Colloid and Interface Science 812033857X, 9788120338579

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Colloid and Interface Science
 812033857X, 9788120338579

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COLLOID AND INTERFACE SCIENCE

PALLAB GHOSH Associate Professor Department of Chemical Engineering Indian Institute of Technology Guwahati

New Delhi-110001 2009

About the cover illustration: The background depicts the waves created at the air–water interface induced by external forces (e.g. when a rain drop falls on a calm water surface); (top left) petroleum oil-spill on sea that can cause extensive environmental hazard; (middle) gas bubbles fascinate us with their shape and colour, and they also constitute an integral part of gas–liquid dispersions; (bottom right) water-drops on hydrophobic tree-leaves depicting the wetting phenomena; (top right) carbon-nanotube catalytic reactor depicting the production of ethanol [taken from the work of Pan, X., Fan, Z., Chen, W., Ding, Y., Luo, H. and Bao, X., “Enhanced Ethanol Production Inside Carbon-Nanotube Reactors Containing Catalytic Particles”, Nature Materials, 6, 507 (2007), reproduced by permission from Macmillan Publishers Ltd., © 2007]

Rs. 425.00

COLLOID AND INTERFACE SCIENCE Pallab Ghosh © 2009 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-3857-9 The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli-131028, Sonepat, Haryana.

Contents

Preface SI Units

xi xiii

1. BASIC CONCEPTS OF COLLOIDS AND INTERFACES ..................................... 1–23 1.1 Introduction .................................................................................................................................. 2 1.2 Examples of Interfacial Phenomena .............................................................................................. 3 1.2.1 Capillary Action .................................................................................................................. 3 1.2.2 Tears of Wine ...................................................................................................................... 4 1.3 Solid–Fluid Interfaces ................................................................................................................... 5 1.4 Colloids ........................................................................................................................................ 6 1.4.1 Colloids and Interfaces ....................................................................................................... 7 1.4.2 Classification of Colloids .................................................................................................... 9 1.4.3 Electrical Charge on Colloid Particles .............................................................................. 10 1.4.4 Stability of Colloids .......................................................................................................... 11 1.4.5 Kinetic and Thermodynamic Stabilities ............................................................................ 13 1.4.6 Preparation of Colloids ..................................................................................................... 13 1.4.7 Parameters of Colloid Dispersions .................................................................................... 17 Summary .............................................................................................................................................. 19 Keywords ............................................................................................................................................. 19 Notation ............................................................................................................................................... 20 Exercises ............................................................................................................................................. 21 Numerical and Analytical Problems ................................................................................................... 21 Further Reading .................................................................................................................................. 22

2. PROPERTIES OF COLLOID DISPERSIONS ....................................................... 24–68 2.1 Introduction ................................................................................................................................ 25 2.2 Sedimentation under Gravity ...................................................................................................... 25 2.3 Sedimentation in a Centrifugal Field .......................................................................................... 28 iii

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2.4 Brownian Motion ........................................................................................................................ 30 2.5 Osmotic Pressure ........................................................................................................................ 32 2.6 Optical Properties ....................................................................................................................... 36 2.6.1 Determination of Molecular Weight by Light Scattering .................................................. 39 2.6.2 Transmission Electron Microscopy (TEM) and Scanning Electron Microscopy (SEM) ............................................................................. 42 2.6.3 Dynamic Light Scattering (DLS) ...................................................................................... 42 2.6.4 Small-Angle Neutron Scattering (SANS) ......................................................................... 43 2.7 Electrical Properties .................................................................................................................... 43 2.7.1 Reciprocal Relationships .................................................................................................. 48 2.7.2 z-potential Obtained from Different Techniques ............................................................... 49 2.8 Some Properties of Lyophilic Sols .............................................................................................. 49 2.8.1 Hofmeister Series ............................................................................................................. 49 2.8.2 Viscosity of Lyophilic Sols ............................................................................................... 50 2.8.3 Gelation ............................................................................................................................ 51 2.8.4 Imbibition ......................................................................................................................... 51 2.8.5 Syneresis ........................................................................................................................... 51 2.8.6 Gold Number .................................................................................................................... 52 2.9 Rheological Properties of Colloid Dispersions ........................................................................... 52 2.9.1 Einstein’s Equation of Viscosity ....................................................................................... 55 2.9.2 Mark–Houwink Equation for Polymer Solutions .............................................................. 56 2.9.3 Deborah Number .............................................................................................................. 57 2.9.4 Peclet Number .................................................................................................................. 57 Summary .............................................................................................................................................. 58 Keywords ............................................................................................................................................. 59 Notation ............................................................................................................................................... 60 Exercises ............................................................................................................................................. 62 Numerical Problems ............................................................................................................................ 63 Appendix .............................................................................................................................................. 66 Further Reading .................................................................................................................................. 66

3. SURFACTANTS AND THEIR PROPERTIES ....................................................... 69–94 3.1 Introduction ................................................................................................................................ 70 3.2 Surfactants and their Properties .................................................................................................. 70 3.2.1 Anionic Surfactants ......................................................................................................... 71 3.2.2 Cationic Surfactants ........................................................................................................ 72 3.2.3 Zwitterionic Surfactants .................................................................................................. 72 3.2.4 Nonionic Surfactants ....................................................................................................... 72 3.2.5 Gemini Surfactants .......................................................................................................... 73 3.2.6 Biosurfactants ................................................................................................................. 73 3.2.7 Micellisation of Surfactants ............................................................................................ 74 3.2.8 Thermodynamics of Micellisation .................................................................................. 83 3.2.9 Krafft Point and Cloud Point ........................................................................................... 84 3.2.10 Liquid Crystals ................................................................................................................ 84 3.2.11 Hydrophilic–Lipophilic Balance ..................................................................................... 86 3.3 Emulsions and Microemulsions .................................................................................................. 88 3.4 Foams ......................................................................................................................................... 89

Contents

v

Summary .............................................................................................................................................. 90 Keywords ............................................................................................................................................. 90 Notation ............................................................................................................................................... 91 Exercises ............................................................................................................................................. 91 Numerical Problems ............................................................................................................................ 92 Further Reading .................................................................................................................................. 93

4. SURFACE AND INTERFACIAL TENSION ......................................................... 95–142 4.1 Introduction ................................................................................................................................ 96 4.2 Surface Tension ........................................................................................................................... 96 4.2.1 Effect of Temperature on Surface Tension .................................................................... 103 4.3 Interfacial Tension .................................................................................................................... 106 4.4 Contact Angle and Wetting ....................................................................................................... 108 4.5 Shape of the Surfaces and Interfaces ......................................................................................... 109 4.5.1 Radius of Curvature ...................................................................................................... 109 4.5.2 Young–Laplace Equation .............................................................................................. 112 4.5.3 Pendant and Sessile Drops ............................................................................................ 115 4.5.4 Capillary Rise or Depression ........................................................................................ 118 4.5.5 The Kelvin Equation ..................................................................................................... 120 4.6 Measurement of Surface and Interfacial Tension ...................................................................... 122 4.6.1 The Drop-Weight Method ............................................................................................. 122 4.6.2 du Noüy Ring Method .................................................................................................. 124 4.6.3 Wilhelmy Plate Method ................................................................................................ 126 4.6.4 Maximum Bubble Pressure Method .............................................................................. 127 4.6.5 Spinning-Drop Method ................................................................................................. 128 4.7 Measurement of Contact Angle ................................................................................................ 130 Summary ............................................................................................................................................ 132 Keywords ........................................................................................................................................... 133 Notation ............................................................................................................................................. 134 Exercises ........................................................................................................................................... 136 Numerical and Analytical Problems ................................................................................................. 137 Appendix ............................................................................................................................................ 138 Further Reading ................................................................................................................................ 140

5. INTERMOLECULAR AND SURFACE FORCES .............................................. 143–200 5.1 Introduction .............................................................................................................................. 143 5.2 van der Waals Forces ................................................................................................................ 145 5.2.1 van der Waals Force between Two Macroscopic Bodies ............................................... 149 5.2.2 Derjaguin Approximation ............................................................................................. 151 5.2.3 The Hamaker Constant .................................................................................................. 152 5.2.4 Estimation of Surface Tension from Hamaker Constant ............................................... 157 5.2.5 van der Waals Force in Electrolyte Solutions ................................................................ 157 5.2.6 Disjoining Pressure ....................................................................................................... 158 5.2.7 Experimental Determination of van der Waals Force .................................................... 158 5.3 Electrostatic Double Layer Force ............................................................................................. 160 5.3.1 The Models of Electrostatic Double Layer ................................................................... 162 5.3.2 Mathematical Modelling of the Diffuse Double Layer ................................................. 164 5.3.3 The Stern Layer ............................................................................................................ 171 5.3.4 Double Layer around Spherical Particles ...................................................................... 173

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5.3.5 Electrostatic Double Layer Repulsion between Two Surfaces ...................................... 174 5.3.6 The Derjaguin Approximation ...................................................................................... 177 5.3.7 The Zeta Potential ......................................................................................................... 177 5.3.8 Limitations of the Poisson–Boltzmann Equation .......................................................... 179 5.4 The DLVO Theory .................................................................................................................... 180 5.5 Non-DLVO Forces .................................................................................................................... 184 5.5.1 Hydration Force ............................................................................................................ 184 5.5.2 Hydrophobic Interaction ............................................................................................... 186 5.6 Steric Forces due to Adsorbed Polymer Molecules ................................................................... 186 Summary ............................................................................................................................................ 188 Keywords ........................................................................................................................................... 189 Notation ............................................................................................................................................. 190 Exercises ........................................................................................................................................... 192 Numerical and Analytical Problems ................................................................................................. 194 Appendix 5A ...................................................................................................................................... 195 Appendix 5B ...................................................................................................................................... 197 Further Reading ................................................................................................................................ 198

6. ADSORPTION AT INTERFACES ..................................................................... 201–238 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

Introduction .............................................................................................................................. 202 The Gibbs Dividing Surface ..................................................................................................... 202 Gibbs Adsorption Equation ....................................................................................................... 204 Langmuir and Frumkin Adsorption Isotherms .......................................................................... 207 Surface Equation of State (EOS) .............................................................................................. 208 Effect of Salt on Adsorption of Surfactants ............................................................................... 210 Adsorption Isotherms Incorporating the Electrostatic Effects ................................................... 215 Calculation of Free Energy of Adsorption ................................................................................ 217 Adsorption of Inorganic Salts at Interfaces ............................................................................... 218 Dynamics of Adsorption of Surfactants at the Interface ............................................................ 219 Adsorption at Solid–Fluid Interfaces ........................................................................................ 221 6.11.1 Henry, Freundlich and Langmuir Adsorption Isotherms ............................................. 222 6.11.2 The Brunauer–Emmett–Teller (BET) Theory .............................................................. 224 6.11.3 Adsorption Hysteresis ................................................................................................. 226 6.12 Direct Experimental Techniques for Measurement of Adsorption ............................................ 228 Summary ............................................................................................................................................ 229 Keywords ........................................................................................................................................... 229 Notation ............................................................................................................................................. 230 Exercises ........................................................................................................................................... 232 Numerical and Analytical Problems ................................................................................................. 232 Appendix ............................................................................................................................................ 235 Further Reading ................................................................................................................................ 236

7. INTERFACIAL RHEOLOGY .............................................................................. 239–269 7.1 Introduction .............................................................................................................................. 239 7.2 Surface Shear Viscosity ............................................................................................................ 240 7.2.1 Molecular Theory of Newtonian Surface Viscosity ....................................................... 242 7.2.2 Boussinesq Number ...................................................................................................... 243 7.2.3 Measurement of Surface Shear Viscosity ...................................................................... 243

Contents

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7.3 Interfacial Tension Gradient and Marangoni Effect .................................................................. 250 7.4 Surface Dilatational Viscosity ................................................................................................... 252 7.4.1 Measurement of Surface Dilatational Viscosity ............................................................ 254 7.5 Boussinesq–Scriven Model ....................................................................................................... 257 7.6 Importance of mS and kS ............................................................................................................ 258 7.7 Interfacial Turbulence ............................................................................................................... 259 7.8 Motions of Drops in a Liquid .................................................................................................... 261 Summary ............................................................................................................................................ 264 Keywords ........................................................................................................................................... 264 Notation ............................................................................................................................................. 264 Exercises ........................................................................................................................................... 266 Numerical and Analytical Problems ................................................................................................. 266 Further Reading ................................................................................................................................ 268

8. MONOLAYERS AND THIN LIQUID FILMS ..................................................... 270–315 8.1 Introduction .............................................................................................................................. 271 8.2 Surfactant Monolayers and their Applications .......................................................................... 271 8.3 Properties of Monolayers .......................................................................................................... 272 8.3.1 Measurement of Surface Pressure ................................................................................. 273 8.3.2 Surface Pressure Isotherm ............................................................................................. 274 8.3.3 Model of Gas-Phase Monolayer .................................................................................... 276 8.3.4 Surface Potential ........................................................................................................... 278 8.3.5 Monolayers at Liquid–Liquid Interfaces ....................................................................... 279 8.4 Langmuir–Blodgett Films ......................................................................................................... 279 8.5 Surface Diffusion ...................................................................................................................... 282 8.5.1 The Apparatus ............................................................................................................... 283 8.5.2 Measurement Procedure ................................................................................................ 284 8.5.3 Determination of Ds ...................................................................................................... 285 8.6 Thin Liquid Films ..................................................................................................................... 286 8.6.1 Critical Film Thickness ................................................................................................. 287 8.6.2 Hydrostatics of Thin Liquid Films ................................................................................ 292 8.6.3 Drainage of Liquid Films .............................................................................................. 293 8.6.4 The Lubrication Flow .................................................................................................... 295 8.6.5 Film-Drainage Time ...................................................................................................... 298 8.6.6 Stability of Thin Liquid Films ....................................................................................... 302 8.6.7 Black Films ................................................................................................................... 303 Summary ............................................................................................................................................ 306 Keywords ........................................................................................................................................... 307 Notation ............................................................................................................................................. 307 Exercises ........................................................................................................................................... 309 Numerical and Analytical Problems ................................................................................................. 310 Appendix ............................................................................................................................................ 311 Further Reading ................................................................................................................................ 312

9. EMULSIONS, MICROEMULSIONS AND FOAMS ........................................... 316–371 9.1 Introduction .............................................................................................................................. 317 9.2 Emulsions ................................................................................................................................. 317 9.2.1 Preparation of Emulsions .............................................................................................. 318

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9.2.2 Stability of Emulsions ................................................................................................... 319 9.2.3 Ostwald Ripening ......................................................................................................... 321 9.2.4 Flocculation and Coalescence of Drops ........................................................................ 324 9.2.5 Phase Inversion ............................................................................................................. 338 9.2.6 Application of Emulsions .............................................................................................. 339 9.3 Microemulsions ........................................................................................................................ 340 9.3.1 Winsor’s Classification of Microemulsions .................................................................. 340 9.3.2 Stability of Microemulsions .......................................................................................... 345 9.3.3 Rheology of Microemulsions ........................................................................................ 351 9.3.4 Applications of Microemulsions ................................................................................... 353 9.4 Foams ....................................................................................................................................... 353 9.4.1 Preparation of Foams and Measurement of their Stability ............................................ 355 9.4.2 Structure of Foams ........................................................................................................ 356 9.4.3 Foam Drainage .............................................................................................................. 358 9.4.4 Applications of Foams .................................................................................................. 360 Summary ............................................................................................................................................ 360 Keywords ........................................................................................................................................... 361 Notation ............................................................................................................................................. 362 Exercises ........................................................................................................................................... 364 Numerical and Analytical Problems ................................................................................................. 366 Further Reading ................................................................................................................................ 368

10. BIOLOGICAL INTERFACES ............................................................................. 372–409 10.1 Introduction .............................................................................................................................. 372 10.2 Protein Adsorption at Interfaces ................................................................................................ 373 10.2.1 Adsorption of Proteins at Solid–Liquid Interfaces ...................................................... 374 10.2.2 Adsorption of Proteins at Fluid–Fluid Interfaces ........................................................ 377 10.3 Biological Membranes .............................................................................................................. 381 10.4 Interfacial Forces ...................................................................................................................... 384 10.4.1 van der Waals Force .................................................................................................... 385 10.4.2 Hydration Force .......................................................................................................... 385 10.4.3 Steric Forces ............................................................................................................... 387 10.4.4 Electrostatic Double Layer Force ................................................................................ 390 10.4.5 Hydrophobic Forces .................................................................................................... 392 10.5 Cell Adhesion ........................................................................................................................... 393 10.6 Pulmonary Surfactant ............................................................................................................... 396 10.7 Biomaterials .............................................................................................................................. 398 Summary ............................................................................................................................................ 400 Keywords ........................................................................................................................................... 401 Notation ............................................................................................................................................. 401 Exercises ........................................................................................................................................... 403 Numerical and Analytical Problems ................................................................................................. 404 Appendix ............................................................................................................................................ 405 Further Reading ................................................................................................................................ 406

11. NANOMATERIALS ............................................................................................ 410–451 11.1 Introduction .............................................................................................................................. 411 11.2 Approaches for the Synthesis of Nanomaterials ....................................................................... 412

Contents

ix

11.3 Self-Assembly and Structure .................................................................................................... 412 11.4 Synthesis of Nanoparticles ........................................................................................................ 414 11.4.1 Homogeneous Nucleation ........................................................................................... 414 11.4.2 Microemulsion-Based Methods .................................................................................. 421 11.4.3 Synthesis of Carbon Fullerenes ................................................................................... 422 11.5 Synthesis of Nanowires, Nanorods and Nanotubes ................................................................... 425 11.5.1 Nanowires and Nanorods ............................................................................................ 425 11.5.2 Nanotubes ................................................................................................................... 429 11.6 Deposition of Thin Films .......................................................................................................... 435 11.7 Synthesis of Microporous and Mesoporous Materials .............................................................. 438 11.8 Lithographic Techniques ........................................................................................................... 442 11.9 Toxic Effects of Nanomaterials ................................................................................................. 444 Summary ............................................................................................................................................ 445 Keywords ........................................................................................................................................... 445 Notation ............................................................................................................................................. 446 Exercises ........................................................................................................................................... 446 Numerical and Analytical Problems ................................................................................................. 448 Further Reading ................................................................................................................................ 448

12. INTERFACIAL REACTIONS ............................................................................. 452–493 12.1 Introduction .............................................................................................................................. 452 12.2 Reactions at Solid–Fluid Interfaces .......................................................................................... 453 12.2.1 Langmuir–Hinshelwood Model .................................................................................. 455 12.2.2 External Transport Processes ...................................................................................... 460 12.2.3 Determination of Mass and Heat Transfer Coefficients .............................................. 464 12.2.4 Internal Transport Processes ....................................................................................... 465 12.3 Interfacial Polycondensation ..................................................................................................... 469 12.4 Gas–Liquid and Liquid–Liquid Interfacial Reactions ............................................................... 476 12.5 Reactions at Biointerfaces ........................................................................................................ 477 12.6 Micellar Catalysis ..................................................................................................................... 481 12.7 Phase Transfer Catalysis ........................................................................................................... 484 Summary ............................................................................................................................................ 485 Keywords ........................................................................................................................................... 486 Notation ............................................................................................................................................. 486 Exercises ........................................................................................................................................... 489 Numerical and Analytical Problems ................................................................................................. 490 Further Reading ................................................................................................................................ 492

INDEX ....................................................................................................................... 495–503

Preface The applications of colloids and interfaces have impacted human civilisation to a very great extent. Beginning with edibles and personal hygiene care products, the applications of colloid and interface science are visible in large-scale industrial undertakings such as petroleum recovery, manufacture of heavy chemicals and coating processes. Traditionally, this branch of science was confined within the precincts of chemistry departments in universities the world over. In recent times, it has been groomed to blossom into a multidisciplinary subject meant for study by chemical engineers, biotechnologists, physicists and environmental scientists. In most of the universities in the United States of America and Europe, colloid and interface science form an integral part at the level of undergraduate or postgraduate curricula of chemistry and chemical engineering. There are specialised research institutes for the advancement of research activities on colloids and interfaces. This has led to a phenomenal development of the subject. One hundred years ago, this subject was mostly permeated with qualitative ideas. However, the works of many brilliant scientists have laid it on a solid quantitative foundation. Many among these scientists have been awarded the Nobel Prize in recognition of their pioneering work in this branch of science. Hundreds of books have been written on the subject, which have immensely contributed to its vigorous expansion and accelerated the emergence of newer frontiers such as nanoscience and technology. The rapid growth of colloid and interface science has been fostered and nurtured by a large number of specialised journals on this subject. In addition, many broad-based journals dealing with chemistry, chemical engineering, biotechnology and physics regularly publish research articles on colloids and interfaces. Some of these journals devote special sections to publication of research work on colloid and interface science. The results of research works published in the journals in course of the past one hundred years may concatenate the subject matter of a tome like this. However, space crunch presents a formidable obstacle to cite all of them. Some of the relevant texts, reference books, and journal articles have been suggested for further reading at the end of the concerned chapters. It is expected that these references will equip the reader with a congenial road-map for advanced studies in the target areas. In many universities, this subject is taught as part of a course on physical chemistry. However, nowadays many chemistry and chemical engineering departments of institutes of higher studies are offering exclusive courses on colloid and interface science. For a sustained growth of this subject, xi

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it is necessary that it be taught at the senior-undergraduate or postgraduate level so that young researchers may be armed with basic tools for future research. This book strives at providing the fundamentals of colloids and interfaces in a simple and straightforward manner with emphasis on their scope and applications. The range of colloid and interface science is so vast that it is impossible to squeeze all the aspects in a single volume. This book does not attempt to achieve this feat. It has been written keeping in view a one-semester course, although even then it may not be possible to disgorge the entire contents of the book within a span of approximately 45 lecture-hours during a four-month period. The superstructure of colloid and interface science is built on the foundation of thermodynamics. Therefore, the readers of this book are expected to possess a working knowledge of elementary classical thermodynamics. The reader will find the application of the principles of thermodynamics all through this book. The analyses, examples and problems given encompass familiarity with aspects of elementary mathematics. However, the solution of many problems will require the aid of a computer. Therefore, the basic knowledge of computer software such as Microsoft Excel or MATLAB, and development of computer programs in a language such as C++ is needed. A few computer programs have been listed in the appendices at the end of some of the chapters. Knowledge of elementary numerical analysis is also necessary for solving some of the problems. SI units have been used throughout this book in keeping with the trends of modern scientific literature in chemical engineering and other disciplines. A brief introduction to the essential features of the SI units has been presented at the very beginning of this book. However, the IUPAC system of nomenclature of chemical compounds has not been uniformly followed throughout. This style has been adopted because it obviates the necessity of tedious descriptions in many cases. Moreover, most of the technical literature on colloid and interface science describes the organic compounds by their generic names. I have been immensely helped by many publishers of books and journals, and equipment manufacturers who have kindly allowed me to use figures and photographs copyrighted by them. I have acknowledged their contribution and unstinted cooperation at appropriate places. I am extremely thankful to the authorities of IIT Guwahati for providing me the environment and necessary infrastructural facilities for preparing this book. I am thankful to my students for their valuable comments and suggestions, which have been found useful. I am particularly indebted to my parents for providing me the inspiration and impetus to take up this challenging task of assembling such an enormous mass of information in one volume. My work would not have been completed without the time-management designed and enforced by my sister Kakali. I am thankful for her help in preparing the index of this book. Finally, I am much thankful and obliged to my publisher, M/S PHI Learning Private Limited, for undertaking to publish this work in view of the fact that this is the first book on colloid and interface science published in India. Their help, management and strict adherence to the schedules have been commendable and a source of inspiration for me. I shall consider myself fully recompensed if this book can successfully impart the elementary concepts of colloid and interface science to my readers. My greatest source of happiness lies in the possible fuelling of interest in this subject in the minds of the young learners and researchers for whom this book is primarily meant. Because this is the first edition, a few errors cannot be averted, for which who else but the writer alone is at fault. I solely rely on the magnanimity of my esteemed readers in drawing my attention to these unintended pitfalls without any reservation or hesitation to enable me to correct the shortcomings in the subsequent editions. All such messages will be duly acknowledged in a spirit of humility and gratitude. Pallab Ghosh

SI Units ‘SI’ is abbreviated from the French Le Système international d'unités. This is the international system of units for commerce as well as science. In the older metric system, several systems of units were used, such as cgs (centimetre-gram-second), fps (foot-pound-second) or mks (metre-kilogramsecond). The SI system was developed in 1960 based on the mks system. Several newly named units were introduced in the SI system. Moreover, units were created and definitions were modified through international agreement among many nations as the technology of measurement progressed. This system is adopted almost universally. All recent books, monographs and journals have used the SI units. Some countries still use the older units in addition to the SI units. Therefore, conversion factors from the older units to the SI system of units are also available in many texts. In this book, SI units have been used throughout. The SI base units and the derived units (including those with special names) have the important advantage of forming a coherent set such that unit conversions are not required when inserting particular values for quantities into equations. The need for inserting large number of zeros is eliminated by the SI prefixes. They are easy to pronounce, and based on natural standards (such as size of earth, and laws of physics). Because the SI is the only system of units that is globally recognised, it also has a very important advantage for establishing a worldwide dialogue. Therefore, it would simplify the teaching of science and technology to the next generation if everyone uses this system.

