conditions and feasibility have been critically reviewed (51). Flow cells like that illustrated in Figure 21.1.7 have been developed to limit beam damage and to refresh the solution in the cell. Windows are usually of Si3 N4 , and the overall cell thickness does not exceed a few hundred nanometers. Successful research has been conducted on systems involving metal deposition, corrosion, electrocatalytic particles, and battery materials (51–55), where morphological changes accompanying electrode reactions are of interest. Figure 21.1.8 shows results from such a case. Events can be observed practically in real time with remarkable spatial resolution, but not with the atomic resolution often achieved with STEM, because the sample and cell walls remain rather thick.

Electron beam Nanoparticle

Counter Reference

Working Si

Si3N4 window

500 nm

Liquid electrolyte Si

Si3N4 window

410 μm

90 μm

90 μm

410 μm

Figure 21.1.7 Liquid flow cell for in situ TEM. Reference- and counter-electrode pads are symmetrically situated on both sides of the working electrode. [Reprinted with permission from Zhu et al. (52)/American Chemical Society.]

(a)

(b)

(c)

(d)

(e)

(f)

50 nm

Figure 21.1.8 STEM images of Pt–Ni catalyst nanoparticles on a carbon working electrode in 0.1 M HClO4 during linear-sweep cycling between 0 and 1.0 V vs. RHE at 100 mV/s. The images were taken at the 0.0-V switching potential during cycles (a) 1, (b) 10, (c) 11, (d) 12, (e) 15, and (f ) 20. A large Ni-rich particle degrades and dissolves at the point of the arrow in each frame. [From Beerman et al. (55) with permission from Royal Society of Chemistry.]

939

940

21 In situ Characterization of Electrochemical Systems

Section 20.4.2 covers a different example in which TEM is used to study an electrochemical process in a captured liquid.

21.2 Quartz Crystal Microbalance In many electrochemical experiments, mass changes occur as material is either deposited on or lost from the electrode. It can be rather helpful to measure those changes in parallel with the electrochemical response, and the quartz crystal microbalance (QCM) is the most important tool for doing so. Commercial instruments are available, and they are easily interfaced to electrochemical systems. 21.2.1

Basic Method

The QCM rests on the piezoelectric properties of a slice of quartz crystal, which cause it to deform when an electric field is applied across it (Figure 21.2.1a) (56–61). The crystal has a mechanical resonant mode depending upon its size and thickness, and it oscillates at its natural frequency, f0 , when used in a circuit that can supply electrical energy to sustain the oscillation. Typical crystals used in QCM have f0 = 5 MHz. If the crystal is altered by placement of a mass per unit area, Δm (g/cm2 ), on its surface, the frequency of oscillation changes by Δf (Hz) according to the Sauerbrey equation: Δf = −2f02 nΔm∕(dq 𝜇)1∕2 = Cf Δm

(21.2.1)

where n is the harmonic number of the oscillation, 𝜇 is the shear modulus of quartz (2.947 × 1011 g cm−1 s−2 ), and dq is the density of quartz (2.648 g/cm3 ). The constants are usually lumped together to yield the sensitivity factor, C f , which is 5.66 × 107 Hz cm2 g−1 (56.6 Hz cm2 μg−1 ) for a 5-MHz crystal in air. The resonant behavior varies with the environment in which the crystal is operating, because an adjacent medium couples to (or “loads”) the crystal surface and affects the shear mode. Thus, f0 and C f values in liquids are lower than Edge view

Au Counter

Node

Reference

Potentiostat

Computer

Shear deformation Au

Quartz

Top view

Working Quartz

Quartz crystal

Au Oscillator

(a)

Frequency counter

(b)

Figure 21.2.1 Apparatus for QCM. (a) Quartz crystal with deposited gold electrodes. The acoustic wave and the deformation (shear) of the crystal under application of an electric field are shown in the edge view. A typical 5-MHz crystal has a 25-mm diameter and disk-shaped contacts of about 12-mm and 6-mm diameter. The active area of the crystal is defined by the applied electric field and is limited by the smaller electrode. (b) Electrochemical cell and measurement system.

21.2 Quartz Crystal Microbalance

those in air or vacuum (62). For usual aqueous solutions, C f is about 42 Hz cm2 μg−1 for a 5-MHz crystal. The frequency also depends on temperature. In an electrochemical experiment (57–60, 63), the quartz crystal is often clamped in an O-ring joint to expose only one of the metal electrodes to the solution (Figure 21.2.1b). This contact (usually Au or Pt) is also the working electrode for electrochemistry and is an element in both the potentiostat and oscillator circuits. Usually, Δf is measured with a frequency counter during a potential step or sweep. A calibration experiment might involve the electrodeposition of Cu or Pb on the electrode, which produces a decrease in frequency. For example, if the Au electrode area is 0.3 cm2 , and a monolayer of Pb (mass about 0.1 μg) is deposited on it, the observed frequency change, taking C f = 42 Hz cm2 μg−1 , would be about 14 Hz. Frequency can normally be measured with a precision of 0.1–1 Hz; hence, fractional monolayer sensitivities are readily achievable. As (21.2.1) shows, the sensitivity increases with the square of f0 and linearly with n; hence, higher sensitivities can be obtained by working with crystals of higher f0 (e.g., 10 MHz) or at higher harmonics. The QCM has been used to measure mass changes at electrodes during underpotential deposition of metals, adsorption/desorption of surfactants, and reactions of redox centers in modification layers. Figure 21.2.2a provides an example involving a film of polyvinylferrocene (PVFc) at an Au electrode on a QCM crystal. Upon oxidation of the PVFc, the QCM frequency . Upon scan decreases (curve 2), signaling the increase in mass caused by incorporation of PF− 6 reversal, reduction of the film occurs, and the frequency increases back to the original value. The mass changes were correlated with the charge passed during the redox processes to show that the redox process did not cause solvent incorporation into the film (64). A second example (Figure 21.2.2b) relates to a working electrode of the high-area carbon YP-50 placed on a gold contact of a QCM crystal also exposed to 0.75 M Et4 PBF4 in MeCN (65).

6 100 μA

5 Δm/μg cm–2

2

1000 Hz

4 3 2

1 1 0

–0.2

0.0

0.2 0.4 E/V vs. Ag/AgCl (a)

0.6

0.8

−1.5

−1.0 −0.5 0.0 Capacitive charge/mC cm–2 (b)

0.5

Figure 21.2.2 Two examples of QCM measurements: (a) Events during a potential scan at a PVFc film on a gold electrode in 0.1 M KPF6 . (1) CV at 10 mV/s. Potential is more positive to the right and anodic currents are up. Scan begins at the negative limit and first moves positively. (2) QCM frequency change. [Reproduced with permission from Varineau and Buttry (64). © 1987, American Chemical Society.] (b) Mass change measured at a QCM of a YP-50F (high-area carbon) electrode vs. charge on the electrode. In MeCN + 0.75 M tetraethylphosphonium tetrafluoroborate. Dashed line is a linear fit. [Adapted from Griffin et al. (65) with permission.]

941

942

21 In situ Characterization of Electrochemical Systems

This system is a model for a supercapacitor electrode made of the same materials. When the carbon electrode is polarized from its PZC, it undergoes charging. Under negative polarization, the carbon charges negatively, and a virtually linear mass change accompanies the charging. However, the change in mass exceeds the amount required for neutralization by adsorbed Et4 P+ , indicating that solvent also enters the porous electrode structure during charging. 21.2.2

QCM with Dissipation Monitoring

The thickness and viscoelasticity of a surface layer often complicate QCM measurements (56–61, 66). Equation 21.2.1 is derived with assumptions that the deposited material is both rigid (i.e., with a large shear modulus like a metal) and present in a very thin film at a shear wave antinode of the crystal. For thicker deposits, especially of materials like polymer films that can undergo viscoelastic shear, the situation is more complex (56, 66). A better approach is to employ a quartz crystal microbalance with dissipation monitoring (QCM-D), which assesses both the mass change and the energy losses to viscoelasticity. A QCM-D instrument operates in the time domain, rather than the frequency domain. When the piezoelectric crystal is operating continuously, as in the basic QCM, an electrical circuit supports the continuous oscillation. In the QCM-D, the driving signal is halted at t = 0, and the oscillatory motion gradually dies away as energy is dissipated, both within the crystal and in its surroundings. The decay can be followed by measuring the voltage difference across the crystal, V x , vs. time. The signal is a damped sinusoid that can be expressed as Vx = Vx0 e−2𝜋ftD sin(2𝜋ft + 𝜙)

(21.2.2)

where f is the frequency, 𝜙 is the phase angle, and Vx0 is the initial sinusoidal amplitude. The parameter D is a dimensionless dissipation factor, embodying the scale and viscoelasticity of the environment. Of the four parameters in (21.2.2), only two, f and D, define the shape of the damped sinusoid; thus, they can be fitted with good precision. Commercial instruments can carry out the measurement process up to 200 times per second, supplying a set of Δf = f − f0 and D every 5 ms. Models are available in the literature for the interpretation of QCM-D results (56, 66). These systems are extensively used in studies of structurally complex electrochemical systems (67–70).

21.3 UV–Visible Spectrometry Optical methods based on UV or visible light have a long history of application in electrochemistry (2–7, 71) and were covered in considerable detail in earlier editions of this book.4 Here we address only three methods in broad, continuing use. 21.3.1

Absorption Spectroscopy with Thin-Layer Cells

Perhaps the simplest spectroelectrochemical experiment is to direct a UV–visible light beam through an electrode surface and to measure absorbance changes resulting from species produced or consumed in the electrode process. The prerequisite is an optically transparent electrode (OTE), many types of which have been reported (2, 7, 71–81). They may be thin films 4 Second edition, Section 17.1; first edition, Section 14.1.

21.3 UV–Visible Spectrometry

For counter electrode

ITO on glass

1.90 cm

Optical axis Optical axis For OTTLE Silicone spacers

1.00 cm

Quartz slide (a)

(b)

For reference electrode (c)

(d)

Figure 21.3.1 (a) Exploded view of an OTTLE with an ITO working electrode. (b) Front view of assembled OTTLE. (c) Cuvette cell for electrochemistry in a spectrophotometer or spectrophotofluorimeter. Top is open for insertion of solution and OTTLE. Other ports receive reference and counter electrodes. The solution in the cuvette is only at the very bottom, below the optical ports. (d) Top view of cuvette cell, showing placements of the OTTLE and the reference and counter electrodes. The OTTLE is at 45∘ vs. either optical axis. [Adapted from Wilson, Pinyayev, Membreno, and Heineman (83) with permission.]

of a conducting oxide (e.g., indium tin oxide), a metal (e.g., Au or Pt), or carbon (e.g., BDD or CNT) deposited on a glass, quartz, or plastic substrate; or they may be fine wire mesh “minigrids” with perhaps several hundred wires per centimeter. A popular mode for transmission experiments involves a thin-layer system (2, 7, 71, 75, 76, 79, 82, 83) (Figure 21.3.1). The working electrode is sealed into a chamber (e.g., between two glass slides spaced 0.05–0.5 mm apart). The chamber is filled from the bottom by capillarity, and contact is maintained with the larger volume, which also includes the reference and counter electrodes. The cell behaves electrochemically like the thin-layer systems covered in Section 12.6. One can carry out CV and coulometry in the ordinary ways, but there is also a facility for obtaining absorption spectra of species in the layer. The advantage of this optically transparent thin-layer electrode (OTTLE) is that bulk electrolysis is achieved inside the layer in a few seconds, so that spectral data can be gathered on a static solution composition. The cell of Figure 21.3.1 has the same size as a standard optical cuvette and can be used for either absorption or fluorescence spectroelectrochemistry. Figure 21.3.2 presents spectra obtained at an OTTLE for a cobalt complex with a Schiff base ligand (84). At −0.9 V vs. SCE, the complex contains Co(II), but at −1.45 V the metal center is reduced to Co(I). If the system is reversible, results like these may be used to obtain precise standard potentials, because the solution in the OTTLE quickly comes to equilibrium with the electrode potential. Spectroelectrochemical methods can be useful for unraveling a complex sequence of charge transfers. Figure 21.3.3 is a display from a classic example (85). The sample is a mixture of cytochrome c and cytochrome c oxidase, which is initially fully oxidized. The experiment is a coulometric titration by the electrogenerated radical cation of methyl viologen (MV2+ ; Figure 1): +

MV2+ + e ⇌ MV ∙ +

(21.3.1)

In solution, an MV ∙ ion can reduce a single heme site in cytochrome c or one of two in the oxidase. The spectra were recorded after increments of charge were passed, and they show that + one of the heme groups in cytochrome c oxidase is reduced first. Then, MV ∙ reduces the heme in cytochrome c before it reacts with the second heme of the oxidase. As in this example, indirect methods must be employed for electrochemical studies of biological macromolecules that are unable to exchange charge directly at an electrode (Section 17.8.2).

943

21 In situ Characterization of Electrochemical Systems

O

j

O Co

C H

N

N

C C H2 H2

a

e f h, i, j

h

C H

i

g

Co(sal2en)

b c d

f 0.1 a.u.

g

e d c b a

350

400

450

500

550

600

650

700

750

λ/nm

Figure 21.3.2 Spectra of the Co(III,II) complex with bis(salicylaldehyde)ethylenediimine, obtained at an OTTLE. Applied potentials: (a) −0.900, (b) −1.120, (c) −1.140, (d) −1.160, (e) −1.180, (f ) −1.200, (g) −1.250, (h) − 1.300, (i) −1.400, and (j) − 1.450 V vs. SCE. [From Rohrbach, Deutsch, and Heineman (84), with permission.] Figure 21.3.3 Coulometric titration of cytochrome c (17.5 μM) and cytochrome c + oxidase (6.3 μM) by MV ∙ generated at an SnO2 OTE. (a) Spectra recorded after uniform increments of charge were passed. (b) Absorbance change vs. total charge. An OTTLE was not used in this experiment, but the cell had a small enough volume that equilibrium could be re-established during a short wait after each charge injection. [From Heineman, Kuwana, and Hartzell (85), with permission from Academic Press, Inc.]

550 nm 100

605 nm 550 nm

Absorbance change/%

0.1 a.u.

Absorbance

944

50

0

0

2

4 Q/mC

(b) 605 nm

Wavelength (a)

6

8

21.3 UV–Visible Spectrometry

One uses smaller molecules to exchange the charge heterogeneously with the electrode and homogeneously with the macromolecules. Such mediators provide a pathway to electrochemical equilibrium and are especially useful in determinations of standard potentials for redox centers in large molecules. 21.3.2

Ellipsometry

The reflection of linearly polarized light from a surface usually produces elliptically polarized light, because the parallel (p) and perpendicular (s) components5 of the light are reflected with different efficiencies and different phase shifts. One can measure the changes of intensity and phase angle and use them to characterize the reflecting system. This method is called ellipsometry (86–92). There are two measured parameters: • The difference in phase angle between the leading and trailing components, Δ. • The ratio of electric field amplitudes expressed as an angle, Ψ = tan−1 |Ep |∕|Es | The values of Δ and Ψ may be recorded as functions of other experimental variables, such as potential or time. Several methods for evaluating Δ and Ψ exist, but the most precise approaches rely on null balance. Light that is polarized linearly at 45∘ with respect to the plane of incidence impinges on the sample. Linear polarization means that |Ep | = |Es | and Δ = 0. After reflection, the beam is passed through an optical compensator that is adjusted to restore the original condition of Δ = 0. The position of the compensator is then a measure of the value of Δ induced by reflection. The resulting linearly polarized beam is then passed through a second polarizer (an analyzer), which is rotated until its axis of transmission is at right angles to the plane of polarization of the oncoming light. Then no light passes through the analyzer to the detector, and the condition of extinction is reached. The angular position of the analyzer provides a measure of Ψ. Extinction will not be achieved unless the compensator and the analyzer are both correctly adjusted. Commercial automated ellipsometers can handle optical analysis of the beam on a millisecond time scale. Ellipsometry is commonly used to study film growth on surfaces. Results from an example concerning an anodic film on aluminum (93) are provided in Figure 21.3.4. The initial measurement is found at the point marked “bare” substrate,6 and measurements made subsequently during film growth are shown as crosses. They trace out a closed loop, and then, with still greater thicknesses (circles), they begin to retrace the loop. With the two measured parameters, Δ and Ψ, and the known optical constants for aluminum, one can determine two fundamental parameters for the film at any stage of growth. In this case, the film is assumed to be nonabsorbing; hence, the refractive index, n, and the thickness, l, can be calculated. The curve in Figure 21.3.4 is the predicted response for n = 1.62. The kinetics of film growth can be studied without removing the electrode from the cell or interrupting the electrolysis. Ellipsometry has also been used to study the growth or alteration of polymer films on electrodes (Chapter 17). Although ellipsometry is usually performed at a single wavelength (typically produced by a laser) and at a single angle of incidence, other modes of operation are possible. For example, the film spectral response can be obtained by making measurements with varying wavelength (spectroellipsometry). 5 In an optical system, the state of polarization is expressed relative to the plane of incidence. For p-polarization, the electric vector is parallel to the plane of incidence. For s-polarization, it is perpendicular. 6 Which is not at 0.0 Å because of the native oxide film on Al.

945

946

21 In situ Characterization of Electrochemical Systems

Figure 21.3.4 Ellipsometric results for anodization of aluminum in 3% tartaric acid (pH 5.5). Numbers along fitted curve indicate film thickness in Å. Crosses: Points measured in the first loop (0–2397 Å). Circles: Points measured in the second loop (>2397 Å). [From Dell’Oca and Fleming (93), reprinted by permission of the publishers, The Electrochemical Society, Inc.]

1400

240

1800

1200

2000 200

1100

Al2O3 n = 1.62 ± 0.02

Δ/deg

Al n = 1.3 ± 0.1 k = 6.5 ± 0.3

2200 160

0.0 and 2397

1000

“bare” substrate 120 900

200 400 80

21.3.3

40

800 45 Ψ/deg

50

55

Surface Plasmon Resonance

The mobile and stationary charges in a metallic conductor comprise a plasma that can be driven into collective oscillation by polarized light. Such a state is called a plasmon. At a surface between a metal and an electronic insulator (including an electrolyte solution), the plasmons behave distinctively and are known as surface plasmons. They are quantized, and light absorption can occur upon excitation, reducing the reflectivity of the metal surface. This effect is called surface plasmon resonance (SPR), and it can be exploited for studies of electrochemical surfaces (94, 95). For a thin-film working electrode (e.g., a 50-nm layer of Au on glass), the reflectivity can be measured as a function of the angle of incidence, 𝜃, as shown in Figure 21.3.5a. The electrode is irradiated from the backside by p-polarized light from a laser, and the front side of the film is exposed to the solution in the cell. As 𝜃 changes, the component of the electric vector of the incident light changes in either surface plane, affecting the probability of light absorption. At a particular angle, the SPR minimum, the reflectivity reaches its lowest point, which can approach zero (Figure 21.3.5a,b). The oscillating electric field associated with a surface plasmon extends into the adjacent solution, but decays exponentially with distance away from the electrode surface. The decay length is on the order of 200 nm. Interactions of this field with charges and dipoles in the solution or adsorbed on the surface modify the wavelength of the surface plasmon; thus, they alter the SPR minimum. Accordingly, the position of the minimum provides information about the surface dielectric constant and layer thickness. It has been used to study the adsorption of biological molecules and self-assembled monolayers on metals, the potential distribution

21.4 Vibrational Spectroscopy

100 80 Reflectivity/%

Figure 21.3.5 Reflectivity curves for clean gold (circles) and sequentially adsorbed monolayers of 11-mercaptoundecanoic acid (triangles), a 22% biotinylated poly-L-lysine monolayer (squares), and the protein avidin (bowties). (a) Wide angular range; (b) magnification of the SPR minimum. The shift in the minimum reflects the thickness of the adsorbed layer. [Reproduced with permission from Frey, Jordan, Kornguth, and Corn (96)/American Chemical Society.]

60 θ

40 20

Au Solution

0 41

42

43

44

θ/deg

45

46

(a)

2.5

Reflectivity/%

2.0 1.5 1.0 0.5 Au 0.0

43.6

+ MUA + PL 43.8

44.0

+ Avidin 44.2

44.4

44.6

θ/deg

(b)

at an electrode surface, and electrochemical processes like underpotential deposition and electrode oxidation. Figure 21.3.5 illustrates the changes in behavior of a clean gold surface after adsorption of successive layers (97). Note the systematic shift in the SPR minimum with increasing film thickness. The physics of surface plasmons have important effects on the optical properties and spectroscopy at electrode surfaces. We will encounter them again in Section 21.4.

21.4 Vibrational Spectroscopy A vibrational spectrum has features arising from the normal modes of the substances in the observed sample (98, 99). Often, these features are narrow, numerous, and readily assignable to particular chemical moieties (e.g., −CH2 , −CH3 , C=O, −Ph). In situ vibrational spectra for electrochemical systems can provide diagnostic molecular information that is extremely difficult to obtain otherwise. They can help an investigator to identify participants in an electrode reaction, even if they are present only in a partial monolayer. A good example is found in Figure 15.3.2. Vibrational spectra can be obtained from operating cells by infrared spectroscopy and Raman spectroscopy. 21.4.1

Infrared Spectroscopy

When the frequency of incident electromagnetic radiation resonates with a molecular vibrational mode, light absorption can occur. The energy changes correspond to the infrared region; hence, one can obtain a vibrational spectrum by recording absorbance vs. frequency—or, more

947

948

21 In situ Characterization of Electrochemical Systems IR window

IR

IR Piston Thin electrolyte layer

Working electrode (a)

IR-transparent electrode (b)

Figure 21.4.1 Optical configurations for IR spectroelectrochemistry: (a) External reflection mode. (b) Attenuated total reflection mode. The window or ATR element must be transparent to IR radiation and insoluble in the solution of interest (e.g., CaF2 , Si, ZnSe). [From Chazalviel and Ozanam (102), with permission.]

commonly, vs. wavenumber7 —of the incident light. Infrared spectroscopy probes molecular vibrations that involve changes in the dipole moment; hence, the vibrations of polar molecular bonds generally correspond to strong infrared bands. In infrared spectroelectrochemistry (100–109), species are probed at the electrode surface and in a thin zone of solution near the surface. The most common configuration is the external reflectance mode (Figure 21.4.1a), where the infrared radiation passes through a window and a thin layer of solution, reflects at the electrode surface, and is detected. The solution layer between the window and electrode must be thin (1–100 μm), because most solvents are good absorbers of IR radiation. The electrode is often placed at the end of a piston that can be used to adjust the spacing between electrode and window. In an alternative mode. the electrode is deposited on one face of an internal reflection element (Figure 21.4.1b). During its passage through the element, the light is reflected several times from the electrode/electrolyte interface, and at each point of reflection its evanescent wave (Section 21.1.3) allows for light absorption in a zone extending a fraction of a wavelength beyond the plane of reflection. Absorption is detectable in the emergent beam. This mode is called attenuated total reflection (ATR). For either mode, the absorbance of the species of interest is usually much smaller than that of other absorbing components, such as solvent and supporting electrolyte; hence, steps must be taken to isolate the absorbances of interest from others that only interfere. Several different approaches have been devised. They are summarized and contrasted in Table 21.4.1.8 These IR methods have been used to study adsorbed species, to examine species produced in the thin layer of solution between electrode and window, and to probe the electrical double layer. In favorable cases, information about the orientation of an adsorbed molecule and the potential dependence of adsorption can be obtained. Figure 15.3.2 provides an example in which ATR-SEIRAS was carried out during a CV scan for a system in which methanol was oxidized at Pt. The spectra show the adsorption of both formate and CO, with the CO existing in two different geometries. The mix of adsorbates changed markedly with electrode potential.

7 The wavenumber, ṽ of light with frequency v is ṽ = v∕c. It expresses the number of wavelengths per unit of distance (usually per cm). The corresponding photon energy is E = hv = hc̃v; thus, the wavenumber of a photon is proportional to its energy. The unit cm−1 is often treated as unit of energy, with 1 eV = 8065.6 cm−1 . The normally observed vibrational spectrum spans about 400–4000 cm−1, or about 0.05–0.5 eV. 8 Names and abbreviations are not standardized in this field and can be confusing. Table 21.4.1 is organized by the name and abbreviation that the authors perceive to be most widely employed. Other names and abbreviations are also identified. The prefix ATR is often added to a name if the ATR mode is used (e.g., ATR-SEIRAS).

21.4 Vibrational Spectroscopy

Table 21.4.1 Principal Forms of Infrared Spectroelectrochemistry Electrochemically Modulated Infrared Reflectance Spectroscopy (EMIRS) Concept

The potential is modulated (usually by stepping) between a value where the species of interest is absent and another value where that species is electrochemically generated or consumed. The technique allows detection of electroreactants or electroproducts while discriminating against the solvent and dissolved species, whose IR absorbances are unaffected by the modulation. Demodulation separates the modulated signal and presents it as a varying dc signal, which can be processed or recorded normally.

Instrument

A broadband IR source and IR monochromator produce the incident beam. A spectrum is produced by scanning the wavelength using the monochromator, while recording the demodulated signal from the detector. Modified versions employ certain types of FTIR instruments.

Limitations

Modulation rates are usually limited to a few Hz, because the high resistance of the thin layer of solution between electrode and window results in a large cell time constant. Measurements in the time domain, including scanning, must occur over periods of seconds or longer. Acquisition times for whole spectra are relatively long. The electrochemical system must be chemically reversible and stable for the period of observation.

Advantages

Spectroscopic needs can be satisfied by a commercial IR spectrometer. The cell and related optics, plus demodulation of the detected signal, must be tailored.

References

(101–103, 106–108, 110)

Infrared Reflection-Absorption Spectroscopy (IRRAS) Concept

The system is observed at a fixed potential using IR radiation that is rapidly modulated with respect to polarization using a photoelastic modulator inserted into the incident beam. Randomly oriented solution molecules absorb p- and s-polarized light to the same extent, so their absorbances are unmodulated. Only p-polarized light is surface-sensitive; therefore, demodulation of the detected signal gives the IR response of the adsorbed layer.

Instrument

This method can be used with either a monochromator-based or an FTIR spectrometer (see SNIFTIRS below). The polarization modulation is at much higher frequency than that used by the interferometer, so it is possible to demodulate the detected signal separately from analysis of the interferogram. A broadband IR source and IR monochromator produce the incident beam. A spectrum is produced by scanning the wavelength using the monochromator, while recording the demodulated signal from the detector, or by conventional FT inversion of an interferogram.

Limitations

Although, spectroscopic needs can be satisfied by a commercial IR spectrometer, there is more tailored instrumentation and signal processing than in most approaches. The modulation and demodulation steps, plus the cell and related optics, must be customized. Time-domain observations are limited spectroscopically by the time required to scan the spectral range of interest (which may be narrow). Depending on instrumental constraints, sequences of spectra can be obtained over periods of perhaps 1 s or longer. Time-domain observations are limited electrochemically by the long cell time constant created by the thin solution layer over the working electrode (see EMIRS above).

Advantages

Observations relate strictly to immobile species in a surface layer. Chemical reversibility is not required, but stability is needed for the period of observation.

Other names

Polarization-modulated internal reflection-absorption spectroscopy (PM-IRRAS)

References

(101–108) (Continued)

949

950

21 In situ Characterization of Electrochemical Systems

Table 21.4.1 (Continued) Surface-Enhanced Infrared Absorption Spectroscopy (SEIRAS) Concept

Surface-enhanced absorption occurs when a vibrational mode of a molecule couples to a surface plasmon on a metal surface of suitable composition and shape. The metal acts, in effect, as an antenna, gathering the incident light and greatly increasing the probability of photon absorption by the molecule (normally an adsorbate). The enhancement is by orders of magnitude, so it becomes possible to observe the surface layer without taking other steps to isolate the signal. However, an ATR element is frequently employed to provide a stronger absorption.

Instrument

This method can be used with either a monochromator-based or an FTIR spectrometer (see SNIFTIRS below).

Limitations

Surface enhancement occurs on surfaces with roughness reflecting three-dimensional granular growth (Section 15.6.3). Time-domain observations are limited by the same factors applying to IRRAS (discussed above).

Advantages

When this method can be applied, it is relatively simple. Optical needs and data acquisition can be satisfied by a commercial IR spectrometer. Only the cell and related optics must be tailored.

References

(100–104, 106–108, 111)

Subtractively Normalized Interfacial FTIR Spectroscopy (SNIFTIRS) Concept

Spectra are obtained separately at two potentials—the first where the electrochemical events of interest have not yet happened, and the second where they are occurring or have been completed. The difference spectrum is the result. IR components that are unaffected by potential change are subtracted out

Instrument

The IR instrument is a Fourier transform infrared (FTIR) spectrometer, in which the sample is illuminated with a continuous spectrum of IR radiation, and an interferogram is recorded in milliseconds. The interferogram can be converted to a spectrum by FT. Usually, multiple interferograms are averaged to improve the signal-to-noise ratio before the transformation is undertaken. The total spectral acquisition time is typically a few tenths of a second to several seconds

Limitations

This method is designed to take a snapshot of changes in the IR spectrum following a potential step. Time-domain observations are limited by the same factors applying to IRRAS (discussed above)

Advantages

Optical and data acquisition can be satisfied by a commercial FTIR spectrometer. Only the cell must be tailored. Spectral subtraction and other data-based operations can be handled in the instrument’s user interface. The electrochemical system being observed need not be chemically reversible or stable

Other names

Potential difference infrared spectroscopy (PDIRS), single-potential alteration infrared spectroscopy (SPAIRS)

References

(101, 103, 105–108, 110)

21.4.2

Raman Spectroscopy

Raman scattering experiments usually involve excitation with light that is not absorbed by the sample. Most of it passes directly through the system or is scattered without a change in photon energy (Rayleigh scattering); however, some photons exchange vibrational quanta with the sample, reflecting the loss or gain in energy by a change in wavelength (Raman scattering) (98, 99). Because the energy changes are quantized, the Raman effect produces light with discrete energy differences relative to the energy of incident light. Usually one studies the Stokes lines, which are Raman emissions at lower energy than the excitation energy (resulting in vibrational excitation

21.4 Vibrational Spectroscopy

of the scatterer). However, a Raman photon can also have more energy than the incident light by being scattered from a system with some initial vibrational activation. This anti-Stokes branch is usually of much lower intensity. Because the energy differences correspond to quanta of the vibrational modes of the scattering species, Raman spectroscopy provides molecular information similar to that from IR spectroscopy. Excitation and detection are in the visible region of the spectrum; therefore, Raman spectroscopy can be employed in electrochemical cells with glass windows and aqueous solutions. Since scattering probabilities and energy shifts are small, an intense monochromatic source is essential. Lasers are universally employed. In-situ measurements in electrochemical systems can be made on dissolved species, but the greatest interest has involved adsorbates. Ordinarily, Raman scattering provides small signals and low sensitivity; therefore, studies of electrochemical systems have relied on techniques that involve large enhancements of the signal (112–121). • Resonance Raman spectroscopy (RRS) is carried out using an excitation wavelength near an electronic transition in the target molecule. The Raman scattering probability can be increased by 104 to 106 , and it becomes practical to monitor dissolved electroreactants and electroproducts in operating cells (122). • Surface-enhanced Raman spectroscopy (SERS) was discovered for molecules adsorbed on roughened surfaces of Ag, Au, and Cu (123–125), where the optical properties give rise to enhancements of Raman scattering by orders of magnitude. Much work has now been reported in which single molecules have been monitored (112–114, 126, 127). The mechanisms of surface enhancement have been of keen interest (114, 115). Most generally, the effect is ascribed to the coupling of a scattering molecule with a surface plasmon (Section 21.3.3) created in a “plasmonic” metal (Ag, Au, or Cu). The plasmon provides an oscillating electric field at the frequency of the light, extending outward from the surface for a distance on the order of 𝜆/2. It increases the efficiency of Raman scattering from a molecule at or near the surface, but to be effective it must have a component perpendicular to the surface. To fulfill this condition, the metal surface must be nonplanar on the molecular scale, in keeping with the common observation that nanoscopically roughened surfaces are generally required for the observation of SERS. An additional resonance enhancement may apply in some systems from chemical interactions between the adsorbate and the host surface (114, 115). Since the oscillating field of the plasmon extends away from an Ag, Au, or Cu surface for a distance, one can cover the SERS-active metal with a very thin layer of another metal and still obtain enhanced signals for adsorbates on the outer surface. This approach greatly broadens the application of SERS and has been widely employed in situ for investigations of electrochemical surfaces (113, 117–120). Tip-enhanced Raman spectroscopy (TERS) is a scanning probe method in which the enhancement is carried out locally by a movable tip (e.g., an AFM tip) made of a plasmonic metal, typically gold (112–114, 116). When this tip is bathed in the incident light, its plasmonic field extends outward and can strongly enhance Raman scattering from molecules on a surface just below the tip. In some manifestations of this method, the tip is relatively simple, e.g., a sharp gold cone, and the excitation is applied separately—from above the surface, from below, or from the side. In other versions, the excitation light is sent through a fiber optic to an apertureless termination in the tip, which acts optically like an NSOM (Section 21.1.3). There are also instruments in which both excitation and emission collection are carried out by fiber optics integrated with the tip. TERS can provide two-dimensional Raman mapping,

951

21 In situ Characterization of Electrochemical Systems

CV scan

591 1354 1421

1641 1492

663 885

1184

*

1542

Raman intensity

0.00 –0.19 –0.60 E/V vs. Ag/AgCl

952

–0.24

0.00 Retr. 1800

1400

1000

600

–1

Raman shift/cm

Figure 21.4.2 In-situ TERS of Nile Blue adsorbed on an ITO working electrode during CV in 50 mM tris buffer + 50 mM NaCl (pH 7.1). v = 10 mV/s. Spectra were acquired every 20 mV during the scan. Those shown here were selected for the initial potential (0.0V vs. Ag/AgCl), E pc (−0.24 V), E 𝜆 (−0.60 V), E pa (−0.19 V), and the return to 0.0 V. The curve labeled Retr. is the spectrum when the tip was retracted from the surface. The asterisk marks a peak originating in the AFM tip. [Reproduced with permission from Kurouski, Mattei, and Van Duyne (128)/American Chemical Society.]

and, since the enhancement is provided by the movable probe, the observed sample can be non-enhancing—even flat or insulating. Spectra from an electrochemical TERS experiment are presented in Figure 21.4.2. The dye Nile Blue was adsorbed on an ITO working electrode, where it was probed by a gold ATM tip illuminated by light from an He–Ne laser. Nile Blue undergoes a reversible reduction to its leuco form, H N

N H2N

O Nile Blue

N+

+ H+ + 2e

H2 N

O

N

Leuco Nile Blue

(21.4.1) which could be followed spectroscopically during a CV scan. The system involved 1.9 × 1012 molecules/cm2 of the dye (about 0.02 monolayer), and the probed area is expected to be a circle of about half the 20-nm radius of curvature of the tip; thus, the investigators concluded that spectra in Figure 21.4.2 were produced by fewer than 10 molecules (128).

21.5 X-Ray Methods

21.5 X-Ray Methods For studies of electrochemical systems, X-rays with energies of several keV offer two important advantages: • Their wavelengths are comparable to the atomic spacings in molecules and solids; hence, their interaction with matter can provide structural information on the atomic scale. • They can penetrate at least a few millimeters of water or many other condensed phases before being lost through scattering; thus, use in situ is practical (if demanding). At electrochemical surfaces, there are relatively few atoms to interact with the X-rays, so signals are much weaker than in experiments with bulk materials. The solution is to employ X-radiation from a synchrotron, which is 8–10 orders of magnitude brighter than a laboratory source. With the availability of time at synchrotrons, in situ X-ray studies of electrochemical systems have blossomed. Many reviews have appeared (129–138). Electrochemical cells must be designed to minimize X-ray absorption losses; thus, cell windows are typically thin films of polyethylene or polyimide, and only thin layers of solution (∼10 μm) are used. The design may feature transmission of the X-ray beam through the working electrode or observation at grazing incidence. If the transmission mode is used, the observed sample must be thin enough to permit appreciable transmission. X-ray absorption spectroscopy (XAS) follows Beer’s law: I = I0 exp(−𝜇x)

(21.5.1)

where I and I 0 are the intensities of transmitted and incident radiation, respectively, x is the distance, and 𝜇 is the linear absorption coefficient, which depends on the photon energy, E. The spectrum of 𝜇 vs. E is characterized by absorption edges, which are the energies just needed to photoionize core electrons, such as 1s electrons (K edge) or 2p3/2 electrons (L3 edge). As examples, K-edge absorption spectra for pure iron and several iron oxides are shown in Figure 21.5.1a. The spectral variations at energies above an absorption edge comprise the X-ray absorption fine structure (XAFS) (139, 140), which has two parts: • The X-ray absorption near-edge structure (XANES)9 makes up a region about 50-eV wide, comprising the absorption edge itself and a bit of the spectrum at higher energies (Figure 21.5.1a,b). The XANES arises from core-electron transitions to high unoccupied orbitals in the atom and is defined by local aspects of the observed atom, especially the oxidation state and the electronegativity of the environment. Figure 21.5.1b illustrates these effects. XANES spectroscopy is one of few methods that can provide information about the state of oxidation for atoms on electrode surfaces. • The extended X-ray absorption fine structure (EXAFS) comprises spectral features in the few hundred eV beyond the XANES (Figure 21.5.1a). The EXAFS results from wave-based interference between the ejected electrons and the atomic structure of the neighboring environment. An EXAFS spectrum can be analyzed to provide information about the average distances of atoms neighboring those responsible for the absorption edge in the sample (e.g., the atoms neighboring Fe in the samples used for Figure 21.5.1a). An example of EXAFS analysis is provided in Figure 21.5.2, which relates to UPD of Cu on Au nanoparticles (141). While we cannot delve into the details of analysis, we can appreciate that it provides the number and spacings of neighboring atoms. The investigators in this study reported Cu–O distances for two axial and four equatorial H2 O ligands for Cu2+ in the system 9 Or near-edge X-ray absorption fine structure (NEXAFS).

953

μ/arbitrary units

21 In situ Characterization of Electrochemical Systems

Figure 21.5.1 (a) Fe K-edge spectra of (solid curve) pure Fe, (dashed curve) 𝛾-Fe2 O3 , (alternating dash-dot curve) Fe3 O4 , and (alternating dash-double-dot curve) 𝛾-FeOOH. (b) XANES for Fe, 𝛾-Fe2 O3 , and 𝛾-FeOOH. The ordinate in both cases is the absorption coefficient due only to the K shell (prior-edge background has been subtracted). The shaded area in (a) corresponds to the spectral range of (b). [Adapted from Long and Kruger (137), with permission.]

EXAFS XANES XAFS = XANES + EXAFS

7070

μ/arbitrary units

954

7210 E/eV (a)

7350

7500

Pure Fe γ–Fe2O3 γ–FeOOH

7100

7120 E/eV (b)

7140

as observed at 0.20 V, as well as Cu–Cu and Cu–Au distances for 2–3 neighboring atoms of each type in the system as observed at −0.51 V. X-ray diffraction (XRD) involves scattering of a monochromatic X-ray beam from an ordered material with measurement of surface reflectivity or determination of the diffraction pattern (131, 132). Diffraction patterns result from interference of beams scattered from surface atoms in accord with Bragg’s law. These patterns provide information about structure and about processes that alter structure, such as reconstruction of a single-crystal electrode or the conversion of a phase.

21.6 Mass Spectrometry Mass spectrometry is commonly coupled to operating electrochemical cells, nearly always for sensitive, selective monitoring of electrolytic products. Specialized commercial devices are available to facilitate the interfacing, which can be carried out in several ways (142–144). Two modes predominate: • Differential electrochemical mass spectrometry (DEMS) (142–146) is widely used to observe volatile species produced at an electrode by sampling them through a porous barrier separating the electrolyte from the mass spectrometer. The electrode itself, if porous, may be the interface (145). Alternatively, one can use a probe with a permeable tip (e.g., a fine

21.7 Magnetic Resonance Spectroscopy

2+

Cu(H2O)6

2.0

|χ(R)|Å–1

0.20 V

1.0 –0.21 V

Cu atom

–0.42 V 0.0

Au atoms

–0.51 V 0

1

2

3 R/Å (a)

4

5

6

C substrate (b)

Figure 21.5.2 (a) Analysis of Cu K-edge EXAFS obtained in situ for a working electrode of Au nanoparticles on carbon, held potentiostatically at the indicated potentials (vs. Hg/Hg2 SO4 ) in 0.5 M H2 SO4 + 2 mM CuSO4 . The analysis involves Fourier transformation of the spectrum, producing the distribution shown here vs. radius from the absorbing atom. The experimental data (black curves) can provide the number of neighboring atoms and their distances from the absorbing atom by fitting against a model (gray curves). (b) Schematic signaling that the upper curves in (a) relate to Cu(H2 O)2+ , but the lower ones apply to UPD Cu on Au. [Adapted with 6 permission from Price, Speed, Kannan, and Russell (141). © 2011, American Chemical Society.]

glass frit or even a polymer film through which the target analyte can diffuse). The probe is placed as near as possible to the working electrode without interfering with electrochemical processes. Often, a porous barrier is treated to make it nonwetting. Differential pumping is used to protect the mass spectrometer and to allow fast transfer of products into the ionization chamber (146). Response times in these systems can be smaller than 50 ms; therefore, real-time analysis of reaction products during CV is possible. The principal limitation of DEMS is that it can be used only for volatile products. While this condition limits application, there are many instances in which small molecules are of primary interest, but are difficult to monitor in other ways (147). Figure 15.3.7 provides an example (148). • Online electrochemistry-mass spectrometry (OEMS) (142–144, 149) involves direct introduction of the sampled electrolyte into the mass spectrometer, most commonly by desorption electrospray ionization (DESI-MS). The sampling is done by placing a capillary probe as close as possible to the electrode and allowing the liquid to flow into the ionizer. The lag between formation of an electroproduct and detection of its mass signal is generally longer than with DEMS. The advantage of OEMS is that it applies to a large range of analytes, even those that are comparatively involatile.

21.7 Magnetic Resonance Spectroscopy 21.7.1

ESR

Electron spin resonance (ESR)10 is frequently used for the detection and identification of electroproducts or intermediates having an odd number of electrons, i.e., radicals, radical ions, and 10 Or electron paramagnetic resonance (EPR).