Basic and Supplementary Quantities and Units Type

Quantity

SI unit

Symbol

Basic

length mass time electric current thermodynamic temperature amount of substance luminous intensity plane angle solid angle

metre kilogram second ampere kelvin mole candela radian steradian

m kg s A K mol cd rad sr

Supplementary

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SI Units

Derived Units Quantity

SI unit

Symbol

Formula

frequency force pressure, stress energy, work power, radiant flux electric charge, quantity of electricity electric potential, potential difference, electromotive force capacitance electric resistance conductance magnetic flux magnetic flux density inductance luminous flux illuminance activity of radionuclides absorbed dose

hertz newton pascal joule watt

Hz N Pa J W

1/s kg·m/s2 N/m2 N·m J/s

coulomb

C

A·s

volt farad ohm siemens weber tesla henry lumen lux becquerel gray

V F W S Wb T H lm lx Bq Gy

W/A C/V V/A A/V V·s Wb/m2 Wb/A cd·sr lm/m2 1/s J/kg

Additional Common Derived Units Quantity acceleration angular acceleration angular velocity area concentration of substance current density density (mass) electric-charge density electric-field strength electric-flux density energy density entropy heat capacity heat-flux density luminance magnetic-field strength molar energy molar entropy

Symbol m/s2 rad/s2 rad/s m2 mol/m3 A/m2 kg/m3 C/m3 V/m C/m2 J/m3 J/K J/K W/m2 cd/m2 A/m J/mol J·mol–1 K–1

Quantity molar heat capacity moment of force permeability permittivity radiance radiant intensity specific heat capacity specific energy specific entropy specific volume surface tension thermal conductivity velocity viscosity (dynamic) viscosity (kinematic) volume wave number

Symbol J·mol–1 K–1 N·m H/m F/m W·m–2 · sr–1 W/sr J·kg–1 · K–1 J/kg J·kg–1 · K–1 m3/kg N/m W·m–1 · K–1 m/s Pa · s m2/s m3 m–1

SI Units

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SI Prefixes Prefix yotta zetta exa peta tera giga mega kilo hecto deca (or deka) deci centi milli micro nano pico femto atto zepto yocto

Symbol

Multiplication factor

Y Z E P T G M k h da d c m m n p f a z y

1024 1021 1018 1015 1012 109 106 103 102 101 10–1 10–2 10–3 10–6 10–9 10–12 10–15 10–18 10–21 10–24

There are several rules for writing the symbols for the SI units. These are: 1. Symbols do not have an appended period or a full stop (.), unless at the end of a sentence. 2. Symbols are written in upright Roman type (e.g. ‘m’ for metre), so as to differentiate from the italic type used for mathematical variables (e.g. m for mass). 3. Symbols for units are written in the lower case, except for symbols derived from the name of a person (e.g. symbol of force, N, is derived from Isaac Newton). 4. The symbols of units are not pluralised (e.g. 10 kg is correct, not 10 kgs). The name of the unit, however, can be pluralised (e.g. 10 kilograms). 5. A space should separate the number and the symbol (e.g. 298 K). 6. Spaces may be used as a thousands separator (e.g. 5 000 000), in contrast to commas or periods (e.g. 5,000,000) in order to reduce confusion resulting from the variation between these forms in different countries. 7. Symbols for derived units formed by multiplication from multiple units are joined with either a space or centre dot (.) (e.g. either ‘N m’ or N·m). 8. Symbols formed by division of two units are joined with a solidus (/), or given as a negative exponent. For example, ‘metre per second’ can be written either as ‘m/s’ or ‘m . s–1’. If the exponent is required for only one basic (or supplementary) symbol in the symbol for the quantity under concern, some authors prefer to use the solidus (i.e. ‘m/s’ is preferred to ‘m . s–1’). The important point is that the use of solidus or exponent must not create any ambiguity. For example, if the required symbol is ‘joule per mole-kelvin’, it should be written either as ‘J. mol–1. K–1’ or as J/(mol . K), but not as ‘J/mol . K’.

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Some Fundamental Physical Constants in SI Units Constant Acceleration due to gravity Avogadro’s number Boltzmann’s constant Electronic charge Faraday’s constant Gravitational constant Mass of electron Mass of hydrogen atom Mass of 1/12 of 12C atom Gas constant Molar volume Permittivity of free space Planck’s constant Speed of light in vacuum

Value in SI unit 9.807 m/s2 6.022 × 1023 mol–1 1.381 × 10–23 J/K 1.602 × 10–19 C 9.649 × 104 C/mol 6.670 × 10–11 N m2 kg–2 9.109 × 10–31 kg 1.673 × 10–27 kg 1.661 × 10–27 kg 8.314 J mol–1 K–1 22.414 × 10–3 m3/mol 8.854 × 10–12 C2 J–1 m–1 6.626 × 10–34 J s 2.998 × 108 m/s

1

Basic Concepts of Colloids and Interfaces

Thomas Graham (1805 – 1869)

Thomas Graham is regarded as the father of colloid science. He was born in Glasgow, Scotland. His father was a prosperous textile manufacturer. Graham entered the University of Glasgow in 1819 where he became interested in chemistry. After receiving an M.A. degree from the same university in 1826, he joined the University of Edinburgh where he worked for two years. He taught mathematics and chemistry privately, and in 1830 became professor of chemistry at the Royal College of Science and Technology, Glasgow. Graham was elected a fellow of the Royal Society in 1834. Later, he moved to London and became professor of chemistry at the University College of London. He founded the Chemical Society of London in 1841. He was the Master of Royal Mint from 1855 to 1869. Some of the famous works of Thomas Graham were on the diffusion of gases, aurora borealis, absorption of hydrogen by palladium, phosphorus compounds and production of alcohol during bread-making. Graham received several awards such as the Royal Medal and the Copley Medal of the Royal Society. His textbook entitled Elements of Chemistry was widely studied in Europe. Graham died in London.

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Colloids and interfaces Applications of interfacial phenomena Some classic examples of interfacial phenomena Colloid dispersions Classification of colloids Electric charge on colloid particles Stability of colloids Effect of polymers on colloid stability Colloid preparation and purification techniques 1

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Colloid and Interface Science

1.1

INTRODUCTION

In our everyday life, we encounter innumerable phenomena where interfaces are involved. Starting from the toothpaste and shaving foam, detergents, beverages, and various food products, modern life has been rendered comfortable with judicious application of these phenomena. Enormous industrial development has been taking place implementing various interfacial phenomena to meet our requirements. As a result, scientific understanding of interfacial phenomena has become essential. In fact, this quest started more than 150 years ago. Several leading scientists such as Thomson, Gauss, Laplace, Newton, Maxwell and Rayleigh had explained some of the naturally-occurring interfacial phenomena. For example, surface tension-driven flows were explained by James Thomson in 1855, and capillary action was explained by Maxwell in 1878. A major scientific development took place in 1891 when Agnes Pockels published her work on surface tension. The interfacial effects benevolent to mankind were suitably utilised much before these seminal works, perhaps without understanding much of the science behind them. For example, seafarers discovered centuries ago that a few tanks of oil would suffice to calm down the rampaging waves of the sea, and the connoisseurs of wine had developed the ‘sparkling wine’ to make their life more colourful. Apart from these applications, there are various natural phenomena such as the fascinating colours of the soap bubbles, which exhibit great perfection of their geometric form and attract the scientific mind spontaneously. The work of Langmuir and subsequent research over the past few decades have improved our understanding of numerous interfacial phenomena. A concise account of some applications of interfacial phenomena in various branches of science and engineering is given in Table 1.1. Table 1.1 Discipline

Applications of interfacial phenomena Examples

Chemistry

Adsorption, chromatography, ion exchange, liquid crystals, membranebased separation, nucleation, nucleic acids and proteins, supercooling, superheating and supersaturation

Chemical industries

Adhesives, beverages, catalysis, cosmetics and personal-care products, dairy products, drugs, emulsions, encapsulated products, foams, lubricants, nanoparticles, paints, paper, pigments, soaps and detergents

Environmental science

Aerosols, cloud seeding, fog, sewage treatment and water purification

Imaging technology

Flat-panel display, photographic emulsions and printing ink

Materials science

Alloys, cement, ceramics, fibres, plastics and powder metallurgy

Petroleum engineering, soil science and ore mining

Dewatering, emulsification, flotation, oil recovery, ore enrichment and soil porosity

Now, the question that spontaneously arises is ‘what is an interface?’ The simplest answer to this question is: interface is the boundary between two phases where the properties or behaviour of material differ from those of the adjoining phases*. The phases may be constituted by two immiscible liquids, a liquid and a solid, a gas and a liquid, or a gas and a solid. The interface between water and carbon tetrachloride is shown in Figure 1.1. If one of the phases is a gas, the interface is called surface. There are two continuum models for the interface: one model represents

*

A phase is defined as any homogeneous and physically distinct part of a system which is separated from other parts of the system by definite bounding surfaces.

Basic Concepts of Colloids and Interfaces

3

it as a three-dimensional region, the thickness of which may be a few molecular diameters, and the other model treats it as a two-dimensional surface. The molecular dynamics simulations have shown that the thickness of the liquid–liquid interfacial region is less than 1 nm. The molecules of the two phases are likely to have special orientations near the interface, which are different from their orientations in the bulk phase (Chang and Dang, 1996).

Figure 1.1

Illustration of the interface between two liquids.

One of the important parameters by which an interface is characterised is interfacial energy. The minimum amount of work required to create unit area of the interface is the equivalent of interfacial energy. The magnitude of this energy is often measured through interfacial tension. There are several techniques to measure interfacial tension, which will be discussed in Chapter 4. This tension apparently reflects the difference between the two phases. For example, the surface tension of water is 72.5 mN/m, and that of carbon tetrachloride is 27 mN/m (at 298 K). The interfacial tension between water and CCl4 is 45 mN/m at the same temperature, which is very close to the difference between the two surface tensions. On the other hand, the interfacial tension between water and cyclohexanol is 4 mN/m, whereas the surface tension of cyclohexanol is 32 mN/m at 298 K. These data indicate that the physical properties of the two liquids in the former case are quite different, but water and cyclohexanol have significant affinity for each other (mainly through the hydrogen bonding between the two liquids).

1.2

EXAMPLES OF INTERFACIAL PHENOMENA

It has been mentioned in Section 1.1 that interfacial phenomena have been known to mankind for centuries. Some of them were observed in daily activities. Two classic examples are presented in this section to provide a glimpse of these phenomena to the reader.

1.2.1 Capillary Action This phenomenon is responsible for the spontaneous rise of water through the trunk of the tree, or transport of water from the wet soil to the dry areas. It can be demonstrated quite easily. Suppose that a capillary is dipped in a vessel containing water. It will be observed that the liquid level inside

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Colloid and Interface Science

the tube is higher than the liquid level outside (Figure 1.2). This phenomenon is known as capillary rise. For some liquids such as mercury the reverse happens and capillary depression is observed.

Figure 1.2

Illustration of capillary rise (water is colourized for illustration).

The reason behind capillary rise is the adhesive force between the liquid and the solid material, and the surface tension of the liquid. The upward movement of the liquid column is driven by surface tension. The liquid continues to rise until the weight of the liquid column balances the upward pull. It can be shown easily (see Chapter 4) that water will rise about 30 mm above the air–water interface inside a 1 mm diameter glass tube. Water will rise higher if a smaller-diameter capillary is used. Theoretical interpretation of the capillary action will be presented in Chapter 4. The capillary action has been used in advanced scientific analyses such as thin layer chromatography. It has also been utilised to manufacture garments used for sports and rigorous outdoor activities.

1.2.2 Tears of Wine This phenomenon was first interpreted scientifically by James Thomson in 1855. It is also known as wine legs. When a wine of high alcohol-content is taken in a glass, a ring of clear liquid is formed in the upper part of the glass. This is caused by the induced flow of liquid driven by the surface tension gradient, and capillary action. Alcohol exhibits surface activity when it is added to water. For example, the surface tension of water is reduced to less than half of its original value when a small amount of ethyl alcohol is added to it (the surface tension of water is 72.5 mN/m at 298 K, whereas the same for ethyl alcohol is 23 mN/m). Capillary action forces the liquid to climb along the glass wall. Therefore, a thin liquid film is formed on the glass wall. However, alcohol has higher vapour pressure and lower boiling point than water. As a result, it evaporates faster and increases the surface tension of the liquid in the film. This creates a surface tension gradient in the film, which draws further liquid from the bulk of the wine, because the wine has lower surface tension due to its higher alcohol content. In this process, a considerable amount of the liquid climbs up along the wall of the

Basic Concepts of Colloids and Interfaces

5

glass. After some time, the liquid in the film begins to fall downwards due to gravity. These falling streams manifest themselves as the ‘tears of wine’ (see Figure 1.3).

Figure 1.3

Illustration of tears of wine (the drops formed above the pool of wine are returning back to the pool).

Some people believe that wines which produce a lot of ‘tears’ are of better quality. However, these phenomena are mainly related to the alcohol-content of the wine. Apparently, if evaporation is reduced, tears should subside, which is actually observed when the glass is covered. The transport of liquid due to the gradient in surface tension is known as Marangoni effect (named after Italian physicist Carlo Marangoni who published this effect ca. 1865). The surface tension gradient is caused by the gradient in the concentration of ethanol in this case. The Marangoni effect is believed to be a very important factor in the stabilisation of foams and emulsions.

1.3

SOLID–FLUID INTERFACES

The interface between a crystalline solid and gas or liquid is important in diverse areas such as processing of ceramic materials, manufacture of catalysts, drilling of petroleum crude oil, liquid coating and thin solid films for electronic devices. The ceramic technology has been using the colloid and interface science extensively. Ceramic materials usually have very high melting points. The processing techniques used for metals such as melting and resolidification cannot be used for ceramic materials. Most ceramic materials are processed by the powder processing technology. The basic raw material is a blend of clay, flint and potash feldspar. Water is added to this mixture in such a manner that the suspension is of right consistency for the potter’s wheel or the mould. Addition of electrolytes such as sodium silicate has a profound effect on the fluidity of the system. A major part of the chemical industry deals with catalytic reactions. Catalysts with controlled surface area, desired pore-size distribution and specific adsorption capabilities for gas–solid and liquid–solid reactions have been manufactured in the past five decades. It is a highly competitive and valued industry where the skill, expertise and know-how are unique, and often key to successful

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Colloid and Interface Science

manufacturing. Very little of the actual technology practised by the manufacturers is exposed in the open technical literature. The essential requirement is that the commercial catalysts must faithfully reproduce the laboratory results. In addition, they must be uniform, consistent and economical. The recovery of petroleum trapped in nearly-inaccessible fractures of sandstone or limestone rocks, and inside the fine pores is a challenging task. The technology to recover such oil is known as enhanced (or tertiary) oil recovery. This improved extraction is achieved by gas injection, thermal recovery or by chemical injection. Gas injection is done by injecting carbon dioxide, nitrogen or natural gas into the oil reservoir whereupon it expands and thereby pushes additional oil to a production wellbore. The gas dissolves in the oil to lower its viscosity and improves the flow-rate of the oil. Another technique is thermal recovery which uses heat to improve the flow-rate. In the chemical injection technique, surfactants are injected to lower the capillary pressure. Polymers are used to enhance the effectiveness of the waterfloods. Coating on solid surfaces has a wide variety of applications such as paints, adhesives and claycoated papers, photographic films (which have as many as 18 layers of coatings of different chemicals in gelatin–water base), and aluminium disks coated with iron oxide which make computer hard disks. A great variety of coating methods is available for the varied coating applications to transfer the liquid from a reservoir onto the solid substrate. The chosen method depends on the rheology of the coating liquid, type of solid surface, desired coating thickness and uniformity. Many modern thin-film devices are made by vapour deposition. Examples are conductors, magnetic recorders, decorative coatings, laser-gyroscope mirrors, magnetic films, metallurgical thin films, and thin-film sensors which are used to detect gases. In such thin films, ultimately a solid–solid interface is created by depositing vapour on the solid surface. A number of sophisticated techniques are used to deposit the material under high vacuum. The thickness of the film can vary significantly depending on the requirement and the manufacturing technique employed. For example, the polycrystalline metallic films can be several micrometers thick. They are used as magnetic recording media. On the other hand, the semiconductor epitaxial films have thickness of the order of a few nanometres. The technology of thin films deals with thickness between 0.1 nm and 1000 nm.

1.4

COLLOIDS

A colloid is defined as a particle that has some linear dimension between 1 nm and 1 mm. Sometimes (especially in applied colloid science) the upper limit of size is extended to much larger values (e.g., several hundreds of micrometres). A dispersion of such particles is called a colloid dispersion. Notwithstanding the fact that the size of the particles is very small, a colloid dispersion is quite different from a solution. Thomas Graham observed that a true solution passed through parchment or cellophane papers, but a colloid dispersion did not, only the continuous medium of the dispersion seeped through. However, many of these dispersions passed through an ordinary filter paper like a true solution. The particles of the dispersed phase can be viewed easily in an ultramicroscope or an electron microscope. They can be discerned by light scattering. Therefore, a definition of a colloid system can be given as follows: “A colloid system is a two-phase heterogeneous system in which one phase is dispersed in a fine state (1 nm to 1 mm) in another medium, called continuous or dispersion medium”. Two micrographs of colloid particles are shown in Figure 1.4. The dispersion medium is not restricted to liquids; it can be a gas or a solid also. The particles can have any shape, not just spherical or cylindrical shapes. Some examples of colloid dispersions of different types are presented in Table 1.2.

Basic Concepts of Colloids and Interfaces

Figure 1.4

7

(a) Field-emission scanning electron micrographs of silver nanoparticles (Lin et al., 2008) [reproduced by permission from Elsevier Ltd., © 2008], and (b) Transmission electron micrograph of TiO2 colloid (Jia et al., 2008) [reproduced by permission from Elsevier Ltd., © 2008]. Table 1.2 Various types of colloid dispersions with examples

Dispersed phase

Dispersion medium

Name

Examples

Liquid

Gas

Liquid aerosol

Fog, cloud, liquid sprays (e.g. hair spray)

Solid Gas

Gas Liquid

Solid aerosol Foam

Smoke, dust Foam on surfactant solutions, fireextinguisher foam

Liquid

Liquid

Emulsion

Milk, mayonnaise

Solid

Liquid

Sol, paste, colloid suspension

Gold sol, silver iodide sol, toothpaste, pigmented ink, paint

Gas

Solid

Solid foam

Insulating foam, expanded polystyrene

Liquid

Solid

Solid emulsion

Pearl, bituminous road paving, ice cream

Solid

Solid

Solid suspension, solid dispersion

Ruby (gold) glass, carbon in steel

Sols and emulsions are probably the most common types among the various colloid systems. If the dispersion medium is aqueous, the sol is called hydrosol.

1.4.1 Colloids and Interfaces So far, we have described the elementary features of the colloid systems and interfaces. The natural question that arises at this juncture is what is the link between these two? The link follows from the fact that small particles have a high surface area per unit mass. That is the key feature of enormous importance. The following example illustrates this.

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Colloid and Interface Science

EXAMPLE 1.1 A spherical particle of 1 cm diameter is broken uniformly into a large number of spherical particles such that the diameter of each of the new particles is 1 × 10–7 m. What is the total surface area of the new particles? Solution

The surface area of a sphere of 0.01 m diameter is A = pD2 = 3.14 × 10–4 m2

The volume of the sphere is

V

S D3 6

If the diameter of the new particles be d and their number be N, then È S d3 Ø NÉ Ù Ê 6 Ú

S D3 6

V

Therefore, the number of particles is given by N

È DØ ÉÊ d ÙÚ

3

È 1 – 10 2 Ø É Ù Ê 1 – 10 7 Ú

3

1 – 1015

Therefore, the total surface area of the new particles is Atot

NS d 2



1 – 1015 – S – 1 – 10 7



2

31.4 m 2

Therefore, the surface area has increased from 3.14 × 10–4 m2 to 31.4 m2. Similarly, for liquids, a spray of fine droplets would generate a huge surface area. To quantify the surface area per unit mass, a term known as specific surface area is used. In the foregoing discussion, let us assume that the droplets of liquid assume spherical shape because of surface tension (see Chapter 4). For atomisation of a liquid in a gas, the specific surface area is related to the diameter of the droplets (d) and the density of the liquid (r) by the relation SSA

Sd2 (S d / 6) U 3

6 dU

(1.1)

For 1000 cm3 water, the total surface area of the droplets of 1 mm diameter produced by atomisation can be larger than a soccer ground! If the droplets are reduced to 10 nm diameter, the specific surface area would be 6 × 105 m2/kg. The fraction of water molecules that would be present on the surface consequently increases with the reduction in the size of the droplets. The work necessary to reduce the size of the droplets to the sub-micron level is also very high, and work equivalent to several hundreds of joules is necessary to generate droplets of size < 10 nm. Thus, we can understand that a huge amount of work needs to be done to disperse water into a fog. In many liquid–liquid chemical reactions, such as nitration of benzene, high interfacial area is necessary for sufficient progress of the reaction leading to industrially-viable yield of the desired product. To disperse one liquid into another in the form of fine droplets, very high speed of agitation is necessary. It is also evident why the effectiveness of a catalyst is greatly increased when it is in a finely-divided form. Another example of high importance is the removal of pollutants from water by adsorption on particulate charcoal, which has very high surface area.

Basic Concepts of Colloids and Interfaces

9

All the technologies of colloids are firmly rooted in a common underpinning science, namely that of interfaces. This subject is essentially concerned with how molecules and their assemblies behave structurally as well as dynamically at the interfaces, as opposed to the bulk.

1.4.2

Classification of Colloids

Colloid systems are subdivided into two categories: (i) lyophilic (or solvent-loving) colloids and (ii) lyophobic (or solvent-fearing) colloids. If water is the medium, the terms hydrophilic and hydrophobic colloids are used respectively. This classification was given by Freundlich (1926). Lyophilic colloid dispersions are formed quite easily by the spontaneous dispersion of the colloid particles in the medium. For example, the swelling of gelatin in water indicates the high affinity between the water and gelatin molecules. Lyophobic colloids do not pass into the dispersed state spontaneously. They are generally produced by mechanical or chemical action. Sometimes these systems are called reversible and irreversible (Kruyt, 1952). For reversible colloids, the dispersed phase is spontaneously distributed in the surrounding medium by thermal energy. For example, a protein crystal dissolves in water spontaneously. Such spontaneous dispersion leads to an equilibrium size distribution corresponding to the minimum value of the thermodynamic potential. If the dispersed phase is thrown out of the colloid state, its redispersion is achieved easily. The irreversible colloid dispersions, on the other hand, are thermodynamically unstable. To illustrate, a gold crystal, if brought in contact with water, will not generate the sol spontaneously. The subdivision of the gold crystal into small particles can be performed only by supplying a considerable amount of energy. The total free energy of the gold–water interface is a positive quantity. The small entropic gain in the subdivision process is not sufficient to make the formation of sol spontaneous. This type of colloids has a natural tendency to be thrown out of the dispersion medium from the colloid state. Their stability in the colloid state is achieved with considerable difficulty. Many colloid systems, however, show intermediate behaviour. For example, the powders of silica and alumina can have varied affinities with liquids. Nonetheless, it is the usual practice to describe the colloid systems in terms of the two extremes as discussed before. Classic examples of the lyophilic colloids are macromolecular proteins, micelles and liposomes. Some lyophilic colloids (e.g. surfactant micelles and liposomes) may consist of clusters of many molecules which assemble together due to various reasons (will be discussed in Chapter 3). These are known as association colloids. The molecules which form the clusters have a lyophilic part (e.g. polar head group) and a lyophobic part (e.g. hydrocarbon tail). Some of the association colloids are very important carriers of nanoparticles, drugs and perfumes. The stability of the lyophilic colloids is a consequence of the strong favourable interaction with the dispersion medium. The stability of lyophilic colloids is governed by two factors: electric charge of the particles and solvation. On the other hand, the stability of the lyophobic colloids is governed by a single factor: electric charge. The lyophobic colloids can be coagulated when a small amount of electrolyte is added to the system. The effectiveness of the electrolyte depends upon its valence and concentration. When a lyophobic colloid dispersion is coagulated, the homogeneous system turns turbid and distinctly non-homogeneous. When viewed with an ultramicroscope, it can be observed that each individual particle is subject to thermal motion before coagulation. This motion is known as Brownian movement. Afterwards, the particles cluster together to form larger aggregates, called coagulum. In the coagulum, the particles retain their individuality. They are held together by weak forces. Sometimes, the coagulum can be redispersed into the original colloid dispersion. This process

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is known as peptisation. The terms coagulation and flocculation are widely used in colloid chemistry to mean aggregation of the particles. Some scientists use the term coagulation to imply the formation of compact aggregates leading to the macroscopic separation of a coagulum, and the term flocculation is used implying the formation of a loose or open network which may or may not separate macroscopically. Industrially, coagulation refers to the first stage when colloid particles begin to stick together forming microflocs. This is usually achieved by adding a chemical known as coagulant. Later, these microflocs grow bigger to form visible flocs. This step is called flocculation. High molecular weight polymers, called coagulant aids, may be added during this step to help formation of bridges between the particles. This is known as bridging flocculation. Coagulation occurs by the change in the electrical properties around the particles, which we will discuss in the next section. As the lyophobic colloids are very sensitive towards electrolytes, Freundlich (1926) called them electrocratic colloids. The lyophilic colloids do not coagulate by addition of a small quantity of electrolyte. However, their physical properties such as viscosity are often affected by the addition of a small amount of electrolyte.

1.4.3 Electrical Charge on Colloid Particles It has been known for over a century that the colloid particles are electrically charged. However, the overall colloid system is electrically neutral. Therefore, it can be concluded that the dispersion medium must contain an equivalent amount of opposite charge. These charges are carried by oppositely charged ions on the surface of the particles and in the solution, as illustrated in Figure 1.5. Let us consider the silver iodide colloid. In a dialysed AgI sol, the particles carry negative charge. This charge is developed by the excess iodide ions. The negative charge is balanced by the H+ ions to maintain electrical neutrality. These positive ions, however, are distributed non-uniformly around the charged particle, as shown in Figure 1.5. The strong electrical attraction attempts to bind the H+ ions, however, the thermal motion of these ions drives them away from the surface of the colloid particle. Therefore, many H+ ions do not adsorb on the surface but remain in the aqueous medium as free ions. The result of these two counteracting forces is that the H+ ions remain in the neighbourhood of the negatively charged AgI particles. The charge of the particles is screened off by an equivalent opposite charge by the swarm of the H+ ions.

Figure 1.5

Charged colloid particle in a dispersion.

This representation of the surface charge and the ionic atmosphere is known as electrostatic double layer, a very important term in the stability of colloids. One layer of the ions is situated on the surface of the particles. The ions in this layer may not be distributed as uniformly on the surface as shown in Figure 1.5, but for the sake of analysis and simplicity, the distribution of ions is assumed to be uniform. The other layer is the ionic atmosphere in the continuous medium near the particle. These ions are known as counterions. These two layers of electrostatic double layer are often referred

Basic Concepts of Colloids and Interfaces

11

to as inner and outer layers. Since the ions of the outer layer are in thermal motion, the charge in this layer decreases with distance from the particle. The effective distance up to which the presence of this diffuse layer is significant is of the order of 100 nm. It is significantly affected by the presence of electrolytes. The details of electrostatic double layer will be discussed in Chapter 5.