955

956

21 In situ Characterization of Electrochemical Systems

certain transition-metal species. The method allows detection of radical ions at the 10−8 M level and produces information-rich spectra. The coupling of electrochemistry and ESR (150) has been reviewed in detail (2, 7, 151–155). ESR is based on the absorption of microwave radiation by a paramagnetic species contained in a magnetic field of strength, H. The field splits the unpaired electron energy levels by ΔE = g𝜇B H, where 𝜇B is the Bohr magneton (5.788 × 10−5 eV/T), and g is a spectroscopic splitting factor dependent on the orbital and the electronic environment of the electron. For a free electron and most organic radical species, g ≈ 2. When the field and radiation frequency are such that h𝜈 = g𝜇B H

(21.7.1)

absorption of the incident radiation produces transitions between the split levels. Spectra are recorded by scanning the field strength and measuring absorption as a function of H. Hyperfine structure arises in the spectra from additional splitting of the energy levels by neighboring atoms having magnetic moments that interact with the unpaired electron (e.g., protons, 14 N, or 31 P). Basic principles and the interpretation of spectra have been presented in detail (156–159). A stand-alone commercial ESR instrument is usually employed, together with a cell designed to allow electrochemistry to be carried out in the ESR cavity. The cells have large A/V , with the working electrode in the sensitive region and smaller reference and counter electrodes positioned somewhat more remotely (160–162). The ESR signal and the current in the cell can be monitored simultaneously vs. potential or time. Varied information is available from ESR: • From detection of a spectrum, confirmation that radical ions or other radical species are produced in an electrochemical reaction. Electrogenerated radicals that are too unstable for direct observation by ESR can often be captured with a suitable spin trap (e.g., phenyl-t-butylnitrone), yielding a stable, observable radical (163–165). • From detailed analysis of hyperfine structure, spin density distributions in the radical and information about ion pairing, solvation, and restricted internal rotation. • From the effect of concentration on widths of spectroscopic lines, electron exchange rates between a radical ion and its parent (166, 167). • From comparative spectra of the same species in different media, information about medium effects (168). An example in the last category comes from the ESR behavior of methyl viologen radical + cation, MV ∙ (169). In solution, the spectrum shows a rich hyperfine structure due to interac+ tions of the unpaired electron with 1 H and 14 N. A similar spectrum is seen for MV ∙ incorTM porated in Nafion , indicating that the radical ion remains free to tumble. However, if the viologen ion is incorporated in a polymer backbone (e.g., PVOS or PXV, Figure 17.4.1), there is no hyperfine structure because of restricted mobility. 21.7.2

NMR

Although nuclear magnetic resonance (NMR) has often been used to analyze products of bulk electrolysis, it has been difficult to use the method for electrochemical investigations in situ. The problem is the comparatively low sensitivity of NMR, which necessitates a sizable number of observable nuclei in the sample. Even so, the method can be effective and chemically informative in two general areas:

21.8 Ex-situ Techniques

• Examination of catalysts, adsorbates, or other species at electrodes of very large area, such as those used for electrocatalysis or for supercapacitors (170, 171). • Examination of bulk phases in electrochemical systems, including materials used for insertion electrodes, electrodeposited phases, or electrolytes (172, 173). Observations have been made operando on whole electrochemical devices.

21.8 Ex-situ Techniques The methods discussed in Sections 21.1–21.7 were presented in the context of in-situ or operando application. Although nearly all are also employed ex situ, separate discussions for that context are unnecessary. This section is dedicated to two groups of methods that are widely used for characterization of electrochemical systems, but nearly always ex situ. 21.8.1

Electron Microscopy

Basic principles of electron microscopy were introduced in Section 21.1.4, where in situ applications were covered. However, nearly all electron microscopy in support of electrochemical research is carried out ex situ. The three primary methodologies are outlined in Table 21.8.1. Table 21.8.1 Principal Forms of Electron Microscopy for Electrochemical Studies Scanning Electron Microscopy (SEM) Measured quantity

An image formed by rastering a primary electron beam over the sample and detecting either (a) secondary electrons produced by the primary beam or (b) backscattered electrons from the primary beam. The image is the intensity of the detected signal vs. primary beam position. Secondary electrons are low-energy electrons ( 1, 1) = 0. In the second Thus, C A (1, 1) = 0 and CB (1,1) = CA A B A iteration, diffusion alters the concentration in box 2, because fluxes of both A and B must cross the boundary between boxes 1 and 2. To maintain the interfacial condition that C A (1, k) = 0 (for k > 0), the incoming flux of A must be converted into B. This yields the current for the second iteration. Continuing the process will generate the concentration profiles and the current as functions of time. C(x, t + Δt) = C(x, t) +

B.1.3 Dimensionless Parameters ∗ , an equal number of simulations If one wanted results for several starting concentrations, CA ∗ . If f (j, k) = C(j, k)∕C ∗ , would be needed. Consider, though, the effects of dividing (B.1.8) by CA A then

f (j, k + 1) = f (j, k) + DM [f (j + 1, k) − 2f (j, k) + f (j − 1, k)]

(B.1.9)

where f (j, k), called a fractional concentration, is an example of a dimensionless parameter. Another is the model diffusion coefficient, DM = DΔt/Δx2 , to be discussed later. Now suppose we simulate the step experiment again, but in place of concentrations we substitute fA and f B . Equation (B.1.9) describes the way in which these parameters are altered by diffusion, and the boundary condition is simply fA (1, k) = 0 (for k > 0). Initially, fA = 1 and f B = 0 everywhere. The simulation is straightforward, and one obtains the time evolution of

987

988

Appendix B Basic Concepts of Simulation

Initial

CA*

After first iteration

After second iteration 3

2 CA

3

0

1

2

3

4

1

2 3 Box no.

4

1

2

3

4

3

4

2

1 CA*

3 2

CB 3

0

1

2

3

4

1

2 3 Box no.

4

1

2

Figure B.1.2 Evolving concentration profiles for a system undergoing the electrode reaction A + ne → B. Arrows show mass flow, with the corresponding iteration number indicated in each case.

the fractional concentration profiles. However, these profiles from a single simulation describe ∗ . To obtain the dimensioned the characteristics of the experiment for every possible value of CA profiles for a specific starting concentration, one need only multiply the values of f (j, k) by that ∗. value of CA Dimensionless parameters are valuable because they enable the compact display of theoretical results. For example, consider the homogeneous reaction A → B with rate constant k 1 . The ∗ exp(−k t), where C ∗ is the familiar solution for the concentration of A at time t is CA = CA 1 A concentration at t = 0. One might first think of making a graphical display of results by plotting ∗ and k . This involves a family of m × n curves, where m is the C A vs. t for different values of CA 1 ∗ number of values of CA , and n is the number of k 1 -values (Figure B.1.3a). If one recognizes that k 1 t is a dimensionless parameter accounting fully for the effects of k 1 and t, then one could plot C A vs. k 1 t and represent the results in just m curves, covering the ∗ values (Figure B.1.3b). different CA ∗ is also a useful dimensionless parameter, then the equation If one further realizes that CA ∕CA ∗ and a = k t. Hence, a plot of f vs. a shows all can be written fA = exp(−a), where fA = CA ∕CA 1 A of the desired information in a single curve (Figure B.1.3c), which is the essential shape function of the system, sometimes called a working curve. Solving differential equations in terms of dimensionless parameters yields results that characterize entire families of specific experimental situations. This is a magnificent asset, especially where numerical solutions are required; consequently, the use of dimensionless variables is a general practice. However, there is a confusing aspect in the way that these parameters combine the effects of more than one observable. One can sometimes have difficulty in mentally separating the effects

B.1 Setting Up the Model

Figure B.1.3 Curves describing exponential decays.

3.0

1

2.0 C*A/mM

(a)

3

1.0 4 0.0

1. C*A = 2.5 mM, k1 = 0.2 s–1 2. C*A = 1.0 mM, k1 = 0.2 s–1 3. C*A = 2.5 mM, k1 = 1 s–1 4. C*A = 1.0 mM, k1 = 1 s–1

0

2

2

4

6

8

10

t/s 3.0

(b) 1. C*A = 2.5 mM 2. C*A = 1.0 mM

1

2.0 * /mM CA 1.0

0.0

2

0

1

2

3

4

5

k1t 1.0

(c)

fA 0.5

0.0

0

1

2

a

3

4

5

of a single variable. For example, the abscissa of Figure B.1.3c deals with changes in k 1 , or in t, or in both. It is easier to understand a working curve by thinking in terms of a given experiment, which normally involves holding some variables constant: • For example, one might study the reaction A → B by looking at the decay of A continuously ∗ and k ; hence, the working with time. A given experiment would involve fixed values of CA 1 curve can be regarded as a scaled decay function of concentration, C A , vs. time, t. We might even call f the dimensionless concentration and a the dimensionless time. • Alternatively, we might study A → B by measuring C A at a fixed elapsed time t. In that experimental context, the working curve would be seen as a plot of the remaining concentration of A (relative to the initial concentration) at the sampling time, t, vs. a scaled rate constant. For that purpose, a could be called a dimensionless rate constant. Thus, the interpretation we apply to a working curve depends on the specific experiment at hand. Dimensionless parameters are normally created by dividing the variables of interest by one or more variables describing some characteristic feature of the system. For example, ∗. fractional concentrations relate actual concentrations to a characteristic concentration, CA The parameter a can be understood similarly by recognizing that 1/k 1 is the average lifetime

989

990

Appendix B Basic Concepts of Simulation

of A (Problem 3.7); thus, a is the ratio of an observation time, t, to that lifetime. Grasped in this way, a is more than a numerical value; it is a guide to the design and interpretation of experiments. This capacity for expressing relationships between observables and characteristic features is a powerful aspect of dimensionless parameters, and, with practice, it can aid one’s intuition in very useful ways. (See, for example, Sections 13.2.2 and 13.4.) B.1.4 Time Let us return now to the model introduced above. Time, t, measured into the simulation is kΔt, where k is the iteration number. Since Δt is a model variable, it is our choice. In setting its value, we make the equivalent statement that some known characteristic time, t k —typically a step width, a scanning time, or some similar experimental duration—will be broken into l iterations in our model. Consequently, Δt = tk ∕ll

(B.1.10)

Either l or Δt can be selected arbitrarily as the model variable, though it usually is more convenient to work in terms of l. Thus, we might carry out simulations of a step experiment with duration t k by breaking t k into 100, 1000, or 10,000 iterations, at will. All of the simulations will yield valid results within their individual abilities to approximate events, but the larger the value of l, the higher the quality of the simulation. However, a larger l also requires more resources (e.g., computation time or spreadsheet size), so an optimal value must be sought. In simple explicit simulations, l is typically 100–1000 iterations per t k . Time is easily expressed in dimensionless terms as the ratio t/t k , t∕tk = k∕ll

(B.1.11)

B.1.5 Distance The center of box j is at a distance (j − 1)Δx from the electrode surface. Since Δx is also our choice, we can apparently also determine the fineness of the model in spatial terms. However, there is a limit to the smallness of Δx, because the model diffusion coefficient, DM , cannot exceed 0.5 for an explicit simulation. For larger values, the finite difference calculation will not be stable, because Δx and Δt are not actually independent. In our treatment of diffusion, we implicitly assumed that within a period Δt, material could diffuse only between neighboring boxes. If we try to set Δx too small for a given Δt, this assumption becomes inadequate, and the simulation diverges from reality. Given Δt, DM becomes an equivalent model variable to Δx. Instead of specifying Δx, it usually is more convenient to specify DM . Then, ( )1∕2 DΔt Δx = (B.1.12) DM The larger we choose DM , the smaller is Δx, and the better is the model; hence, DM is usually set at a high constant value, often 0.45. Substituting for Δt, one obtains ( Δx =

Dt k DM l

)1∕2 (B.1.13)

Now it becomes clear that l is the important determinant of both temporal and spatial resolution for any simulation.

B.1 Setting Up the Model

The distance of the center of box j from the electrode can now be written ( )1∕2 Dt k x(j) = (j − 1) DM l

(B.1.14)

The convenient dimensionless distance, 𝜒(j), is obtained by placing the real system variables on the left and the model variables on the right: 𝜒(j) =

x(j) (Dt k

)1∕2

=

j−1

(B.1.15)

(DM l)1∕2

The expression (j − 1)∕(DM l)1∕2 allows one to calculate 𝜒(j) easily from the simulation parameters, and x(j)/(Dt k )1/2 allows one to correlate the properties of box j with the properties of the experimental solution segment situated at a real distance, x, from a physical electrode. Given the characteristic time t k , the parameter 𝜒(j) is the ratio of actual distance to the diffusion length (Dt k )1/2 . Recalling the step simulation considered above, we see that the most efficient way to use the calculation is to report the concentration profiles as fA and f B vs. 𝜒 for various values of t/t k . These curves would then fully characterize every possible electrochemical experiment satisfy∗ , t , and D, ing the initial conditions and the boundary conditions. Given specific values of CA k it is a simple matter to convert the curves into functions of C A and C B vs. x for various values of t. B.1.6 Current Usually there is a flux of electroactive species across the boundary between boxes 1 and 2. For species A in iteration k + 1, it is −JA1,2 (k + 1) = DC ∗A

[fA (2, k) − fA (1, k)]

(B.1.16) Δx In the absence of other processes, this flux will cause a concentration change in box 1. However, there is always a boundary condition that dictates the circumstances at the electrode surface, and it must be maintained. Consider the example used above, where the surface concentration of A is zero, so that A always moves from box 2 into box 1. Maintenance of the surface condition requires that the arriving A be eliminated in box 1 by faradaic conversion to B. For iteration k + 1, then, the current is defined by JA1,2 (k + 1). We can write it as i(k + 1) =

nFADC∗A fA (2, k)

(B.1.17)

Δx Substituting for Δx from (B.1.13) yields i(k + 1) =

∗ f (2, k)(D l)1∕2 nFAD1∕2 CA A M

(B.1.18)

1∕2

tk

We obtain a dimensionless current, Z(k), by following the standard recipe—placing experimental variables on the left and model variables on the right: (

1∕2

Z(k + 1) =

i(k + 1)tk

∗ nFAD1∕2 CA

= (DM l)

1∕2

fA (2, k) =

l DM

)1∕2 DM fA (2, k)

(B.1.19)

991

992

Appendix B Basic Concepts of Simulation

The product of the last two quantities on the right, DM f A (2, k), is the fractional concentration that would exist in box 1 after iteration k + 1 if it were not eliminated by electrolysis. This definition of Z relates the actual current to the Cottrell current expected at time t k (Problem B.1). The current for the first iteration1 is calculated differently because there is no flux. Instead, the current flows because we first establish the surface condition by eliminating A from box 1 at that time. The number of moles electrolyzed in the first interval, Δt, is ΔxAC ∗A ; hence, the current is i(1) =

nFAC∗A Δx Δt

=

nFAC∗A D1∕2 l1∕2 1∕2

1∕2

(B.1.20)

tk D M

Thus, Z(1) = (ll∕DM )1∕2

(B.1.21)

To what time should we assign Z(k)? Since our current calculation really involves dividing the integral charge passed during an iteration by the duration of the iteration, it is appropriate to assign the current to the midpoint, rather than to the end, of the iteration. Thus, we say that the dimensionless current Z(k) flowed at t∕tk = (k − 0.5)∕ll. (See also Section B.2.3.) B.1.7 Thickness of the Diffusion Layer How many boxes are needed for a practical simulation? A rule of thumb provides a safe answer: Any electrode reaction that has proceeded for time t under conditions of linear diffusion can alter the solution for a distance no larger than 6(Dt)1/2 [Section 6.1.1(c)]. The longest time covered by the simulation is k max Δt, where k max is the last iteration in the model; therefore, the highest box number required is [ ] 6(Dk max Δt)1∕2 jtot = Int +1 (B.1.22) Δx where the Int function denotes truncation to the highest integer. Upon substitution for Δx and Δt, one has jtot = Int[6(DM kmax )1∕2 + 1]

(B.1.23)

which defines the sizes of the concentration arrays needed for the model. A sound routine policy is to set jtot for DM = 0.5 (the largest possible value), so that 1∕2

jtot = Int[4.24kmax + 1]

(B.1.24)

If kmax = l, as is often true, the model requires 43 boxes for l = 100 or 135 boxes for l = 1000. However, k max is sometimes a multiple of l, in which case larger arrays are needed. In most iterations, it is unnecessary to carry out diffusion calculations for much of the array. Iteration k corresponds to an elapsed time kΔt, so the largest box number that can be affected by diffusion is, jmax = Int[6(DM k)1∕2 + 1]

(B.1.25)

One need not calculate array elements beyond this index. This is a time-saving consideration in simulations written as sequential programs, but it is usually irrelevant for simulations based on spreadsheets. 1 Because the early computations in any finite difference simulation are inaccurate, a computation of the current for the first iteration is usually not of importance. Recall that in the Cottrell experiment, i → ∞ as t → 0.

B.2 An Example

B.1.8 Diffusion Coefficients The parameter DM exists in diffusion expressions for each species, and in each case, DM contains the diffusion coefficient for that species. But if DA ≠ DB ≠ DC · · ·, the DM values cannot all be equal. In general, we must distinguish them as DM, A , DM, B , DM, C , … This complicates the model; hence, one frequently simplifies by assuming that all diffusion coefficients are indeed equal, so that a single value of DM suffices. If it is essential to take account of differences in D values, then one can employ different DM values. One of them is a model variable and can be chosen at will. Usually, this is DM, A . The others then are determined by the fact that (B.1.26)

DM,i ∕DM,A = Di ∕DA This practice ensures that the model behaves diffusively as the real system does.2

B.2 An Example In prior editions of this book, simulation was illustrated using a program written in the FORTRAN language.3 In this edition, we shift to an Excel spreadsheet, which is just as effective and now more convenient.4 The example is a 100-iteration model for the Cottrell experiment, [Section 6.1.1(a)]. An electroreactant, A, is initially uniformly distributed, but a potential step is applied at t = 0 to force the surface concentration of A to zero by faradaic conversion to species B. Semi-infinite linear diffusion applies, and species B is not present in the bulk.

®

B.2.1 Organization of the Spreadsheet Any simulation involves a set of sizable arrays, so thoughtful layout is important for implementation by spreadsheet. In this discussion, we follow the map shown for the main page in Figure B.2.1. Figure B.2.2 shows selected portions, complete with numeric results, of both the finished main page and a second page. The core of the layout in Figure B.2.1 is the two-dimensional concentration array for species A, containing all values of f A (j, k). Distance from the electrode increases with box number, j, across the columns, and time increases with iteration number, k, down the rows. This is a simulation for l = 100 and kmax = l, so there are 101 rows, one for each iteration, plus one more for iteration 0, where initial conditions are set. There are columns for boxes 1–43, which is the number needed for k max = 100 according to (B.1.24). The three spaces shaded in white—for box 1, box 43, and iteration 0—all belong to this array, but require special handling. The main worksheet also contains several other arrays: • • • •

∗ , l, k A table defining model variables, including DM, A , DM, B , CB∗ ∕CA max , and jtot . Arrays indicating iteration numbers, box numbers, and values of 𝜒(j). One-dimensional arrays reporting functions of time, including t/t k and Z(k). A mass-conservation array to be discussed below.

2 Models are normally built on the assumption that diffusion coefficients are not functions of x; however, spatial variations in diffusion coefficients can be accommodated. 3 Section B.2 in either earlier edition. 4 Although the discussion below addresses some details of an Excel-based simulation, no effort is made here to teach Excel. A functional knowledge is presumed. Among many self-teaching tools are references (9) and (10).

993

Appendix B Basic Concepts of Simulation

Mass conservation array

Parameters

Row 5

Box no. array

Row 9 Row 14 Box 1

Iteration 0

Box 43

Fractional concentration array (Species A)

Box number Time arrays

994

Iteration no. array

Iteration number

Iteration 100

Row 114 Col A

Col F

χ(j) Array

Col AV

Figure B.2.1 Map of the main worksheet in a spreadsheet-based 100-iteration simulation of the Cottrell experiment.

Any simulation should explicitly include all participants in the modeled electrode process; therefore, a fractional concentration array of identical size and structure must exist for each of them. Arrays for participants other than species A do not appear in Figure B.2.1, which maps only the main worksheet. Concentration arrays are large and unwieldy, so it is better to place them on separate worksheets. Figure B.2.2b shows part of the array for species B, which is on the second page of the Excel workbook.

(a) Main Page: Time Profiles and Diffusion Layer for Species A

no

(b) Diffusion Layer for Species B no

®

Figure B.2.2 Results of a 100-iteration simulation of the Cottrell experiment using Excel . Boxes 5–41 and iterations 4–98 are omitted to permit compact presentation. (a) Main page: Time profiles and diffusion layer for species A. (b) Diffusion layer for species B.

996

Appendix B Basic Concepts of Simulation

B.2.2 Concentration Arrays Each concentration array involves four zones, of which three are handled without special reference to events at the electrode: ∗ ∕C ∗ for each • Values for iteration 0 represent the initial conditions and are set uniformly at Cm A species, m. In our example, all values for species A are unity and all for species B are zero. • Each box in the gray-shaded part of the concentration array (G15:AU114) receives a formula equivalent to (B.1.9). For example, the cell at column G, row 15 would have

= G14 + DM A∗ (H14 − 2 ∗ G14 + F14)

(B.2.1)

where DM_A is an Excel name given to DM, A . This cell can be replicated across and down to fill (G15:AU114). • The column for the last box also needs a diffusion formula like (B.2.1); however, it differs because there is no box to the right. The diffusion field ends here, and the only flux is across the single interface between boxes 42 and 43. Thus, cell AV15 would have = AV14 + DM A ∗ (−AV14 + AU14)

(B.2.2)

This can be replicated down the column to AV114. The final zone to be addressed in each concentration array is the column for box 1. This is a more delicate matter because the formulas in this zone must • Maintain the required boundary condition at the electrode surface. • Account for diffusion between boxes 1 and 2 • Account for electrolytic conversion. For the present example, the surface boundary condition is that f A (1, k) = 0 for all k greater than 1. Therefore, F15:F114 are all set to zero in the array for species A. Things are more complex in box 1 for species B, because one must deal with diffusion and electrolytic conversion. Many elements on the main page are not required on the worksheet for B; therefore, its array is mapped differently than the concentration array for species A. The cell for f B (1, 1) is B7. In that cell, the required formula is = Main!F14 + DM A ∗ (Main!G14 − Main!F14) + (B6 + DM B ∗ (C6 − B6))

(B.2.3)

The label “Main!” identifies a cell on the main worksheet, where the concentration array for species A is carried. The first term in (B.2.3) addresses any A remaining in box 1 after the prior iteration, all of which must be converted to B. The second term describes the amount of A diffusing into box 1 from box 2 during the current iteration, all of which must be converted to B. The third term expresses the diffusion of B between boxes 1 and 2 during the current iteration. (B.2.3) can be replicated down the column to the last iteration. B.2.3 Results and Error Detection With the steps in the preceding section, the core of the simulation becomes complete; however, other steps must be taken to draw results and to assure reliability. As results are reported, one must recognize that quantities computed in any iteration, k, may correspond to slightly different times, depending on their nature: • Outcomes of an iteration relate to the end of the iteration at t∕tk = k∕ll. Important examples include all elements of a concentration array, such as f A (j, k). • Average values for an iteration are assigned to the midpoint of the iteration at t∕tk = (k − 0.5)∕ll. Parameters expressing rates, including the dimensionless current, Z(k), generally fall in this category, as discussed earlier in Section B.1.6.

B.2 An Example

The following additional steps are needed to complete this project: 1) The dimensionless current, Z(k), must be computed for each iteration in C15:C114 using (B.1.21) for k = 1 and (B.1.19) for k > 1. 2) The dimensionless time, t∕tk = (k − 0.5)∕ll, corresponding to each value of Z(k) must be placed in B14:B114. 3) For this particular simulation, we want to compare the simulated Z(k) with the analytical result, which is obtained by rearrangement of the Cottrell equation, (6.1.12), as ZCott = [𝜋 1∕2 (t∕tk )1∕2 ]−1

(B.2.4)

Values of ZCott are computed for each iteration in D15:D114. 4) The ratio R = Z(k)/ZCott is computed for each iteration in E15:114 to provide a running index of the simulation’s quality. 5) Finally, the mass-conservation array must be completed. Many coding errors cause changes in the total mass of participants, so it is always valuable to assure that no mass is gained or lost. Errors in simulations can be subtle and difficult to detect. It is important to use every possible safeguard. Let us now explore how one can check for conservation of mass. There are two parts to that effort in the present example, the first comprising cells F9:AV9, in which the fractional concentrations of all participants are summed box by box for the last iteration. One can achieve this goal in cell F9 through the following formula:5 = INDEX(FAarray, ELL + 1, F13) + INDEX(FBarray, ELL + 1, F13)

(B.2.5)

where FAarray, FBarray, and ELL are Excel names for f A (j, k), f B (j, k), and l. This formula is replicated through AV9. When all diffusion coefficients are equal, the total concentration of all participants at any location in the diffusion layer is always equal to the sum of bulk concentrations. Thus, for every box in all iterations, ∑ ∑ ∗ ∕C ∗ fm (j, k) = 1 + Cm (B.2.6) A m

m≠A

where the index m runs over all species. One can readily inspect the mass-conservation array to see if this condition is fulfilled after the last iteration. In the present example, DM, A = DM, B and ∗ = 0; therefore, each box in that array should have a value of precisely 1, which, indeed, CB∗ ∕CA it does (Figure B.2.2). One can always run a model with equal diffusion coefficients to check its integrity in this way, even if ultimate application requires that diffusion coefficients differ. When they do differ, (B.2.6) does not hold. The full mass accountability array is not uniform; so visual inspection is not useful. Even so, one can still check the total mass in the system by examining cell F10, which computes the average of all values in the full mass-conservation array, F9:AV9. This average should precisely equal the right side of (B.2.6), because all mass originally in the system must remain somewhere in the system at the end. In our example, this cell is unity, as it should be (Figure B.2.2). B.2.4 Performance Figure B.2.3 shows the ratio R = Z(k)/ZCott as a function of time through the simulation. Ideally, R is always exactly unity. The figure shows that large errors occur in the first few iterations, 5 The INDEX function in Excel refers to rows and columns relative to the top left cell in the named array, which is at row 1, column 1. In the concentration arrays for A and B, (B.2.5) specifies row l + 1 = 101 (corresponding to the last iteration) and the column identified in cell F13 (identifying the box number for mass-conservation cell F9).

997

998

Appendix B Basic Concepts of Simulation

t/tk 0.195

2.0

0.395

0.595

0.795

0.995

60

80

100

1.8 1.6 Z/ZCott 1.4 1.2 1.0 0.8

0

20

40

Iteration number, k

Figure B.2.3 Z(k) from a simulated Cottrell experiment divided by the analytical solution. l = 100, DM, A = DM, B = 0.45, and CB∗ ∕CA∗ = 0. Z(k) assigned to t∕tk = (k − 0.5)∕l, leading to the values on the upper abscissa. 1.0

0.8 fA 0.6 f(j,50) 0.4 fB 0.2

0.0

0

1

2

χ(j)

3

4

5

Figure B.2.4 Concentration profiles for t/tk = 0.5 from a simulated Cottrell experiment using l = 100, DM, A = DM, B = 0.45, and CB∗ ∕CA∗ = 0. Curves from simulation; points from (6.1.14) and its complement.

as one must expect from the coarse nature of the model at that stage. However, by the 10th iteration, the error has fallen to 3%, and it falls further afterward. At the end, for t/t k = 0.9995, the error is 0.2%. With a larger l, smaller errors would have been seen at any given value of t/t k . Figure B.2.4 displays the concentration profiles for t/t k = 0.5. They are indistinguishable from the analytical solution of this problem.

B.3 Incorporating Homogeneous Kinetics

B.3 Incorporating Homogeneous Kinetics When the electrochemical process is coupled to one or more homogeneous chemical reactions (Chapter 13), the differential equations describing the system can easily become too difficult for an analytical solution, and simulation shows its value.

B.3.1 Unimolecular Reactions Consider a system in which an electrode reaction is followed by a unimolecular conversion: A+e→B k1

B −−−→ C

(at the electrode)

(B.3.1)

(in solution)

(B.3.2)

The differential equations describing B and C must account for both diffusion and reaction (Section 13.2.4). For species B, we have 𝜕CB (x, t)

𝜕 2 CB (x, t)

− k1 CB (x, t) (B.3.3) 𝜕t 𝜕x2 The first term on the right is Fick’s second law, for which (B.1.7) is the finite-difference representation. Thus, we can immediately write the finite difference analog to (B.3.3) as = DB

CB (x, t + Δt) = CB (x, t) + DM,B [CB (x + Δx, t) − 2CB (x, t) + CB (x − Δx, t)] − k1 Δt ⋅ CB (x, t)

(B.3.4)

∗ and introducing the notation of the simulation, we obtain Dividing by CA

fB (j, k + 1) = fB (j, k) + DM,B [fB (j + 1, k) − 2fB (j, k) + fB (j − 1, k)] −

k1 tk

(B.3.5) f (j, k) l B which allows a one-step accounting of diffusion and kinetic effects for the B array during iteration k + 1. In practice, a two-step procedure is usually employed, in which one first calculates the effects of diffusion, then adds the effect of kinetics. We can diagram it in this way: [fB( j,k)]

Diffusion

[fB′ ( j,k+1)]

Kinetics

[fB( j,k+1)]

Iteration k+1

The diffusion calculation can be written as fB′ (j, k + 1) = fB (j, k) + DM,B [fB (j + 1, k) − 2fB (j, k) + fB (j − 1, k)]

(B.3.6)

and the effect of the kinetic process becomes fB (j, k + 1) = fB′ (j, k + 1) −

k1 tk l

fB (j, k)

(B.3.7)

999

1000

Appendix B Basic Concepts of Simulation

The sum of (B.3.6) and (B.3.7) is (B.3.5); therefore, the result is the same whether the calculation is carried out in one step or two. However, there are practical advantages to the stepwise calculation: • Kinetic effects appear separately from diffusion, so it is easier to make programming changes for different reaction schemes. • Equation (B.3.5) can easily produce negative values of f B (j, k + 1) when kinetic effects are large. Sometimes it is difficult to predict this behavior, and it is messy to apportion the available mass when it does happen. Mass allocation is more straightforward in the sequential approach because the total elimination of a reactant can happen only in a kinetic step. If the simulation is implemented by spreadsheet, one can accomplish the stepwise calculations by creating two worksheets for each species involved in the kinetics (species B and C in the example here). Each worksheet must carry a diffusion-layer array covering iterations 0 − k max for all boxes, exactly as described in Section B.2.2. On the first worksheet for each species, the formulas in each box would manifest the diffusion process. On the second, they would manifest the kinetics. For any species involved in kinetics, the fractional concentrations at the end of an iteration would be those on the kinetic page, and they would be the values of fm (j, k) used in (B.3.6) and (B.3.7) to obtain fm (j, k + 1). If the simulation is implemented by a sequential program, the kinetics would be manifested in a section following that for diffusion. The dimensionless kinetic parameter in (B.3.7) is k 1 t k . It must be given a numeric value for any particular simulation, and the results of that simulation are valid for all those experiments, but only those experiments, for which the product of k 1 and t k is equal to the chosen value. This parameter is the ratio of the characteristic experimental time, t k , to the lifetime of B, which is 1/k 1 . In general, the unimolecular decay of B will hardly be felt in the experiment if k 1 t k ≪ 1, and it will be completely manifested for k 1 t k ≫ 1. The finite difference method is based on an approximation of true derivatives; therefore, one can expect (B.3.7) to account accurately for the kinetic effect only when k1 tk ∕ll is not too large. Otherwise, the extent of decay per unit of time resolution becomes excessive. The upper limit of the most useful modeling range is k1 tk ≈ l∕10. The lower limit is where the kinetic perturbation no longer has an experimentally significant impact. B.3.2 Bimolecular Reactions Now consider a bimolecular complication to an electrode process: A+e→B

(at the electrode)

k2

B + B −−−→ C

(in solution)

(B.3.8) (B.3.9)

For B, we have (Section 13.2.4) 𝜕CB (x, t)

𝜕 2 CB (x, t)

− k2 CB (x, t)2 (B.3.10) 𝜕t 𝜕x2 Transforming this to the notation of the simulation exactly as above, we obtain the analogue to (B.3.5): = DB

fB (j, k + 1) = fB (j, k) + DM,B [ fB (j + 1, k) − 2fB (j, k) + fB (j − 1, k)] −

∗ k2 tk CA

l

[fB (j, k)]2

(B.3.11)

B.4 Boundary Conditions for Various Techniques

Again, it is advantageous to handle diffusion and homogeneous reaction sequentially, so we split (B.3.11) into two parts. The diffusion effects are registered by (B.3.6), and the changes in concentration due to reaction are given by fB (j, k + 1) =

fB′ (j, k

+ 1) −

∗ k2 tk CA

l

[fB (j, k)]2

(B.3.12)

∗ . Its value must The dimensionless parameter pertaining to the second-order process is k2 tk CA be fixed for a given simulation, and several successive simulations must be carried out to show ∗ . For reasons outlined above, the most useful modeling range the effects of variations in k2 tk CA ∗ is k2 tk CA ≤ l∕10.

B.4 Boundary Conditions for Various Techniques So far, we have considered only a step to a potential where the electroreactant is brought to the electrode at the mass-transfer-limited rate. The boundary condition in that case is particularly simple: C A (0, t) = 0, so that f A (1, k) = 0. Other situations demand other conditions, and we introduce some of them here. B.4.1

Potential Steps in Nernstian Systems

Suppose the electrode reaction A + ne ⇌ B

(B.4.1)

is nernstian, so that ′

E = E0 +

RT CA (0, t) ln nF CB (0, t)

(B.4.2)

always applies. In terms of fractional concentrations, we obtain ′

E = E0 +

RT fA (1, k) ln nF fB (1, k)

(B.4.3)

which can be rearranged to give a dimensionless potential parameter: Enorm

′ f (1, k) (E − E0 )nF = = ln A RT fB (1, k)

(B.4.4)

or fA (1, k) fB (1, k)

= exp(Enorm )

(B.4.5) ′

The normalized potential, Enorm , is the energy difference n(E − E0 ) in units of kT. To simulate an experiment in which we begin with a uniform solution of A and then apply a step to potential E, we would first set up the initial conditions just as we did earlier. However, in subsequent iterations, the ratio fA (1)/fB (1) would be maintained in the first box at the value dictated through (B.4.5), where the value of Enorm corresponds to the step potential E. This value of Enorm is a model variable, and a separate simulation would have to be carried out for each desired value of E.

1001

1002

Appendix B Basic Concepts of Simulation

Maintaining (B.4.5) usually causes a diffusive flux across the boundary between boxes 1 and 2. If the flux alters fA (1)/fB (1) after the diffusive step in the simulation; the ratio must be re-established by converting species A into B or vice versa. The amount converted gives rise to a dimensionless current, Z(k), calculated as described in Section B.1.6. B.4.2 Heterogeneous Kinetics For the one-step, one-electron heterogeneous reaction, kf

−−−−−−− ⇀ A+e ↽ − B

(B.4.6)

kb

the current is always given by i = kf CA (0, t) − kb CB (0, t) nFA

(B.4.7)

In terms of simulation variables, we obtain 1∕2

it k

1∕2

∗ nFADA CA

⎛ k t 1∕2 ⎞ ⎛ k t 1∕2 ⎞ f k ⎟ b ⎜ = Z(k + 1) = f (1, k) − ⎜ k ⎟ fB (1, k) ⎜ D1∕2 ⎟ A ⎜ D1∕2 ⎟ ⎝ A ⎠ ⎝ A ⎠ 1∕2

1∕2

1∕2

(B.4.8)

1∕2

where the clusters (kf tk ∕DA ) and (kb tk ∕DA ) are dimensionless rate constants. The dimensionless current, Z(k), can be calculated for any iteration from (B.4.8) if those parameters are specified. For Butler–Volmer kinetics, 1∕2

kf tk

1∕2

DA

1∕2

kb tk

1∕2

DA

⎛ k 0 t 1∕2 ⎞ = ⎜ k ⎟ exp(−𝛼Enorm ) ⎜ D1∕2 ⎟ ⎝ A ⎠

(B.4.9)

⎛ k 0 t 1∕2 ⎞ = ⎜ k ⎟ exp[(1 − 𝛼)Enorm ] ⎜ D1∕2 ⎟ ⎝ A ⎠

(B.4.10)

′

with Enorm = (1)F(E − E0 )/RT. Thus, both rate constants can be calculated from Enorm when two model variables are supplied, viz. the transfer coefficient, 𝛼, which is dimensionless in itself, 1∕2 1∕2 and a dimensionless standard rate constant, k 0 tk ∕DA . The fractional concentrations in box 1 must be obtained by considering diffusion into or out of box 2 as well as the change in box 1 caused by passage of current. (D )1∕2 M,A fA (1, k + 1) = fA (1, k) + DM,A [fA (2, k) − fA (1, k)] − Z(k + 1) (B.4.11) l (D )1∕2 M,A fB (1, k + 1) = fB (1, k) + DM,B [fB (2, k) − fB (1, k)] + Z(k + 1) (B.4.12) l For iteration k + 1, the current parameter Z(k + 1) is first calculated from (B.4.8), which implies 1∕2 1∕2 that an amount of species A equivalent to (kf tk ∕DA )fA (1, k) is converted to B, while a quan1∕2

1∕2

tity of species B equivalent to (kb tk ∕DA )fB (1, k) is converted to A. Then, the fractional concentrations in box 1 for iteration k + 1 can be calculated from (B.4.11) and (B.4.12).

B.4 Boundary Conditions for Various Techniques

B.4.3 Potential Sweeps If we want to apply the program (B.4.13)

E = Ei ± vt to the system expressed in (B.4.6), then we have ′

F(Ei − E0 )

Fv t ± (B.4.14) RT RT The first term is a normalized initial potential Ei, norm , which would have to be specified as a model variable to the simulation. The second term, describing the effects of the sweep, has a changing value as the simulation evolves (i.e., the second term varies with the iteration number k). The specific function is obtained by substitution from (B.1.11): Enorm =

Enorm = Ei,norm ±

Fv tk RT

⋅

k l

(B.4.15)

We have yet to define the known time, t k , corresponding to l iterations. Several choices could be made, but a convenient one is to let t k be the time required to scan from Ei to the final potential, Ef . Then t k = (Ei − Ef )/v, and Enorm = Ei,norm +

(Ei − Ef ) k k ⋅ = Ei,norm + (Ei,norm − Ef,norm ) (RT∕F) l l

(B.4.16)

Sweep experiments are often simulated to study the effects of heterogeneous kinetics or coupled homogeneous kinetics, which are added into the model as described in Sections B.4.2 or B.3, respectively. When heterogeneous kinetics are included, the value of Enorm calculated from (B.4.16) is used in (B.4.9) and (B.4.10) to find the dimensionless rate constants for determination of Z(k + 1) in (B.4.8). Calculation of concentrations for the first box then proceeds from (B.4.11) and (B.4.12). B.4.4 Controlled Current For the electrode reaction (B.4.1), the application of a controlled current is equivalent to controlling the gradient in the concentration of A at the electrode surface: ( ) 𝜕CA (x, t) i = −JA (0, t) = DA (B.4.17) nFA 𝜕x x=0 To convert this expression to the finite-difference notation of the simulation, we assume that the concentration profile is linear from the center of box 1 (the electrode surface) to the center of box 2. Then, [f (2, k) − fA (1, k)] i ∗ A = DA CA (B.4.18) nFA Δx Controlling the current in a real experiment is, therefore, equivalent to controlling the difference in fractional concentrations between boxes 1 and 2 in the model. Now we rearrange (B.4.18) to obtain the usual current parameter: 1∕2

Z=

it k

1∕2

∗ nFADA CA

1∕2 1∕2

=

DA tk Δx

[fA (2, k) − fA (1, k)]

(B.4.19)

1003

1004

Appendix B Basic Concepts of Simulation

Substituting from (B.1.13) gives Z = (DM,A l)1∕2 [fA (2, k) − fA (1, k)]

(B.4.20)

If the current has a constant magnitude, it is most convenient to define the known time, t k , as the transition time given by the Sand equation (9.2.11) for species A. Thus, l1∕2 iterations correspond to 1∕2

𝜏

1∕2

=

1∕2 tk

=

∗ 𝜋 1∕2 nFADA CA

2i and the current parameter is

(B.4.21)

𝜋 1∕2 = (DM,A l)1∕2 [fA (2, k) − fA (1, k)] (B.4.22) 2 In carrying out an actual simulation, one must hold the difference in fractional concentration between the first two boxes at a constant value. With t k = 𝜏, the required difference is given from (B.4.22) as Z=

fA (2, k) − fA (1, k) =

𝜋 1∕2 2(DM,A l)1∕2

(B.4.23)

In each iteration, one allows diffusion to occur; then the value of fA (1) is adjusted downward so that (B.4.23) is maintained. Since this adjustment corresponds to a faradaic conversion, fB (1) must be adjusted upward by an equal amount. These steps give the final values, fA (1, k) and fB (1, k), for iteration k. If the system is the uncomplicated case of (B.4.1), the transition time, which is found when f A (1, k) = 0, will be reached ideally in the l th iteration. Deviations from this result will occur when complications, such as homogeneous kinetics, are introduced into the electrode process. The potential–time curve for a reversible system can be obtained by reporting the value of Enorm calculated with equation (B.4.4) at each iteration. Equivalent data for a quasireversible system would require the specification of heterogeneous rate parameters in a fashion like that outlined in Section B.4.2.

B.5 More Complex Systems In previous editions, this appendix also introduced the simulation of • • • •

Convection, including treatment of the RDE and RRDE. Electrical migration and effects of the diffuse double layer. Thin-layer cells and resistive effects. Two-dimensional mass transfer in multielectrode systems

The earlier presentations6 remain valid, and the interested reader may wish to consult them; however, explicit simulation is no longer an optimal approach for modeling in such cases. Section 4.5.4 introduces commercial electrochemical simulators and general simulator/solvers, which are much better tools for demanding problems. The second edition also contained a brief introduction to more sophisticated simulation methods,7 which are sometimes employed in the software packages just mentioned. 6 Prior editions, Sections B.5 and B.6. 7 Second edition, Section B.6.4.

B.7 Problems

B.6 References 1 B. Speiser, Electroanal. Chem., 19, 1 (1996). 2 S. W. Feldberg, Electroanal. Chem., 3, 199 (1969). 3 D. Britz and J. Strutwolf, “Digital Simulation in Electrochemistry,” 4th ed., Springer,

Switzerland, 2016. 4 J. T. Maloy in “Laboratory Techniques in Electroanalytical Chemistry,” P. T. Kissinger and

W. R. Heineman, Eds., 2nd ed., Marcel Dekker, New York, 1996, Chap. 20. 5 M. Rudolph in “Physical Electrochemistry,” I. Rubinstein, Ed., Marcel Dekker, New York,

1995, Chap. 3. 6 S. W. Feldberg in “Computers in Chemistry and Instrumentation,” Vol. 2, “Electrochem-

7 8 9 10 11

istry,” J. S. Mattson, H. B. Mark, Jr., and H. C. MacDonald, Jr., Eds., Marcel Dekker, New York, 1972, Chap. 7. K. B. Prater, ibid., Chap. 8. J. T. Maloy, ibid., Chap. 9. B. Jelen, “Power Excel by MrExcel,” Holy Macro! Books, Merritt Island, FL, 2019. B. Jelen, “Excel 2016 in Depth,” Que, Indianapolis, 2016. R. S. Nicholson and I. Shain, Anal. Chem., 36, 706 (1964).