1.4.4

Stability of Colloids

A colloid dispersion contains many particles. Each of these particles has an electrostatic double layer surrounding it. The particles retain their individual entity (rather than agglomeration with the other particles) by the repulsion between the electrostatic double layers. When two particles approach each other, their diffuse double layers are compressed and confined in a small space, which is entropically unfavourable. This is the origin of the repulsion between the two particles, which does not allow them to come closer than a certain separation. The strength of the double layer repulsion depends on the charge on the surface of the particles and the concentration of electrolyte in the dispersion. If the double layer is of sufficient strength, the colloid remains stable. A stable dispersion can be formed by simply shaking the particles in the liquid. When the particles are stabilised, the process is known as peptisation. The opposite phenomenon, i.e. the destabilisation of the colloid (which can be done by simply adding some salt to it) is coagulation. Therefore, it is evident that certain ions are necessary to cause peptisation or coagulation. The amount of electrolyte required to induce coagulation depends upon the valence of the counterion in the salt. This concentration of electrolyte is known as critical coagulation concentration or flocculation value. Some values are presented in Table 1.3 for the AgI sol. Table 1.3

Critical coagulation concentration of some salts for the AgI sol

Type of electrolyte

Formula of the electrolyte

Critical coagulation concentration (mol/m3)

1:1

LiNO3 NaNO3 KNO3

165.00 140.00 136.00

2:1

Mg(NO3)2 Ca(NO3)2 Ba(NO3)2

2.53 2.37 2.20

3:1

Al(NO3)3 La(NO3)3 Ce(NO3)3

0.07 0.07 0.07

These values indicate that the valence of the counterion is a very important parameter for coagulating a sol. The amount of divalent counterion required to coagulate the AgI colloid is much smaller than the amount of the monovalent counterion. The amount of trivalent counterion required is even less. The effects of these ions are related to their ability to reduce the electrostatic double layer repulsion which stabilises the colloid particles. It can also be observed from the data presented in Table 1.3 that the values of critical coagulation concentration are quite similar for the different counterions of the same valence. In general, a larger counterion of the same valence is somewhat more effective for coagulation than a smaller counterion. One well-known rule about the effect of valence of counterion on coagulation is known as Schulze–Hardy rule. It states that the critical coagulation concentration varies with the inverse sixth-

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Colloid and Interface Science

power of the valence of the counterion. The specific nature of these ions is less important. Also, the effect of the valence of the coions (i.e. the ions of the electrolyte having the same charge as the colloid particles) is less significant on coagulation. EXAMPLE 1.2 The critical coagulation concentrations for NaCl, MgCl2 and AlCl3 for As2S3 colloids (the potential determining ion is S–2) are 0.06 mol/dm3, 7 × 10–4 mol/dm3 and 9 × 10–5 mol/dm3, respectively. Check whether these values are consistent with the Schulze–Hardy rule or not. Solution Since the As2S3 colloid particles are negatively charged, the concentrations of the cations are important. If we respresent the concentrations of the monovalent, divalent and trivalent cations by c1, c2 and c3, then from the given data on critical coagulation concentration, we have

c1 : c2 : c3

0.06 : 7 – 10 4 : 9 – 10 5

1 : 0.012 : 0.0015

According to Schulze–Hardy rule c1 : c2 : c3

1:

1 2

6

:

1 36

1 : 0.016 : 0.0014

Therefore, the critical coagulation concentration data agree with the Schulze–Hardy rule. Apart from the electrical stabilisation, colloids can be stabilised by polymers also, which is known as steric stabilisation. This method of stabilisation has been used in industry for a long time, although the mechanism of stabilisation has been studied only in the past few decades. Polymer molecules can adsorb on the surface of the lyophobic colloid particles as shown in Figure 1.6(a). When two such particles approach each other, the polymer layers on their surfaces are compressed, which causes reduction in entropy. As a result, the particles repel each other. The polymeric stabilisation is effective when the concentration of polymer is moderate, and the polymer molecules are adsorbed and anchored on the particle surface. The steric stabilisation will be discussed in detail in Chapter 5. When the concentration of polymer is small, the molecules cause flocculation of the colloid particles. The polymer chain on one particle attaches to the bare surface of another particle [see Figure 1.6(b)]. This process is known as polymer bridging. It causes destabilisation of the dispersion. One very important use of polymers is in water purification where a small amount of solid is to be removed from a large volume of water. A variety of polymers are used for this purpose. Polymers of high molecular weight (e.g. 20,000 kg/kmol) are commonly used because they have long tails through which they can form the bridge between the colloid particles. Polyacrylamides are widely used for flocculation. These polymers can hydrolyse in aqueous media (depending upon the pH) to form polyacrylic acid derivatives. Sometimes calcium salts are added to act as bridging agents.

Figure 1.6

Effect of polymers on colloids: (a) steric stabilisation, and (b) bridging flocculation.

Basic Concepts of Colloids and Interfaces

13

For lyophilic colloid systems, a major source of stabilisation is hydration. It is possible to obtain stable lyophilic sols without charge. Some very interesting observations have been reported in the literature to suggest the role of hydration in imparting stability. For example, the behaviour of an agar-agar sol resembles the properties of lyophobic colloids if 50% or more alcohol is added to it: the Tyndall scattering increases, viscosity decreases and a small amount of electrolyte causes coagulation showing the distinct effect of the valence of the counterions as observed with the lyophobic colloids. It has been suggested that alcohol dehydrates the lyophilic colloid particles. The stability of the bare particles then depends upon their electric charge so that the role of the counterions on coagulation becomes significant.

1.4.5 Kinetic and Thermodynamic Stabilities The stability of a colloid system can be divided into two categories: kinetic and thermodynamic stability. The equilibrium thermodynamics is silent about the rate at which a process occurs. Therefore, some two-phase systems are unstable in a thermodynamic sense (e.g. diamond–graphite system) but the change is so slow that the thermodynamic instability has little practical significance. Many colloid systems described in Table 1.2 are thermodynamically unstable. However, they are kinetically stable. A system which shows resistance to coagulation is called kinetically stable. Therefore, the frequently-used term colloid stability generally implies kinetic stability. A classic example of thermodynamic stability is the formation of surfactant micelles. When the concentration of surfactant in the solution is increased beyond the critical micelle concentration, the surfactant molecules assemble to form micelles. At higher concentrations of the surfactant, the shape of the micelles changes (e.g. spherical micelles change to cylindrical micelles). Charged lyophobic colloid particles may exist in colloid crystalline phases (body-centred or face-centred cubic structures) to reduce the free energy. Other important examples of thermodynamically stable colloid systems are microemulsions and polymeric microstructures.

1.4.6

Preparation of Colloids

The preparation techniques of a colloid dispersion depend on the properties of the materials. The techniques for the preparation of sols are different from the preparation techniques for association colloids or emulsions. In this section, we will discuss some of the well-known techniques for preparation of various colloids. Some more methods will be discussed in Chapters 9 and 11.

Preparation of Sol The preparation of sol involves three steps: (i) the substance is converted to the desired state of fine division in the dispersion medium, (ii) stabilising agents are added to maintain the stability of the system unless such agents are already present in the reagents used to make the dispersion, and (iii) purification of the colloid. Two methods are employed to create a dispersed phase in a medium: condensation method and dispersion method. The condensation method involves generation of colloid particles in situ either by a chemical reaction or by physical changes under controlled conditions so that insoluble colloid particles are generated. The chemical reactions can be of various types, such as oxidation, reduction, hydrolysis and metathesis. In order to generate the desired particles, temperature, concentration and the speed of agitation are some of the parameters which need to be controlled. For example, a gold sol is prepared by reduction of chloroauric acid, a sulphur sol is prepared by the oxidation of a hydrogen

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Colloid and Interface Science

sulphide solution, and a silver iodide sol is prepared by the reaction of potassium iodide with silver nitrate. A sulphur sol can also be prepared by pouring an alcohol-solution of sulphur in water. Similarly, if small quantities of ferric chloride are added to boiling water, a ferric hydroxide sol is obtained. Many such examples are known where the condensation method is used to prepare colloid dispersions. Some of the typical chemical reactions for the production of sols are: Hydrolysis:

FeCl3  3H 2 O  Fe(OH)3 [colloid]  3HCl

(1.2)

Metathesis:

KI + AgNO3  AgI[colloid]+ KNO3

(1.3)

Reduction:

2AuCl3  3H 2 O  3CH 2 O  2Au[colloid]  3HCOOH  6HCl

(1.4)

In the dispersion method, the substance is pulverised into a fine state by a colloid mill, or by ultrasonic methods. However, sometimes the extent to subdivision reaches a limit below which the size of the particles cannot be reduced any further, because the smaller particles reunite under the influence of the mechanical forces, or due to the attractive forces between the particles. Sometimes an inert diluent or a surface-active material is added to prevent the particles from coming into contact during grinding. An example is, sulphur and glucose are ground together and dispersed in water to form the sol. The glucose is then removed by dialysis. Sometimes the pulverisation can be effected easily. For example, when gelatin is heated in water with stirring, the gelatin sol is prepared. In Bredig’s process, the pulverisation is achieved by electric sparking. Dilute sols of gold, silver and platinum are prepared by bringing two electrodes of the metal close together in water (see Figure 1.7). The electric discharge between them generates fine colloid particles. In the Svedberg method, an alternating current is applied between conducting metal electrodes or between partiallyconducting electrodes of inorganic compounds dipped in water or an organic solvent to prepare various hydrosols and organosols.

Figure 1.7

Bredig’s arc process.

The preparation of colloid involves the generation of a new phase. Two distinct stages are involved: nucleation and crystal growth. The relative rates of these processes determine the particlesize of the dispersion and the stability of the particles. A high degree of dispersion is obtained when

Basic Concepts of Colloids and Interfaces

15

the rate of nucleation is high and the rate of crystal-growth is slow. The rate of nucleation in the initial stages depends upon the degree of supersaturation. Colloid sols are easily prepared when the substance has low solubility. If the material is appreciably soluble, there is a tendency for the smaller particles to redissolve and recrystallise on the larger particles in the precipitate. The rate of growth of the particles depends mainly on the concentration and viscosity of the medium, orientation of the particles, presence of impurities, and aggregation between the particles. The coarsening of the particles in which a larger particle grows in expense of smaller particles is known as Ostwald ripening. The small particles dissolve and ultimately disappear completely. The sols produced by the methods described above usually have a wide distribution in particle size. Such sols are known as polydispersed sols. However, in terms of usefulness, often a narrow range of particle-size or monodispersed colloid system is preferred. For example, monodispersed polystyrene sols are used as calibration standards for electron microscopes, light scattering photometers and Coulter counters. Monodispersed silica is used in the coatings of antireflection lens. Monodispersity is also useful in photographic films, magnetic devices, pharmaceutics and in the preparation of catalysts. Monodispersed colloids can be prepared by setting appropriate conditions so that the nucleation step is restricted to a relatively short period at the start of the sol formation. A well-known technique is seeding a supersaturated solution with very small particles. Monodispersed gold sols and polymer latex dispersions have been prepared by this technique. Monodispersed sulphur sols have been prepared by mixing very dilute aqueous solutions of hydrochloric acid and sodium thiosulphate. Similarly, silver bromide sols have been produced by controlled cooling of hot saturated aqueous solution of silver bromide. The concentration of the solute increases steadily with time and reaches the supersaturation point. This is followed by the rapid relief of supersaturation. Virtually, all nuclei are born during this brief period. The nuclei then grow uniformly by a diffusion-controlled process and a sol of monodispersed particles is formed. The final stage of sol-preparation is its purification. All colloid systems do not require purification, and purification is seldom practised beyond a certain extent to avoid instability of the colloid. The sols have an important property that the dispersed phase does not percolate through certain membranes. This is the basis of the purification procedure known as dialysis. The membranes that are most widely used for dialysis are prepared from cellulose acetate products such as collodion, Cellophane and Visking. The sol to be purified is taken in a bag which is kept suspended in a vessel containing sufficient amount of the dispersion medium (e.g. water), as shown in Figure 1.8(a). The undesirable ions in the sol move out through the wall of the membrane. Some amount of solvent from outside may enter the bag. The solvent in the vessel is replaced from time to time and the dialysis is continued. The purity of the sol is checked at regular intervals by chemical analysis (e.g. analysis of conductance). Dialysis is particularly useful for removing small dissolved molecules from colloid dispersions, e.g. KNO3 from AgI sol. The dialysis process is hastened by stirring. To accelerate the purification process and to achieve a greater degree of purity, sometimes the dialysis is carried out with the aid of an electric field. This process is known as electrodialysis. The electrodialyser is shown schematically in Figure 1.8(b). The middle compartment (with the membrane on its two sides) contains the colloid dispersion. This compartment is surrounded on both sides by the solvent. Two platinum electrodes are inserted in the solvent compartments. When the electric current is passed, the ionic impurities in the dispersion travel outward towards the respective electrodes. To enhance the rate of dialysis, the solvent is circulated through the two sidecompartments. An indicator (e.g. a lamp) is put in the circuit which indicates the increase in resistance when the concentration of the ions falls. The process is stopped when a certain degree of

16

Colloid and Interface Science

purity is achieved. Since extreme purification of the sol can lead to its instability, often purification is not made beyond a certain level.

Figure 1.8

(a) Dialysis, and (b) electrodialysis.

Preparation of Association Colloids Some organic molecules have a lyophobic and a lyophilic part. Examples of such compounds are the sodium and potassium salts of the fatty acids, which are commonly known as soap. If the concentration of such a compound is increased in a solution beyond a certain value, they assemble spontaneously to form particles of colloidal dimensions. These are known as micelles. The concentration is known as critical micelle concentration (CMC). The number of molecules in a micelle can vary widely (e.g. 50–1000). Micelles are thermodynamically stable. They can form in aqueous solutions as well as in organic liquids. In the latter case, the aggregate is known as reverse micelle. In aqueous medium, the presence of electrolytes influences the CMC significantly. We will discuss the details of these phenomena in Chapters 3 and 6. At surfactant concentrations higher than the CMC, the shape of the micelles changes. Some of these colloidal systems are used as templates for the preparation of mesoporous solids of well-organised structure having size between 2 nm and 50 nm. These materials have high demand in molecular separation processes and catalysis.

Preparation of Emulsions Emulsions are very important colloidal systems in which one liquid is dispersed in another immiscible liquid in the form of tiny droplets, e.g. water droplets in crude petroleum. In most of the emulsions, an emulsifier is required to stabilise the droplets. Commercial emulsifiers are usually a mixture of surfactants. Apart from the petrochemical-derivatives (e.g. aryl sulphonates), a variety of biological surfactants are used for stabilising emulsions. The biosurfactants are especially suitable for food emulsions. The surface-active molecules adsorb at the interface between the liquids and prevent them from coalescence (Figure 1.9). The most common methods of preparation of emulsions are ultrasonication and high-speed stirring. A special category of emulsion is microemulsion in which the diameter of the droplets is between 1 nm and 100 nm. They are formed spontaneously if the interfacial tension is very low (~0.001 mN/m). The size of microemulsion droplets is similar to the

Basic Concepts of Colloids and Interfaces

17

size of the micelles. They are sometimes called swollen micelles. In emulsions, the distribution of droplet-size can be wide. Smaller droplets coalesce to form the larger droplets. Microemulsions, however, have narrow distribution of droplet size.

Figure 1.9

Oil droplets in water stabilised by a water-soluble surfactant.

Preparation of Polymeric Colloids Colloid scientists often use the emulsion polymerisation method to synthesise colloidal polymers. In this type of polymerisation, the control of temperature is easy as compared to bulk polymerisation. The monomer is distributed in the form of emulsion droplets with the help of an emulsifier. The emulsion droplets of the monomer act as the reservoir to supply the monomer to the polymerisation sites by diffusion through the aqueous phase. The polymer particles formed in the reaction mixture are stabilised by the emulsifier. The rate of polymerisation and the size of the polymer particles formed depend upon the emulsifier concentration. The number of polymer particles may far exceed the monomer droplets present initially. Monodispersed sols containing spherical polymer particles (such as polystyrene latexes) are prepared by emulsion polymerisation. These latexes are widely used as model systems for studying colloidal behaviour.

1.4.7 Parameters of Colloid Dispersions All colloidal systems are more or less polydispersed with regard to their size. A statistical distribution of particle size is always observed. Suppose that the total number of particles in the dispersion is Ntot, and Ni represents the number of particles of diameter di. Therefore, the fraction of particles having diameter di is fi

Ni , i N tot

1, 2,", N tot

(1.5)

The values of Ni and di can be determined directly from particle-size analysis. From elementary statistics, we know that

6 fi

1

i

(1.6)

The average particle diameter is d

6 f i di i

(1.7)

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Colloid and Interface Science

The variance is given by

V2

Ç fi (di  d )2

(1.8)

i

The variance is a measure of the broadness of the particle size distribution. The standard deviation is the square root of variance (i.e. s). Its unit is same as that of particle size. The particle concentration in a dispersion is another very important parameter. If there are Ntot particles dispersed in a total volume of V, then the number density is N tot V If the particle volume is vp, then the volume fraction (f) is n

I

(1.9)

nv p

(1.10)

È Up Ø IÉ Ù Ê Us Ú

(1.11)

The weight fraction (w) is given by w

where rp and rs are the densities of the particle and the solvent respectively. The total interfacial area in a colloidal dispersion is given by A

N tot S d 2

6V I d

(1.12)

EXAMPLE 1.3 From the droplet-size distribution of nitrobenzene in 92% sulphuric acid (at 298 K) shown in Figure 1.10, the following data were obtained (Rahaman et al., 2007). di (nm) fi

2.0 0.017

2.2 0.048

2.6 0.111

2.9 0.199

3.4 0.253

3.8 0.214

4.4 0.114

5.0 0.037

5.8 0.007

Calculate the mean and standard deviation of the distribution.

Figure 1.10

Size distribution obtained by dynamic light scattering for nitrobenzene droplets in 92% H2SO4 (Rahaman et al., 2007) [adapted by permission from John Wiley and Sons, Inc., © 2007].

Basic Concepts of Colloids and Interfaces

Solution

19

The mean diameter is, d

6 fi di = 0.017 – 2  0.048 – 2.2  0.111 – 2.6  0.199 – 2.9  0.253 – 3.4  0.214 – 3.8  i

= 0.114 – 4.4  0.037 – 5  0.007 – 5.8 \ d = 3.406 nm The variance is V2

6 fi ( di  d )2 = 0.017 – (2  3.406)2  0.048 – (2.2  3.406)2  0.111 – (2.6  3.406)2 i

0.199 – (2.9  3.406)2  0.253 – (3.4  3.406)2  0.214 – (3.8  3.406) 2 0.114 – (4.4  3.406)2  0.037 – (5  3.406)2  0.007 – (5.8  3.406)2 \ s 2 = 0.5065 nm2 The standard deviation is s = 0.7117 nm

SUMMARY This chapter presents an overview of the colloids, interfacial phenomena and their applications. In the beginning of the chapter, several applications of interfacial phenomena are explained with examples. The definition of a colloid system is given and some fundamental properties of colloid dispersions are explained. Various types of colloid dispersions are explained with examples. The relation between colloid and interface science is explained. The Freundlich and Kruyt’s classifications of colloid systems are presented. The terms such as coagulation and flocculation are explained. The origin of electric charge on colloid particles and the role of this charge on colloid stability are discussed. The stability of colloid particles is discussed and the Schulze–Hardy rule is illustrated with example. The role of polymers in stabilisation and flocculation of colloid particles is discussed. The difference between thermodynamic and kinetic stability of colloid systems is explained. Some methods of preparation of colloids are discussed. The purification methods such as dialysis and electrodialysis are explained. The chapter ends with a brief discussion on a few parameters of colloid dispersions which are used to measure the size of the particles and surface area.

KEYWORDS Association colloid Bredig’s process Capillary action Capillary depression Capillary rise Cellophane Coagulant Coagulant aid Coagulation Coagulum Collodion

Colloid Colloid dispersion Colloid mill Condensation method Counterion Critical coagulation concentration Critical micelle concentration Crystal growth Crystal nucleation Dialysis Dispersion method

20

Colloid and Interface Science

Ostwald ripening Peptisation Polydispersed sol Polymer bridging Polymeric colloid Reverse micelle Reversible colloid Schulze–Hardy rule Sol Solid–liquid interface Specific surface area Steric stabilisation Surface Surface tension Swollen micelle Tears of wine Thermodynamic stability Tyndall effect Visking

Electrostatic double layer Electrocratic colloid Electrodialysis Emulsion Flocculation Hydrophilic colloid Hydrophobic colloid Hydrosol Interface Interfacial energy Interfacial phenomena Interfacial tension Irreversible colloid Kinetic stability Lyophilic colloid Lyophobic colloid Marangoni effect Microemulsion Monodispersed sol Oil recovery

NOTATION A Atot c d d D f n N Ntot SSA vp V w

surface area, m2 total surface area, m2 concentration, mol/m3 diameter of daughter particle, m average diameter, m diameter of parent particle, m fraction number density, m–3 number of particles total number of particles specific surface area, m2/kg particle volume, m3 volume, m3 weight fraction

Greek Letters rp rs s f

density of the particle, kg/m3 density of the solvent, kg/m3 standard deviation, m volume fraction

Basic Concepts of Colloids and Interfaces

21

EXERCISES 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. 29. 30. 31. 32. 33. 34.

Discuss five applications of interfacial phenomena in modern life. Explain the interfacial phenomena associated with capillary action and tears of wine. Define interface and surface. Explain their difference. Explain surface tension and interfacial tension. What is interfacial energy? What is the unit of surface tension? Explain the application of colloid and interface science in ceramic processing. What is enhanced oil recovery? Explain three applications of the chemical vapour deposition technology. Define a colloid system. Why do the colloid particles fail to permeate through a collodion membrane? Find out the dispersed phase and dispersion medium for the following colloid dispersions: (i) shaving foam, (ii) butter milk, (iii) sunscreen lotion, (iv) ice cream, (v) insecticide spray, and (vi) pearl. Explain how colloid and interface science are related to each other. Define specific surface area. Explain the difference between lyophilic and lyophobic colloids with examples. What are reversible and irreversible colloids? Explain what you understand by association colloids. Explain the terms coagulation and flocculation. What is the difference between them? What is coagulant? What is bridging flocculation? How is it achieved? Explain how a colloid particle acquires charge and how the electrostatic double layer is formed. What is peptisation? What is critical coagulation concentration? Explain Schulze-Hardy rule. What is steric stabilisation? Explain the difference between kinetic and thermodynamic stabilities. Explain briefly how a sol is prepared. Explain Bredig’s process. What is metathesis? What is Ostwald ripening? Explain why a sol is usually polydispersed. Explain how a near-monodispersed sol can be prepared. What is critical micelle concentration? Explain how polymeric colloids are prepared by emulsion polymerisation. Explain dialysis and electrodialysis.

NUMERICAL AND ANALYTICAL PROBLEMS 1.1 A 1 mm3 drop of carbon tetrachloride in water is put under shear so that it breaks up into 100 droplets of equal size. Calculate the increase in interfacial area by this break-up process.

22

Colloid and Interface Science

1.2 A batch reactor of 10 cm diameter and 20 cm height contains benzene and mixed acid (a mixture of concentrated sulphuric and nitric acids). The mixture is stirred slowly such that the interface between benzene and the aqueous phase is flat. Calculate the interfacial area. If the stirrer speed is raised, benzene disperses into the aqueous phase in the form of 10000 droplets of 1 mm diameter. Calculate the interfacial area in that case. 1.3 Verify whether the data on critical coagulation concentration for LiNO3, Mg(NO3)2 and Al(NO3)3 given in Table 1.3 agree with Schulze–Hardy rule or not. 1.4 The size distribution in a sample of finely-divided particles is given below. di (nm) Ni

87.3 5

100.0 16

114.5 52

131.2 131

150.3 234

172.1 268

197.1 188

225.8 80

258.6 22

296.2 4

The sample size is 1000. Calculate the average diameter and the standard deviation. 1.5 Derive Eq. (1.12). 1.6 For a dispersion of alumina in water the number of colloid particles is estimated to be 2 × 1020 per dm3 of the dispersion. The approximate diameter of the particles is 0.1 mm. Calculate the total interfacial area in the dispersion.

FURTHER READING Books Adamson, A.W. and A.P. Gast, Physical Chemistry of Surfaces, John Wiley, New York, 1997. Boys, C.V., Soap Bubbles: Their Colors and Forces which Mold Them, Dover, New York, 1959. Dobiáš, B. (Ed.), Coagulation and Flocculation (Surfactant Science Series, Vol. 47), Marcel Dekker, New York, 1993. Freundlich, H., Colloid and Capillary Chemistry, Methuen, London, 1926. Holmberg, K. (Ed.), Handbook of Applied Surface and Colloid Chemistry, Vol. 1, John Wiley, Chichester (England), 2002. Isenberg, C., The Science of Soap Films and Soap Bubbles, Dover, New York, 1992. Kruyt, H.R. (Ed.), Colloid Science, Elsevier, Amsterdam, 1952. Stokes, R.J. and D.F. Evans, Fundamentals of Interfacial Engineering, Wiley-VCH, New York, 1997. Verwey, E.J.W. and J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Dover, New York, 1999.

Articles Abbott, N.L., “New Horizons for Surfactant Science in Chemical Engineering”, AIChE J., 47, 2634 (2001). Chang, T.-M. and L.X., Dang, “Molecular Dynamics Simulations of CCl4–H2O Liquid–Liquid Interface with Polarisable Potential Models”, J. Chem. Phys., 104, 6772 (1996). Lin, H.-W., W.-H. Hwu, and M.-D. Ger, “The Dispersion of Silver Nanoparticles with Physical Dispersal Procedures”, J. Mat. Proc. Tech., 206, 56 (2008).

Basic Concepts of Colloids and Interfaces

23

Jia, Y., W. Han, G. Xiong, and W. Yang, “Layer-by-Layer Assembly of TiO2 Colloids onto Diatomite to Build Hierarchical Porous Materials”, J. Coll. Int. Sci., 323, 326 (2008). Neogi, P., “Tears-of-Wine and Related Phenomena”, J. Coll. Int. Sci., 105, 94 (1985). Rahaman, M., B.P. Mandal, and P. Ghosh, “Nitration of Nitrobenzene at High Concentrations of Sulfuric Acid”, AIChE J., 53, 2476 (2007).

2

Properties of Colloid Dispersions

Richard Adolf Zsigmondy (1865 – 1929)

Richard Zsigmondy was born in Vienna. His father was a famous dentist in Austria. Zsigmondy’s interest in chemistry and physics developed in his childhood. After his basic studies in Vienna on quantitative analysis, he went to Munich in 1887 for higher studies. He received doctorate from the University of Munich in 1889. From 1907 to 1929, he was the director of the Institute for Inorganic Chemistry at the University of Göttingen. Sometime in 1897, Zsigmondy directed his attention to colloidal gold present in ruby glass, and he discovered water-dispersion of gold. To facilitate his study, he designed and built an ultramicroscope in 1903. He used it to investigate the various aspects of colloids, including Brownian motion. His work proved particularly helpful in biochemistry and bacteriology. He was awarded the Nobel Prize in Chemistry in 1925 for the elucidation of heterogeneity of colloids. Zsigmondy died in Göttingen.