B.7 Problems B.1 Show that Z(t) is proportional to the ratio of the current at time t to the Cottrell current at time t k . What is the proportionality factor? B.2 Using Excel or another spreadsheet application, develop the entire 100-iteration simulation of the Cottrell experiment, as outlined in Section B.2. Take care with organization and be systematic. This simulation can be a starting point for others, perhaps as needed for Problems B.5 and B.6 or for your own exploration. (a) Check the numbers in your simulation against all corresponding data in Figure B.2.2. Comment on any differences. (b) Do you have perfect mass conservation in your simulation? (c) Use the plotting features of your spreadsheet application to make graphs equivalent to those in Figures B.2.3 and B.2.4. (d) Change DM, B to 0.20 and let the simulation develop. What is the effect on Z(k)? On the mass-conservation array? Explain. (e) Change both DM, A and DM, B to 0.20 and allow the simulation to develop. What are the effects on Z(k)? On the value of R(100). On the graph of R(k)? On the mass-conservation array? Comment. B.3 Suppose one desires a simulation of chronocoulometry. Derive a dimensionless charge parameter analogous to Z(k). In carrying out a simulation, to what time should the dimensionless charge calculated for iteration k be assigned? B.4 Consider the following mechanism: A+e⇌B k2

(at the electrode)

B + C −−−→ D (in solution)

1005

1006

Appendix B Basic Concepts of Simulation

Derive the diffusion-kinetic equations analogous to (B.3.11) and (B.3.12) and identify the dimensionless kinetic parameter involving k 2 . B.5 Develop a spreadsheet (perhaps by modification of the one created in Problem B.2) and carry out simulations of cyclic voltammetry for a quasireversible system. Let l = 100 and DM = 0.45. Take 𝛼 = 0.5 and use equal diffusion coefficients for all species. Cast your dimensionless intrinsic rate parameter in terms of the function 𝜓 defined in (7.3.6), and carry out calculations for 𝜓 = 20, 1, and 0.1. Compare the peak splittings in your voltammograms with those in Table 7.3.1. B.6 To the simulation spreadsheet devised for Problem B.5, add a provision for first-order homogeneous decay of the reduction product B, that is, kf

−−−−−−− ⇀ A+e ↽ − B (quasireversible) kb

k1

B −−−→ D

(in solution)

Simulate 𝜓 = 20 and k 1 t k = 1. Compare the results with those predicted by Nicholson and Shain (11).

1007

Appendix C Reference Tables

Table C.1 Selected Standard Electrode Potentials in Aqueous Solutions at 25 ∘ C(a) E 0 /V vs. Old NHE(b)

Reaction

E 0 /V vs. NHE(c)

Ag+ + e ⇌ Ag

0.7991

0.7989

AgBr + e ⇌ Ag + Br−

0.0711

0.0709

AgCl + e ⇌ Ag + Cl− AgI + e ⇌ Ag + I− Ag2 O + H2 O + 2e ⇌ 2Ag + 2OH–

0.2223

0.2221

−0.1522

−0.1524

0.342

0.342

Al3+ + 3e ⇌ Al

−1.676

−1.676

Au+ + e ⇌ Au

1.83

1.83

Au3+ + 2e ⇌ Au

1.36

1.36

BQ + 2H+ + 2e ⇌ HQ

0.6992

0.6990

Br2

(aq) + 2e ⇌ 2Br–

CO2 + 8H+ + 8e ⇌ CH4 + 2H2 O

1.0874

1.0872

0.169

0.169

Ca2+ + 2e ⇌ Ca

−2.84

−2.84

Cd2+ + 2e ⇌ Cd

−0.4025

−0.4027

Cd2+ + 2e ⇌ Cd(Hg)

−0.3515

−0.3517

1.72

1.72

1.3583

1.3581

Ce4+ + e ⇌ Ce3+ Cl2

(g) + 2e ⇌ 2Cl–

HClO + H+ + e ⇌ 1/2Cl2 + H2 O

1.630

1.630

Co2+ + 2e ⇌ Co

−0.277

−0.277

Co3+ + e ⇌ Co2+

1.92

1.92

−0.90

−0.90

−0.424

−0.424

1.36

1.36

0.520

0.520

Cr2+ + 2e ⇌ Cr Cr3+ + e ⇌ Cr2+ Cr2 O2− + 14H+ 7 Cu+ + e ⇌ Cu

+ 6e ⇌

2Cr3+

+ 7H2 O

(Continued)

Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

1008

Appendix C Reference Tables

Table C.1 (Continued) E0 /V vs. Old NHE(b)

Reaction

E 0 /V vs. NHE(c)

Cu2+ + 2CN− + e ⇌ Cu(CN)− 2

1.12

1.12

Cu2+ + e ⇌ Cu+

0.159

0.159

Cu2+ + 2e ⇌ Cu

0.340

0.340

Cu2+ + 2e ⇌ Cu(Hg)

0.345

0.345

Eu3+ + e ⇌ Eu2+ 1/2F

2

−0.35

+ H+ + e ⇌ HF

Fe2+ + 2e ⇌ Fe Fe3+ + e ⇌ Fe2+ Fe(CN)3− +e⇌ 6 + 2H + 2e ⇌ H2

3.053 −0.44

Fe(CN)4− 6

2H2 O + 2e ⇌ H2 + 2OH−

−0.35 3.053 −0.44

0.771

0.771

0.3610

0.3608

0.0000

0.0000

−0.828

−0.828

H2 O2 + 2H+ + 2e ⇌ 2H2 O

1.763

1.763

Hg2+ 2

0.7960

0.7958

+ 2e ⇌ 2Hg

Hg2 Cl2

+ 2e ⇌ 2Hg + 2Cl–

0.26816

0.26799

HgO + H2 O + 2e ⇌ Hg + 2OH–

0.0977

0.0975

Hg2 SO4 + 2e ⇌ 2Hg + SO2− 4

0.613

0.613

I2

(c) + 2e ⇌ 2I–

0.5355

0.5353

I2

(aq) + 2e ⇌ 2I–

0.621

0.621

0.536

0.536

K+ + e ⇌ K

−2.925

−2.925

Li+ + e ⇌ Li

−3.045

−3.045

Mg2+ + 2e ⇌ Mg

−2.356

−2.356

Mn2+ + 2e ⇌ Mn

−1.18

−1.18

Mn3+ + e ⇌ Mn2+

1.5

1.5

MnO2 + 4H+ + 2e ⇌ Mn2+ + 2H2 O MnO− + 8H+ + 5e ⇌ Mn2+ + 4H2 O 4 + Na + e ⇌ Na

1.23

1.23

1.51

1.51

−2.714

−2.714

Ni2+ + 2e ⇌ Ni

−0.257

−0.257

I− + 2e ⇌ 3I− 3

Ni(OH)2

+ 2e ⇌ Ni + 2OH–

O2 + e ⇌ O2 −∙ O2 + 2H+ + 2e ⇌ H2 O2 O2 + 4H+ + 4e ⇌ 2H2 O O2 + 2H2

O + 4e ⇌ 4OH–

−0.72

−0.72

−0.284

−0.284

0.695

0.695

1.229

1.229

0.401

0.401

Appendix C Reference Tables

Table C.1 (Continued) E0 /V vs. Old NHE(b)

Reaction

O3 + 2H+ + 2e ⇌ O2 + H2 O

E 0 /V vs. NHE(c)

2.075

2.075

Pb2+ + 2e ⇌ Pb

−0.1251

−0.1253

Pb2+ + 2e ⇌ Pb(Hg)

−0.1205

−0.1207

1.468

1.468

PbO2 + 4H+ + 2e ⇌ Pb2+ + 2H2 O PbO2 + SO2− + 4H+ + 2e ⇌ PbSO4 4 PbSO4 + 2e ⇌ Pb + SO2− 4 Pd2+ + 2e ⇌ Pd Pt2+ + 2e ⇌ Pt PtCl2− 4 PtCl2− 6

−

+ 2e ⇌ Pt + 4Cl + 2e ⇌

PtCl2− 4

−

+ 2Cl

+ 2H2 O

1.698

1.698

−0.3505

−0.3507

0.915

0.915

1.188

1.188

0.758

0.758

0.726

0.726

0.10

0.10

S + 2e ⇌ S2–

−0.447

−0.447

Sn2+ + 2e ⇌ Sn

−0.1375

−0.1377

Ru(NH3 )3+ + e ⇌ Ru(NH3 )2+ 6 6

Sn4+ + 2e ⇌ Sn2+

0.15

0.15

Tl+ + e ⇌ Tl

−0.3363

−0.3365

Tl+ + e ⇌ Tl(Hg)

−0.3338

−0.3340

Tl3+ + 2e ⇌ Tl+

1.25

1.25

−1.66

−1.66

−0.52

−0.52

U3+ + 3e ⇌ U U4+ + e ⇌ U3+ UO2+ + 4H+ + 2e ⇌ 2 UO2+ + e ⇌ UO+ 2 2 V2+ + 2e ⇌ V

U4+

+ 2H2 O

V3+ + e ⇌ V2+

0.273

0.273

0.163

0.163

−1.13

−1.13

−0.255

−0.255

0.337

0.337

1.000

1.000

VO2+ + 2H+ + e ⇌ V3+ + H2 O VO+ + 2H+ + e ⇌ VO2+ + H2 O 2 Zn2+ + 2e ⇌ Zn

−0.7626

−0.7628

ZnO2− + 2H2 O + 2e ⇌ Zn + 4OH− 2

−1.285

−1.285

(a) Data mainly from Bard et al. (1) (prepared under the auspices of the Electrochemistry and Electroanalytical Chemistry Commissions of IUPAC). Other sources of standard potentials and thermodynamic data include Bard and Lund (2) and Milazzo and Caroli (3). (b) Before 1982, the NHE was based on a standard state of 1 atm for H2 . All measurements leading to these tabulated values relate to this definition of the NHE. (c) Since 1982, the NHE has been defined in terms of a standard state of 1 bar for H2 (Section 2.1.6). The present NHE is +0.169 mV vs. the “old” NHE. If the value in the preceding column has more than three decimal places, the entry in this column is adjusted by subtraction of 0.169 mV.

1009

1010

Appendix C Reference Tables

Table C.2 Selected Formal Potentials in Aqueous Solution at 25 ∘ C(a) ′

E0 /V vs. NHE

Reaction

Conditions

Cu(II) + e ⇌ Cu(I)

1 M NH3 + 1 M NH+ 4

0.01

Ce(IV) + e ⇌ Ce(III)

1 M HNO3

1.61

1 M HCl

1.28

1 M HClO4

1.70

1 M H2 SO4

1.44

Fc+ + e ⇌ Fc

0.1 M NaClO4

0.148(b)

FcA + e ⇌ FcA−

0.20 M Na2 HPO4 , pH 9.2

0.288(c)

FcMeOH+ + e ⇌ FcMeOH

0.1 M phosphate buffer, pH 8

0.21(d)

FcTMA2+ + e ⇌ FcTMA+

0.2 M KCl

0.35(e)

Fe(III) + e ⇌ Fe(II)

1 M HCl

0.70

10 M HCl

0.53

1 M HClO4

0.735

1 M H2 SO4

0.68

2 M H3 PO4

0.46

0.1 M HCl

0.56

1 M HCl

0.71

1 M HClO4

0.72

Fe(CN)3− 6

+e⇌

Ru(NH3 )3+ 6

Fe(CN)4− 6

+e⇌

Ru(NH3 )2+ 6

Sn(IV) + 2e ⇌ Sn(II)

0.02 M NaOAc, pH 5.5

−0.174(f )

0.2 M NaOAc, pH 5.5

−0.190(f )

2.0 M NaOAc, pH 5.5

−0.225(f )

1 M HCl

0.14

(a) Data mainly from Charlot (4). Additional values are found in Lingane (5) and Meites (6). Abbreviated chemical species are identified in Table 5 (front of book). Structures are in Figure 1, accompanying Table 5. (b) Noviandri et al. (7). Measurements were vs. Ag/AgCl at 22 ∘ C, but have been converted here to SCE by subtraction of 47 mV. (c) Matsue et al. (8). At 20 ∘ C. (d) Bourdillon et al. (9). Measured in 0.0365 M KH2 PO4 adjusted with 1 M NaOH to pH 8, leading to 0.1 M ionic strength. (e) Watkins et al. (10). (f ) Tsou and Anson (11). Measured vs. SSCE and converted here to the SCE by subtraction of 5 mV.

Appendix C Reference Tables

Table C.3 Estimated Formal Potentials vs. Aqueous SCE in Aprotic Solvents(a), (b)

E 0 /V

An + e ⇌ An−∙

DMF, 0.1 M TBAI

−1.92

+e⇌

DMF, 0.1 M TBAI

−2.5

Reaction

An

An−∙

An2−

+

AB

An ∙ + e ⇌ An

MeCN, 0.1 M TBAP

AB + e ⇌

DMF, 0.1 M TBAP

−1.36

AB−∙

BP

BQ

Me10 Fc

NB

O2

Ru(bpy)n3

AB−∙

+e⇌

AB2−

1.3

DMF, 0.1 M TBAP

−2.0

AB + e ⇌

AB−∙

MeCN, 0.1 M TEAP

−1.40

AB + e ⇌

AB−∙

PC, 0.1 M TBAP

−1.40

BP + e ⇌

BP−∙

MeCN, 0.1 M TBAP

−1.88

BP + e ⇌

BP−∙

THF, 0.1 M TBAP

−2.06

BP + e ⇌

BP−∙

NH3 , 0.1 M KI

−1.23(d)

BP−∙ + e ⇌ BP2−

NH3 , 0.1 M KI

−1.76(d)

BQ + e ⇌ BQ−∙

MeCN, 0.1 M TEAP

−0.54

+e⇌

MeCN, 0.1 M TEAP

−1.4

BQ−∙ Fc

′

Conditions(c)

Substance(c)

BQ2−

Fc+ + e ⇌ Fc

DCE, 0.1 M TBAP

0.478(e)

Fc+ + e ⇌ Fc

DMF, 0.1 M TBAP

0.497(e)

Fc+ + e ⇌ Fc

MeCN, 0.1 M TBAP

0.460(e)

Fc+ + e ⇌ Fc

MeCN, 0.1 M TBAPF6

0.382(f )

Me10 Fc+ + e ⇌ Me10 Fc

DCE, 0.1 M TBAP

−0.054(e)

Me10 Fc+ + e ⇌ Me10 Fc

DMF, 0.1 M TBAP

0.039(e)

Me10 Fc+ + e ⇌ Me10 Fc

MeCN, 0.1 M TBAP

−0.046(e)

Me10 Fc+ + e ⇌ Me10 Fc

MeCN, 0.1 M TBAPF6

−0.125(f )

NB + e ⇌ NB−∙

MeCN, 0.1 M TEAP

−1.15

NB + e ⇌

NB−∙

DMF, 0.1 M NaClO4

−1.01

NB + e ⇌

NB−∙

NH3 0.1 M KI

−0.42(d)

NB−∙ + e ⇌ NB2−

NH3 0.1 M KI

−1.241(d)

O2 + e ⇌ O2 −∙

DMF, 0.2 M TBAP

−0.87

O2 + e ⇌ O2

− ∙

MeCN, 0.2 M TBAP

−0.82

O2 + e ⇌ O2

− ∙

DMSO, 0.1 M TBAP

−0.73

Ru(bpy)3+ + e ⇌ Ru(bpy)2+ 3 3 Ru(bpy)2+ + e ⇌ Ru(bpy)+ 3 3 + 0 Ru(bpy)3 + e ⇌ Ru(bpy)3 Ru(bpy)03 + e ⇌ Ru(bpy)− 3

MeCN, 0.1 M TBABF4

1.32

MeCN, 0.1 M TBABF4

−1.30

MeCN, 0.1 M TBABF4

−1.49

MeCN, 0.1 M TBABF4

−1.73 (Continued)

1011

1012

Appendix C Reference Tables

Table C.3 (Continued) ′

Reaction

TCNQ

TCNQ + e ⇌ TCNQ−∙

MeCN, 0.1 M LiClO4

0.13

+e⇌

MeCN, 0.1 M LiClO4

−0.29

TCNQ−∙

Conditions(c)

E 0 /V

Substance(c)

TCNQ2−

+

TMPD ∙ + e ⇌ TMPD

TMPD

+

TTF ∙ + e ⇌ TTF

TTF

TTF2+

+

+ e ⇌ TTF ∙

+

TH ∙ + e ⇌ TH

TH

TH2+

+

+ e ⇌ TH ∙

+

TH ∙ + e ⇌ TH +

TH2+ + e ⇌ TH ∙ +

TPTA ∙ + e ⇌ TPTA

TPTA

DMF, 0.1 M TBAP

0.21

MeCN, 0.1 M TEAP

0.30

MeCN, 0.1 M TEAP

0.66

MeCN, 0.1 M TBABF4

1.23

MeCN, 0.1 M TBABF4

1.74

SO2 , 0.1 M TBAP

0.30(g)

SO2 , 0.1 M TBAP

0.88(g)

THF, 0.2 M TBAP

0.98

(a) See footnote (a) in Table C.1. (b) Problems arise in reporting potentials in nonaqueous solvents. Use of an aqueous reference electrode introduces an unknown, occasionally irreproducible, junction potential. Sometimes a reference electrode made up in the solvent of interest (e.g. Ag/AgClO4 ) or a QRE is employed. Results here are reported vs. an aqueous SCE unless noted otherwise. A frequent practice for reporting potentials in nonaqueous solvents is to reference them to the potential of a particular reversible couple in the same solvent. This couple (sometimes called the “reference redox system”) may have been chosen on the basis of an assumption that the redox potential of this system is only slightly affected by the solvent system. Commonly used are various ferrocene/ferrocenium couples. See reference (e) below. (c) Abbreviated chemical species are identified in Table 5 (front of book). Structures are in Figure 1, accompanying Table 5. (d) vs. Ag/Ag+ (0.01 M) in NH3 at −50 ∘ C. (e) Noviandri et al. (7). Measurements were vs. Ag/AgCl at 22 ∘ C, but have been converted here to SCE by subtraction of 47 mV. This source provides data for Fc+ /Fc and Me10 Fc+ /Me10 Fc in 29 solvents. (f ) Aranzaes et al. (12). At 20 ∘ C. (g) vs. Ag/AgNO3 (sat’d) in SO2 at −40 ∘ C.

Table C.4 Selected Diffusion Coefficients(a)

Substance

Medium

T (∘ C)

105 D (cm2 s−1 )

References

Cd

Hg

25

1.5

(13)

Cd2+

0.1 M KCl

25

0.70

(14)

Fc

MeCN, 0.5 M TBABF4

RT

1.7

(15)

Fc

MeCN, 0.1 M TEAP

RT

2.4

(16)

Fc

MeCN, 0.6 M TEAP

RT

2.0

(17) (8)

FcA−

0.20 M Na2 HPO4 , pH 9.2

20

0.54(b)

FcTMA+

0.2 M KCl

RT

0.75

(10)

Fe(CN)3− 6

0.1 M KCl

25

0.76

(18)

Fe(CN)3− 6

1.0 M KCl

25

0.76

(18)

References

Table C.4 (Continued)

Substance

Medium

T (∘ C)

105 D (cm2 s−1 )

References

Fe(CN)4− 6

0.1 M KCl

25

0.65

(18)

Fe(CN)4− 6

1.0 M KCl

25

0.63

(18)

H2

0.1 M KNO3

25

5.0

(19)

H+

0.1 M KNO3

25

7.9

(19)

Ru(NH3 )3+ 6 Ru(NH3 )3+ 6 Ru(NH3 )3+ 6

0.1 M KNO3

RT

0.74

(20)

0.1 M NaTFA

RT

0.67

(17)

0.09 M phosphate

RT

0.53

(21)

Stilbene

DMF, 0.5 M TBAI

RT

0.80

(22)

(a) NaTFA, sodium trifluoroacetate; RT, room temperature (exact temperature not specified). Abbreviated chemical species are identified in Table 5 (front of book). Structures are in Figure 1, accompanying Table 5. (b) At 20 ∘ C.

References 1 A. J. Bard, J. Jordan, and R. Parsons, Eds., “Standard Potentials in Aqueous Solutions,”

Marcel Dekker, New York, 1985. 2 A. J. Bard and H. Lund, Eds., “The Encyclopedia of the Electrochemistry of the Elements,”

Marcel Dekker, New York, 1973–1986. 3 G. Milazzo and S. Caroli, “Tables of Standard Electrode Potentials,” Wiley-Interscience, 4 5 6 7 8 9 10 11 12 13 14 15 16 17

New York, 1977. G. Charlot, “Oxidation-Reduction Potentials,” Pergamon, London, 1958. J. J. Lingane, “Electroanalytical Chemistry,” Interscience, New York, 1958. L. Meites, Ed., “Handbook of Analytical Chemistry,” McGraw-Hill, New York, 1963. I. Noviandri, K. N. Brown, D. S. Fleming, P. T. Gulyas, P. A. Lay, A. F. Masters, and L. Phillips, J. Phys. Chem. B, 103, 6713 (1999). T. Matsue, D. H. Evans, T. Osa, and N. Kobayashi, J. Am. Chem. Soc., 107, 3411 (1985). C. Bourdillon, C. Demaille, J. Moiroux, and J.-M. Savéant, J. Am. Chem. Soc., 117, 11499 (1995). J. J. Watkins, J. Chen, H. S. White, H. D. Abruña, E. Maisonhaute, and C. Amatore, Anal. Chem., 75, 3962 (2003). Y.-M. Tsou and F. C. Anson, J. Electrochem. Soc., 131, 595 (1984). J. R. Aranzaes, M.-C. Daniel, and D. Astruc, Can. J. Chem., 84, 288 (2006). I. M. Kolthoff and J. J. Lingane, “Polarography,” 2nd ed., Interscience, New York, 1952, p. 201. D. J. Macero and C. L. Rulfs, J. Electroanal. Chem., 7, 328 (1964). M. V. Mirkin, T. C. Richards, and A. J. Bard, J. Phys. Chem., 97, 7672 (1993). A. M. Bond, T. L. E. Henderson, D. R. Mann, T. F. Mann, W. Thormann, and C. G. Zoski, Anal. Chem., 60, 1878 (1988). D. O. Wipf, E. W. Kristensen, M. R. Deakin, and R. M. Wightman, Anal. Chem., 60, 306 (1988).

1013

1014

Appendix C Reference Tables

18 19 20 21 22

M. von Stackelberg, M. Pilgram, and W. Toome, Z. Elektrochem., 57, 342 (1953). J. V. Macpherson and P. R. Unwin, Anal. Chem., 69, 2063 (1997). J. Kim and A. J. Bard, J. Am. Chem. Soc, 138, 975 (2016). R. M. Wightman and D. O. Wipf, Electroanal. Chem., 15, 267 (1989). H. Kojima and A. J. Bard, J. Electroanal. Chem., 63, 117 (1975).

1015

Index a Absolute potential, Fermi level and, interfacial potential differences 88–91 Absolute rate theory. See Transition state theory Ac circuits 446–450 Accumulation layer, semiconductor electrodes 895, 899 Ac polarography 473 Activation energy: nucleation on an electrode 693–696 potential energy surfaces 122 Activation overpotential, i-𝜂 equation 136 Ac voltammetry 443, 470 chemical analysis by 481–482 cyclic ac voltammetry 477 large amplitude 479–481 quasireversible and irreversible systems 473–476 reversible systems 470–473 second harmonic 478–479 Adder potentiostat: described 731–732 refinements to 732–733 Adders, operational amplifier 726–727 Adiabaticity, kinetics, microscopic theories 145–146 Adsorbed monolayer responses: chronocoulometry 775–777 chronopotentiometry 779 coulometry in thin-layer cells 777–778 impedance measurements 778–779

Adsorption: electrical double-layer structure 599–645 (See also Electrical double-layer structure) electroinactive species 635 practical aspects of 634–636 irreversible, monolayers 756 nonspecific 627 specific 627–629 Adsorption energy 657 Adsorption isotherms, electrical double-layer structure 630–632 Adsorptive stripping voltammetry 530 Amperometric methods, electrometric end-point detection 509–511 Amperometric sensors 803 Analog circuits, electrochemical instrumentation 721 Analog-to-digital converter (ADC) 737 Angular frequency 446 Anion-induced adsorption 777 Anode 18 Anodic current 18 Anodic stripping voltammetry, stripping analysis 527 Anodic stripping voltammograms (ASVs) 707 Anson plot 297 Anti-Stokes, Raman spectroscopy 951 Apparent diffusion coefficient 787 Aptamer 805 Arrhenius equation, potential energy surfaces and 122–123

Electrochemical Methods: Fundamentals and Applications, Third Edition. Allen J. Bard, Larry R. Faulkner, and Henry S. White. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/BardElectrochemical3e

1016

Index

Arrhenius rate expression nucleation, CNT 691–693 Associative mechanism ORR 669 Atomic force microscopy (AFM) 828, 934–937 Attenuated total reflection (ATR) 948 Automated potentiostats. See Electrochemical workstations Auxiliary electrode. See Counter (auxiliary) electrode

b Background limits, defined 11–12 Band gap, photoelectrochemistry 885 Band model, photoelectrochemistry 885–886 Band UME 221–222 Bandwidth, operational amplifiers 723–724 Biconductive films 782 Bioelectrocatalysis 799–803 Bimolecular reactions, digital simulations of homogeneous kinetics 1000–1001 Biologically related materials, electroactive layers 782 Biological systems, SECM 840–841 Bipolar electrode, electrogenerated chemiluminescence 921 Bipotentiostats, described 733–734 Blocking collision, nanoparticle collision behavior 857–861 Blocking layers 791–798 permeation through pores and pinholes 792–796 tunneling through blocking films 796–798 Blocking polymers 780 Bode plots 449, 450 operational amplifiers 723–724 Bohr magneton, electron spin resonance 956 Booster amplifier, potentiostats 733 Boundary conditions: mass transfer, diffusion 200–201

simulation 1001–1004 controlled current 1003–1004 heterogeneous kinetics 1002 nernstian systems 1001–1002 potential sweeps 1003 Bubble nucleation, CNT 698–699 Bulk electrolysis methods 489–531. See also Electrolysis classification of 489–490 controlled-current methods: characteristics of 501–502 constant-current electrolysis 506–507 coulometric titrations 502–506 controlled-potential methods 495–501 compliance-limited electrolysis 497 controlled potential is impractical 498 coulometry 498–500 current–time behavior 495–497 electrogravimetry 500–501 electroseparations 501 current efficiency 491 electrode process, completeness of 490–491 electrometric end-point detection 507–510 amperometric methods 509–511 current–potential curves during titration 507–508 potentiometric methods 508–509 experimental concerns 491–495 cell design 492 cell resistance 494–495 effect of long time scale 495 geometry of electrode placement 493–494 isolation of the counter electrode 494 mass transport 493 working electrodes 493 flow electrolysis 510–521 dual-electrode flow cells 515–516 mathematical treatment 510, 512–515

Index

microfluidic flow cells 516–521 overview 510 overview 489 stripping analysis 527–531 applications and variations 529–531 overview 527 principles and theory 528–529 thin-layer electrochemistry 521–526 applications of 526 chronoamperometry and coulometry 521–524 dual-electrode thin-layer cells 526 overview 521 potential sweep in a nernstian system 524–526 Butler–Volmer equation, HER 660–661 Butler–Volmer kinetics 231–232, 281–282, 1002 Butler–Volmer model 126–132 energy barriers, effects of potential on 127 implications of 132–142 current-overpotential equation 133–134 exchange current plots 139 i- 𝜂 equation 135–142 limits of 141–142 mass transfer effects 140–141 reversible behavior 139–140 one-step, one-electron process 127–130, 132–142 SECM feedback 823 standard rate constant 130–131 transfer coefficient 131–132

c Capacitance: charge and, electrode/solution interface 41–44 charging current and electrical double layer 44–51 excess charge and, electrical double-layer structure 603–606 Capillary electrophoresis, microfluidic flow cells, flow electrolysis 516, 521

Carrier generation-tip collection mode (CG-TC mode) 844–845 Catalytic (EC′ ) reaction, coupled homogeneous electrode reactions 542 Cathode 18 Cathodic current 12, 18 Cathodic depolarizer 503 Cathodic diffusion-limited steady-state current 211 Cathodic peak current 316 Cathodic peak potential 316 Cell emf: concentration and, thermodynamics 69–71 free energy and, thermodynamics 64–66 Cell equivalent circuits, EIS: Faradaic impedance (See Faradaic impedance) Randles equivalent circuit 451–452 Cell impedance measurement 444–445 Cell potential 4–5 Cell resistance: bulk electrolysis 494–495 measurement of potential 34–41 components of 35–37 three-electrode cells 37–38 two-electrode cells 37 uncompensated resistance 38–41 Cell time constants 300–303 CE reaction, coupled homogeneous electrode reactions 542 Characteristic dimension, UME 215 Charge, capacitance and electrode/solution interface 41–44 Charge carriers, photoelectrochemistry 887 Charge density, electrode/solution interface and charging current 42 Charge step (coulostatic) methods 403–406 coulostatic perturbation by temperature jump 405–406

1017

1018

Index

Charge step (coulostatic) methods (contd.) large excursions 405 small excursions 404–405 Charge-transfer electrodes 17 Charge transfer, microscopic theories of 142–168. See also Microscopic theories (kinetics) Charge-transfer overpotential 23, 34 Charging current: capacitance and, electrical double layer 44–51 in electrochemical measurements, electrical double layer capacitance 49–50 electrode/solution interface 41–44 Chemically Modified Electrodes 755. See also Electroactive layers; Blocking layers Chemical reversibility, thermodynamics 61–62 Chemical state of electrode surface 304 Chemiluminescence. See Electrogenerated chemiluminescence Chlorine evolution, inner-sphere reactions 670 Chronoamperometric reversal techniques: approaches to problem 292–293 current–time responses 293–294 Chronoamperometry 262 information from 270–271 linear diffusion 262–265 measurement in 264 microscopic and geometric areas 271–275 other ultramicroelectrodes 267–270 spherical electrode 265–267 thin-layer electrochemistry 521–524 Chronocoulometry: adsorbed monolayer 775–777 effects of heterogeneous kinetics 299–300 large-amplitude potential step 295–296 reversal experiments under diffusion control 296–299

Chronopotentiometry 389–391 adsorbed monolayer responses 779 programmed current, controlled-current techniques 394 transient voltammetry and, coupled homogeneous electrode reactions (See also Coupled homogeneous electrode reactions) Cis-trans isomerization, potential energy surface 122 Clark oxygen electrode, gas-sensing ISEs 111 Classical nucleation theory (CNT): activation energy, electrode surface properties 693–696 Arrhenius rate expression for nucleation 691–693 bubble nucleation 698–699 nucleation overpotential 689–691 3D solid nucleation 696–698 Clays, inorganic films 781 Cluster deposition, single atoms and atomic clusters 871 CNT. See Classical nucleation theory (CNT) Collision frequency, particle collision experiments 854–855 Compact layer. See Inner layer Comparative simulation 551 Complex notation, mathematical methods 979–980 Composites, multilayer assemblies and, electroactive layers 782 Comproportionation 250, 543 204 COMSOL Multiphysics Concentration arrays, simulation 996 Concentration, cell emf and, thermodynamics 69–71 Concerted vs. stepwise reaction, coupled homogeneous electrode reactions 584–590 diagnostic criteria 586–588 mechanistic switching 589 proton-coupled electron transfers 589–590

®

Index

reaction diagrams 585–586 reductive and oxidative elimination reactions 588 Conductance, liquid junction potentials 92–96 Conducting phases, interactions between, interfacial potential differences 82–84 Conduction band 885 Constant-current chronopotentiometry 389 Constant-current electrolysis: bulk electrolysis 506–507 potential-time curves in, controlled-current techniques 394–398 (See also Controlled-current techniques) Constant current mode, scanning tunneling microscopy 932 Contact mode, atomic force microscopy 937 Contact potential, electrochemical potentials 87 Continuity equation, convective systems 413 Controlled-current methods, bulk electrolysis methods 501–506 characteristics of 501–502 coulometric titrations 502–506 Controlled-current techniques 389–406 boundary conditions, simulations 1003–1004 bulk electrolysis 489 chronopotentiometry 389–391 galvanostatic double pulse method 401–402 multicomponent systems and multistep reactions 400–401 potential-time curves in constant-current electrolysis 394–398 irreversible waves 394–395 quasireversible waves 395 reversible (nernstian) waves 394

transition time measurement 396–398 programmed current chronopotentiometry 394 reversal techniques 398–399 current reversal 398–399 response function principle 398 theory 391–394 constant-current electrolysis 392–394 programmed current chronopotentiometry 394 semi-infinite linear diffusion 391–392 Controlled-potential bulk electrolysis 495–501 compliance-limited electrolysis 497 controlled potential, when impractical 498 coulometry 498–500 current–time behavior 495–497 electrogravimetry 500–501 electroseparations 501 Control of current vs. control of potential 16–17 Convection, mass-transfer-controlled reactions 24 Convective-diffusion equation: convective systems theory 412 solution of, rotating disk electrode 416–418 Convective systems, hydrodynamic methods 411–414 Conventional polarography 355 Coordinating (ligand-bearing) polymers 780 Coreactant, electrogenerated chemiluminescence 916–917 CO2 reduction, inner-sphere reactions 673–674 Cottrell equation 263–264 Cottrell slope 270–271 Coulometry: bulk electrolysis 502–506 electrochemical cells 19

1019

1020

Index

Coulometry (contd.) thin-layer cells, adsorbed monolayer responses 777–778 Coulostatic methods. See Charge step (coulostatic) methods Coulostatic perturbation by temperature jump, charge step (coulostatic) methods 405–406 Counter (auxiliary) electrode, electrochemical cells and cell resistance 37 Coupled homogeneous reactions, electrode processes with 539–593 electrochemical methods 591–593 impact of coupled reactions on cyclic voltammetry: characteristic times 547 comparative simulation 551 diagnostic criteria 545–546 example 547–548 kinetics in theory 548–551 reaction classification 539–545 multiple E-step reactions 542–545 one E-step reactions 541–542 simplified introduction 31–34 survey of behavior: bidirectional following reaction 558–561 concerted vs. stepwise reaction 584–590 Cr Er 564–568 ECE/DISP reactions 576–584 ECi systems 556–558 Er Ci mechanism 552–556 Er Ci′ systems 561–564 multistep electron transfers 569–576 reaction schemes 590–591 Covalent attachment, monolayers 756–757 Cross-reaction current, modified electrode 788

Current amplifier, low current measurements 746 Current compliance, potentiostat 738 Current efficiency, bulk electrolysis 491 Current feedback 725–728 adders 726–727 current follower 725–726 integrators 727–728 scaler/inverter 726 Current follower, operational amplifier 725–726 Current function 317 Current interruption, electronic resistance compensation 742–744 Current-overpotential equation, Butler–Volmer model 133–134 Current-potential curves: background i–E curve 10–12 change of working electrode 12–13 open-circuit potential 9–10 precedence of electrode reactions 14–15 redox couple 15–16 Current reversal: chronopotentiometry 390 controlled-current techniques 398–399 Current, simulation 991–992 Current-time behavior, controlled-potential methods 495–497 Current-to-voltage converter, operational amplifier 725 Curve fitting 237 Curve splitting 145 Cyclic ac voltammetry (CACV) 477 Cyclic chronopotentiometry 390 Cyclic square-wave voltammetry (CSWV) 381, 477 Cyclic voltammetry (CV) 311–347 adsorbed monolayer: electric-field-driven acid–base chemistry 771–775

Index

fundamentals 757–758 irreversible adsorbate couples 763–765 more complex systems 770–771 nernstian processes involving adsorbates and solutes 766–770 reversible adsorbate couples 758–763 basic experimental conditions 344–345 charging-current background 324 choice of initial and final potentials 345–347 convolutive transformation 336–339 coupled homogeneous electrode reactions: characteristic times 547 comparative simulation 551 diagnostic criteria 545–546 example 547–548 kinetics in theory 548–551 deaeration 347 E1/2 , estimation of 323 fast cyclic voltammetry 334–336 information from reversible 324–325 liquid/liquid interfaces 339–344 mode 214 more general notation for LSV and CV 325 multicomponent systems 332–333 multistep charge transfers 333–334 nernstian (reversible) systems 313–325 peak current ratio 322 peak separation 322–323 quasireversible systems 325–329 theory 321 totally irreversible systems 329–332 transient responses to potential sweep 311–313 uncompensated resistance, effect of 323–324 Cylindrical electrode 270 Cylindrical UME 221

d Dahms–Ruff equation 787 Dark-field microscopy 938 Data logging 305 Deaeration 347 Debye–Hückel theory 68 Debye length 611 Decoration, electrode surface 755 Degeneracy, semiconductor electrodes 895, 899 Density functional theory (DFT), electrocatalytic reactions 679 Depletion layer, semiconductor electrodes 895 Dc polarography. See also Polarography Differential capacitance 603 Differential electrochemical mass spectrometry (DEMS) 954–955 Differential equations, Laplace transform technique 967–976. See also Laplace transform technique Differential pulse voltammetry (DPV) 369–376 concept of 370–371 definition 375 renewal vs. pre-electrolysis 374–375 residual currents 375 theory 371–374 Diffuse double-layer effects on mass transport 640–645 Diffuse layer, electrical double layer 44, 609–614 Diffusion: mass transfer 193–199 boundary conditions 200–201 Fick’s laws 196–198 flux of an electroreactant, electrode surface 199 microscopic view 193–196 mass-transfer-controlled reactions 24 microscopic view, mass transfer 193–196

1021

1022

Index

Diffusion (contd.) migration and, mass transfer, mixed near an active electrode 187–192 Diffusion coefficients, simulation 993 Diffusion controlled currents, at ultramicroelectrodes, potential step methods 303–305. See also Potential step methods Diffusion layer 208 Diffusion layer thickness, simulation 992 Diffusion length 212 Diffusion-limited current-time response 268, 269 Diffusion-limited steady-state current 219, 271 Diffusion potentials 105 liquid junction potentials 92 Diffusion, simulation 986–987 Diffusive renewal, normal pulse voltammetry 364 Digital simulation 203 comparative 551 Digital-to-analog converter (DAC) 484, 737 Dimensionally stable anode (DSA) 891 Dimensionless parameters, simulations 987–990 Direct methanol fuel cell (DMFC) 670 Direct process monitoring 252 Discrete energy, Raman spectroscopy 950 Disk electrode 267–270 Disk UME: diffusion-limited steady-state current 219 steady-state diffusion 216–219 surface concentrations of O and R 220–221 variation of current density 220 Display system, electrochemical instrumentation 721 DISP reaction, coupled homogeneous electrode reactions 581–583 Disproportionation 543 Dissipation factor, quartz crystal microbalance 942

Dissociative mechanism ORR 669 Distance, simulation 990–991 Dopants 889 Doped semiconductors 889–890 Double junction 112 Double layer. See Electrical double layer Double-layer capacitance, effect of 319–320 Double potential step chronoamperometry 291 Dropping mercury electrode (DME) 355–356, 358. See also Polarography differential pulse polarography 375 normal pulse polarography 364–366 reverse pulse polarography 368 Dual-electrode cells, microfluidic flow cells, flow electrolysis 516–517, 521 Dual-electrode flow cells, flow electrolysis 515–516 Dual-potential generation, radical-ion annihilation 912–913 Dummy cells 56 Dye sensitization, semiconductor electrodes 905–906 Dynamic diffuse layer 640 Dynamic equilibrium, kinetics 121–122

e Earth ground, operational amplifiers 721 891 Ebonex ECE reaction, coupled homogeneous electrode reactions 544 Er Ci Er Reactions 578–580 Er Cr Er reactions 580–581 EC reaction, coupled homogeneous electrode reactions 541 ′ EC reaction, coupled homogeneous electrode reactions 542 EE reaction, coupled homogeneous electrode reactions 542–543 EIS. See Electrochemical impedance spectroscopy (EIS)

®

Index

Electrical double layer: capacitance and charging current 44–51 conducting phases, interactions between 83 electrode/solution interface and charging current 42–44 Electrical double-layer structure 599–645 electrode reaction rates 636–645 diffuse double-layer effects on mass transport 640–645 principles 636–638 specific adsorption of electrolyte: absence of 638–639 presence of 639–640 experimental evaluations: electrocapillarity 602 excess charge and capacitance 603–606 relative surface excesses 606–607 models for 606–619 Gouy–Chapman–Stern model 614–617 Gouy–Chapman theory 609–614 Helmholtz model 607–609 specific adsorption 617–619 solid electrodes 619–627 double layer at 623–627 well-defined single-crystal surfaces 620–623 specific adsorption 627–634 adsorption isotherms 630–632 electrical double-layer structure 628–629 rate of 633–634 thermodynamics 599–602 electrocapillary equation 601 Gibbs adsorption isotherm 599–600 relative surface excesses 601–602 Electrical insulator 888 Electroactive domain 466–470 Electroactive layers 755–812 adsorbed monolayer responses 775–779 chronocoulometry 775–777

chronopotentiometry 779 coulometry in thin-layer cells 777–778 cyclic voltammetry 757–775 impedance measurements 778–779 biologically related materials 782 composites and multilayer assemblies 782 dynamics in modification layers 782–791 interplay of dynamical elements 789–791 principal dynamic processes in modifying films 784–788 steady state at a rotating disk 783–784 electrochemical methods 798–812 bioelectrocatalysis based on enzyme-modified electrodes 799–803 electrocatalysis 799 electrochemical sensors 803–809 faradaic electrochemical measurements in vivo 809–812 inorganic films 780–782 monolayers 756–757 covalent attachment 756–757 irreversible adsorption 756 reversible adsorption 756 transferred layers 757 underpotential deposition 757 overview 755–756 polymers 780 substrates 755 thick modification layers on electrodes 780–782 Electroactive polymers 780 Electrocapillarity, electrical double-layer structure 602 Electrocapillary equation, electrical double layer structure 601 Electrocapillary maximum (ECM) 603 Electrocatalysis 628, 653

1023

1024

Index

Electrocatalytic amplification collision, nanoparticle collision behavior 861–864 Electrocatalytic correlations 684–688 Electrocatalytic reaction mechanisms: HER kinetics, Tafel analysis 660–667 hydrogen evolution reaction 657–660 Electrochemical cells: definition 3 electrochemical experiment and variables 18–21 reactions and, electrode processes 2–4 types and definitions of 17–18 Electrochemical impedance spectroscopy (EIS) 443. See also Ac voltammetry ac circuits 446–450 applications 470 cell equivalent circuits: faradaic impedance (See Faradaic impedance) Randles equivalent circuit 451–452 cell impedance measurement 444–445 electroactive domain 466–470 instrumentation for: frequency-domain measurements 482–483 time-domain measurements 483–485 Laplace plane analysis 485 measurement conditions 458–460 nonlinear responses: large amplitude ac voltammetry 479–481 second harmonic ac voltammetry 478–479 resistance and capacitance measurement 465–466 system with simple faradaic kinetics 460–465 Electrochemical instrumentation. See Instrumentation Electrochemical methods, coupled homogeneous electrode reactions 591–593

Electrochemical methods, electroactive layers 798–812 bioelectrocatalysis based on enzyme-modified electrodes 799–803 electrocatalysis 799 electrochemical sensors 803–809 faradaic electrochemical measurements in vivo 809–812 Electrochemical Ostwald ripening 707 Electrochemical phase transformations: classical nucleation theory (See Classical nucleation theory (CNT)) electrodeposition 699–707 gas evolution 707–713 nucleation and growth 688–689 Electrochemical potentials, interfacial potential differences 85–88 Electrochemical renewal, normal pulse voltammetry 363–364 Electrochemical scanning tunneling microscopy (EC-STM) 706, 932–933 Electrochemical thermodynamics. See Thermodynamics Electrochemical transient techniques 262 Electrochemical workstations 721 Electrochemistry: definition 1 literature on 52 Electrode kinetics effect 230 Electrodeposition 699–707 electrochemical growth 703–705 E0 dependence on particle size 705–707 Frank–van der Merwe 701, 702 Stranski–Krastanov 701, 702 underpotential deposition 702–703 Volmer–Weber 701 Electrode processes 1–52 current–potential curves 9–16 electrochemical cells and reactions 2–4 electrode/solution interface and charging current 41–44