TOPICS COVERED © © © © © © © © © ©

Sedimentation of colloid particles Brownian motion Osmotic pressure Light scattering Electron microscopy Dynamic light scattering Small-angle neutron scattering Electrokinetic phenomena Properties of lyophilic sols Rheological properties of colloid dispersions 24

Properties of Colloid Dispersions

2.1

25

INTRODUCTION

Colloid dispersions are characterised by many unique properties. For example, some of the colloids are coloured. Their colour depends upon the size and shape of the particles in the dispersion. One of the most important properties of many colloids is their charged nature. Due to their charge, they move in definite directions under the application of an electric field. The thermal motion of colloid particles, and their motion in gravitational and centrifugal fields are of considerable importance. Many separation techniques, particularly in biochemistry, depend on the centrifugal force. The lyophobic and lyophilic colloids differ in some of their properties (e.g. light scattering, gelation, and the effect of salt on coagulation and viscosity). These differences are discussed in this chapter. The rheological properties of colloid dispersions are important in a variety of industrial processing. A wide range of rheological behaviour is displayed by these dispersions. For example, the dispersions containing charged particles can show significant amount of elasticity. If the interparticle interactions are not strong, the dispersions show simple viscous behaviour. A few properties of colloidal dispersions have been discussed in Chapter 1. In this chapter, we will discuss some of the most important properties of colloid dispersions, such as sedimentation under gravitational and centrifugal forces, Brownian motion, osmotic pressure, light scattering, electrical and rheological properties. The difference in the properties of the lyophobic and lyophilic sols will be elucidated.

2.2

SEDIMENTATION UNDER GRAVITY

Let us consider the movement of an uncharged spherical particle through a liquid under the gravitational force. For small colloid particles, the flow occurs at very low velocities relative to the sphere. This is known as creeping flow. Two other forces act on the particle: buoyant force and drag force (see Figure 2.1). The buoyant force acts parallel with the gravity force, but in the opposite direction. The drag force appears whenever there is relative motion between a particle and the fluid. It opposes the motion and acts parallel with the direction of the movement, but in the opposite direction. Let us represent the gravity, buoyant and drag forces by Fg, Fb and Fd respectively. Let the mass of the particle be m, and and its velocity relative to the fluid be v. The resultant force on the particle is Fg – Fb – Fd. The acceleration of the particle is dv/dt. Therefore, we can write the following force balance: m

dv dt

Figure 2.1

Fg  Fb  Fd

Forces acting on a particle falling downwards under gravity.

(2.1)

The gravity force is Fg = mg

(2.2)

26

Colloid and Interface Science

By Archimedes’ principle, the buoyant force is the product of the mass of fluid displaced by the particle and the acceleration under gravity. Fb

mU g Up

(2.3)

where r is the density of the liquid and rp is the density of the particle. The drag force on the particle is given by Stokes’ law, Fd

3SP vd

(2.4)

where m is the viscosity of the liquid and d is the diameter of the particle. Equation (2.4) is applicable when the particle Reynolds number œ dv U P is low (say, < 0.1). Two-thirds of the total drag on the particle is due to skin friction, and the rest is due to form drag. From Eqs. (2.1)–(2.4), we get dv dt

Ë È U p  U Ø 3SP vd Û Ìg É Ü Ù m ÝÜ ÍÌ Ê U p Ú

(2.5)

When the particle settles under gravity, the drag always increases with velocity. The acceleration decreases with time and approaches zero. A small colloid particle quickly reaches a constant velocity after a very brief accelerating period. This is the maximum attainable velocity under the circumstances. It is known as the terminal velocity, vt. The equation for vt can be obtained by setting dv/dt = 0. Therefore, from Eq. (2.5), we get

vt

È Up  U Ø mg É Ù Ê Up Ú 3SP d

(2.6)

Putting m = pd3grp / 6 in Eq. (2.6), we get vt

d 2 g( U p  U ) 18 P

(2.7)

This is the expression for terminal velocity based on Stokes’ law. This derivation is based on the following assumptions: (i) Settling is not affected by the presence of other particles in the fluid. This condition is known as free settling. When the interference of other particles is appreciable, the process is known as hindered settling. (ii) The walls of the container do not exert an appreciable retarding effect. (iii) The particle is large compared with the mean free path of the molecules of the fluid. Otherwise, the particles may slip between the molecules and thus attain a velocity that is different from that calculated. It is evident that when U p ! U the particle settles or sediments to the bottom (e.g. paint pigments settle to the bottom of a paint container). When U p  U the converse is true and the particle rises, which is known as creaming (e.g. cream rises to the top of a bottle of milk). If the particles are very small, Brownian movement becomes important. This is a random motion imparted to the particle by collision between the particle and the molecules of the surrounding liquid. This effect becomes appreciable when the diameter of the particle is about 1 mm. When the diameter of the particle becomes less than 0.1 mm, the random movement of the particle tends to suppress the effect of

Properties of Colloid Dispersions

27

the force of gravity, and settling does not occur. The relative effect of Brownian movement can be reduced by applying centrifugal force. The combined effects of gravity and Brownian movement on particles suspended in a liquid have been analysed by Davies (1949). EXAMPLE 2.1 Finely divided particles of a solid are settling in water. The average diameter of the particles is 1 mm. The density of the particles is 8000 kg/m3. Calculate the terminal velocity of a particle assuming free settling. Given: density of water = 1000 kg/m3 and viscosity of water = 1 mPa s. Solution

The terminal settling velocity obtained from Eq. (2.7) is vt

d 2 g( U p  U )

(1 – 10 6 )2 (9.8)(8000  1000) 3

18 P

(18)(1 – 10 )

3.8 – 10 6 m/s

The particle Reynolds number is, dvt U P

1 – 10 6 – 3.8 – 10 6 – 1000

3.8 – 10 6 1 – 10 3 The concentrations of dispersions used in industrial applications are usually high so that there is significant interaction between the particles. As a result, hindered settling is observed. The sedimentation rate of a particle in a concentrated dispersion may be considerably less than its terminal falling velocity under free settling conditions. Robinson (1926) suggested a modification of Stokes equation and used the density and viscosity of the dispersion in place of the properties of the fluid. Therefore, Re p

vs

K1 d 2 ( U p  Ud ) g Pd

(2.8)

where P d is the viscosity of the dispersion, U d is its density and K1 is a constant. The density and viscosity of the dispersion can be calculated from the following equations: Ud

Pd

and

U p  Hv (Up  U)

(2.9)

P (1  K 2 c)

(2.10)

where H v is the voidage of the dispersion, c is the concentration of the particles, and K2 is a constant for a given shape of the particle. For very concentrated dispersions, P d can be calculated from the equation Pd

È K2 c Ø P exp É Ê 1  K 3 c ÙÚ

(2.11)

where P is the viscosity of the liquid, and K3 is another constant. Steinour (1944) studied the sedimentation of small particles of uniform size. He gave the following correlation: vs

Ë H v2 d 2 ( U p  U ) g Û Ì Ü (10) 1.82(1 H v ) P 18 ÌÍ ÜÝ

(2.12)

It has been assumed in these correlations that the upward thrust acting on the particles is

28

Colloid and Interface Science

determined by the density of the dispersion rather than the density of the liquid. The rate of sedimentation of fine colloid particles is further complicated by flocculation. The ionic solute present in the liquid and the nature of the surface of the particles influence the degree of flocculation, and the size and density of the flocs. Flocculation of colloid particles is usually spontaneous and rapid in water. A deflocculating agent is frequently used to maintain the dispersion of the particles. The type of settling exhibited by flocculated dispersions depends on the initial concentration of the solid and the chemical environment.

2.3

SEDIMENTATION IN A CENTRIFUGAL FIELD

It is easy to produce high acceleration in a centrifugal field so that the effects of gravity become negligible. The equation of motion for the particles in a centrifugal field is similar to the equation for gravitational field, except that the acceleration due to gravity needs to be replaced by the centrifugal acceleration rZ 2 where r is the radius of rotation and Z is angular velocity. Therefore, in this case the acceleration is a function of the position r. For a spherical particle obeying Stokes’ law, the equation of motion is È S d3 Ø 2 É 6 Ù ( U p  U )rZ Ê Ú

3SP d

dr dt

(2.13)

As the particle moves outwards, the accelerating force increases. Therefore, it never acquires an equilibrium velocity in the fluid. Equation (2.13) can be simplified to give

dr dt

d 2 ( U p  U)rZ 2 18P

È rZ 2 Ø vt É Ù Ê g Ú

(2.14)

Thus, the instantaneous sedimentation velocity dr/dt can be considered as the terminal velocity in the gravity field that is increased by a factor equal to the acceleration ratio, rw 2/g. Let us define a sedimentation coefficient (s) as s

dr dt

(2.15)

rZ 2

The sedimentation coefficient represents the sedimentation velocity per unit centrifugal acceleration. It can be evaluated by measuring the location of a particle along its settling path. Integrating Eq. (2.15), we get s

ln(r2 / r1 )

(2.16)

Z 2 (t2  t1 )

EXAMPLE 2.2 A spherical galena particle suspended in water is placed in a centrifugal field. The approximate diameter of the particle is 0.1 mm. What should be the rotational speed so that the particle moves from 6.5 cm to 7 cm in 60 s? Density of galena is 7500 kg/m3. Solution

From Eqs. (2.7), (2.14) and (2.15), the sedimentation coefficient is s

d2 (Up  U)

(1 – 10 7 ) 2 (7500  1000)

18 P

(18)(1 – 10 3 )

3.6 – 10 9 s

Properties of Colloid Dispersions

29

The speed of rotation can be calculated from Eq. (2.16) as 12

Z

Ë ln(r2 r1 ) Û Ì s 't Ü Í Ý

12

Ë ln(7 6.5) Û Ì Ü Í 3.6 – 10 9 – 60 Ý

585.7 rad/s

Note that this rotation-speed is equivalent to 5600 revolutions per minute. Fine colloid particles sediment at a very slow rate under gravity. Swedish chemist Theodor Svedberg (Nobel Prize in Chemistry, 1926) developed a centrifuge which operates at a very high speed (e.g. 5000 – 10000 rad/s) generating an enormous centrifugal force. This force is much greater than the gravitational force. It is known as ultracentrifuge (Figure 2.2). The dispersion is taken in a specially designed cell and vigorously whirled by special motors. The speed of rotation is so high that an acceleration as high as 106 times that of the gravitational acceleration can be attained. Ultracentrifuge not only hastens the sedimentation process, but it also separates the molecules on the basis of differences in mass, density or shape. The ultracentrifuge has proved to be very effective for macromolecular solutions, especially the proteins. Sedimentation of macromolecular solutions in an ultracentrifuge may be studied in two ways: sedimentation equilibrium method and sedimentation velocity method. In the sedimentation equilibrium method, sedimentation is allowed to proceed until an equilibrium distribution throughout the cell is reached. After a long time of centrifuging, the concentration of particles is measured in the ultracentrifuge cell as a function of distance from the centre of rotation. The concentration is usually measured by some optical method (e.g. measurement of refractive index). In the sedimentation velocity method, centrifuging is done at a high speed (e.g. 5000 rad/s). Starting with a well-defined layer from near the centre of rotation, its movement towards the outside wall of the cell is carefully followed. The change in position of the layer is usually detected by the measurement of refractive index or ultraviolet absorption.

Figure 2.2 Beckman Coulter TL 100 ultracentrifuge (Photograph courtesy: Beckman Coulter India Pvt. Ltd.).

30

2.4

Colloid and Interface Science

BROWNIAN MOTION

In 1827, the English botanist Robert Brown observed under a microscope that pollen grains suspended in water had a ceaseless chaotic movement. Pollen grains which had been stored for a century moved in the same way. Colloid particles when examined under an ultramicroscope through Tyndall scattering were also found to execute ceaseless random motion. This motion is known as Brownian motion. A sketch of Brownian motion of colloid particles is shown in Figure 2.3. Even when all possible causes which may lead to motion (such as heat, light, mechanical vibrations and other disturbances) are carefully eliminated, the particles still move about ceaselessly. Therefore, it can be concluded that the Brownian motion is due to the molecular impacts from the medium on all sides of these dispersed particles. Brownian motion becomes less vigorous with the increase in size of the particles. Increase in viscosity of the medium can also reduce the vigour of the movement. Brownian motion is often cited as an indirect evidence of the existence of the molecules and their incessant thermal motion.

Figure 2.3

Brownian movement of colloid particles of 1 mm diameter. The successive positions after every 30 s are joined by straight lines. The mesh size is 3.2 mm (Perrin, 1990) [reproduced by permission from Ox Bow Press, © 1990].

At any instant, the numerous impacts suffered by a colloid particle on all sides are not evenly matched. This results in a net displacement. This displacement is, however, random. The random movements result in self-diffusion of the particles in the fluid. The works of Albert Einstein (Nobel Prize in Physics, 1921) and Marian Smoluchowski about a century ago have formed the basis of the theory of Brownian motion. They derived a relationship between the distance moved by a particle (as a result of Brownian motion) and its diffusion coefficient (D) by treating Brownian movement as a random-walk process. They obtained

Ãx2 Ó

2 Dt

(2.17)

where à x Ó is the mean square displacement of the particle during a period t. Equation (2.17) is known as Einstein–Smoluchowski equation. The details of the derivation of this equation have been presented by Van Kampen (1981) and Hunter (2005). This equation provides us with a means to calculate the diffusion coefficient for particles which can be viewed under a microscope. The actual displacement of a particle in time t is measured by a microscope. From a large number of observations, the mean square displacement is calculated statistically. 2

Properties of Colloid Dispersions

31

The diffusion coefficient of a spherical dispersed particle can be related to its diameter and the viscosity of the liquid as D

RT 3SP N A d

(2.18)

where R is gas constant, NA is Avogadro’s number and T is the temperature. Equation (2.18) is known as Stokes–Einstein equation. This equation has been found to be quite successful for describing the diffusion of large spherical molecules. The diffusion coefficient can vary widely depending upon the medium. For simple molecules (e.g. air and water), the typical value of D is about 0.1 cm2/s in gases, 10–5 cm2/s in liquids, 10–10 cm2/s in solids, and 10–8 cm2/s in polymers and glasses (Cussler, 1997). For macromolecules and other colloid particles, the diffusion coefficient can be much smaller than these values. For example, Eq. (2.18) predicts that a 10 nm diameter colloid particle will have D = 4.4 × 10–7 cm2/s in water at 298 K, which is 100 times lower than the value for the small molecules. The value of the mean square Brownian displacement, therefore, can vary significantly with particle size. Some examples are presented in Table 2.1. Table 2.1 Values of D and à x 2 Ó1 2 for uncharged spherical particles in water at 298 K Diameter (nm) 1 10 100

D (cm2/s)

à x 2 Ó1 2 after 3.6 ks (cm)

4.4 × 10–6 4.4 × 10–7 4.4 × 10–8

0.177 0.056 0.018

EXAMPLE 2.3 The diffusion coefficient of a polymeric surfactant in water at 293 K is 4.5 × 10–10 m2/s. Estimate its diffusion coefficient in ethylene glycol at 313 K. Given: viscosity of ethylene glycol at 313 K is 12.5 mPa s. Solution

\

From Stokes–Einstein equation, D — T P . Therefore, 3 Ø È 313 Ø È 1 – 10 ÉÊ 293 ÙÚ É Ù Ê 12.5 – 10 3 Ú

D2 D1

È T2 Ø È P1 Ø ÉÊ T ÙÚ ÉÊ P ÙÚ

D2

4.5 – 10 10 – 0.0855 3.85 – 10 11 m 2 /s

1

2

0.0855

Equation (2.18) can be used to determine the value of Avogadro’s number. In 1908, Jean Perrin (Nobel Prize in Physics, 1926) and his collaborators performed studies with monodispersed spheres of natural colloids (e.g. gamboge). The mean square displacement was measured as follows. The particles of the dispersion were focused under a microscope, which carried the design of a previously-calibrated square paper within its eyepiece. The Brownian displacement along the x-axis was read by noting its position on the square design (see Figure 2.3). A large number of observations were made on a single particle, noting its displacement at every 30-second interval. This was repeated for a large number of particles. The magnitude of à x 2 Ó in 30 s was thus determined. The value of NA obtained by them varied between 5.5 × 1023 mol–1 and 8 × 1023 mol–1. Later, Svedberg studied monodispersed gold sols of known particle size in ultramicroscope and obtained NA = 6.09 × 1023 mol–1. These results are considered as strong evidences in favour of the kinetic theory.

32

2.5

Colloid and Interface Science

OSMOTIC PRESSURE

French physicist Jean Antoine Nollet discovered in 1748 that when alcohol and water were separated by a pig’s bladder membrane the water passed through the membrane into the alcohol, causing an increase of pressure. However, the alcohol was not able to pass out into the water. This happened because the bladder was semi-permeable. It allowed the water to pass through it, but the alcohol was not allowed to pass. The phenomenon of transport of a solvent through a semi-permeable membrane from the solvent to a solution, or from a dilute solution to a concentrated solution is known as osmosis. In biology, there are many instances of semi-permeable membrane, e.g. inner walls of eggshells, potato-skin, and intestinal walls of some animals. To demonstrate the origin of osmotic pressure, let us perform a thought experiment in the set-up shown in Figure 2.4. A semi-permeable membrane is placed at the centre of the U-tube as shown in the figure. The right part of the tube is filled with a solution and the left part is filled with the pure solvent. In absence of any external field (e.g. electric field) the solvent spontaneously passes through the membrane into the solution (osmosis). The transport of the solvent into the solution can be counteracted by applying a pressure (po) on the solution. This excess pressure on the solution which would just prevent osmosis is called the osmotic pressure of the solution. Therefore, the osmotic pressure of a solution is the pressure required to prevent osmosis when the solution is separated from the pure solvent by a semipermeable membrane. If the pressure on the solution side is greater than po, the solvent will flow in the reverse direction (i.e. from the solution to the solvent). In this case, the semi-permeable membrane functions like a filter that separates the solvent from the solution. This process is known as reverse osmosis, which has large-scale industrial applications (e.g. desalination, concentration of fruit juice and removal of pollutants from water).

Figure 2.4

Illustration of osmotic pressure.

It is important, however, to note that the concept of osmotic pressure is more general than that discussed above. In fact, one does not have to invoke the presence of a membrane to define osmotic pressure. The osmotic pressure is a property of the solution. For example, in electrostatic double layer (Chapter 5), the concentration of the counterions in the vicinity of the surface is larger than their concentration in the bulk solution. This difference in concentration generates osmotic pressure which maintains the double layer. The van’t Hoff’s law of osmotic pressure is S o cRT

(2.19)

Properties of Colloid Dispersions

33

where c is the concentration of the solute in the solution, R is gas constant and T is the temperature. This relationship is very similar to the ideal gas law. Therefore, van’t Hoff stated that the osmotic pressure of a solution is the same as the solute would exert if it existed as a gas in the same volume as that occupied by the solution at the same temperature. It was also found by van’t Hoff that the dilute aqueous solutions of electrolytes such as NaCl showed considerable departure from the above law. The observed osmotic pressure of salt solutions was found to be much higher than that predicted from Eq. (2.19). To account for this deviation from ideality, van’t Hoff introduced a factor i, which is defined as the ratio of the observed and ideal osmotic pressures. Therefore, the modified van’t Hoff’s law is: po = icRT. The origin of van’t Hoff factor greater than unity is the dissociation of the electrolyte in solution. In dilute solution, the dissociation is almost complete. Thus, one NaCl molecule generates two ions in a dilute solution and the value of i approaches 2. Similarly, for H2SO4, the value of i approaches 3 in a dilute solution. The osmotic pressure of a non-electrolyte solution may be represented as So

È RT Ø 2 3 ÉÊ M ÙÚ c  B „c  C „c  "

(2.20)

where c is the concentration, M is the molecular weight of the solute, and B¢ and C¢ are constants related to the second and third virial coefficients respectively. In dilute solutions (small c), we can drop the terms beyond the second term on the right side of Eq. (2.20). Therefore, S o RT  B „c (2.21) c M Therefore, when po /c is plotted against c, a straight line is obtained. From the slope and intercept, we can calculate B¢ and M. The colloid dispersions are usually polydispersed. For such a system, the average molecular weight can be determined by osmometry. Let us assume that the dispersion behaves ideally. If the experimentally observed values of concentration and osmotic pressure are cexp and S oexp respectively, then S oexp cexp

RT M

(2.22)

where M is the average molecular weight. For each of the molecular weight fractions (e.g. the jth fraction), we have

S oj cj

RT Mj

(2.23)

The experimental osmotic pressure is the sum of the pressures exerted by each of the fractions. S oexp

Ç S oj

(2.24)

Also, the experimental concentration is the sum of the concentrations of the fractions.

cexp

Çcj

(2.25)

34

Colloid and Interface Science

Therefore, from Eqs. (2.22)–(2.25), we get

Ç cj

M

(2.26)

Çcj M j

The concentration cj is defined as nj M j

cj

(2.27)

Vd

From Eqs. (2.26) and (2.27), we obtain

Ç nj M j

M

(2.28)

Ç nj

Therefore, the average molecular weight determined from osmometry is the number-average molecular weight. EXAMPLE 2.4 The variation of osmotic pressure with the concentration of nitrocellulose in methanol is given below (Dobry, 1935). c (kg/m3)

0.7

1.9

6.7

So (mol/kg) RTc

0.011

0.012

0.015

12.1 0.021

Determine the molecular weight of nitrocellulose from these data. Solution

From Eq. (2.21), we can write So RTc

1 È B„ Ø  c M ÉÊ RT ÙÚ

The given data were plotted as shown in Figure 2.5. Intercept

\

1 M

0.0102 mol/kg

M = 98.04 kg/mol

Sometimes the molecular weight determined by osmometry of a colloid containing macroion is found to be significantly lower than that obtained by other methods. Since the macroion cannot pass through the membrane, its counterion(s) also do not pass through the membrane to maintain the electroneutrality of the solution. Osmotic pressure depends upon the number of solute particles. For a molecule that dissociates as Na+ and I–, the osmotic pressure is associated with two particles. If the colloidal electrolyte were A+z I z  , the osmotic pressure would be associated with (z + 1) particles. The observed molecular weight from osmometric measurement is the number-average molecular weight of the macroion and the ions dissociated from it. So, if we do not consider the presence of the counterions, the molecular weight calculated from Eq. (2.21) would be lower than the actual molecular weight.

Properties of Colloid Dispersions

Figure 2.5

35

Determination of molecular weight from osmometry.

It has been observed that the osmotic pressure of the solution of a substance containing the membrane-impermeable macroion is lowered if the other side of the membrane contains the solution of a salt that has the same counterion as that of the macroion. The reason for this behaviour was explained by Irish chemist Frederick Donnan in 1911. Suppose that the charge of the macroion Iz– is balanced by Na+ ions. When the solution of Iz– is kept separated from a NaCl solution by a semipermeable membrane, it is expected that only the Na+ and Cl– ions would diffuse through the membrane because the membrane acts as an impermeable barrier towards Iz–. This perturbs the concentration distributions of the small ions and gives rise to an ionic equilibrium which is different from the equilibrium that would result if Iz– were absent. The resulting equilibrium is known as Donnan equilibrium, which is very important for biological membranes. The following example illustrates how Donnan equilibrium influences the osmotic pressure. EXAMPLE 2.5 Suppose that a semi-permeable membrane separates an aqueous NaCl solution from an aqueous solution of an organic salt, Na+I–. All ions except I– are permeable through the membrane. Assuming that both NaI and NaCl are fully dissociated in the solutions, determine how the osmotic pressure of the solution of NaI will be affected by the presence of NaCl. Solution

The distribution of the ions in the two compartments is Initial state Na+

I–

Na+

Cl–

c1

c1

c2

c2

(1)

(2)

The chloride ions can permeate through the membrane from compartment 2 to compartment 1. However, the I– ions cannot transfer from compartment 1 to compartment 2. In order to maintain

36

Colloid and Interface Science

electrical neutrality, sodium ions must permeate through the membrane from compartment 2 to compartment 1. Equilibrium state Na+

I–

Cl–

(c1 + x) c1

x

Na+

Cl–

(c2 –x)

(c2 –x)

(1)

(2)

where c1, c2 and x are the concentrations of the ions. At equilibrium

[Na  ]1 [Cl  ]1 \

(c1  x ) x

\

x

[Na  ]2 [Cl  ]2

(c2  x )(c2  x )

c22 c1  2c2

The true osmotic pressure of NaI is given by the van’t Hoff equation po = 2c1RT The counter pressure caused by NaCl is p

2 RT (c2  x  x )

2 RT (c2  2 x )

The observed osmotic pressure, therefore, is

S oobs

So  p

2 RT [c1  (c2  2 x)]

2 RT (c1  c2  2 x )

Therefore, S oobs So

c1  c2  2 x c1

c1  c2 

2c22 c1  2c2

c1

c1  c2 c1  2c2

If c1 !! c2 , S oobs / S o  1. On the other hand, if c2 !! c1 , S oobs / S o  1 / 2. If c1 = c2, it is easy to see that S oobs / S o  2 / 3. Therefore, due to the unequal distribution, addition of an electrolyte with common counterion will reduce the observed osmotic pressure of the salt which is completely dissociated in the solution, but contains a non-permeating ion. The results of Example 2.5 show that a large amount of salt can ‘swamp out’ the effect associated with the macroion.

2.6

OPTICAL PROPERTIES

When a beam of light falls on a colloid dispersion, some of the light may be absorbed, some part may be scattered, and the remaining part is transmitted undisturbed through the sample. If light of certain wavelengths is selectively absorbed by the particles, the dispersion appears to be coloured. The wavelength of visible light lies between 400 nm (violet) and 700 nm (red). The colour of the

Properties of Colloid Dispersions

37

lyophobic colloids depends on the size of the particles as well as the distance between them. For example, gold sols have rich variety in their colour tones. A gold sol is red when the particles are very fine, but the sol is blue when the dispersed particles are bigger. When light impinges on matter, the electric field of the light induces an oscillating polarisation of the electrons in the molecules. The molecules then serve as secondary sources of light, and subsequently radiate light. The shifts in frequency, angular distribution, polarisation, and the intensity of the scattered light are determined by the size, shape and molecular interactions in the scattering material. The most ubiquitous manifestation of light scattering is observed when a beam of light passes through a dark room. The air-borne dust particles are responsible for this scattering. A photograph of light scattering is shown in Figure 2.6.

Figure 2.6

Natural light scattering.

The colloid particles show intense scattering of light. When a strong narrow beam is allowed to pass through a colloid system placed against a dark background, bright specks or flashes of light can be observed against the darkness when viewed from above through a microscope. The specks of light change continuously in the dark field. This proves the heterogeneous nature of the colloid dispersions, light-scattering ability of the colloid particles, and their continuous motion. This effect is known as Tyndall effect (named after the Irish physicist John Tyndall). The scattered beam is polarised and its intensity depends upon factors such as the position of the observer, properties of the system, and the wavelength of light. Quantitave studies of Tyndall effect and other optical properties of colloid dispersions have become possible with the ultramicroscope developed by Siedentopf and Zsigmondy. Particles smaller than about 200 nm cannot be seen directly in ordinary microscopes. However, these particles can scatter light which is visible in the ultramicroscope. The presence of particles as small as 5 – 10 nm can be discerned by ultramicroscope. The principle of the slit ultramicroscope is illustrated schematically in Figure 2.7. A narrow beam of parallel or slightly convergent light (from a powerful source such as an arc-lamp) is passed at right angles to the direction of the microscope through a cell on which the instrument is focused. If no particle is present in the cell, the field will appear completely dark. On the other hand, if the cell contains a colloidal dispersion, the particles will scatter light, some of which will pass vertically into the microscope. Each particle will appear as a small disc of light on a dark background.

38

Colloid and Interface Science

Figure 2.7 Slit ultramicroscope.

The light-scattering by colloid particles can be divided into three categories as described below. (i) Rayleigh scattering: In Rayleigh scattering, the particles are small so that they can act as point sources of scattered light. The size of the particles is much smaller than the wavelength of light. (ii) Debye scattering: In Debye scattering, the particles are relatively large, however, the refractive indices of the dispersed phase and the dispersion medium are similar. (iii) Mie scattering: In Mie scattering, the particles are relatively large (diameter/wavelength ratio is close to unity), and the refractive index of the dispersed phase is significantly different from the dispersion medium. Rayleigh developed the light-scattering theory by applying the electromagnetic theory of light to the scattering by small, non-absorbing spherical particles in a gaseous medium. According to the Rayleigh theory, when an electromagnetic wave falls on a small particle, oscillating dipoles are induced in the particle. The particle then serves as a secondary source for the emission of scattered radiation of the same wavelength as the incident light. According to the Rayleigh theory, the scattering intensity is proportional to 1/l4. Therefore, blue light is scattered more than red light. If the incident light is white, a colloid will appear to be blue if viewed at right angles to the incident beam, and red when viewed from end-on. This phenomenon is responsible for the blue colour of the sky and the yellowish-red colour of the rising and setting sun. Colloidal dispersions sometimes appear turbid due to the scattering of light. Solutions of some macromolecular materials may appear to be clear, but actually they are slightly turbid due to weak scattering of light. When light is incident on a perfectly homogeneous system, there is no scattering. Pure liquids, dust-free gases and true solutions can approach this limit of no-scattering. The turbidity of a material is defined by the expression I I0

exp ( W l)

(2.29)

where I0 is the intensity of the incident light beam, I is the intensity of the transmitted light beam, l is the length of the sample and t is the turbidity. Equation (2.29) is valid for a non-absorbing system.

Properties of Colloid Dispersions

39

The Lambert–Beer law describes the intensity of the transmitted light when only absorption takes place, but no scattering. I I0

exp ( H a l)

(2.30)

where ea is the absorbance of the material. In a system where both absorption and scattering are important, the following composite formula applies: I I0

exp > (H a  W )l @

(2.31)

The experimental value of extinction is the sum of ea and t. EXAMPLE 2.6 An aqueous dispersion taken in a test cell at 300 K and 1 atm pressure is found to absorb and scatter 91% of the incident monochromatic radiation at l = 0.42 mm. The path length is 10 cm. Calculate the total extinction coefficient. Solution

From Eq. (2.31),

I 0 exp > (H a  W )l @

I

Fraction of the incident radiation transmitted = 1 – 0.91 = 0.09 Path length, l = 10 cm = 0.1 m Therefore, I I0

\

2.6.1

0.09

exp > (H a  W ) – 0.1@

ea + t = 24.1 m–1

Determination of Molecular Weight by Light Scattering

In liquid dispersions, the scattering of light is due to fluctuations in the solvent density and fluctuations in the particle concentration. The total intensity of the scattered light in all directions is given by

Is

S Èi Ø I 0 Ô É s Ù (2S r 2 sin I dI ) Ê I0 Ú 0

(2.32)

The second term inside the integral in Eq. (2.32) represents an area on the surface of a sphere, where r is the radius of the sphere and f is the angle with the horizontal axis. From the theory of Rayleigh scattering in a solution, is/I0 is given by (see Appendix)

is I0

Kc(1  cos2 I ) r 2 (1 M  2 Bc)

(2.33)

where is is the intensity of the light scattered per unit volume of solution, M is the molecular weight of the solute, c is the concentration, B is the second virial coefficient and K is a constant. From

40

Colloid and Interface Science

Eqs. (2.32) and (2.33), we obtain

Is I0

S

Ë 2S Kc(1  cos2 I ) sin I dI Û Ü Ô ÌÌ 1 M  2 Bc ÝÜ 0Í

(2.34)

S

The value of the integral:

Ô sin I (1  cos

2

I ) dI is 8/3. Therefore, Eq. (2.34) becomes

0

Is I0

16S Kc 3(1 M  2 Bc)

Hc 1 M  2 Bc

(2.35)

where H œ 16S K 3 is a constant. It is related to the refractive index, its gradient and wavelength of light in the medium by the following equation: H

32S 3 nr2 (dnr dc)2

(2.36)

3N A O 4

where nr is the refractive index of the solution, dnr/dc is the refractive index gradient, l is the wavelength of light in the solution and NA is Avogadro’s number. The quantity Is/I0 is the turbidity, t. Therefore, from Eq. (2.35), we have Hc 1  2 Bc (2.37) W M Therefore, Eq. (2.37) predicts that the plot of Hc/t versus c should be a straight line. From the intercept, the molecular weight M can be calculated. Equation (2.37) is known as Debye equation. It was derived for dilute solutions of macromolecules. The advantage of the light-scattering method is that the measurements become easier as the particle size increases. For spherical particles, the Debye equation is applicable when the diameter of the particles is less than about 20 nm (for l » 500 nm). For asymmetric molecules, this limit is lower.

EXAMPLE 2.7 is given below.

The variation of Hc/t with concentration for poly(methyl methacrylate) in benzene

c (kg/m3) Hc/t (mol/kg)

2.5 0.0056

4.0 0.0061

6.5 0.0066

8.0 0.0072

10.0 0.0080

From these data, calculate the molecular weight of the polymer and the second virial coefficient. Solution The plot of Hc/t vs. c is shown in Figure 2.8. The data were fitted by a straight line. The intercept is 1 0.0048 mol/kg M \ M = 208.333 kg/mol Slope = 2B = 0.0003 mol m3 kg–2 \ B = 1.5 × 10–4 mol m3 kg–2

Properties of Colloid Dispersions

Figure 2.8

41

Determination of molecular weight from light scattering.

If the system is polydispersed, Eq. (2.37) is applicable for each molecular weight fraction. For dilute solutions, we can neglect the second term on the right side of Eq. (2.37). For the jth fraction, we have

Hc j

1 Mj

Wj

(2.38)

The experimentally-measured concentration, turbidity and the average molecular weight are correlated by Eq. (2.37) as

Hcexp W exp

1 M

(2.39)

cexp

Ç cj

(2.40)

W exp

ÇW j

(2.41)

where

and From Eqs. (2.38)–(2.41), we have

M

W exp

ÇW j

H Ç cj M j

Ç cj M j

Hcexp

H Ç cj

H Ç cj

Ç cj

(2.42)

Since cj = njMj/Vd, we have

M

Ç n j M 2j Ç nj M j

This molecular weight is known as weight-average molecular weight.

(2.43)

42

Colloid and Interface Science

2.6.2 Transmission Electron Microscopy (TEM) and Scanning Electron Microscopy (SEM) We have discussed at the beginning of Section 2.6 that the colloid particles are too small to be viewed in an optical microscope. The numerical aperture of an optical microscope is generally less than unity, which can be increased up to 1.5 with oil-immersion objectives. Therefore, for light of 600 nm wavelength, the resolution limit is of the order of 200 nm. To increase the resolving power of a microscope so that particles of colloidal dimensions can be observed directly, the wavelength of the radiation must be reduced considerably below that of visible light. Electron beams can be produced which have wavelengths of the order of 0.01 nm. These are focused by electric or magnetic fields, which act as the equivalent of lenses. A resolution of the order of 0.2 nm can be attained after smoothing the noise. Single atoms appear to be blurred irrespective of the resolution, owing to the rapid fluctuation of their location. The TEM can be used for measuring particle size between 1 nm and 5 mm. Due to the complexity of calculating the degree of magnification directly, calibration is done using pre-characterised polystyrene latex particles. The use of electron microscopy for studying colloid systems is limited by the fact that electrons can travel without any hindrance only in high vacuum. Therefore, the samples need to be dried before observation. A small amount of the sample is deposited on an electron-transparent plastic or carbon film (10–20 nm thick) supported on a fine copper mesh grid. The sample scatters electrons out of the field of view, and the final image can be viewed on a fluorescent screen. The amount of scattering depends on the thickness and the atomic number of the atoms of the sample. The organic materials are relatively transparent for electrons whereas, the heavy metals make ideal samples. To enhance contrast and obtain three-dimensional effect, various techniques (e.g. shadow-casting) are generally employed. In scanning electron microscopy, a fine beam of medium-energy electrons scans across the sample in a series of parallel tracks. These electrons interact with the sample to produce various types of signals such as secondary electron emission, back-scattered electrons, cathodoluminescence and X-rays. These are detected, displayed on a fluorescence screen and photographed. In the secondaryelectron-emission mode, the particles appear to be diffusely illuminated. Their size can be measured and the aggregation behaviour can be studied. In the back-scattered-electron mode the particles appear to be illuminated from a point source and the resulting shadows can provide good impressions of height. The resolution limit in SEM is about 5 nm, and the magnification achieved is generally less than that in a TEM. However, the depth of focus is large, which is important for studying the contours of solid surfaces, particle shape and orientation.

2.6.3

Dynamic Light Scattering (DLS)

If the light is coherent and monochromatic (e.g. a laser), it is possible to observe time-dependent fluctuations in the scattered intensity using a suitable detector such as a photomultiplier capable of operating in photon-counting mode. These fluctuations arise because the particles are small, and they undergo Brownian movement. The distance between them varies continuously. Constructive and destructive interference of light scattered by the neighbouring particles within the illuminated zone gives rise to the intensity fluctuation at the detector plane. From the analysis of the timedependence of the intensity fluctuation, it is possible to determine the diffusion coefficient of the particles. Then, by using the Stokes–Einstein equation [Eq. (2.18)], the hydrodynamic radius of the particles can be determined.

Properties of Colloid Dispersions

43

Dynamic light scattering is also known as photon correlation spectroscopy (PCS) or quasielastic light scattering (QELS). It is now a well-established technique for the measurement of the size distribution of proteins, polymers, micelles, carbohydrates, nanoparticles, colloidal dispersions, emulsions and microemulsions. An example of the size distribution measured by DLS was illustrated in Figure 1.10. The details of dynamic light scattering have been discussed by Berne and Pecora (2000).

2.6.4 Small-Angle Neutron Scattering (SANS) In this method, neutrons from reactors or accelerators are slowed down in a moderator to kinetic energies corresponding to room temperature or less. The wavelength probed by SANS is quite different from the visible light. A typical range of wavelength is 0.3–3 nm. This range is much smaller than that of visible light (400–700 nm). The usefulness of SANS to colloid and polymer science becomes evident when one considers the length scales and energy involved in neutron radiation. Light scattering is indispensable for studying particles having size in the micrometre range. For very small particles, neutrons are useful. The energy of a neutron with 0.1 nm wavelength is 1.3 × 10–20 J. Due to such low energy, neutron scattering is useful for sensitive materials. For colloid dispersions, we are interested in small scattering angle (~p/100 rad or even less), which corresponds to a small scattering vector. Light and X-rays are both scattered by the electrons surrounding atomic nuclei, but neutrons are scattered by the nucleus itself. There is, however, no systematic variation of the interaction with the atomic number. The isotopes of the same element can show significant difference in scattering. For example, neutrons can differentiate between hydrogen and deuterium. This has useful applications in biological science in the technique known as contrast matching. The interaction of neutrons with most substances is weak, and the absorption of neutrons by most materials is very small. Neutron radiation, therefore, can be very penetrating. Neutrons can be used to study the bulk properties of samples with path-length of several centimetres. They can also be used to study samples with somewhat shorter path-lengths but contained inside an apparatus (e.g. cryostat, furnace, pressure cell or shear apparatus). The SANS technique provides valuable information over a wide variety of scientific and technological applications such as chemical aggregation, defects in materials, surfactant assemblies, polymers, proteins, biological membranes, and viruses. A review of the experimental techniques using neutron scattering for biological structures has been presented by Teixeira et al. (2008).

2.7

ELECTRICAL PROPERTIES

The electrical properties of colloidal dispersions lead to some of the most important phenomena in colloid and interface science. In Chapter 1, we have briefly discussed the origin of electrical charge on colloid particles. The presence of electrostatic double layer surrounding the particles results in their mutual repulsion so that they do not approach each other closely enough to coagulate. An increase in the size of the particles by coagulation would lead to a decrease in total area, and hence to a decrease in free energy of the system. Therefore, union of colloid particles would be expected to occur, were it not for the repulsion caused by the electrostatic double layer. The stability of the charged colloid particles depends on the presence of electrolytes in the dispersion. The theory of electrostatic double layer will be discussed in details in Chapter 5.

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Colloid and Interface Science

The phenomena associated with the movement of charged particles through a continuous medium or with the movement of a continuous medium over a charged surface are known as electrokinetic phenomena. There are four major types of electrokinetic phenomena: electrophoresis, electroosmosis, streaming potential and sedimentation potential. There is a common origin for all the electrokinetic phenomena: the electrostatic double layer. Electrophoresis refers to the movement of particles relative to a stationary liquid under the influence of an applied electric field. If a dispersion of positively charged particles is subjected to an electric field, the particles move towards the cathode. Electrophoresis is perhaps the most important electrokinetic phenomenon. Three types of electrophoresis are usually used: microelectrophoresis, moving-boundary electrophoresis and zone electrophoresis. The details of these techniques have been presented by Shaw (1992), and Hiemenz and Rajagopalan (1997). Electroosmosis refers to the movement of the liquid of an electrolyte solution past a charged surface (e.g. a capillary tube or a porous plug) under the influence of an electric field. The pressure necessary to balance the electroosmotic flow is known as electroosmotic pressure. To understand how electroosmosis occurs, consider a glass capillary containing an aqueous electrolyte solution. The charge on the wall of the tube can develop from either the dissociation of the surface –SiOH groups or adsorption of the OH– ions on the wall. This charge is balanced by an equal and opposite charge in the solution (Figure 2.9). When the electric field is applied, the ions in the diffuse part of the double layer move towards one of the electrodes depending on their charge. The motion of these hydrated ions imparts a body-force on the liquid in the double layer. This force sets the liquid in motion.

Figure 2.9

Illustration of electroosmotic flow through a capillary.

Now, suppose that the electrolyte solution is forced to pass through a capillary under pressure applied from outside. An electrical potential is generated between the ends of the capillary. This is called streaming potential. The same phenomenon can be observed when the solution is forced through a porous medium. If a dispersion of charged particles is allowed to settle, the resulting motion of the particles causes the development of a potential difference between the upper and lower parts of the dispersion. It is known as Dorn effect and the potential is known as sedimentation potential. Therefore, the situations which give rise to streaming potential and sedimentation potential are opposite to those of electroosmosis and electrophoresis respectively. Electrophoresis is widely used in biochemical analysis for separation of proteins. The pioneering works of Arne Tiselius (Nobel Prize in Chemistry, 1948) on moving-boundary electrophoresis is very important in this regard. Another very important application of electrophoresis is electrodeposition. Electroosmosis has been used in many applications related to environmental pollution abatement. The surface charge of the colloid particles is expressed in terms of the zeta (z ) potential. It is the potential at the ‘surface of shear’. When a particle moves in an electric field, the liquid layer immediately adjacent to the particle moves with the same velocity as the surface, i.e. the relative

Properties of Colloid Dispersions

45

velocity between the particle and the fluid is zero at the surface. The boundary located at a very short distance from the surface at which the relative motion sets in is known as the surface of shear. The precise location of this surface cannot be determined, but it is presumed that it is located very close to the surface of the particle, may be a few molecular-diameters apart. The magnitude of z-potential provides an indication of the stability of the colloid system. Some values of z-potential at the critical coagulation concentration of a colloid system for different electrolytes are shown in Table 2.2. Table 2.2

z-potentials at the critical coagulation concentration (CCC) for positively charged Fe2O3 sol

Electrolyte

CCC (mol/m3)

z-potential at CCC (mV)

KCl CaSO4 K2CrO4 K3Fe(CN)6

100.00 6.60 6.50 0.65

33.7 32.5 32.5 30.2

It is observed that the values are approximately same for the different electrolytes. The pH of the medium strongly affects the z-potential. The details of z-potential will be discussed in Chapter 5. Let us consider the motion of a small spherical colloid particle moving with velocity u in an electric field E. In a dilute dispersion, the mobility is given by the Hückel equation

u E

2HH 0] , N Rs  0.1 3P

(2.44)

where k is the Debye-Hückel parameter, e is the dielectric constant of the medium, e0 is permittivity of the free space, m is the viscosity of the liquid and Rs is the radius of the sphere. The derivation of this equation has been presented in Chapter 5. For large values of kRs, the relationship between the electrophoretic mobility and z-potential is given by Smoluchowski equation (also known as Helmholtz-Smoluchowski equation)

u E

HH 0] , N Rs ! 100 P

(2.45)

Equation (2.45) is applicable for relatively high salt concentrations for which k is large. An experimental set-up used for the determination of z-potential of small air bubbles in protein solutions is shown in Figure 2.10 (Phianmongkhol and Varley, 2003). A 2 mm diameter cylindrical microelectrophoresis cell was used in their study. Air bubbles were produced by gently mixing the protein solution with air using a pipette. The protein solution, including the air bubbles, was then introduced into the electrophoresis cell using the pipette. Using this technique, a number of small air bubbles could be placed within the cell. Thereafter, the platinum electrodes were gently inserted into the electrophoresis cell. A voltage was applied across the electrodes. To minimise the polarisation effect, the voltage was periodically alternated (thus converting the anode to the cathode and vice versa) during the experiment. Movement of air bubbles (i.e. electrophoretic mobility) was observed in both directions under a microscope. The diameters of the air bubbles were determined by image analysis. The z-potential of air bubbles was calculated from the measured electrophoretic mobilities using the Smoluchowski equation.

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Colloid and Interface Science

Figure 2.10 Microelectrophoresis cell used to measure z-potential of air bubbles in protein solutions (Phianmongkhol and Varley, 2003) [reproduced by permission from Elsevier Ltd., © 2003].

So far we have discussed how the z-potential can be calculated for the two limiting conditions represented by Eqs. (2.44) and (2.45). However, there is a wide range of values of kRs in which these two equations are not applicable. The general electrophoresis equations have been developed for spherical and rod-shaped particles. For spherical particles, if the charge density is unaffected by the applied field and if the z-potential is small, Henry’s equation can be used for calculating the electrophoretic mobility in the intermediate range of kRs.

u E

2HH 0] f (N Rs ) 3P

(2.46)

where f(kRs) is known as Henry’s function. It is given by (Henry, 1931),

and

f (N Rs )

3 9 75  (N Rs ) 1  (N Rs ) 2  330(N Rs ) 3  ", N Rs ! 25 2 2 2

(2.47)

f (N Rs )

1 5 1 1 Ë 2 3 4 5Û Ì1  16 (N Rs )  48 (N Rs )  96 (N Rs )  96 (N Rs ) Ü Ì Ü , N R  25 N Rs s Ì Î1 Ü 1 exp ( t ) 4 6Þ dt Ü Ì  Ï (N Rs )  (N Rs ) ß exp(N Rs ) Ô 96 t à ÍÌ Ð 8 ÝÜ ‡

(2.48)

For kRs ® 0, f (kRs) ® 1, and Eq. (2.46) reduces to the Hückel equation [Eq. (2.44)]. On the other hand, if kRs ® ¥, f (kRs) ® 3/2, and Eq. (2.46) reduces to the Smoluchowski equation [Eq. (2.45)]. As Henry stated, “there is thus a hiatus in the course of computations”, and for values of kRs between 5 and 25 one needs an interpolation formula. To eliminate this inconvenience, Ohshima (1994) proposed the following simple expression for f(kRs):

f (N Rs ) 1 

1 Ë Û 2.5 2 Ì1  Ü Í N Rs ^1  2 exp( N Rs )` Ý

3

This equation is valid for any value of kRs with maximum relative error less than 1%.

(2.49)

Properties of Colloid Dispersions

47

The z-potential can also be determined from electroosmotic flow. When an electric field is applied across a capillary containing electrolyte solution, the double layer ions begin to migrate. After some time, a steady state is reached when the electrical and viscous forces balance one another, i.e. the force exerted on the medium by the ions is balanced by the force exerted by the medium on the ions. If the steady state volumetric flow rate in a fine capillary due to electroosmosis is V, then the z-potential is given by ]

VP , N Rc !! 1 HH 0 EA

(2.50)

where Rc is the radius and A is the cross-sectional area of the capillary. Therefore, by measuring the volumetric flow rate through the capillary, the z-potential can be determined. EXAMPLE 2.8 An aqueous solution of sodium chloride is placed inside a capillary in an electroosmosis apparatus and subjected to an electric field of 90 V/m. The electroosmotic velocity in the capillary is observed to be 5 mm/s. Calculate z-potential from these data. Solution The electroosmotic velocity is given by, V A

H

5 – 10 6 m/s

8.854 – 10 12 C2 J 1m 1 , P = 1×10 3 Pa s

78.5, H 0

Therefore, from Eq. (2.50), we have

]

VP HH 0 EA

5 – 10 6 – 1 – 10 3 78.5 – 8.854 – 10 12 – 90

0.08 V = 80 mV

The z-potential can be correlated with streaming potential as follows. A pressure difference (Dp) across a capillary is applied which sets the liquid in motion inside it. The charge of the double layer moves with the surrounding liquid generating an electric current, which is known as the streaming current. On the other hand, the charge transferred downstream generates an electric field in the opposite direction. After a short time, the two currents due to pressure gradient and reverse electric field balance each other. The streaming potential (Es) is the potential drop associated with this electric field. The following equation gives the z-potential (Hunter, 2005). ]

P k s È Es Ø , N Rc !! 1 HH 0 ÉÊ 'p ÙÚ

(2.51)

where ks is the conductivity of the electrolyte solution. This equation is valid for large values of kRc, where Rc is the radius of the capillary. Therefore, it is likely to give erroneous results when the concentration of salt is low (which would result in a low value of k). It was observed more than a century ago that an electric field develops during the settling of charged particles. This is known as sedimentation potential. Smoluchowski presented the first theoretical estimate of the magnitude of the field. For a dispersion of solid non-conducting spheres of radius Rs, immersed in an electrolyte solution of conductivity ks, dielectric constant e, and viscosity m, the sedimentation potential is predicted to be (Booth, 1954; Davies and Rideal, 1961)

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Colloid and Interface Science

Esed

4SHH 0]'U gnRs3 3 P ksVd

(2.52)

where n is the number of particles and Vd is the volume of dispersion. This equation is valid in those situations where the thickness of electrostatic double layer is small with respect to the radius of the particles (kRs >>1). It can be observed from Eq. (2.52) that the sedimentation potential is proportional to the amount of the dispersed phase. EXAMPLE 2.9 Calculate the sedimentation potential for 50 mm radius water drops in an oil (e = 2, m = 0.5 × 10–3 Pa s, ks = 1 × 10–9 W–1 m–1) if the z-potential is 25 mV. The density of the oil is 700 kg/m3 and the volume fraction of the dispersed aqueous phase is 0.05. Solution

e = 2, e0 = 8.854 × 10–12 C2 J–1 m–1, g = 9.8 m/s2 Rs = 50 × 10–6 m, m = 0.5 × 10–3 Pa s, ks = 1 × 10–9 W–1 m–1 Dr = 1000 – 700 = 300 kg/m3, z = 25 × 10–3 V Volume fraction of the aqueous phase

4S Rs3 n 3Vd

0.05

Therefore, from Eq. (2.52), the sedimentation potential is

Esed

4SHH 0]'U gnRs3 3P ksVd

(0.05) (2) (8.854 – 10 12 ) (25 – 10 3 ) (300) (9.8) (0.5 – 10 3 ) (1 – 10 9 )

130.2 V m

The settling of fine droplets can generate high sedimentation potentials. For example, the settling of water drops in the gasoline storage tanks can produce a very high sedimentation potential owing to the low conductivity of the oil phase, which can be dangerous. The value of sedimentation potential can be as high as 1000 V/m, even for a moderate value of the z-potential (e.g. 25 mV). If the diameter of the droplets is larger than 100 mm, they settle down completely and the sedimentation potential becomes zero. If they are smaller than 1 mm, the sedimentation potential gradient reduces the rate of settling and a haze of water drops floats in the electric field. The sedimentation velocity is reduced by the sedimentation potential gradient. If the steady state settling velocity (i.e. terminal velocity) of the particle is vt when the particle is uncharged, and vc is the velocity when the particle carries a surface charge, then for a single sedimenting particle vc

2 Ë 1 È HH 0] Ø Û Ì Ü , N Rs !! 1 vt 1  Ì P ks ÉÊ Rs ÙÚ Ü Í Ý

(2.53)

The experimental data agree with this equation within an order of magnitude (Tiselius, 1932). If Rs = 0.1 mm, z = 25 mV and ks = 1 × 10–4 W–1 m–1, it can be shown that the velocity of the particle in water can be reduced by 30%.

2.7.1 Reciprocal Relationships It can be noted from what we have discussed in this section that the cause and effect are interchanged in streaming potential and electroosmosis. Similarly, the situation in sedimentation potential is the opposite of electrophoresis. From Eq. (2.50), putting E = Ic/(Aks), we can write

Properties of Colloid Dispersions

HH 0] P ks

V Ic

49 (2.54)

From Eqs. (2.51) and (2.54), we can write Es 'p

V Ic

HH 0] P ks

(2.55)

The coupling of two different electrokinetic ratios, viz. Es/Dp and V/Ic through Eq. (2.55) is an example of the law of reciprocity of Lars Onsager (Nobel Prize in Chemistry, 1968).

2.7.2

z-potential Obtained from Different Techniques

The z-potential depends only on the properties of the phases in contact. Therefore, its value must be independent of the experimental method employed for its determination. Several scientists, however, have found that z-potential obtained from streaming potential or electroosmotic measurements is quite smaller than the value obtained from electrophoretic mobility or sedimentation potential measurements (Rastogi et al., 1981; Chowdiah et al., 1983; Peace and Elton, 1960). These discrepancies can be due to the influence of surface conductivity on streaming potential and electroosmotic flow. Inclusion of surface conduction modifies Eq. (2.51) to ]

P (ks  2 ksurf Rc ) È Es Ø ÉÊ 'p ÙÚ HH

(2.56)

0

where ksurf is the surface conductivity (which is measured per unit length of the perimeter of the capillary tube). Furthermore, if the condition kR >>1 (where R is the radius of sphere or the radius of capillary) is not satisfied, then the Smoluchowski equation can yield inaccurate values of z-potential. Two conditions must be satisfied to justify comparisons between the values of z-potential obtained by different electrokinetic experiments: (i) the effect of surface conductivity must be taken care of, unless it is negligible, and (ii) the surface of shear must divide comparable double layers in all the cases. Fulfilment of the second requirement depends upon the experimental procedure. For example, if the same capillary is used for electroosmosis and streaming potential studies, the second condition can be satisfied. On the other hand, the surfaces of a capillary and a migrating particle can be quite different. Sometimes, the surfaces are coated with a protein, and hence the characteristics of both surfaces are governed by the adsorbed protein. A comparison of z-potentials measured by the sedimentation potential and electrophoretic methods is shown in Figure 2.11 for dispersion of haematite in aqueous medium. It can be observed that these results compare well with each other.