Index

capacitance and charge at an electrode 41–42 electrical double layer 42–44 electrical double layer capacitance and charging current 44–51 ideally polarizable electrode 41 electron energy, potential 6 faradaic and nonfaradaic processes 17 interfacial potential differences and cell potential 4–5 magnitudes in electrochemical systems 8–9 mass-transfer-controlled reactions 23–31 modes of 24–25 steady-state mass transfer 25–31 nernstian reactions with coupled chemical reactions 31–34 irreversible reactions 32–34 reversible reactions 31–32 overview 1–2 reaction rate 6–8 reference electrodes and control of potential, working electrode 5–6 Electrode reaction 3 Electrode reaction rate and current, factors affecting, faradaic processes 21–23 Electrode reactions, kinetics of 121–176. See also Kinetics Electrode/solution interface and charging current: electrical double layer 42–44 electrical double layer capacitance and charging current 44–51 ideally polarizable electrode 41 Electrode surface reactions 653–654 Electrogenerated chemiluminescence 910–922 applications 918–922 chemical fundamentals 910–912 quantum dots 917–918 radical-ion annihilation 912–916 dual-potential generation 912–913

modes of exploration 913–916 single-potential generation 916–917 solution phase 922 Electrogravimetry, bulk electrolysis 489, 500–501 Electrohydrodynamic phenomena 436–438 Electrolysis 18. See also Bulk electrolysis methods balance sheet for mass transfer during, mixed migration and diffusion 187–192 electrolysis of copper complexes with excess electrolyte 191 electrolysis of copper complexes without added electrolyte 190 electrolysis of HCl 188–189 constant-current: controlled-current techniques 392–394 potential-time curves in, controlled current techniques 394–398 potential-time curves in, controlled current techniques (See also Controlled-current techniques) Electrolysis, nanoparticle collision behavior 864–870 Electrolyte/electrolyte boundary, liquid junction potentials 91 Electrolyte, excess, mixed migration and diffusion 191 Electrolytic cells 17, 18 Electrometric end-point detection 507–510 amperometric methods 509–511 current–potential curves during titration 507–508 potentiometric methods 508–509 Electron and ion spectrometry 958–960 Electron diffusion current 787 Electronically conductive polymers 780 Electronic resistance compensation 740–744 Electron microscopy 957–958 Electron spectrometry 958–960

1025

1026

Index

Electron spillover 617 Electron spin resonance (ESR) 955–956 Electron-transfer-catalyzed substitution 544 Electron tunneling, charge transfer 143–144 Electroosmotic flow 437 Electrophoresis 438 Electrophotonic sensors 803 Electroseparations: bulk electrolysis 489 controlled-potential methods 501 Electrosorption valency 629–630 Ellipsometry 945–946 Emf. See Cell emf Energy barriers, effects of potential on, Butler–Volmer model 127 Energy-deficient, electrogenerated chemiluminescence 911 Energy state distributions, kinetics, microscopic theories 162–168 Gerischer model 163–166 Marcus–Gerischer Kinetics 166–167 reorganization energy effect of 167–168 Energy-sufficient, electrogenerated chemiluminescence 911 Enzyme-Modified Electrodes 799–803 Equilibrium potential 10 Equivalent circuits, cell: faradaic impedance (See Faradaic impedance) Randles equivalent circuit 451–452 Error detection, simulation 996–997 Error function, gaussian distribution and, mathematical methods 977–978 Esin–Markov coefficient, electrical double layer structure 618 Esin–Markov effect 618 Evanescent wave, near-field scanning optical microscopy 938 Excess charge, capacitance and, electrical double-layer structure 603–606

Exchange current, Butler–Volmer model, equilibrium conditions 133 Exchange current plots, Butler–Volmer model 139 Exchange velocity, dynamic equilibrium 122 Ex-situ techniques 957–960 electron and ion spectrometry 958–960 electron microscopy 957–958 Extended charge transfer, tunneling and, kinetic microscopic theories 143–144 Extended X-ray absorption fine structure (EXAFS) 953 External reflectance mode, infrared spectroscopy 947–950 Extrinsic semiconductors 889

f Faradaic impedance method 445 interpretation of 452–455 mean potential effect 457 multistep electrode reactions 458 obtaining kinetic parameters 455–456 phase angle 458 reversible electrode reactions 456–457 Faradaic processes: definition 17 electrochemical cells 17–18 electrochemical experiment and variables 18–21 electrode reaction rate and current 21–23 Faradaic response 305 Faraday constant 7 Faraday’s law, defined 7 Fast cyclic voltammetry, cyclic voltammetry (CV) 334–336 Fast-scan cyclic voltammetry (FSCV) 810 Feedback. See Current feedback; Voltage feedback Felici instability, electrohydrodynamic phenomena 438 Fermi energy 890–891

Index

Fermi energy, absolute potential and, interfacial potential difference 88–91 Ferrocenylmethyltrimethylammonium (FcTMA+ ) 234, 235 Fick’s laws of diffusion 208, 209, 263, 265, 466 mass transfer 196–198 partial differential equations 967 Flat-band potential, semiconductor electrodes 894 Floating input or output, operational amplifiers 721 Flow electrolysis 510–521 bulk electrolysis 492 dual-electrode flow cells 515–516 mathematical treatment 510, 512–515 microfluidic flow cells 516–521 overview 510 Fluorescence microscopy 938 Fluorine-doped tin oxide (FTO) 892 Forbidden region in solids 885 Forced convection. See Hydrodynamic methods Formal potentials 71–72 Four-electrode configuration, potentiostats 734 Fourier series and transformation, mathematical methods 981–982 Fourier transformation (FT) 483 Frank–van der Merwe electrodeposition 701, 702 Free energy, cell emf and, thermodynamics 64–66 Frequency domain, Fourier series 981 Frequency-domain instruments, EIS 482–483 Frequency factor, potential energy surfaces 122 Frequency response analyzer (FRA) 482–483 Frumkin correction 638 Frumkin effect 636

Frumkin isotherm 632 Function generator, electrochemical instrumentation 721 Fundamental frequency 477

g Gain–bandwidth product (GBP) 723, 724 Galvanic cell 17, 18 Galvanostatic double pulse method 401–402 Galvanostatic experiments, electrochemical cells 19 Galvanostatic techniques 389 Galvanostats 389, 721, 734–736 Gas-evolution reactions 689 electrochemical phase transformations 707–713 Gas-sensing electrodes, ion-selective electrodes 111–112 Gaussian distribution, error function and, mathematical methods 977–978 General formulation, linear diffusion problems 391–392 Generator–collector mode 244 Geometric area, electrode 272 Gerischer model, energy state distributions, kinetics, microscopic theories 163–166 Gibbs adsorption isotherm, electrical double layer structure 599–600 Gibbs free energy, reversibility and, thermodynamics 64 Gibbs–Thomson relationship 706 Glass electrodes, ion-selective 102–106 Glassy carbon (GC) disk electrode 55 Glucose sensors 803–805 Gouy–Chapman–Stern model, electrical double-layer structure 614–617 Gouy–Chapman theory, electrical double-layer structure 609–614 Ground, operational amplifiers 721

1027

1028

Index

h Half-peak potential 316 Half-reactions: electrochemical cells 3 standard electrode potentials and, thermodynamics 66–67 Half-wave potential 237, 243, 275 Hanging mercury drop electrode (HMDE) 528. See also Static mercury drop electrode (SMDE) Helmholtz layer. See Inner layer Helmholtz model, electrical double-layer structure 607–609 Helmholtz–Smoluchowski equation 437 Henry’s law 710 Heterogeneous electron-transfer reactions: kinetic measurements in SECM 828–831 surface topography image 826–827 Heterogeneous kinetics, simulations, boundary conditions 1002 Heterogeneous rate constant 12 Heterogeneous reactions 8 Highly oriented pyrolytic graphite (HOPG) 622 Hole transport material (HTM) 900 Homogeneous kinetics, simulation 999–1001 bimolecular reactions 1000–1001 unimolecular reactions 999–1000 Homogeneous reactions 8 SECM 831–835 Homogeneous nucleation 693 Hydrodynamic methods 411–438 convective systems theory 411–414 convective-diffusion equation 412 velocity profile determination 412–414 electrohydrodynamic phenomena 436–438 overview 411–412 rotating disk electrode 414–426 concentration profile 418–419 convective-diffusion equation, solution of 416–418

current distribution at 423–426 i–E curves 419–420 Koutecký-Levich method 420–423 practical considerations for application 426 transients at 432–433 velocity profile at 414–416 rotating disk electrode (modulated) 435–436 rotating ring-disk electrode 428–431 transients at 433–435 rotating ring electrode 426–428 Hydrogen evolution reaction (HER) electrocatalytic reaction mechanisms 657–660 hydrogen atom reductive adsorption 658–659 Tafel plot analysis 660–667 Volmer–Tafel and Volmer–Heyrovský mechanisms 659–660 Hydrogen peroxide oxidation reduction 677–678 Hyperfine structure, electron spin resonance 956

i Ideally polarizable electrode (IPE) 21 Ideal nonpolarized electrode, electrochemical experiments 21 Ideal polarized electrode, capacitance and electrode/solution interface 41–42 i- 𝜂 equation, Butler–Volmer model 135–142 Ilkoviˇc equation, polarography 356–357 Immiscible liquids, liquid junction potentials 101 Impedance bridge 445 Impedance measurements, adsorbed monolayer responses 778–779 Increasing electrolyte conductivity, cell properties and electrode placement 740 Indicator electrode. See Working electrode Indium tin oxide (ITO) 892

Index

Infrared spectroscopy 947–950 Inner Helmholtz plane (IHP) 43, 617 Inner layer, electrical double layer 42 Inner potential, physics of, interfacial potential differences 80–82 Inner-sphere electrode reactions: adsorption energies 657 chlorine evolution 670 CO2 reduction 673–674 density functional theory 679 electrode surface role 653–654 1e electron-transfer reactions 654–657 hydrogen evolution reaction 679–681 hydrogen peroxide oxidation, reduction 677–678 methanol oxidation 670–673 microscopic theories 142–143 organic halide reduction 676–677 oxidation of NH3 to N2 674–676 oxygen reduction reaction 667–669, 681–684 Inorganic films, electroactive layers 780–782 Input bias current, operational amplifiers 724 Instantaneous current efficiency, bulk electrolysis 491 Instrumentation 721–751 current feedback 725–728 adders 726–727 current follower 725–726 integrators 727–728 scaler/inverter 726 galvanostats 734–736 integrated 736–737 low current measurements 744–748 current amplifier 746 fundamental limits 744–746 practical considerations 746 simplified instruments and cells 746–748 operational amplifiers 721–725 ideal properties 721–723 nonidealities 723–725 potential control difficulties 737–744

cell properties and electrode placement 740 resistance, electronic compensation of 740–744 types of 737–740 potentiostats 730–734 adder potentiostat 731–732 adder potentiostat (refinements) 732–733 basics 730–731 bipotentiostats 733–734 four-electrode potentiostats 734 practical use of 749–751 electrochemical workstations 749 troubleshooting 749–751 short time scales 748 voltage feedback 728–730 Integral capacitance 603 Integrated electrochemical systems 737 Integrators, operational amplifier 727–728 Interdigitated array 245 Interfacial potential differences 80–91 conducting phases, interactions between 82–84 electrochemical potentials 85–88 Fermi energy and absolute potential 88–91 measurement of 84–85 phase potentials, physics of 80–82 Intrinsic semiconduction 887 Inverted region: anodic process 160 cathodic process 159 Inverter, operational amplifier 726 Inverting input, operational amplifiers 722 In vivo electrochemistry 809–812 Ion-exchange polymers (polyelectrolytes) 780 Ionic current rectification (ICR) 438 Ionization energy (IE) 654 Ion-selective electrodes: commercial devices 109 detection limits 109

1029

1030

Index

Ion-selective electrodes (contd.) gas-sensing electrodes 111–112 glass electrodes 102–106 interfaces 101–102 plastic membranes 107–109 solid-contact ion-selective electrodes 109–111 solid-state membranes 106–107 Ion spectrometry 958–960. See Electron and ion spectrometry Ion transfer across a liquid/liquid interface (ITIES) 840 Irreversible adsorption, monolayers 756 Irreversible electrode reactions 230–239 applications of irreversible i–E curves 287–289 Irreversible processes, multistep mechanisms 174–176 Irreversible reactions, coupled, nernstian reactions with 32–34 Irreversible waves, potential-time curves in constant-current electrolysis 394–395 Isoenergetic electron transfer, Marcus microscopic model 146

k Kinematic viscosity 413 Kinetics 121–176 Arrhenius equation and potential energy surfaces 122–123 Butler–Volmer model 126–132 energy barriers, effects of potential on 127 one-step, one-electron process 127–130 standard rate constant 130–131 transfer coefficient 131–132 dynamic equilibrium 121–122 electrode reactions, essentials of 125–126 microscopic theories 142–168 energy state distributions 162–168 extended charge transfer and adiabaticity 143–146

implications of the Marcus theory 152–168 inner-sphere electrode reactions 142–143 Marcus microscopic model 146–152 outer-sphere electrode reactions 142–143 multistep mechanisms 171–176 at equilibrium 173–174 inner-sphere electrode reactions 172 multiple outer-sphere heterogeneous electron transfers 172 nernstian multistep processes 174 outer-sphere heterogeneous electron transfer coupled to homogeneous reactions 172 primacy of one-electron transfers 172–173 quasireversible and irreversible multistep processes 174–176 rate-determining, outer-sphere electron transfer 173 open-circuit potential (OCP): definition 168–169 establishment or loss of nernstian behavior at an electrode 170–171 in multicomponent systems 169–170 multiple half-reaction currents in i–E curves 171 transition state theory 123–125 Koutecký–Levich method 237–239, 420–423

l Langmuir–Hinshelwood mechanism 671 Langmuir isotherm 631 Laplace’s equation 209, 217 Laplace transform technique 967–976 fundamentals of 969–970 ordinary differential equations 970–972 overview 968–969

Index

partial differential equations 967–968, 973–975 simultaneous linear ordinary differential equations 972–973 zero-shift theorem 975–976 Large amplitude ac voltammetry 479–481 Large-amplitude potential step 295–296 Large A/V conditions 207 Leibnitz rule 979 Levich equation 418 Ligand-bearing polymers 780 Ligand bridge, microscopic theories 142 Limiting current 26 Linear diffusion: concentration profile 264–265 semi-infinite and Cottrell equation 263–264 Linear potential sweep chronoamperometry 311 Linear sweep voltammetry (LSV) 311–347 anodic case 331–332 correction for charging current 320–321 current–potential curves 329–330 double-layer capacitance, effect of 319–320 mass-transfer problem 314–316 mode 213 peak current and potential 316–317, 331 quasireversible systems 326 spherical electrodes and UMEs 317–318 uncompensated resistance, effect of 320 Liquid chromatography with electrochemical detection (LCEC) 517–521 Liquid junction 36 Liquid junction potentials 36, 91–101 calculation of 96–100 conductance, transference numbers, and mobility 92–96

electrode/electrolyte boundary 91 immiscible liquids 101 minimization of 100 types of 91–92 Liquid/liquid interfaces: cyclic voltammetry (CV) 339–344 effect of interfacial potential 341 experimental approach to voltammetry 340–341 voltammetric behavior 341–344 Logarithmic Temkin isotherm 632 Low current measurements, instrumentation 744–748 current amplifier 746 fundamental limits 744–746 practical considerations 746 simplified instruments and cells 746–748 Low-energy electron diffraction (LEED) 621 Luggin–Haber capillary, electrochemical cells and cell resistance 39 Luminophores, electrogenerated chemiluminescence 921

m Maclaurin series, Taylor expansions 977 Magnetic resonance spectroscopy 955–957 electron spin resonance 955–956 nuclear magnetic resonance 956–957 Marcus–Gerischer kinetics, model, semiconductor electrodes 896 Marcus–Gerischer kinetics, energy state distributions, kinetics, microscopic theories 166–167 Marcus microscopic model: conformational change and activation energy 148–152 precursor state 146–147 rate constant, form of 147–148 Marcus theory, electrogenerated chemiluminescence 912 Mass spectrometry 954–955

1031

1032

Index

Mass transfer: Butler–Volmer model 140–141 diffusion 193–199 boundary conditions 200–201 Fick’s laws 196–198 flux of an electroreactant, an electrode surface 199 microscopic view 193–196 formulation and solution of 199–204 analytical solution 202–203 boundary conditions at the electrode surface 201 electrochemical simulators 203–204 initial conditions 200 linear diffusion 201 migration or convection 202 semi-infinite boundary conditions 200 simulator/solvers 204 generally 183–186 migration and diffusion, mixed near an active electrode 187–192 migration in bulk solution 186–187 Mass-transfer coefficient, steady-state mass transfer 26 Mass-transfer-controlled reactions 23–31 modes of 24–25 steady-state mass transfer 25–31 Mass-transfer overpotential 23 i- 𝜂 equation 136 Mathematical methods 967–982 complex notation 979–980 error function and Gaussian distribution 977–978 Fourier series and transformation 981–982 Laplace transform technique 967–976 fundamentals of 969–970 ordinary differential equations 970–972 overview 968–969 partial differential equations 967–968, 973–975

simultaneous linear ordinary differential equations 972–973 zero-shift theorem 975–976 Leibnitz rule 979 Taylor expansions 976–977 Mechanistic switching, coupled homogeneous electrode reactions 589 Mercury film electrode (MFE), stripping analysis 528 Metal-deposition reactions 689 Metallic conduction 886 Metal oxides, inorganic films 781 Methanol oxidation, inner-sphere reactions 670–673 Method of finite differences, simulations 985 Microelectrodes 207 cell time constants at 300–303 Microfluidic ECL cells, electrogenerated chemiluminescence 921 Microfluidic flow cells, flow electrolysis 516–521 Microscopic area, electrode 272 Microscopic theories (kinetics) 142–168 energy state distributions 162–168 extended charge transfer and adiabaticity 143–146 adiabatic and nonadiabatic reactions 145–146 electron tunneling 143–144 implications of the Marcus theory 152–168 vs. Butler–Volmer Kinetics 152–155 effect of the electrode material 161–162 inverted regions 157–161 predictions for homogeneous reactions 162 rate constants and current–potential curves 155–157 transfer coefficient behavior 155 inner-sphere electrode reactions 142–143 Marcus microscopic model 146–152

Index

conformational change and activation energy 148–152 precursor state 146–147 rate constant, form of 147–148 outer-sphere electrode reactions 142–143 Microscopy 931–940 atomic force microscopy 934–937 optical microscopy 937–938 scanning tunneling microscopy 932–934 transmission electron microscopy 938–940 Migration: diffusion and, mass transfer, mixed near an active electrode 187–192 mass transfer 186–187 mass-transfer-controlled reactions 24 Minority carriers, photoelectrochemistry 889 Mirkin and Bard method 237 Mixed metal oxide (MMO) electrode 891 Mobility, liquid junction potentials 92–96 Modes of exploration, radical-ion annihilation 913–916 Modified Electrodes 755. See also Electroactive layers; Blocking layers Modified Nernst–Planck equation 185 Monolayers 756–757 covalent attachment 756–757 irreversible adsorption 756 reversible adsorption 756 transferred layers 757 underpotential deposition 757 Mott–Schottky relationship, semiconductor electrodes 896, 897 Multicomponent systems: cyclic voltammetry (CV) 332–333 and multistep charge transfers 289–290 multistep reactions and, controlled-current techniques 400–401

Multilayer assemblies, composites and, electroactive layers 782 Multiple E step reactions, coupled homogeneous electrode reactions 542–545 Multiplex advantage 483 Multistep charge transfers: cyclic voltammetry (CV) 333–334 multicomponent systems and 289–290 Multistep electron transfers, coupled homogeneous electrode reactions 569–576 EE scheme 574–576 Eq Er reactions 574 Er Eq reactions 573, 574 Er Er reactions 569–573 (Er )n reactions 573 Multistep mechanisms (kinetics) 171–176 at equilibrium 173–174 inner-sphere electrode reactions 172 multiple outer-sphere heterogeneous electron transfers 172 nernstian multistep processes 174 outer-sphere heterogeneous electron transfer coupled to homogeneous reactions 172 primacy of one-electron transfers 172–173 quasireversible and irreversible multistep processes 174–176 rate-determining, outer-sphere electron transfer 173 Multistep reactions, multicomponent systems and, controlled-current techniques 400–401

n Nanocrystals 888 Nanoparticle collision behavior 857–870 blocking 857–861 electrocatalytic amplification 861–864 electrolysis 864–870 Navier–Stokes equation 413

1033

1034

Index

Near-field scanning optical microscopy (NSOM) 938 Nernst diffusion layer, steady-state mass transfer 25 Nernst–Einstein equation 185 Nernst equation 5, 660 Nernst–Planck equation, mass-transfer controlled reactions 25 Nernstian reactions. See also Reversible Reactions with coupled chemical reactions: irreversible homogeneous reaction 32–34 reversible homogeneous reaction 31–32 cyclic voltammetry 321–325 adsorbed monolayer 766–770 linear sweep voltammetry (LSV) 313–321 multistep mechanisms 174 potential steps, simulations 1001–1002 Nernst–Planck equation 245 Neutral carrier, ion-selective electrodes 107 Nicholson method 328 Nonadiabatic reactions, charge transfer 145–146 Non-contact mode, atomic force microscopy 937 Nonfaradaic processes, defined 17 Noninverting input, operational amplifiers 722 Nonspecific adsorption 627 electrical double layer structure 617–619 Nonspecifically absorbed ions, electrical double layer 44 Normal calomel electrode (NCE) 74 Normal hydrogen electrode (NHE). See also Standard hydrogen electrode (SHE) definition 66 electrochemical cells 5 half-reactions 66 interfacial potential differences 88, 89