2.8 2.8.1

SOME PROPERTIES OF LYOPHILIC SOLS Hofmeister Series

Small amounts of electrolytes diminish the z-potentials of lyophilic sols, but coagulation does not take place. If a large amount of electrolyte is added to a lyophilic system, the dispersed substance precipitates. This is known as salt-out, which is different from the coagulation observed with the

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Colloid and Interface Science

Figure 2.11

Variation of z-potential of 0.2 mm diameter haematite spheres with pH (Ozaki et al., 1999) [reproduced by permission from Elsevier Ltd., © 1999].

lyophobic colloids. It is believed that the stability of the lyophilic systems depends upon their charge and hydration. Large amounts of electrolytes dehydrate the particles in addition to reducing their zpotentials. It is generally accepted that salt-out depends on the nature of the ions. Based on experimental observations, a series has been created for anions and cations reflecting their ability to salt out. This series is known as Hofmeister series (Glasstone, 1986). Cations: Mg++ > Ca++ > Sr++ > Ba++ > Li+ > Na+ > K+ > Rb+ > Cs+ Anions: Citrate– – > Tartrate– – > SO4– – > Acetate– > Cl– > NO3– > ClO3– > I– > CNS– Salt-out of a hydrophilic sol frequently produces a liquid aggregate in place of a solid. This sometimes appears in the form of viscous drops. This phenomenon is known as coacervation and the droplets are known as coacervates. The coacervates are formed most readily when two hydrophilic sols carrying opposite charges (e.g. gelatin at pH < 4.7 and gum acacia sol) are mixed in suitable proportions. It is believed that the shells of tightly-bound water molecules surrounding the particles prevent them from coagulation. On the other hand, the electrostatic attraction holds a number of particles together. Redispersion of coacervates can be achieved by the addition of iodide or thiocyanate ions which favour hydration of the particles. Otherwise, ions of high valence can be added which would diminish the z-potential of either sol.

2.8.2

Viscosity of Lyophilic Sols

The lyophilic sols have significantly higher viscosity than the lyophobic sols. This is due to the extensive solvation of the dispersed-phase particles, which increases the resistance to flow. The viscosity of very dilute lyophilic sols (e.g. gamboge, mastic and protein) obeys the Einstein's equation of viscosity (discussed in Section 2.9.1). The addition of a small amount of an electrolyte to a lyophilic sol produces a marked decrease in viscosity. For example, KCl at 2 mol/m3 concentration can decrease the viscosity of an agar-agar sol by 20%. Experiments with salts containing ions of different valence have shown that the influence increases with the increase in the

Properties of Colloid Dispersions

51

valence of the counterion. Therefore, it can be surmised that the decrease in viscosity is associated with the double layer surrounding the colloid particles. A small amount of electrolyte cannot appreciably affect the solvation of the particles. Therefore, it is evident that the z-potential is an important factor in determining the viscosity of the dispersion. The role of electrical charge of the particles on the viscosity of the sol is known as electroviscous effect.

2.8.3 Gelation The cooling of a not-too-dilute lyophilic sol (e.g. gelatin or agar-agar) results in the formation of a gel. A gel can also be obtained by the addition of electrolytes under suitable conditions to lyophobic sols which exhibit some lyophilic character (e.g. silicic acid and ferric oxide). The gels do not differ fundamentally in their structure and properties from gelatinous precipitates. The gelation process is accompanied by a large increase in viscosity. Two types of gels can be distinguished: elastic and rigid gels. Partial dehydration of an elastic gel (e.g. gelatin) leads to the formation of an elastic solid. The original sol can be generated readily by the addition of water (and subsequent warming, if necessary). Rigid gels (e.g. silica gel) become glossy or powdery and lose their elasticity on drying. Furthermore, mere addition of water may not regenerate the sol. The walls of the capillaries formed upon the dehydration of elastic gels are supple, but they are stiff for the rigid gels. Dehydrated silica gel has a honeycomb structure with fine capillaries, which renders it a valuable adsorbing agent. The dehydration and rehydration of the partially dried gels is the most important distinction between the elastic and nonelastic gels.

2.8.4

Imbibition

A gel imbibes a considerable amount of liquid when placed in a suitable liquid. The volume of the liquid imbibed can be several times the volume of the original gel. Imbibition is thus accompanied by a large increase of volume. For this reason, this phenomenon is often called swelling. This can produce a large amount of pressure. For example, if dry gelatin is placed in a tight porous vessel and kept in water, swelling can break the vessel! However, the net volume decreases in the imbibition process. Sulphate, tartrate, citrate and acetate inhibit the swelling of gelatin and similar gels, and their inhibiting effect decreases in the sequence mentioned. On the other hand, chloride, chlorate, nitrate, bromide, iodide and thiocyanate favour the imbibition of water, and the effect of chloride is the least. For iodide solutions, the gel often disperses at the room temperature and forms the sol spontaneously. Otherwise, the swollen gel is warmed to form the sol. The Hofmeister series gives the order of temperature to which the gel must be heated in presence of the anions to convert it into sol. For this reason the term ‘lyotropic’ (which in Greek means change to liquid) is used for the series of anions.

2.8.5

Syneresis

Many gels exude small amounts of liquid on standing (e.g. concentrated silicic acid gels and dilute gels of gelatin and agar-agar). This phenomenon was first observed by Thomas Graham and he called it syneresis. Examples of syneresis are drainage of lymph from a contracting clot of blood, and collection of whey on the surface of yoghurt. Syneresis is believed to be due to the exudation of water held by the capillary forces between the heavily-hydrated particles constituting the framework of the gel.

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Colloid and Interface Science

2.8.6 Gold Number When a lyophilic sol is added to a lyophobic sol, the sensitivity of the latter towards electrolytes is altered. Some lyophilic sols enhance the stability of the lyophobic sols and render them less sensitive towards electrolytes. The protective effect of a lyophilic sol is different for different lyophobic sols. Zsigmondy set up a standard by which the protective capacity can be expressed. It is known as gold number. The gold number of a lyophilic sol is the amount (in mg) of the dispersed phase of the lyophilic sol (in dry condition) which ought to be added to a 10 cm3 standard gold sol (0.0053%– 0.0058%) to prevent it from turning to blue (from red) upon the addition of 1 cm3 of a 10% solution of sodium chloride. The change in colour reflects the onset of coagulation. Therefore, a low value of the gold number indicates good protective ability of the lyophilic colloid. Gold numbers of some lyophilic sols are presented in Table 2.3. Table 2.3 Sol Gelatin Haemoglobin Albumin Gum arabic Dextrin Potato starch

Gold numbers of some lyophilic sols Gold number 0.005–0.01 0.03–0.07 0.1–0.2 0.15–0.25 6–20 >25

Historically, gold numbers have some very important applications, although many of them have been replaced by more accurate analytical techniques. For example, albumins have higher gold numbers than globulins. Consequently, the changes occurring in blood plasma in tetanus, where the albumins are converted into globulins, can easily be detected by determining the gold number. The protective effect of lyophilic colloids has been attributed to a homogeneous encircling of the suspended particle by the particles of the protective colloid, forming a protective sheath. According to Zsigmondy, this theory is applicable to coarse particles only, not to the fine colloids such as the red gold sols. Zsigmondy observed that particles of gelatin did not form a sheath around the gold particles. However, one particle of gelatin adsorbed several gold particles. He, therefore, suggested that the stability was due to the mutual adsorption of the particles of hydrophobic and hydrophilic colloids. The union of the particles of the two colloids formed a complex which imparted the stability.

2.9

RHEOLOGICAL PROPERTIES OF COLLOID DISPERSIONS

The flow behaviour of the colloid dispersions is important in numerous applications in our daily life and in chemical industries. The particles present in the dispersion exert a significant influence on its flow properties (Quemada and Berli, 2002). Therefore, the colloid dispersions can display a wide range of rheological behaviour. Charged dispersions and sterically-stabilised colloids show elastic behaviour. If the interparticle interaction is not important, the behaviour of the dispersion resembles ordinary viscous liquids. Before we discuss the rheology of dispersions, let us discuss some of the elementary properties of Newtonian and non-Newtonian fluids. Most of the colloid dispersions exhibit these behaviours.

Properties of Colloid Dispersions

53

The rheological behaviour of fluids can be expressed by their profiles of shear stress (ts) versus shear rate (J) at a constant temperature and pressure (Figure 2.12). The simplest relation between shear stress and shear rate is Ws

PJ

(2.57)

Figure 2.12 Shear stress versus shear rate for Newtonian and some non-Newtonian fluids.

This is known as Newton's law of viscosity, which is depicted in Figure 2.12 by the straight line passing through the origin. The proportionality constant (m) is known as viscosity. Gases and many liquids (e.g. water) obey Newton's law of viscosity, and these fluids are known as Newtonian fluids. The flow behaviour of most colloidal dispersions, however, does not follow this simple law. The deviations occur mainly due to two factors: (i) interparticle hydrodynamic interactions, and (ii) colloidal forces (electrostatic or steric) between the particles. The hydrodynamic effects exist even for neutral particles when the concentration is large. The electrical and steric interactions can be important in dilute dispersions if the range of the forces is large enough. For a Newtonian fluid, the slope of the ts versus J curve is constant. Therefore, the viscosity does not vary with shear rate. However, for non-Newtonian fluids such a plot can be non-linear, and the slope varies from point to point. Therefore, the viscosity of such fluids depends upon the shear rate even at constant temperature and pressure. The non-Newtonian fluids can be divided into three categories: (i) Fluids whose W s – J behaviour is independent of time (ii) Fluids whose W s – J behaviour depends upon time, and (iii) Viscoelastic fluids. The non-Newtonian fluids shown in Figure 2.12 (i.e. Bingham plastic, pseudoplastic and dilatant) belong to the first category. Their W s – J behaviour does not depend on the history of the fluid, and a given sample of the material shows the same behaviour, no matter how long the shear stress has been applied. Examples of Bingham plastic fluid are clay dispersions, drilling mud and toothpaste. Polymer solutions and paper pulp in water are examples of pseudoplastic fluid. Dispersions of starch and sand are examples of dilatant fluid. On the other hand, the W s – J curves for the non-Newtonian fluids belonging to the second category depend upon how long the shear stress has been applied. These fluids can be divided into

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Colloid and Interface Science

two categories: (a) thixotropic and (b) rheopectic. In these colloid systems, there is a time-dependent aligning to match the induced flow. The thixotropic fluids show a decrease in viscosity with time under a constant applied shear stress. The classic example of thixotropic behaviour is displayed by flocculated sols of iron (III) oxide, alumina and bentonite clay gels, which can be ‘liquefied’ on shaking and ‘solidified’ on standing. Many paints show thixotropic behaviour. The behaviour of rheopectic fluids is opposite: the apparent viscosity increases with time by the application of shear stress. Certain clay suspensions and gypsum pastes, which thicken or solidify while shaken but sets slowly on standing are examples of rheopectic fluid. The relation between shear stress and shear rate for many non-Newtonian fluids can be expressed by the power-law model: Ws

k p (J )D

(2.58)

where a is called flow-behaviour index and kp is called consistency index. This model is also known as Ostwald–de Waele model. For a = 1 and kp = m, this model predicts Newtonian flow behaviour. Equation (2.58) can be written as KJ ,

Ws

K

k p (J )D 1

(2.59)

where h is known as apparent viscosity. For pseudoplastic fluids, a < 1, and the apparent viscosity decreases with the increase in the rate of shear. For dilatant fluids (a > 1), the apparent viscosity increases with the increase in the shear rate. The Bingham plastic fluid behaves like a solid until a minimum yield stress (ts0) is applied, and subsequently the shear stress varies linearly with shear rate. Ws

W s 0  P BJ

(2.60)

If the dispersion is such that the W  J behaviour is non-linear once the flow starts, the Hershel– Bulkley model can be used.

Ws

W s 0  PHJ E

(2.61)

Example 2.10: The data on the variation of viscosity with shear rate for a mixture of anionic surfactant N-dodecylglutamic acid (neutralised by L-lysine in the molar ratio 1:1) and nonionic surfactant tri(oxyethylene) monotetradecyl ether (C14EO3) are given below (Shrestha et al., 2008).

J (s 1 )

K (Pa s)

J (s 1 )

K (Pa s)

0.11 0.18 0.41 0.67 1.04 1.61 2.49 4.06

680.2 482.8 184.7 114.8 76.3 46.8 31.1 20.5

6.62 10.80 16.80 27.30 42.40 65.75 102.10

13.6 9.1 6.0 3.7 2.0 1.2 0.8

Determine kp and a from these data.

Properties of Colloid Dispersions

Solution:

From Eq. (2.59), we have K

\

55

ln(K )

k p (J )D 1

ln( k p )  (D  1) ln(J )

The plot of ln(h) versus ln (J) is shown in Figure 2.13. From the plot, the slope and the intercept are –0.98 and 4.4 respectively. Therefore, a = 0.02 and kp = 81.45 Pa sa.

Figure 2.13 Determination of a of the power-law model.

2.9.1

Einstein's Equation of Viscosity

Albert Einstein derived an equation for the viscosity of an infinitely-dilute dispersion of very small rigid spheres. His theory is based upon the following assumptions: (i) density and viscosity of the fluid are constant, (ii) the flow velocity is low, (iii) no slippage of the liquid at the surface of the spheres, and (iii) the spheres are large enough compared to the liquid molecules so that the liquid can be regarded as continuum. The Einstein’s equation is,

Pd P

1  2.5 F

(2.62)

where m is the viscosity of the pure liquid, md is the viscosity of the dispersion and c is the volume fraction of the spheres. Equation (2.62) has been found to agree well with many experimental results reported in the literature (Hiemenz and Rajagopalan, 1997). Deviation from Einstein’s equation can result from factors such as electroviscous effects, swelling, flocculation, and adsorption of water on the surface of the particles (Greenberg et al., 1965). For dispersions with higher volume fractions of the particles, Einstein’s equation can be extended as,

Pd P

1  2.5 F  k1 F 2  k2 F 3 "

(2.63)

Equation (2.63) gives a better fit to the data for dispersions which are not very dilute. Several attempts have been made to extend Einstein’s equation for concentrated dispersions (Mooney, 1951).

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Equation (2.62) can be written in terms of the molar volume of the colloid as

Psp

Pd  1 2.5 vm c P

(2.64)

where msp is the specific viscosity, vm is the molar volume and c is the concentration. The variations of msp with c for chromium laurate and chromium stearate are shown in Figure 2.14. The values of molar volume calculated from the slopes of the lines are 0.9 dm3/mol for chromium laurate and 2.75 dm3/mol for chromium stearate.

Figure 2.14

Variation of specific viscosity of the solution with the concentration of two chromium soaps at low concentrations (Mehrotra and Jain, 1994) [reproduced by permission from Elsevier Ltd., © 1994]. The soap solutions were prepared in a 4:1 volumetric mixture of benzene and dimethylformamide.

2.9.2 Mark–Houwink Equation for Polymer Solutions The polymer solutions have particle-size in the colloidal dimensions. Often there is considerable interaction between the solute (i.e. the macromolecule) and the solvent. The viscosity of polymer solutions is distinct because such solutions, even at a dilute concentration, can display high viscosity. This may be caused by the very extensive solvation of the solute molecules, which immobilises the bound liquid, or the long polymer molecules may be intertangling with each other as they move. Many polymers show strong propensity to imbibe the solvent. The average molecular weight of polymer can be determined by dilute solution viscometery. The Mark–Houwink equation correlates the viscosity-average molecular weight ( M v ) with the intrinsic viscosity of the solution [m] as,

[P] where the intrinsic viscosity [m] is defined as

K m ( Mv )a

(2.65)

Properties of Colloid Dispersions

[P]

Ë1 È P  P Ø Û lim Ì É d Ü c0 c Ê P ÙÚ Ý Í

57

(2.66)

The viscosity average molecular weight M v is defined as,

Mv

È 6 ni Mi1 a Ø É 6n M Ù Ê i i Ú

1a

(2.67)

The parameters Km and a are the characteristics of a particular polymer-solvent combination. In ordinary ‘good’ solvents, the constants Km and a are valid only within a rather limited range of molecular weight. The values of Km and a for many polymer systems are tabulated in the literature (Brandrup et al., 1999). To determine these two parameters, [m] is experimentally determined for several known molecular weight fractions of the polymer by dilute solution viscometry.

2.9.3

Deborah Number

We have noted before that the colloidal materials show a wide variety of rheological behaviour. Deborah number is a dimensionless number used to describe the influence of time on the observed flow properties. Whether a material deforms under applied stress or not depends upon the magnitude of the stress and the time of observation. Even some apparently-solid materials flow if they are observed long enough (e.g. silicone putty). In interacting dispersions, it is important to consider the time over which the flow behaviour is observed relative to the time scales over which the shear force alters the local structure of the dispersion. Deborah number is the ratio of the relaxation time of the material (tr) and the observation time (to). De

tr to

(2.68)

The difference between solids and liquids is defined by the magnitude of De. If the time of observation is very large, or, conversely, if the time of relaxation is very small, the material is seen to be flowing. On the other hand, if the time of relaxation of the material is larger than the time of observation, the material, for all practical purposes, can be considered a solid. This name was coined by Markus Reiner (of Israel Institute of Technology, Haifa) in 1964 from a biblical quote. Prophetess Deborah sang after the victory over the Philistines, “The mountains flowed before the Lord.” It symbolises the fact that the mountains flow, as everything flows, and they flowed before the Lord (not before man) for the simple reason that man in his short lifetime cannot see them flowing, but the time of observation of the Lord is infinite.

2.9.4

Peclet Number

The Peclet number is another very important dimensionless number for the colloid dispersions. It compares the effect of shear with the effect of diffusion of the particles. In a quiescent colloidal dispersion, the particles move randomly due to Brownian motion. An equilibrium statistical distribution of the particles is established, which depends on the factors such as the volume fraction occupied by the particles and the force of interaction between the particles. Now, if a shear is imposed on the dispersion, the distribution of the particles is altered. The Brownian motion of the

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particles tries to restore the equilibrium structure. The time taken by a particle to diffuse a distance equal to its radius Rs is of the order of Rs2 / D, where D is the diffusion coefficient. This is the timescale for diffusion (td). The time-scale for shear flow (ts) is J 1 , where J is the shear rate. The Peclet number is defined as

Pe

td ts

Rs2 D 1 J

Rs2J D

(2.69)

This ratio specifies the relative importance of shear and diffusion. If the Peclet number is small, the Brownian motion dominates and the behaviour of the particles is dominated by the diffusional relaxation of the particles. On the other hand, if the Peclet number is large, the shear-effect dominates the distribution.

SUMMARY This chapter presents an overview of the various properties of colloid dispersions. The kinetic, optical and electrical properties of colloids are discussed. The chapter begins with sedimentation of uncharged colloid particles in a gravity field. The forces which act on a settling particle are discussed. The expression for the terminal settling velocity is derived. The effect of the neighbour particles in a concentrated dispersion is explained. Next, sedimentation in a centrifugal field is discussed. The use of ultracentrifuge for settling colloidal particles from a dispersion is illustrated. In the next section, Brownian motion is discussed. The Einstein–Smoluchowski and Stokes– Einstein equations are discussed. Calculation of Avogadro's number from Brownian motion is explained. In the next section, osmosis and osmotic pressure are discussed. Calculation of molecular weight of macromolecular colloids by osmometry is explained. The Donnan effect is discussed with an example. Next, the scattering of light by colloid particles is discussed. Calculation of turbidity of a dispersion is explained. The Tyndall effect and ultramicroscope are discussed. Determination of molecular weight by light scattering is explained with an example. The use of transmission electron microscopy and scanning electron microscopy are discussed. This was followed by a discussion on dynamic light scattering and small-angle neutron scattering. In the next section, the four electrokinetic phenomena: electrophoresis, electroosmosis, streaming potential and sedimentation potential are discussed. The Hückel and Smoluchowski equations for electrophoresis are presented. The methods of calculation of z-potential by these four techniques are discussed. The precautions to be taken while comparing the values of z-potentials obtained from different techniques are explained. In the next section, some special properties of the lyophilic sols are discussed. These include the Hofmeister series, electroviscous effect, gelation, imbibition and syneresis. The application of gold number is explained. In the last section of this chapter, the rheological properties of colloid dispersions are discussed. A brief discussion on Newtonian and non-Newtonian fluids is presented. Einstein’s equation of viscosity is discussed with examples, and its limitations are also discussed. The Mark–Houwink equation correlating the viscosity-average molecular weight of polymer with the intrinsic viscosity of the solution is presented. Finally, the significance of Deborah number and Peclet number is explained.

Properties of Colloid Dispersions

KEYWORDS Absorbance Bingham plastic fluid Brownian motion Centrifugal sedimentation Coacervation Consistency index Deborah number Debye equation Debye scattering Diffusion coefficient Dilatant fluid Donnan effect Dorn effect Dynamic light scattering Einstein's equation of viscosity Einstein–Smoluchowski equation Electrokinetic phenomena Electroosmosis Electrophoresis Electroviscous effect Flow-behaviour index Free settling Gelation Gold number Gravitational sedimentation Hindered settling Hofmeister series Hückel equation Imbibition Lambert–Beer law Light scattering Lyotropic series Mark–Houwink equation Microelectrophoresis Mie scattering Molecular weight from light scattering Molecular weight from osmotic pressure Moving-boundary electrophoresis Newton's law of viscosity Newtonian fluid Non-Newtonian fluid Onsager's reciprocal relationship

Osmosis Osmotic pressure Ostwald–de Waele model Peclet number Power-law model Pseudoplastic fluid Rayleigh scattering Reverse osmosis Rheopectic fluid Salt-out Scanning electron microscopy Sedimentation Sedimentation coefficient Sedimentation equilibrium method Sedimentation potential Sedimentation velocity method Shear rate Shear stress Small-angle neutron scattering Smoluchowski equation Stokes’ law Stokes–Einstein equation Streaming current Streaming potential Surface of shear Swelling Syneresis Terminal velocity Thixotropic fluid Transmission electron microscopy Turbidity Tyndall effect Ultracentrifuge Ultramicroscope van't Hoff equation van't Hoff factor Viscoelastic fluid Viscosity Viscosity-average molecular weight Zeta potential Zone electrophoresis

59

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NOTATION a A B B¢ c CCMC C¢ d D De E Es Esed Fb Fd Fg g H i is I I0 Ic Is k k1, k2 kp ks ksurf K K1 K2 K3 Km l m M M Mv n nr NA p Pe r

constant in Eq. (2.65) cross-sectional area, m2 second virial coefficient, mol m3 kg–2 constant in Eq. (2.20), N m4 kg–2 concentration, kg/m3 (or mol/m3) critical micelle concentration, mol/m3 constant in Eq. (2.20), N m7 kg–3 diameter of the particle, m diffusion coefficient, m2/s Deborah number electric field, V/m streaming potential, V sedimentation potential, V/m buoyant force, N drag force, N gravity force, N acceleration due to gravity, m/s2 constant in Eq. (2.35), m2 kg–2 mol van’t Hoff factor intensity of the light scattered per unit volume of solution, W sr–1 m–3 intensity of the transmitted radiation, W/sr intensity of the incident radiation, W/sr electric current, A total intensity of the scattered light in all directions, W sr–1 m–1 Boltzmann’s constant, J/K constants in Eq. (2.63) consistency index, Pa sa conductivity of solution, W–1 m–1 surface conductivity, W–1 constant in Eq. (2.33), m2 kg–2 mol constant in Eq. (2.8) constant in Eq. (2.10), m3/kg constant in Eq. (2.11), m3/kg Mark–Houwink parameter, (m3/kg) (kg/mol)–a length of the sample, m mass of the particle, kg molecular weight, kg/mol average molecular weight, kg/mol viscosity-average molecular weight, kg/mol number of colloid particles refractive index Avogadro’s number, mol–1 pressure, Pa Peclet number radial distance, m

Properties of Colloid Dispersions

R Rep Rc Rs s t td to tr ts T u v vc vm vs vt V Vd Ãx2 Ó z

gas constant, J mol–1 K–1 particle Reynolds number radius of capillary, m radius of a spherical particle, m sedimentation coefficient, s time, s time-scale for diffusion, s observation time, s relaxation time, s time-scale for shear flow, s temperature, K velocity of the particle in electric field, m/s settling velocity of a particle, m/s settling velocity of a charged particle, m/s molar volume, m3/mol settling velocity in a concentrated dispersion, m/s terminal velocity, m/s volumetric flow rate, m3/s volume of dispersion, m3 mean square displacement, m2 valence

Greek Letters a b J Dp Dr e e0 ea ev z h k l m mB md mH msp [m] po r rd rp

flow-behaviour index index in Eq. (2.61) shear rate, s–1 pressure difference, N/m2 density difference, kg/m3 dielectric constant permittivity of free space, C2 J–1 m–1 absorbance, m–1 voidage of dispersion zeta potential, V apparent viscosity, Pa s Debye–Hückel parameter, m–1 wavelength, m viscosity of the continuous medium, Pa s constant in Eq. (2.60), Pa s viscosity of the dispersion, Pa s constant in Eq. (2.61), Pa s specific viscosity intrinsic viscosity, m3/kg (or m3/mol) osmotic pressure, Pa density of the liquid, kg/m3 density of the dispersion, kg/m3 density of the particle, kg/m3

61

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t ts ts0 f c w

turbidity of the dispersion, m–1 shear stress, Pa minimum yield stress, Pa angle with horizontal axis, rad volume fraction angular velocity, rad/s

EXERCISES 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. 29. 30. 31. 32. 33.

Mention the forces which act on a colloid particle when it settles in a liquid under gravity. What is terminal velocity? What is the difference between free settling and hindered settling? What is creaming? When does it occur? What is the advantage of centrifugal sedimentation over the gravity-induced sedimentation? Define sedimentation coefficient. Explain the principles of operation of an ultracentrifuge. What is Brownian movement? What type of particles exhibit such movement? Explain Einstein–Smoluchowski equation for Brownian movement. How does the diffusion coefficient of a colloid particle depend upon the viscosity of the liquid according to Stokes–Einstein equation? Explain how you would determine Avogadro’s number from Brownian motion. What is the origin of osmotic pressure? What is reverse osmosis? Explain how you will determine the molecular weight of a macromolecular colloid from osmotic pressure measurements. What is the difference between number-average molecular weight and weight-average molecular weight? Explain Donnan equilibrium. Explain why some colloids are coloured. What is the difference between Rayleigh scattering and Mie scattering? What type of light scattering would you expect in a pure liquid? What is Tyndall effect? Explain the main features of an ultramicroscope. Explain why the blue light is more scattered than the red light. Explain how the molecular weight of a macromolecule is related to the turbidity of its solution. What are the advantages and limitations of transmission electron microscopy? What are the different types of signals produced in a scanning electron microscope? Explain how dynamic light scattering can be used to measure the size of a colloid particle. For what type of colloids does dynamic light scattering have advantage over TEM or SEM? What are the advantages of neutron scattering over visible light scattering? What is small-angle neutron scattering? Where is it used? Explain what you understand by electrokinetic phenomena. What are the four major electrokinetic phenomena? Explain electrophoresis. What are the major uses of electrophoresis? What is z-potential? Explain its significance.

Properties of Colloid Dispersions

34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58.

63

Discuss the applicability criteria of Hückel and Smoluchowski equations. What is electroosmosis? How does it differ from osmosis? Explain how you would calculate the z-potential from electroosmosis. What is electroosmotic pressure? What is streaming potential? How is it developed? What is sedimentation potential? Explain Onsager’s reciprocal relationship. Explain why the z-potentials measured from the four electrokinetic phenomena sometimes differ in magnitude. What is salt-out? How does it differ from coagulation? Explain the significance of Hofmeister series. What is coacervation? Explain why the viscosity of lyophilic sols is higher than the viscosity of lyophobic sols. Why does the viscosity of a lyophilic sol reduce significantly when a small amount of salt is added to it? What is electroviscous effect? Explain the terms gelation and imbibition. Explain the significance of gold number. Explain the difference between a Newtonian fluid and a non-Newtonian fluid. What is the difference between pseudoplastic and dilatant fluids? What is the difference between a thixotropic fluid and a rheopectic fluid? Give one example each of (i) pseudoplastic, (ii) Bingham plastic, and (iii) dilatant fluids. Explain the power-law model. Explain Einstein’s equation of viscosity and its limitations. Explain Mark–Houwink equation. Explain the significance of Deborah number. What is the significance of Peclet number?

NUMERICAL PROBLEMS 2.1 Calculate the terminal velocity of a 10 mm diameter carbon tetrachloride drop in water. Given: density of carbon tetrachloride = 1600 kg/m3; viscosity of water = 1 mPa s. 2.2 A 1 mm diameter quartz particle (rp = 2650 kg/m3) suspended in water is placed in a centrifuge. The centrifuge rotates at a speed of 1000 rad/s. Calculate the sedimentation coefficient. What will be the radial position of the particle with respect to its initial position after operating the centrifuge for 60 s? 2.3 Calculate the diffusion coefficient of a reverse micelle of Span 80 of 5 nm diameter in benzene (m = 0.6 × 10–3 Pa s) at 300 K. What would be its RMS displacement based on Brownian motion after 60 s? Estimate the diffusion coefficient of the reverse micelle at 310 K. Given: viscosity of benzene at 310 K is 0.5 mPa s. 2.4 The osmotic pressure of a polystyrene solution in toluene at 298 K varies with concentration as shown below. c (kg/m3) po (cm of toluene)

0.2 0.04

0.4 0.09

0.6 0.16

0.8 0.22

0.9 0.28

Determine the molecular weight of the polymer from these data. Given: density of toluene at 298 K is 860 kg/m3.

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2.5 The variation of osmotic pressure with concentration in benzene (at 298 K) for a poly(isopropyl acrylate) fraction is given below (Mark et al., 1966). c (kg/dm3) (po/c)1/2 (cm)1/2

4.4 × 10–3 17.5

4.7 × 10–3 17.7

6.8 × 10–3 18.9

10.2 × 10–3 19.9

13.4 × 10–3 22.1

Determine the molecular weight from these data. 2.6 The light scattering data from the aqueous solutions of octyltrimethylammonium octanesulphonate are given below (Anacker, 1953). H (c  cCMC ) – 10 5 (kmol/kg) W 4.64 4.94 5.19 5.09 5.21 6.01

c – cCMC (kg/dm3) 9.32 4.16 5.84 1.02 1.14 2.04

2.7 2.8

2.9

2.10

2.11

× × × × × ×

10–4 10–3 10–3 10–2 10–2 10–2

where t is the turbidity in excess of that of the solvent, c is the concentration of the surfactant and cCMC is the critical micelle concentration. Calculate the micellar molecular weight from these data. Calculate the z-potential of a spherical particle of 10 nm diameter that shows a mobility of 1.3 × 10–8 m2 V–1 s–1 in water at 295 K. Given: m = 1 mPa s and e = 78.5. In a streaming potential experiment using an aqueous NaCl solution, application of 101.325 kPa pressure produces a streaming potential of 0.3 V. Calculate the z-potential. Given: ks = 0.01 W–1 m–1 and m = 1 × 10–3 Pa s. Calculate the electrophoretic mobility of a 50 nm diameter spherical particle in an aqueous solution of NaCl at 298 K. The z-potential is 20 mV. The concentration of NaCl in the solution is 0.1 mol/dm3 (Hint: The Debye length is about 1 nm at this concentration of the salt). Fine droplets of water (diameter = 30 mm) are settling in an oil (e = 1.5) tank. The volume fraction of water in the oil is 0.06. The density, viscosity and conductivity of the oil are 800 kg/m3, 0.7 mPa s and 1 × 10–10 W–1 m–1 respectively. If the z-potential is 30 mV, calculate the sedimentation potential. The shear stress versus shear rate data for a 3% (by volume) dispersion of colloidal silica Aerosil R805 (mean diameter = 32 nm) in polypropylene triol are given below (Michel et al., 2003) at 298 K.

J (s1 ) 1 3 1 3 1 3 1

× × × × × × ×

10–4 10–4 10–3 10–3 10–2 10–2 10–1

ts (Pa) 9.4 10.6 11.6 12.6 14.8 16.8 19.8

Fit a suitable rheological model to these data.

J (s1 ) 3×

10–1

1 3 10 30 100

ts (Pa) 24.3 31.1 41.6 65.5 107.5 216.9

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Properties of Colloid Dispersions

2.12 The specific viscosity of a dispersion of glass spheres (diameter = 160 mm) in water varies with the volume fraction of the spheres as given below (Eirich et al., 1936). c msp

0.01 0.02

0.02 0.04

0.03 0.07

0.04 0.08

0.06 0.14

0.08 0.19

0.10 0.24

Verify whether the data obey Einstein’s equation of viscosity or not. 2.13 Kose and Hachisu (1974) have measured the viscosity of a dispersion of spheres of poly(methyl methacrylate) cross-linked with divinylbenzene in benzene. The variation of relative viscosity with the volume fraction of the spheres (with 5% cross-linking) is given below. c md /m

0.042 1.11

0.053 1.14

0.070 1.22

0.084 1.28

0.105 1.36

0.137 1.59

0.168 1.77

0.209 2.16

It has been suggested that md/m can be correlated with c by Eq. (2.63). If so, use the polynomial up to the term containing c3 and determine the parameters k1 and k2. Discuss the importance of swelling of the particles. 2.14 The Mark–Houwink parameters for poly(methyl methacrylate) in benzene at 303 K are Km = 5.2 × 10–3 (dm3/kg)(kmol/kg)a and a = 0.76 in the molecular weight range 6 × 104–2.5 × 106 kg/kmol (Brandrup et al., 1999). The molecular weights of the polymer samples collected during its bulk polymerisation carried out at 323 K in a Parr® reactor at different reaction times are given below (Ghosh et al., 1998). t (ks) M v (kg/kmol)

5.7 6.4 ×

11.3

105

8.6 ×

14.4

105

9.2 ×

105

15.6 1.5 × 106

Calculate the intrinsic viscosities of the polymer solutions. 2.15 The intrinsic viscosities (at 298 K) of eleven molecular weight fractions of cellulose nitrate dissolved in acetone are given below (Holtzer et al., 1954). Mol. wt. (kg/kmol) 77000 89000 273000 360000 400000 640000

[m] (dm3/kg) 123 145 354 550 650 1060

Mol. wt. (kg/kmol) 846000 1270000 1550000 2510000 2640000

Determine the Mark–Houwink parameters from these data.

[m] (dm3/kg) 1490 2450 3030 3100 3630

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APPENDIX The Rayleigh scattering equation for a solution (at constant temperature) is expressed by (Hiemenz and Rajagopalan, 1997) 2

Ë È dn Ø Û 2S 2 Ì nr É r Ù Ü kTc Í Ê dc Ú Ý 1  cos2 I 2 4 È dS o Ø r O É Ê dc ÙÚ



is I0



(2A.1)

where is is the intensity of the light scattered per unit volume of solution. The gradient of refractive index of the solution nr is given by dnr/dc. The osmotic pressure gradient is given by dpo/dc. From Eq. (2.21), we have So

B„ 2 Ø È c RT É  c Ê M RT ÙÚ

È c Ø RT É  Bc 2 Ù , ÊM Ú

B

B„ RT

(2A.2)

Therefore, dS o dc

È 1 Ø RT É  2 Bc Ù ÊM Ú

(2A.3)

From Eq. (2A.1) 2

Ë È dn Ø Û 2S Ì nr É r Ù Ü c Í Ê dc Ú Ý (1  cos2 I ), k Ø 2 4È 1 N Ar O É  2 Bc Ù ÊM Ú 2

is I0

Let

K

Ë È dn Ø Û 2S 2 Ì nr É r Ù Ü Í Ê dc Ú Ý N AO 4

R /N A

(2A.4)

2

(2A.5)

Therefore, from Eq. (2A.4), we get is I0

Kc(1  cos 2 I ) È 1 Ø r 2 É  2 Bc Ù ÊM Ú

(2A.6)

FURTHER READING Books Berne, B.J. and R. Pecora, Dynamic Light Scattering, Dover, New York, 2000. Brandrup, J., E.H. Immergut, and E.A. Grulke (Eds.), Polymer Handbook, John Wiley, New York, 1999.

Properties of Colloid Dispersions

67

Chhabra, R.P., Bubbles, Drops, and Particles in Non-Newtonian Fluids, CRC Press, Boca Raton, 1993. Cussler, E.L., Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, New York, 1997. Davies, J.T. and E.K. Rideal, Interfacial Phenomena, Academic Press, London, 1961. Einstein, A., Investigations in the Theory of the Brownian Movement, Dover, New York, 1956. Evans, D.F. and H. Wennerström, The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet, VCH, New York, 1994. Glasstone, S., Textbook of Physical Chemistry, Macmillan, Delhi, 1986. Hiemenz, P.C. and R. Rajagopalan, Principles of Colloid and Surface Chemistry, Marcel Dekker, New York, 1997. Hunter, R.J., Foundations of Colloid Science, Oxford University Press, New York, 2005. Perrin, J., Atoms, Ox Bow Press, Woodbridge, 1990. Richardson, J.F., J.H. Harker, and J.R. Backhurst, Coulson & Richardson's Chemical Engineering, (Vol. 2), Elsevier, New Delhi, 2003. Shaw, D.J., Introduction to Colloid and Surface Chemistry, Butterworth-Heinemann, Oxford, 1992. Van Kampen, N.G., Stochastic Processes in Physics and Chemistry, Elsevier, New York, 1981.

Articles Anacker, E.W., “Light Scattering by Solutions of Octyltrimethylammonium Octanesulfonate and Octyltrimethylammonium Decanesulfonate”, J. Coll. Sci., 8, 402 (1953). Booth, F., “Sedimentation Potential and Velocity of Solid Spherical Particles”, J. Chem. Phys., 22, 1956 (1954). Chowdiah, P., D.T. Wasan, and D. Gidaspow, “On the Interpretation of Streaming Potential Data in Nonaqueous Media”, Coll. Surf. 7, 291 (1983). Davies, C.N., “The Sedimentation and Diffusion of Small Particles”, Proc. Roy. Soc., A200, 100 (1949). Dobry, A., “Osmotic Pressures of Solutions of Nitrocellulose”, J. Chim. Phys., 32, 50 (1935). Donnan, F.G., “Theory of Membrane Equilibria and Membrane Potentials in the Presence of NonDialysing Electrolytes. A Contribution to Physical–Chemical Physiology”, J. Membr. Sci., 100, 45 (1995). Eirich, F., M. Bunzl, and H. Margaretha, “The Viscosity of Suspensions and Solutions. IV. The Viscosity of Suspensions of Spheres”, Kolloid Z., 74, 275 (1936). Ghosh, P., S.K. Gupta, and D.N. Saraf, “An Experimental Study on Bulk and Solution Polymerization of Methyl Methacrylate with Responses to Step Changes in Temperature”, Chem. Eng. J., 70, 25 (1998). Greenberg, S.A., R. Jarnutowski, and T.N. Chang, “The Behavior of Polysilicic Acid: II. The Rheology of Silica Suspensions”, J. Coll. Sci., 20, 20 (1965). Henry, D.C., “The Cataphoresis of Suspended Particles: I. The Equation of Cataphoresis”, Proc. Roy. Soc. A, 133, 106 (1931). Holtzer, A.M., H. Benoit, and P. Doty, “The Molecular Configuration and Hydrodynamic Behavior of Cellulose Trinitrate”, J. Phys. Chem., 58, 624 (1954).

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Kose, A. and S. Hachisu, “Kirkwood–Alder Transition in Monodisperse Latexes: I. Nonaqueous Systems”, J. Coll. Int. Sci., 46, 460 (1974). Mark, J.E., R.A. Wessling, and R.E. Hughes, “Stereoregularity in Poly(isopropyl acrylate): I. Chain Dimension by Viscometry and Osmometry”, J. Phys. Chem., 70, 1895 (1966). Mehrotra, K.N. and M. Jain, “Viscometric and Spectrophotometric Studies of Chromium Soaps in a Benzene-Dimethylformamide Mixture”, Coll. Surf. (A), 85, 75 (1994). Michel, F.S., F. Pignon, and A. Magnin, “Fractal Behavior and Scaling Law of Hydrophobic Silica in Polyol”, J. Coll. Int. Sci., 267, 314 (2003). Mooney, M., “The Viscosity of a Concentrated Suspension of Spherical Particles”, J. Coll. Sci., 6, 162 (1951). Nollet, J.A., “Investigations on the Causes for the Ebullition of Liquids”, J. Membr. Sci., 100, 1 (1995). Ohshima, H., “A Simple Expression for Henry’s Function for the Retardation Effect in Electrophoresis of Spherical Colloidal Particles”, J. Coll. Int. Sci., 168, 269 (1994). Onsager, L., “Reciprocal Relations in Irreversible Processes I.”, Phys. Rev., 37, 405 (1931). Ozaki, M., T. Ando, and K. Mizuno, “A New Method for the Measurement of Sedimentation Potential: Rotating Column Method”, Coll. Surf. (A), 159, 477 (1999). Peace, J.B. and G.A.H. Elton, “Sedimentation Potentials: Part II. The Determination of the Zeta Potentials of Some Solid Surfaces in Aqueous Media, by Use of Sedimentation Potential Measurements”, J. Chem. Soc., 2186 (1960). Phianmongkhol, A. and J. Varley, “z-Potential Measurement for Air Bubbles in Protein Solutions”, J. Coll. Int. Sci., 260, 332 (2003). Quemada, D. and C. Berli, “Energy of Interaction in Colloids and its Implications in Rheological Modeling”, Adv. Coll. Int. Sci., 98, 51 (2002). Rastogi, R.P., R. Shabd, and B.M. Upadhyay, “Electrokinetic Studies on Estrone/Aqueous Solutions of Nonelectrolyte and Electrolyte Systems”, J. Coll. Int. Sci., 83, 41 (1981). Reiner, M., “The Deborah Number”, Physics Today, 17, 62 (1964). Robinson, C.S., “Some Factors Influencing Sedimentation”, J. Ind. Eng. Chem., 18, 869 (1926). Shrestha, R.G., L.K. Shrestha, and K. Aramaki, “Worm-Like Micelles in Mixed Amino Acid Based Anionic/Nonionic Surfactant Systems”, J. Coll. Int. Sci., 322, 596 (2008). Smoluchowski, M., “Zur Kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen”, Ann. Phys., 21, 756 (1906). Steinour, H.H., “Rate of Sedimentation. Non-Flocculated Suspensions of Uniform Spheres”, J. Ind. Eng. Chem., 36, 618 (1944). Tartar, H.V. and A.L.M. Lelong, “Micellar Molecular Weights of Some Paraffin Chain Salts by Light Scattering”, J. Phys. Chem., 59, 1185 (1955). Teixeira, S.C.M. et al., “New Sources and Instrumentation for Neutrons in Biology”, Chem. Phys., 345, 133 (2008). Tiselius, A., “The Effect of Charge on the Sedimentation Velocity of Colloids, Especially in the Ultracentrifuge”, Kolloid Z., 59, 306 (1932).

3 Surfactants and their Properties

James William McBain (1882 – 1953)

James McBain was born in Chatham, New Brunswick (Canada). He studied at the University of Toronto, graduating with B.A. degree in 1903, and M.A. degree in 1904. He received Ph.D. degree from the University of Heidelberg in 1906. McBain began his academic career at the University of Bristol, England, in 1906. From 1906 to 1919 he was lecturer in physical chemistry, and from 1919 to 1926 he was the first occupant of the Leverhulme Chair of Chemistry. McBain worked in many areas of colloid science. His most famous work was on surfactant micelles and adsorption. He was one of the major contributors who imparted the science stature to colloid science. Professor McBain was elected a fellow of the Royal Society in 1925. In 1939, he was awarded the Humphrey Davy Medal. In 1926 McBain became professor of chemistry at Stanford University, and he held this position until he became emeritus professor in 1947. He was the director of National Chemical Laboratory (Pune, India) during 1950– 1952. McBain died in California.

TOPICS COVERED © © © © © © © © ©

Classification and properties of surfactants Micellisation of surfactants Critical micelle concentration Properties of micelles Thermodynamics of micellisation Krafft point and cloud point Liquid crystals Hydrophobic–lipophilic balance Emulsions, microemulsions and foams 69

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INTRODUCTION

A large number of phenomena at gas–liquid and liquid–liquid interfaces are controlled by a special class of compounds known as surface-active agents or surfactants. These compounds have the unique property that they tend to adsorb at the interface. A surfactant is added in rather small quantities to the system. Nonetheless, they alter the interfacial free energy significantly. When surfactant molecules adsorb at the interface, a significant reduction in the tension results in most cases. Inorganic electrolytes can strongly influence the adsorption of certain surfactants. Various electrochemical phenomena are responsible for these effects, which are discussed in this chapter and later chapters. The traditional use of surfactants has been in household products, coating processes, petroleum industries, and chemical reactions such as emulsion polymerisation. In addition to these important applications, surfactants have found widespread use in various applications mentioned in Table 1.1. At present, novel techniques are being used to design new surfactants with specific chemical, biological, electrochemical and photochemical properties. Theoretical understanding of the surface activity has also progressed significantly in the past decade, with high-speed computer simulations. The present research activities on surfactants are driven by the enormous technological opportunity that exists today, which did not exist in the past decade. At the same time, it cannot be concluded that everything is known about the traditional and much-used surfactants. In fact, some extensivelystudied conventional surfactants are still revealing unexpected properties in multisurfactant systems, electrolyte solutions and concentrated surfactant solutions. In this chapter, a discussion on various surfactants and their properties will be presented. The theory of formation of micelles and their structures will be discussed. The critical micelle concentration and the factors influencing it will be explained. A brief introduction to the use of surfactants in emulsions, microemulsions and foams will be given.

3.2

SURFACTANTS AND THEIR PROPERTIES

As mentioned in Section 3.1, a wide range of interfacial applications are controlled by the surfactants. A surfactant molecule has two parts: one part is soluble in the solvent, and the other part is insoluble. The part of a surfactant molecule that has unfavourable interaction with the solvent is known as the lyophobic part. On the other hand, the part which has favourable interaction with the solvent is called the lyophilic part. For example, consider a sodium stearate molecule [molecular formula: CH3(CH2)16COO– Na+]. In water, the lyophobic part is the hydrocarbon chain, C17H35–, whereas the lyophilic part is –COO– . This type of structure is known as amphipathic structure. The part of a surfactant molecule that is lyophilic in a polar solvent such as water, may be lyophobic in a hydrocarbon solvent such as cyclohexane. Therefore, a surfactant that is useful in water may not be effective in another solvent. In fact, it may not dissolve in another solvent. When the solvent is water, the lyophobic and lyophilic parts are called hydrophobic and hydrophilic parts respectively (see Figure 3.1). Examples of hydrophilic head-groups are –COO– and –N(CH3)3+. The hydrocarbon tail can be made of straight-chain alkyl groups, branched-chain alkyl groups, long alkyl benzene chain, rosin derivative, alkene-oxide polymer, polysiloxane group or lignin derivative. If the hydrophobic group is very long, its solubility in water can be small. For example, let us consider the sodium salts of fatty acids, which are the main ingredients of soap. When the number of carbon atoms is less than 10, the soap is readily soluble in water (however, its surface activity would be low). As the number of carbon atoms increases (i.e. the hydrophobic tail becomes

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longer) the solubility decreases. When the number of carbon atoms reaches 20, the soap is almost insoluble in water.

Figure 3.1

Structure of a surfactant molecule.

When the surfactant molecules come in contact with water, they disrupt the hydrogen bonds between the water molecules. This increases the free energy of the system. Since this is thermodynamically unfavourable, the surfactant molecules are sent away towards the air–water interface. The air molecules are similar to the hydrophobic groups of the surfactant because they are essentially nonpolar. This encourages the surfactant molecules to adsorb at the air–water interface putting their hydrophobic tails in air, and hydrophilic head-groups in water. This is illustrated in Figure 3.2. If a sufficient number of surfactant molecules is present in the medium, the interface becomes covered with a closely-packed monolayer. The long hydrophobic groups favour formation of assemblies, known as micelles (see Section 3.2.7), and liquid crystalline phases (Section 3.2.10). If an aromatic ring is present in the hydrophobic part of the surfactant molecule, the surfactant gets a higher tendency to adsorb at certain interfaces. However, these surfactants have low biodegradability. They usually adsorb at the interfaces with loose packing.

Figure 3.2

Orientation of surfactant molecules at air–water interface: unfavourable and favourable configurations.

A common method of classification of surfactants is by the type of head-groups they possess. The elementary classification is made into four types: anionic, cationic, zwitterionic and nonionic. Some of their main features are discussed in Sections 3.2.1–3.2.11.

3.2.1 Anionic Surfactants The head-group of an anionic surfactant is negatively charged, which is electrically neutralised by an alkali metal cation. The soaps (RCOO– Na+), alkyl sulphates (RSO4– Na+) and alkyl benzene sulphonates (RC6H4SO3– Na+) are some of the well known examples of the anionic surfactants. These surfactants readily adsorb on the positively charged surfaces. The anionic surfactants are the most widely used surfactants in industrial practices. The linear alkyl benzene sulphonates have the highest consumption. Some of the anionic surfactants (e.g. salts of fatty acids) are precipitated from the aqueous solution in presence of salts containing Ca+2 and Al+3 ions. Therefore, their use may be restricted in certain media (e.g. hard water). The calcium and magnesium salts of alkyl benzene sulphonates are soluble in water. Therefore, they are much less sensitive to hard water.

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3.2.2 Cationic Surfactants The cationic surfactants are useful for adsorption on negatively charged surfaces. The head-group of a cationic surfactant has a positive charge. Some of the common uses of the cationic surfactants are in ore floatation, textile industries, pesticide applications, adhesion, corrosion inhibition and preparation of cosmetics. Most of the cationic surfactants have good stability in a wide range of pH. The relatively less use of cationic surfactants in industry is due to their rather poor detergency, lack of suspending power for carbon, and higher cost. Some well-known cationic surfactants are, long chain amines (RNH3+X–), quaternary ammonium salts [RN(CH3)3+X–] and quaternary salts of polyethylene oxide-amine derivatives [RN(CH3){(C2H4O)xH}2+Cl–].

3.2.3 Zwitterionic Surfactants These surfactants have both positive and negative charges on the surface-active part of the molecule. The long chain amino acids (RN+H2CH2COO–) are the well-known examples of the zwitterionic surfactants. Their main advantage is that they are compatible with both anionic and cationic surfactants due to the presence of both positive and negative charges. They are less irritating to eye or skin. Therefore, they find wide use in cosmetics. They are also used as fabric softeners and bactericides. Most of these surfactants are sensitive to pH. They show the properties of anionic surfactants at high pH whereas they behave as cationic surfactants at low pH. The sulphobetaine-type of surfactants [RN+(CH3)2(CH2)xSO3–] remain zwitterionic in a wide range of pH.

3.2.4 Nonionic Surfactants The nonionic surfactants are second most widely used surfactants in the industry. They do not have any significant electric charge on their surface-active part. Therefore, there is very little or no electrical interaction between the head-groups. These surfactants are stable in presence of electrolytes. They are compatible with most other types of surfactants. These surfactants disperse carbon well. Therefore, they have a large number of industrial uses. Most of the nonionic surfactants are available in the form of viscous liquids. They usually generate less foam than the ionic surfactants. Some nonionic surfactants are virtually insoluble in water, but soluble in organic solvents. However, some nonionics are soluble both in water and organic liquids, although the extent of solubility differs. The solubility depends on the structure of the surfactant molecules. The alkyl phenol ethoxylate [RC6H4(OC2H4)xOH] category of surfactants is widely used in emulsions, paints and cosmetics. The alcohol ethoxylates [R(OC2H4)xOH] are biodegradable. They are quite resistant to hard water. Therefore, in the applications involving saline media where the anionic surfactants are salted out of the solution, these surfactants find extensive use. The polyoxypropylene glycols are used in a wide range of molecular weights (e.g. 1000–30000). They are mainly used as dispersing agents for pigments in paints, foam-control agents and for removing scales of boilers. The polyoxyethylene mercaptants [RS(C2H4O)xH] are stable in hot and alkaline solutions. They are used in textile detergents, metal cleaning and shampoos. They are also used with quaternary ammonium-type of cationic surfactants to enhance the effectiveness of the latter. The long chain esters of carboxylic acids have very good emulsifying properties. However, they are unstable to acid and alkali, especially under hot conditions. The sorbitol esters are edible. They are used in food products such as ice creams, beverages, desserts and various confectionary products. These surfactants are also used in pharmaceutical products. The alkanolamines have good stability in the alkaline media. They are

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mainly used as laundry detergents, thickeners for liquid detergents, shampoos, rust inhibitors and fuel oil additives. The polyoxyethylene silicones are used as wetting agents. Apart from the classification of surfactants based on the type of charge present on the surfaceactive part of the molecule, the surfactants are also classified based upon their origin, special structural features, or properties in solution. In the following sections, gemini surfactants and biosurfactants are discussed with examples. The surfactants belonging to these categories can be any of the anionic, cationic, zwitterionic or nonionic types.