Normal pulse polarography 364. See also Polarography Normal pulse voltammetry 361–366. See also Pulse voltammetry implementation 362–363 normal pulse polarography 364–366 practical application 366 renewal at stationary electrodes 363–364 Normal pulse voltammetry (NPV) 275 Notations, electrochemical cells 3 n-type semiconductor 889 Nuclear magnetic resonance (NMR) 956–957 Nucleation overpotential, CNT 689–691 Nyquist plot 449, 450, 459

o Offset voltage, operational amplifiers 724–725 Ohmic control error 738 Ohmic drop 35 Ohm’s law 447, 448 One-electrode amperometry, electrometric end-point detection 509–510 One-electron heterogeneous reaction 1002 One E step reactions, coupled homogeneous electrode reactions 541–542 One-step, one-electron process, Butler-Volmer model 127–130, 132–142 Online electrochemistry-mass spectrometry (OEMS) 955 Open-circuit potential (OCP) 9–10 kinetics: definition 168–169 establishment/loss of nernstian behavior at an electrode 170–171 in multicomponent systems 169–170 multiple half-reaction currents in i–E curves 171

Index

Open-loop gain, operational amplifiers 723 Operational amplifiers 721–725 ideal properties 721–723 nonidealities 723–725 Optically transparent electrode (OTE) 942 Optically transparent thin-layer electrode (OTTLE) 942 Optical microscopy 937–938 Optimizing faradaic current, cell properties and electrode placement 740 Organic halide reduction, inner-sphere reactions 676–677 Ostwald ripening 705, 707 Outer Helmholtz plane (OHP) 43, 614 Outer-sphere reactions, microscopic theories 142 Output limits: operational amplifiers 724 potential control difficulties 737–738 Overpotential: defined 12–13 electrochemical experiments 21 Overpotential deposition (OPD) 700 Oxidation current 6 Oxidation of NH3 to N2 inner-sphere reactions 674–676 Oxidoreductase enzymes 779 Oxygen reduction reaction (ORR) inner-sphere electron-transfer reactions 681–684 inner-sphere reactions 667–669

p Partial differential equations, Laplace transform technique 967–968, 973–975 Particle characterization, single atoms and atomic clusters 871–872 Particle collision experiments 852–854 frequency 854–855 time of first arrival 856–857 variance in number 855–856 Passivation 628

Permeation current 786 Phase potentials, physics of, interfacial potential differences 80–82 Photoanodic current, at semiconductor electrodes 902 Photocatalytic cells, semiconductor electrodes 905 Photocurrent, photoelectrochemistry 901 Photoeffects, at semiconductor electrodes 901–903 Photoelectrochemistry 885–910 radiolytic products 908–910 detection and use 909 photoemission 908–909 photogalvanic cells 909–910 semiconductor electrodes 892–908 conducting polymer 899–901 current–potential curves 896–899 dye sensitization 905–906 interface 892–896 photocatalytic cells 905 photoeffects 901–903 photoelectrosynthetic cells 904–905 photovoltaic cells 903–904 surface photocatalytic processes 906–908 solids 885–892 band model 885–886 categories 886–889 conductive metal oxide 891–892 doped semiconductors 889–890 Fermi energy 890–891 Photoelectrosynthetic cells, semiconductor electrodes 904–905 Photoemission, radiolytic products 908–909 Photogalvanic cells: photoelectrochemistry 910 radiolytic products 909–910 Photovoltaic cells, semiconductor electrodes 903–904 Plane of acid dissociation (PAD) 771 Plane of electron transfer (PET) 762 Plasmon 946

1035

1036

Index

Plastic membranes, ion-selective electrodes 107–109 Point-by-point evaluation 236 Poisson–Boltzmann equation 610 Poisson–Nernst–Planck (PNP) model 641 Polarization curve, electrochemical experiment 21 Polarization, electrochemical experiments 21 Polarization, of semiconductor electrodes 894–895 Polarographic waves 357–358 Polarography 311, 355–361. See also Pulse voltammetry analysis 358–359 dropping mercury electrode 355–356 Ilkoviˇc equation 356–357 polarographic waves 357–358 residual current and detection limits 359–361 Polyelectrolytes (ion-exchange polymers ) 780 Polymer films, electroactive layers 780 Positive feedback compensation, electronic resistance compensation 740–742 Potential control difficulties 737–744 cell properties and electrode placement 740 resistance, electronic compensation of 740–744 types of 737–740 Potential differences, liquid junction potentials 91. See also Interfacial potential differences; Liquid junction potentials Potential energy surfaces, Arrhenius equation and 122–123 Potential–pH diagrams: E–pH diagram for iron 78–79 E–pH diagram for water 77–78 limitations of 80 Potentials, and electron energy 6

Potential step methods. See Voltage (potential) step diffusion controlled currents at ultramicroelectrode large-amplitude potential step (See also Large-amplitude potential step) interference from charging current 305–306 nernstian system, digital simulations 1001–1002 preparation of electrode surface 303–305 Potential sweep. See Voltage ramp (potential sweep) in a nernstian system, thin-layer electrochemistry 524–526 Potential sweep methods: boundary conditions, simulations 1003 simulation 1003 Potential-time curves, in constant-current electrolysis, controlled-current techniques 394–398. See also Controlled-current techniques Potential window 12 Potentiometric methods, electrometric end point detection 508–509 Potentiometric sensors 803 Potentiometry, electrochemical cells 19 Potentiostat 38, 54–55, 730–734 adder potentiostat 731–732 adder potentiostat (refinements) 732–733 basics 730–731 bipotentiostats 733–734 electrochemical instrumentation 721 four-electrode potentiostats 734 Pourbaix diagrams 76. See also Potential-pH diagrams Practical reversibility, thermodynamics 63 Primary (nonrechargeable) cells 17 Primary current distribution, rotating disk electrode (RDE) 424

Index

Programmed current chronopotentiometry 390, 394–398 Projected area, electrode 272 Proton-coupled electron transfers, coupled homogeneous electrode reactions 589–590 p-type semiconductor 90 Pulse voltammetry 355. See also Polarography analysis by 383–385 differential 369–376 normal 361–366 reverse 367–369 square wave 376–383

q Quantum dots (QDs): electrogenerated chemiluminescence 917–918 photoelectrochemistry 889 Quartz crystal microbalance (QCM) 940–942 basic method 940–942 with dissipation monitoring 942 Quasireference electrodes (QRE) 75, 112–113 Quasireversible electrode reactions 293 applications of irreversible i–E curves 235–237, 287–289 cyclic voltammetry (CV) 325–329 electrode kinetics, effect of 230–232, 280–282 influence of electrode shape 234–235 kinetic regimes 234 limit of total irreversibility 232–234, 285–287, 473–476 mass-transfer rates 237–239 multistep mechanisms 174–176 sampled transient voltammetry: for oxidation of R 284–285 for reduction of O 282–284 Quasireversible waves, potential-time curves in constant-current electrolysis 395 Quasi-steady state 221

r Radical-ion annihilation 912–916 dual-potential generation 912–913 modes of exploration 913–916 Radiolytic products 908–910 detection and use 909 photoemission 908–909 photogalvanic cells 909–910 Raman spectroscopy 950–952 Ramp generator, operational amplifier 728 Randles equivalent circuit 451–452 Randles–Ševcík equation 316 Rate-determining steps, electrode reaction rate and current 23 Rate-limiting Heyrovský reaction, HER 663–664 Rate-limiting Tafel reaction, HER 664–665 Rate-limiting Volmer reaction, HER 662–663 Rayleigh scattering, Raman spectroscopy 950 RC networks 56 Reactant diffusion current 785 Reaction coordinate, potential energy surfaces 122, 123 Reconstruction, electrode surface 621 Recording system, electrochemical instrumentation 721 Redox competition, SECM 831 Redox couple 10 Redox cycling 244 single-molecule electrochemistry 875, 877 Reducing double-layer capacitance, cell properties and electrode placement 740 Reduction (cathodic) current 6, 10 Reduction wave 12 Reference electrode 5–6 cell resistance and the measurement of potential 34–35 thermodynamics 72–76

1037

1038

Index

Reference electrode (contd.) aqueous reference electrodes based on solubility equilibria 74 attributes of practical reference electrodes 73–74 interconversion of scales 75–76 leakage, reference tip 112 NHE and classic hydrogen electrodes 72–73 quasireference electrodes 112–113 reference electrodes for nonaqueous systems 75 reversible hydrogen electrode 74–75 Reference electrode tip, cell properties and electrode placement 740 Reference tables 1007–1013 Relative surface excesses, electrical double-layer structure 601–602, 606 Resistance, electronic compensation of, potential control difficulties 740–744 Resonance Raman spectroscopy (RRS) 951 Response function principle, controlled current techniques 398 Rest potential 9 Reversal chronocoulometry 297 Reversal techniques, controlled-current technique 398–399 Reverse pulse voltammetry 367–369 Reversibility (thermodynamics) 61–63 chemical 61–62 Gibbs free energy and 64 practical 63 thermodynamic 63 Reversible adsorption, monolayers 756 Reversible hydrogen electrode (RHE) 75 Reversibility (kinetics). See also Nernstian reactions ac voltammetry 470–473 applications of reversible i–E curves 279 Butler-Volmer model 139–140 concentration profiles 278–279

coupled homogeneous reactions 31–34 mass-transfer-controlled reactions 24 potential sweep methods (See also Potential sweep methods) potential-time curves in constant-current electrolysis 394–398 simplified current–concentration relationships 279 step to arbitrary potential 276–277 voltammogram shape 277–278 wave shape 224–226 Reynolds number 413 Root-mean-square displacement 195 Rotating disk electrode (RDE): hydrodynamic methods 414–426 concentration profile 418–419 convective-diffusion equation 412 current distribution at 423–426 i-E curves 419–420 Koutecký–Levich method 420–423 practical considerations for application 426 transients at 433–435 velocity profile at 414–416 steady-state mass transfer 25 Rotating ring-disk electrode (RRDE): hydrodynamic methods 428–431 modulation 435–436 transients at 433–435 Rotating ring electrode, hydrodynamic methods 426–428 Roughness factor 272

s Sabatier principle 684 Sampled-transient voltammetry (STV) 275 Sand equation, controlled-current techniques 392–394 Saturated calomel electrode (SCE) 5, 74 Scaler, operational amplifier 726 Scanning electrochemical cell microscopy (SECCM) 841–846, 878

Index

Scanning electrochemical microscopy (SECM) 234, 819 applications: biological systems 840–841 probing inside a layer 841 species released from surfaces, films, or pores 839–840 approach curves 821–825 kinetic measurements: heterogeneous electron-transfer reactions 828–831 homogeneous reactions 831–835 potentiometric tips 839 principles 819–821 surface interrogation 835–838 surface topography and reactivity image: conductivity of substrate 825 heterogeneous electron-transfer reactivity 826–827 simultaneous images 827–828 Scanning transmission electron microscopy (STEM) 938 Scanning tunneling microscopy (STM) 932–934 SECM. See Scanning electrochemical microscopy (SECM) SECCM. See Scanning electrochemical cell microscopy (SECCM) Secondary current distribution, rotating disk electrode (RDE) 425 Second harmonic ac voltammetry 478–479 Self-assembled monolayer (SAM) 757 Self-assembly, organized assemblies, monolayers 757 Semiconductor electrodes 892–908 conducting polymer 899–901 current–potential curves 896–899 dye sensitization 905–906 interface 892–896 photocatalytic cells 905 photoeffects 901–903 photoelectrosynthetic cells 904–905 photovoltaic cells 903–904

surface photocatalytic processes 906–908 Semiconductors, photoelectrochemistry at. See also Photoelectrochemistry Semiempirical treatment, steady-state mass transfer, mass-transfer controlled reaction 25–31 Semi-infinite linear diffusion, controlled-current techniques 391–392 Semi-integral techniques, potential sweep methods. See also Potential sweep methods Semimetal 887 Sensing line, potentiostats 734 Sensors based on DNA aptamers 805–807 Shielding mode, SECM 831 Short time scales, instrumentation 748 Shot noise 744 Silver-silver chloride electrode 6, 74 Simulation 985–1004 boundary conditions 1001–1004 controlled current 1003–1004 heterogeneous kinetics 1002 nernstian systems 1001–1002 potential sweeps 1003 comparative 551 example 993–998 concentration arrays 996 error detection 996–997 performance 997–998 spreadsheet organization 993–995 homogeneous kinetics 999–1001 bimolecular reactions 1000–1001 unimolecular reactions 999–1000 model 985–993 current 991–992 diffusion 986–987 diffusion coefficients 993 diffusion layer thickness 992 dimensionless parameters 987–990 discrete system 985–986 distance 990–991 time 990

1039

1040

Index

Simultaneous linear ordinary differential equations, Laplace transform technique 972–973 Single-molecule detection 876 Single-molecule electrochemistry 875–879 Single-particle electrochemistry 851–879 nanoparticle collision behavior 852–854, 857–870 blocking 857–861 electrocatalytic amplification 861–864 electrolysis 864–870 frequency 854–855 time of first arrival 856–857 variance in number 855–856 single atoms and atomic clusters 870–875 Single-potential generation, electrogenerated chemiluminescence 916–917 Slew rate, operational amplifiers 724 Small A/V conditions 207 Small excursions, charge step (coulostatic) methods 404–405 Smith–White model 762 Solid-contact ion-selective electrodes 109–111 Solid electrolytes, electrochemical cells 95 Solid polymer electrolyte (SPE) 782 Solid-state membranes, ion-selective electrodes 106–107 Space charge region, semiconducting electrode 892–894 Specific adsorption 627 electrical double layer structure 617–619 electrical double-layer structure 43, 627–634 (See also Electrical double-layer structure) adsorption isotherms 630–632 nature and extent of 628–629 of electrolyte:

absence of, electrical double-layer structure 638–639 presence of, electrical double-layer structure 639–640 Spectroelectrochemistry 526, 948, 949 Spectroellipsometry 945 Spherical electrode: applicability of linear approximation 267 chronoamperometric transient 266 concentration profile 266 Square schemes, coupled homogeneous electrode reactions 544–545 Square wave pulse voltammetry 376–383 applications 381–383 background currents 380–381 concept and practice 376–377 response predictions 377–380 Secondary (rechargeable) cells 17 Standard electrode potentials and, thermodynamics 66–67 Standard enthalpy of activation, potential energy surfaces 123 Standard entropy of activation, potential energy surfaces 123 Standard free energy of activation, potential energy surfaces 123 Standard hydrogen electrode (SHE). See also Normal hydrogen electrode (NHE) definition 66 electrochemical cells 5 Standard potentials 6 Standard rate constant, Butler–Volmer model 130–131 Standard states and activity 67–69 Static mercury drop electrode (SMDE). See also Hanging mercury drop electrode (HMDE); Polarography differential pulse voltammetry 375 normal pulse voltammetry 365–366 reverse pulse voltammetry 368 Static resistance 247 Stationary electrode polarography 311

Index

Steady-state cyclic voltammetry 214 Steady-state diffusion 216–219 Steady-state mass transfer, semiempirical treatment of, mass-transfercontrolled reactions 25–31 Steady-state methods 19 Steady-state renewal time 212 Steady-state resistance 247 Steady-state voltammetry (SSV): analysis at high analyte concentrations 251–253 described 212–214 diffusion–migration layer 246–247 effects of migration 248–251 electrolyte concentration 248 mathematical approach to problems 245–246 multicomponent systems and multistep charge transfers 239–241 quasireversible and irreversible electrode reactions 230–239 reversible electrode reactions 224–230 at spherical UME: convergence on steady state 211–212 steady-state current 211 steady-state diffusion 208–211 Steady-state voltammogram 213 Stern layer 43. See also Inner layer Stern’s modification, Gouy–Chapman theory 614–617 Stokes lines, Raman spectroscopy 950 Stranski–Krastanov electrodeposition 701, 702 Stripping analysis 490, 527–531 applications and variations 529–531 overview 527 principles and theory 528–529 Substrate generation/tip collection (SG/TC) 835 Substrate, electroactive layer: platform 755 primary reactant 783 Superequivalent adsorption 618 Superposition, principle of 292

Supporting electrolyte 13, 25, 192 migration and diffusion, mixed near an active electrode 192 Surface-enhanced IR absorption spectroscopy (SEIRAS) 671 Surface-enhanced Raman microscopy 938 Surface-enhanced Raman spectroscopy (SERS) 951 Surface interrogation mode (SI-SECM) 835–838 Surface plasmon resonance 946–947 Surface splitting 145 Surface states, semiconductor electrodes 895 Surface topography and reactivity images, SECM: conductivity of substrate 825 heterogeneous electron-transfer reactivity 826–827 simultaneous images 827–828 Switching potential 321

t Tafel behavior, Butler–Volmer model, i- 𝜂 equation 136–137 Tafel equation, electrode reactions 126 Tafel plot analysis, HER: Butler–Volmer equation 660–661 kinetic analysis based Tafel slopes 665–667 rate-limiting Heyrovský reaction 663–664 rate-limiting mass transport of H+ 665 rate-limiting Tafel reaction 664–665 rate-limiting Volmer reaction 662–663 Tafel plots, Butler–Volmer model, i- 𝜂 equation 137 Tapping mode, atomic force microscopy 937 Taylor expansions 976–977 Tetracyanoquinodimethane (TCNQ) 214, 868–869 Tetra-n-butylammonium azide (TBAN3 ) 324

1041

1042

Index

Tetrathiafulvalene (TTF) 250–251 Thermodynamic radius 693 Thermodynamic reversibility 63 Thermodynamics 61–80 cell emf and concentration 69–71 electrical double-layer structure 599–602 electrocapillary equation 601 Gibbs adsorption isotherm 599–600 relative surface excesses 601–602 formal potentials 71–72 free energy and cell emf 64–66 half-reactions and standard electrode potentials 66–67 potential–pH diagrams and thermodynamic predictions: E–pH diagram for iron 78–79 E–pH diagram for water 77–78 limitations of 80 reference electrode 72–76 aqueous reference electrodes based on solubility equilibria 74 attributes of practical reference electrodes 73–74 interconversion of scales 75–76 NHE and classic hydrogen electrodes 72–73 reference electrodes for nonaqueous systems 75 reversible hydrogen electrode 74–75 reversibility 61–63 chemical 61–62 Gibbs free energy and 64 practical 63 standard states and activity 67–69 Thin-layer electrochemistry 521–526 applications of 526 chronoamperometry and coulometry 521–524 dual-electrode thin-layer cells 526 overview 521 potential sweep in a nernstian system 524–526 Thin-layer electrolysis 490 3D solid nucleation, CNT 696–698

Three-electrode cells, electrochemical cells and cell resistance 37–38 Time domain, Fourier series 981 Time-domain instruments, EIS 483–485 Time of first arrival, particle collision experiments 856–857 Time, simulation 990 Tip-enhanced Raman spectroscopy (TERS) 951 Tip generation/substrate collection (TG/SC) mode 831 Titration efficiency 504 Tomeš criterion 225, 237 Totally irreversible reactions 137 anodic currents 286 cathodic currents 285–286 cyclic voltammetry (CV) 329–332 Transfer coefficient: Butler–Volmer model 131–132 Marcus model 154–155 Transference numbers 92–96 Transferred layers, monolayers 757 Transient and steady-state data 271 Transient methods 19, 261–306 Transient reversal technique 291 Transition-metal solids, inorganic films 781–782 Transition state theory, kinetics 123–125 Transition time 389 Transition time constant 393 Transmission electron microscopy (TEM) 938–940 Triplet–triplet annihilation, electrogenerated chemiluminescence 911 Tunneling and extended charge transfer 143–146 Two-electrode amperometry, electrometric end-point detection 510 Two-electrode cell, electrochemical cells and cell resistance 37 Two-electrode potentiometry, electrometric end-point detection 509

Index

u Ultrahigh vacuum (UHV) 958 Ultramicroelectrode (UME) 25, 207–257 additional attributes of 241–245 band 221–222 characteristic dimension 215 cylindrical 221 diffusion-controlled currents at, potential step methods (See also Potential step methods) disk 215–221 effects of conductivity on voltammetry 242–243 instrumentation, low current measurements 746 lab note: integrity of 254–256 preparation and characterization of 254 size of 256–257 migration in steady-state voltammetry 245–251 spatial resolution 243–245 spherical or hemispherical 215 steady-state behavior at 222–224 uncompensated resistance 241–242 Uncompensated ohmic drop, potential control dificulties 738 Uncompensated resistance: effect of 320 electrochemical cells and cell resistance 38–41 Underpotential deposition (UPD) 659, 689, 700 monolayer adsorbates 702–703 monolayers 757 Underpotential deposition, rotating ring-disk electrode 435 Unimolecular reactions: homogeneous kinetics, digital simulations 999–1000 incorporating homogeneous kinetics 999–1000

UV–visible spectrometry 942–947 absorption spectroscopy with thin-layer cells 942–945 ellipsometry 945–946 surface plasmon resonance 946–947

v Vacuum electric permittivity 608 Valence band 885 Variance in number, particle collision experiments 855–856 Vibrational spectroscopy 947–952 infrared spectroscopy 947–950 Raman spectroscopy 950–952 Volmer–Heyrovský mechanism 660 Volmer–Tafel mechanism 659–660 Volmer–Weber electrodeposition 701 Voltage compliance, potentiostat 738 Voltage feedback, instrumentation 728–730 Voltage follower, operational amplifier 728–729 Voltage ramp (potential sweep), electrical double layer capacitance 48–49. See also Potential sweep methods Voltage (potential) step, electrical double layer capacitance 45–49. See also Potential step methods Voltammetric sensors 803 Voltammetry, electrochemical cells 19 Volume ionic strength, standard states and activity 68

w Waveform generator 208 Wave-slope plot 237 Wheatstone bridge 444, 445 Working electrode: cell resistance and the measurement of potential 36 definition 5 electrical double layer capacitance 50–51

1043

1044

Index

potential of 6 reference electrode and control of potential 5–6 Working range 12

X-ray diffraction (XRD) 954 X-ray methods 953–954

y Young–Laplace equation

710

x X-ray absorption fine structure (XAFS) 953 X-ray absorption near-edge structure (XANES) 953 X-ray absorption spectroscopy (XAS) 953

z Zeolites, inorganic films 781 Zero-current baseline 322 Zero-current potential 9, 11, 12 Zero-shift theorem, Laplace transform technique 975–976