3.2.5 Gemini Surfactants These surfactants belong to a relatively new class of surfactants as compared to the conventional surfactants discussed before. They have two or three hydrophobic, and usually two hydrophilic groups (Figure 3.3). The hydrophobic groups are connected by a linkage that is close to the hydrophilic groups. The properties of these surfactants vary greatly depending upon the structure of these three parts of the molecule. The interfacial effects of these surfactants may be much stronger than the surfactants having a single hydrophilic and hydrophobic group. The gemini surfactants can have negative, positive or both types of charges. They can be Figure 3.3 Structure of a gemini surfactant. nonionic as well. Since these surfactants have a large number of carbon atoms in their hydrophobic part, they have a penchant for adsorbing at the interface. However, at the same time, their solubility in water may be less. The hydrophilic groups prevent this difficulty. These surfactants require only a small amount to saturate the interface.

3.2.6

Biosurfactants

Petrochemicals are often the first choice of the surfactant manufacturers because the surfactants can be produced at low cost, the raw materials are easily available, and the performance of the surfactants is good. However, the major problem with many petrochemical-based surfactants is that they are not easily biodegradable. Sometimes toxic by-products are formed during the manufacture of these surfactants. To avoid such problems, the biosurfactants are good alternatives. The biosurfactants are made of biological components such as carbohydrates. In recent times, they have found widespread use in petroleum engineering (such as oil recovery), food industries, pharmaceuticals, cosmetics and environmental pollution abatement. The main advantages of biosurfactants over the petroleum-based surfactants are their lower toxicity, biodegradable nature, and effectiveness at low as well as high temperatures. They are also usable in a wide range of pH and salinity of the medium. The lowmolecular-weight biosurfactants are glycolipids. On the other hand, the high-molecular weight surfactants are generally either polyanionic heteropolysaccharides or complexes containing both polysaccharides and proteins. Generally, biosurfactants are microbial metabolites. A variety of microorganisms produce biosurfactants. Among the microbes, bacteria produce a majority of biosurfactants. The major types of biosurfactants produced by microorganisms are presented in Table 3.1.

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Table 3.1 Biosurfactants produced by microorganisms 1. Glycolipids • Trehalose mycolates and esters • Mycolates of mono-, di-, and trisaccharide • Sophorolipids • Rhamnolipids 2. Fatty acids 3. Phospholipids 4. Lipopeptides and lipoproteins • Gramicidens • Polymyxins • Ornithine–lipid • Cerilipin • Lysin–lipid • Surfactin, subtilysin • Peptide–lipid 5. Polymeric surfactants • Lipoheteropolysaccharide • Heteropolysaccharide • Polysaccharide-protein • Manno-protein • Carbohydrate–protein • Mannan–lipid complex • Mannose/erythrose–lipid • Carbohydrate–protein–lipid complex 6. Particulate biosurfactants • Membrane vesicles • Fimbriae • Whole cells

The protein-based surfactants have good prospects in the pharmaceutical formulations and personal-care products, where safety, mildness to skin, surface activity, antimicrobial activity and biodegradability are required. Peptide or amino acid constitutes the hydrophilic part. The hydrophobic part is a long-chain acyl, amide, alkyl or an ether linkage. The natural proteins are not used as commercial surfactants. They are modified by chemical or enzymatic means to develop strong surface-active properties.

3.2.7 Micellisation of Surfactants We have already discussed one fundamental property of the surfactants, i.e. to adsorb at the interfaces. Surfactants have another very important property—the property of self-assembly. When sufficient amount of a surfactant is dissolved in water, the surfactant molecules form colloidal clusters. For many ionic surfactants, typically 40–100 surfactant molecules assemble to form such clusters. These are called micelles, and the process of the formation of micelles is known as micellisation. The number of surfactant molecules in a micelle is known as aggregation number. These clusters can have a wide variety of shape, such as spherical, cylindrical and lamellar, as shown in Figure 3.4. The threshold concentration at which the formation of micelle begins is known as critical micelle concentration (CMC). The CMC of some surfactants in water are given in Table 3.2.

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75

Figure 3.4 Spherical, cylindrical and lamellar micelles. Table 3.2 CMC of some surfactants at 298 K CMC (mol/m3)

Surfactant C12H25N(CH3)3+ C14H29N(CH3)3+ C16H33N(CH3)3+ C10H21SO4– Na+ C12H25SO4– Na+ C14H29SO4– Na+

Br– Br– Br–

C58H114O26 C14H22O(C2H4O)9.5

15.00 3.50 0.90 33.20 8.10 2.00 0.05 0.20

Several properties of the surfactant solution such as surface tension, equivalent conductivity, osmotic pressrue and turbidity show sharp change in the vicinity of this concentration (Figure 3.5). Near the CMC, the surface is almost saturated by the adsorption of the surfactant molecules. Therefore, the surface tension ceases to decrease when the surfactant concentration is increased beyond the CMC. The equivalent conductivity of the solution decreases at the CMC owing to the lower mobility of the micelles as compared to the surfactant molecules. When the critical micelle concentration is approached, the slope of the osmotic pressure curve decreases and the slope of the turbidity curve increases due to the increase in the average molecular weight of the solute [see Eqs. (2.21) and (2.37)]. The forces that hold the surfactant molecules in the micelles are not due to covalent or ionic bonding. They mainly arise from weaker forces such as van der Waals force, hydrophobic force, electrostatic force and hydrogen bonding. Therefore, the size and shape of the micelles depend on the properties of the solution such as the concentration of electrolyte and the pH. To illustrate, the aggregation number of sodium dodecyl sulphate micelles depends significantly on the concentration of sodium chloride. Some examples are presented in Table 3.3. With the addition of electrolyte (such as NaCl), the critical micelle concentration of ionic surfactants decreases. The electrolytes mask the electrostatic repulsion between the ionic head-groups of the surfactant molecules. This favours more adsorption of the surfactant molecules at the surface, which causes the reduction in CMC. Sometimes, increase in pH favours ionisation. If the charge density increases by changing the pH, the CMC may increase.

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Figure 3.5

Table 3.3

Variation of the properties of the surfactant solution near CMC.

Effect of electrolyte on the aggregation number of sodium dodecyl sulphate in water at 298 K

Concentration of NaCl (mol/dm3)

Aggregation number

CMC (mol/m3)

0.00 0.02 0.10 0.20 0.40

80 100 112 118 126

8.1 3.8 1.4 0.8 0.5

The aggregation number increases with increase in length of the hydrocarbon chains of the surfactant molecules. This is illustrated in Table 3.4 for cationic and anionic surfactants whose hydrocarbon chains belong to the same homologous series. The critical micelle concentration decreases with the increase in chain length, which is evident from the data presented in Table 3.2. The CMC also depends on the size of the hydrophilic group. As the size of the hydrophilic group gets larger, the repulsion between them increases. Table 3.4

Effect of the length of the hydrophobic part on aggregation number in water at 298 K Surfactant C10H21N(CH3)3+ C12H25N(CH3)3+ C14H29N(CH3)3+ C10H21SO4– Na+ C12H25SO4– Na+

Aggregation number Br– Br– Br–

36 50 75 50 80

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The structures of the micelles of anionic and cationic surfactants are essentially the same. The micelles of nonionic surfactants are sometimes very large, and the number of surfactant molecules in these clusters can be much greater than 100. McBain (1913) proposed that soap molecules form micelles which can have lamellar and spherical shapes. Hartley proposed the ‘core model’ in 1936, in which a liquid-like hydrocarbon core is surrounded by a hydrophilic surface layer which is constituted by the head-groups of the surfactant molecules. The central core is mainly hydrocarbon. The hydrocarbon tails of the surfactant molecules are hydrophobic, which distort the hydrogen bonds of water. This increases the free energy of the system. As discussed earlier, this is the driving force behind the surfactant molecules to adsorb at the surface. Another way the free energy can be reduced when the surface is completely occupied by the surfactant molecules is to form clusters in which the hydrophobic part is directed inside the cluster, and the hydrophilic part is directed towards the solvent. Therefore, micellisation occurs when the surface is saturated and the surfactant molecules have a sufficiently long hydrophobic part that disrupts the structure of the solvent. The formation of micelles depends upon factors such as electrostatic repulsion between the surfactant head-groups and interaction between the hydrocarbon parts of the surfactant molecules. The hydrophilic head-groups of the surfactant molecules repel each other electrically whereas the hydrophobic groups attract each other by hydrophobic force at the hydrocarbon–water interface. Therefore, two opposing forces act in the interfacial region: one tends to increase and the other tends to decrease the head-group area, as shown in the energy diagram in Figure 3.6. The optimal area is the area corresponding to the intersection of the two energy curves. When the surfactant molecules pack together to assume a geometrical structure, the relative size of the head-group and the hydrophobic chain determines the size and shape of the micelle. The effects of hydration, repulsion between the ions and the effects of the counterion are also important in the packing of the molecules. The structure of the micelles is classified by a ‘packing parameter’, defined as v /( al) , where v is the volume occupied by the hydrophobic group in the micellar core (i.e. the chain volume), l is the length of the hydrophobic group in the micellar core and a is the optimal (cross-sectional) area occupied by the hydrophilic group at the micelle–solution interface (Figure 3.7). The magnitude of

Figure 3.6 Energy diagram for optimal head-group area.

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l is similar to the fully-extended molecular length of the chain, but somewhat less. Furthermore, the optimal radius of the micelle (r) must be less than l in order to maintain the liquid-like core of the micelle. The variation of the structure of the micelles with the packing parameter is illustrated in Table 3.5.

Figure 3.7 Illustration of v, a and l for a spherical micelle. Table 3.5 Structure of the micelles and the packing parameter Packing parameter,

v

Shape of the micelle

al 0 – 1/3 1/3 – 1/2 1/2 – 1 >1

EXAMPLE 3.1 Solution

Spherical Cylindrical Lamellar Reverse micelles

Show that spherical micelles are formed when v /(al) is less than 1/3.

If the spherical micelle has radius r and aggregation number N, then we have,

N

4 3 Sr 3 v v a

4S r 2 a r 3

v r l al 3 However, r should be less than l so that the core of the micelle is liquid-like. Therefore, v al  1 3 .

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The values of v (in nm3) and l (in nm) can be calculated from the following equations (Tanford, 1980): v

0.0274  0.0269n

(3.1)

(3.2) l … 0.154  0.1265n where n is the number of carbon atoms of the saturated hydrocarbon chain embedded in the core of the micelle. From Eqs. (3.1) and (3.2), we can observe that as n becomes large, the v/ l ratio approaches 0.21 nm2. This defines the minimum cross-sectional area that a hydrocarbon chain can have. The maximum value of l (œ lmax ) is given by the equality in Eq. (3.2). If the length of the chains extends beyond this limit significantly, their aggregation may not be considered liquid-like. The optimal area occupied by the hydrophilic group at the surface of the micelle (a) depends on the structure of the group, electrolyte concentration, pH and the presence of any additive (such as alcohol) in the solution. For ionic surfactants, addition of electrolyte causes the value of a to reduce. The diameter of the cylindrical micelles is of the order of a few nanometres. However, their length can be large. For example, the micelles of an ethoxylated C16 alcohol and ethylene oxide were found to have diameter in the range of 3–8 nm, but their length was ~1000 nm. The micelles of nonionic surfactants may have several hundreds of molecules. and

EXAMPLE 3.2 The aggregation number (N) of sodium dodecyl sulphate (SDS) micelle in water is 80. Compute v and l from Eqs. (3.1) and (3.2). From these values, calculate the packing parameter. Comment on the shape of the SDS micelles in water. Solution For SDS, the number of carbon atoms in the hydrophobic chain (n) is 12. From Eqs. (3.1) and (3.2), we get, v = 0.3502 nm3 lmax 0.154  0.1265n 1.672 nm Given, N = 80 The aggregation number is defined as,

N

4S r 3 (assuming the micelle to be spherical) 3v

r

È 3vN Ø ÉÊ 4S ÙÚ

13

È 3 – 0.3502 – 80 Ø ÉÊ ÙÚ 4 –S

13

1.884 nm

Therefore, the radius is larger than the maximum value of the length of the hydrophobic group in the micellar core (lmax).

a

4S r 2 N

4 – S – (1.884)2 80

0.558 nm2

v 0.3502 0.375 al 0.558 – 1.672 This value is slightly higher than the upper limit of v /( al) for the structure of the micelle to be spherical. Therefore, the SDS micelles will be non-spherical to some extent. The results of this example show that to fit into a spherical micelle, the surfactant molecules should have large surface area and their hydrocarbon-volume should be small.

\

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The aggregation of surfactant molecules is believed to be reversible. The surfactant molecules join and leave the micelle very rapidly (e.g. 10–6 s). The counterions at the surface of the micelles exchange at even faster rates. The water molecules which are bound to the micelles are also highly mobile. The typical lifetime of water molecules in the micelle is about 10–8 s. The shape of some micelles changes with surfactant concentration. Some micelles change their structure from spherical to cylindrical to lamellar with increase in surfactant concentration. The transformation can be facilitated by electrolytes. When the effective areas of the hydrophilic and hydrophobic groups are nearly equal, the micelles can take up lamellar structure (see Figure 3.4). A well-known example of this type of shape is lipid bilayer, which is made of double-chained lipid. The surfactant becomes practically insoluble in water. The lamellar micelles have a tendency to form multilayers. The vesicles are lamellar micelles bent around and joined up in a sphere (Figure 3.8). An aqueous solution core remains inside the sphere. Formation of the closed bilayer of the vesicles is favourable because the energetically unfavourable edges of the planar structure are eliminated, and a finite number of surfactant molecules aggregate. Usually, surfactants having two alkyl chains (e.g. the doublechained lipids) with large head-group areas form vesicles (see Chapter 10). A mixture of singlechain anionic and cationic surfactants of similar hydrophobic size can also form vesicles. It is likely that a two-tailed salt having a large hydrophobic part is formed from these surfactants, which encourages the formation of the vesicle. The fatty acids of sucrose can form vesicles in aqueous solution by sonification. Israelachvili et al. (1976) have shown that if 1/2 < v/(al)< 1, the radius of the smallest vesicle is given by

rc

l 1  v /(al)

Figure 3.8 Vesicle and liposome.

(3.3)

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81

This is the critical radius below which a bilayer cannot curve without introducing unfavourable packing strains on the surfactants. Therefore, the surfactants in both the inner and outer layers can pack with their optimal head-group area and the length of the hydrocarbon chains not exceeding l, as long as the radius of the vesicle does not fall below rc. The aggregation number in the vesicle is given by,

N

2 Ë 2v Ø Û È 4S Ì rc2  É rc  Ù Ü Ê a Ú ÝÜ ÍÌ

(3.4)

a

EXAMPLE 3.3 For egg-lecithin vesicles, a = 0.72 nm2, v = 1.06 nm3 and l = 1.75 nm (Israelachvili, 1997). Calculate the aggregation number. Solution

From the given data, v al

1.06 0.72 – 1.75

0.84

From Eq. (3.3),

rc

l 1  v / (al )

1.75 1  0.84

10.94 nm

From Eq. (3.4), the aggregation number is given by

N

2 Ë 2v Ø Û È 4S Ì rc2  É rc  Ù Ü Ê a Ú ÜÝ ÌÍ a

2 Ë 2 – 1.06 Ø Û È (4S ) Ì(10.94)2  É10.94  Ü Ê 0.72 ÙÚ ÜÝ ÌÍ  3205 0.72

Biologists have used the vesicles as models for the cell membranes. It is believed that the vesicles represent the prototypes of early living cells. It has been demonstrated that certain lipids as well as synthetic surfactants can spontaneously self-assemble to form vesicles. Concentric spheres of vesicles are known as liposomes (Figure 3.8). The interactions in the bilayers of vesicles and liposomes are different. It is believed that hydration force provides them stability. The liposomes have been used as ‘containers’ for drugs and genetic materials. When the value of the packing parameter exceeds unity, some surfactants form reverse micelles in non-polar media. The surfactant molecules assemble in structures in which the head-groups are oriented inwards and the hydrophobic groups are oriented towards the solvent (Figure 3.9). The aggregation number in reverse micelles is usually much smaller than the aggregation number in the aqueous micelles. The negative enthalpy change during micellisation is believed to be an important stabilising factor for the reverse micelles. Surfactants soluble in organic liquids can form reverse micelles. There are some surfactants such as Aerosol OT which can form normal as well as reverse micelles. The micelles present in water can dissolve organic molecules. Conversely, the reverse micelles can solubilise water molecules. The Figure 3.9 Reverse micelle.

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liquid dissolves in the micelle. This depends on the chemical nature of the liquid as well as the surfactant. The extent to which a liquid can be solubilised by the micelles depends on the concentration of the surfactant in the solution. The amount of surfactant necessary to solubilise an organic liquid is large. Micelles have been used as reaction-vessels for the manufacture of nanoparticles. The nanoparticles formed inside the micelles are organised inside them. Metal nanoparticles (e.g. gold, silver and platinum) have been synthesised by this technique. The micelles created from block copolymers such as polystyrene–ethylene oxide have been used to generate well-ordered compartments. The tetrachloroaurates (e.g. LiAuCl4) penetrate inside the micelles and form thermodynamically stable dispersions inside the holes. Ultrasound irradiation is used to support the solubilisation process. Finally, the gold salt is reduced to form the nanoparticles. The structure of the micelles controls the dispersion of the nanoparticles. Block copolymer micelles can act like watersoluble biocompatible nanocontainers with great potential for delivering hydrophobic drugs. Micelles have also been used to remove pollutants from wastewater. The pollutant molecules are trapped inside the micelles. These micelles are then separated by ultrafiltration. This method is known as micelle-enhanced ultrafiltration (MEUF). EXAMPLE 3.4 Determination of aggregation number by light scattering. The light scattering data from aqueous solutions of the cationic surfactant dodecyltrimethylammonium bromide [C12H25(CH3)3NBr] are given below (Tartar and Lelong, 1955). c (kg/dm3)

0.006

0.010

0.015

0.020

0.025

0.030

H(c  cCMC ) – 104 (kmol/kg) W

0.83

1.07

1.63

2.02

2.35

2.76

where t is the turbidity in excess of that of the solvent, c is the concentration of the surfactant and cCMC is the critical micelle concentration. The turbidity below the CMC is essentially the same as that for the solvent, and the light-scattering centres are the micelles of the surfactant. Calculate the molecular weight of the micelle and the aggregation number from these data. Given: cCMC = 4.4 × 10–3 kg/dm3. Solution

Let us write the Debye equation [Eq. (2.37)] as

H (c  cCMC ) W

1  2B(c  cCMC ) M

where M is the weight-average molecular weight of the micelle and B is the second virial coefficient. The surfactant solution of concentration c is considered to consist of monomers of concentration cCMC and micelles of concentration c – cCMC. The solution at the CMC is designated as the solvent. The plot is shown in Figure 3.10. The intercept is 1 M

6.93 – 10 5 kmol/kg

The molecular weight is, M = 14430 kg/kmol Therefore, the aggregation number is 47.

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83

Figure 3.10 Determination of molecular weight of micelle.

3.2.8 Thermodynamics of Micellisation Let us consider a surfactant which is represented as S. If the surfactant is ionic, it represents the surface-active part. The effects associated with counterions (e.g. their binding effects) are not considered in the following derivation for simplicity. When the micelle (SN) forms, the clustering can be represented by the following reaction NS  SN

(3.5)

where N is the aggregation number. The aggregation number actually has a statistical distribution rather than a single value as used here. The reaction represented by Eq. (3.5) is reversible. The equilibrium constant for this reaction is given by amicelle

K

aSN

(3.6)

where a represents activity in Eq. (3.6), expressed in mole fraction scale. The standard Gibbs free energy change for micelle formation per mole of surfactant is given by 'G 0



RT ln K N



RT ln amicelle  RT ln aS N

(3.7)

At the critical micelle concentration, aS = aCMC. Since N is large, Eq. (3.7) becomes

'G0  RT ln aCMC

(3.8)

The standard Gibbs free energy change due to micellisation can be calculated from Eq. (3.8). The activity is expressed as the product of mole fraction and activity coefficient. For most surfactants, the critical micelle concentration is small (< 1 mol/m3). Setting the activity coefficient to unity under the assumption of ideal behaviour of the surfactant solution at CMC, Eq. (3.8) becomes

'G0  RT ln xCMC

(3.9)

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where xCMC is the mole fraction of surfactant in the solution at CMC. The experimentally determined 0 value of CMC expressed as mole fraction can be put in Eq. (3.9) to calculate 'G . EXAMPLE 3.5 Solution

Estimate 'G0 for sodium dodecyl sulphate if its CMC is 8 mol/m3.

The CMC in mole fraction is xCMC 

Therefore, 'G 0

8 – 10 3 1000 18

8.314 – 298 – ln(1.44 – 10 4 )

1.44 – 10 4 21915.9 J/mol = –21.9 kJ/mol

The negative value of 'G0 indicates the spontaneity of micellisation.

3.2.9

Krafft Point and Cloud Point

We have discussed in the preceding sections that the solubility of the surfactant molecules in water decreases with increase in the length of the hydrophobic part, and the solubility increases if the hydrophilic part is more soluble. The solubility of surfactant is also dependent on temperature. The solubility of ionic surfactants increases very rapidly after a temperature known as the Krafft point. At this temperature, the micelles are formed and the solubility is significantly increased. This temperature is important in industrial preparations, especially where concentrated surfactant solutions are required. The Krafft temperature increases with the increase in the number of carbon atoms in the hydrophobic part. Krafft points of some ionic surfactants are presented in Figure 3.11(a). The Krafft point decreases linearly with the logarithm of CMC for many anionic surfactants. It is strongly dependent on the addition of electrolyte, the head-group and the counterion. Electrolytes usually raise the Krafft point. There is no general trend for the dependence on counterions. However, the Krafft point is typically much higher in presence of divalent counterions than monovalent counterions. For alkali alkanoates, Krafft point increases as the atomic number of the counterion decreases. The opposite trend is observed for alkali sulphates or sulphonates. For cationic surfactants, the Krafft point is usually higher for bromides than chlorides, and still higher for the iodides. The solubility of some nonionic surfactants (such as the ethoxylates) decreases dramatically above a certain temperature. This temperature is known as cloud point. These surfactants are quite soluble at the low temperatures (273–278 K). However, they come out of solution upon heating. These surfactants dissolve in water by hydrogen bonding. With increase in temperature, the hydrogen bonds disrupt. This causes reduction in solubility. Cloud point decreases with the increase in chain length of the hydrophobic part. The cloud points of some nonionic surfactants are presented in Figure 3.11(b). The cloud points of many surfactants have been presented by Rosen (2004).

3.2.10 Liquid Crystals Below the CMC, the surfactant solution contains only the surfactant monomer. At CMC, the solution contains a mixture of the micelles and the monomer at the concentration equal to the CMC. The micelles can have various shapes as discussed before. When there is sufficient number of micelles in the solution, they start to pack together in a number of geometric arrangements, depending upon the shape of the individual micelles. These packed-arrangements are known as liquid crystals. The liquid crystalline phases are also known as lyotropic mesomorphs and lyotropic

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85

Figure 3.11 (a) Variation of Krafft point with the number of carbon atoms in the alkyl chain, and (b) Variation of cloud point with the number of oxyethylene units (Gu and Sjöblom, 1992) [adapted by permission from Elsevier Ltd., © 1992].

mesophases. The reverse micelles can also form liquid crystals. The liquid crystals are ordered like solid crystals, but they are mobile like liquids. The spherical micelles pack into cubic liquid crystals, cylindrical micelles form hexagonal liquid crystals, and lamellar micelles form lamellar liquid crystals. With increase in surfactant concentration, some cylindrical micelles become branched and interconnected leading to the formation of a bicontinuous liquid crystalline phase (Figure 3.12). The hexagonal phase appears at a lower surfactant concentration than that for the lamellar phase. The usual sequence of the phases is: micellar ® hexagonal ® lamellar (Fontell, 1992). In between the changes from one phase to another, cubic phases can be detected. The hexagonal and lamellar phases are optically anisotropic. They can be detected under polarising microscope. The cubic phase is isotropic. It can be identified by using dyes (Kunieda, 2003).

Figure 3.12

Bicontinuous liquid crystal structure (Myers, 2006) [reproduced by permission from John Wiley and Sons, Inc., © 2006].

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Because of the ordered arrangement of the molecules in the liquid crystals, the viscosity of the solution increases considerably. The hexagonal phases are more viscous than the lamellar phases. The cubic liquid crystalline phases formed from bicontinuous structures and spherical micelles at high surfactant concentrations are very high-viscosity gels. They are useful in cosmetic and pharmaceutical industries.

3.2.11

Hydrophilic–Lipophilic Balance

A major commercial use of the surfactants is to formulate emulsion-stabilising agents, or emulsifiers (see Section 3.3). Emulsions can be divided into two types: oil-in-water (O/W) emulsions, and waterin-oil (W/O) emulsions. In the oil-in-water type of emulsions, oil droplets are dispersed in the continuous aqueous phase. Such emulsions show high electrical conductivity. On the other hand, in water-in-oil type of emulsions, the aqueous phase is dispersed in the continuous oil phase. These emulsions show low electrical conductivity. Some surfactants stabilise the first type of emulsions whereas the other surfactants are more efficient in stabilising the second type. A rule of thumb is that the most stable emulsion is formed when the surfactant has higher solubility in the continuous phase. For example, a water-soluble surfactant should stabilise oil-in-water emulsions more than water-inoil emulsions; the reverse is expected for a surfactant that is soluble in oil. This rule is known as Bancroft’s rule. Griffin (1949) developed a method to correlate the structural properties of the surfactants with their ability to act as emulsifiers. This method is known as hydrophilic–lipophilic balance (HLB) method. The solubility of surfactants in water varies depending on their HLB value. This variation, and the type of emulsion (i.e. O/W or W/O) produced by these surfactants are presented in Table 3.6. Table 3.6 HLB values, solubility in water and the type of emulsion produced Range of HLB value

Solubility in water

Emulsion type

1–4 4–7 7–9 10–13 13 and higher

Insoluble Poor unstable dispersion Stable opaque dispersion Hazy solution Clear solution

Water-in-oil Water-in-oil – Oil-in-water Oil-in-water

As the name of the method suggests, the balance between the hydrophilic and lipophilic parts of the surfactant molecule is of main concern, and values have been assigned to these parts for various surfactants. A group-number method has been suggested for calculating the HLB value of a surfactant from its chemical formula. HLB = S(hydrophilic group-numbers) – S(group-number per –CH2– group) + 7

(3.10)

The group-numbers for various hydrophilic and lipophilic groups are presented in Table 3.7. EXAMPLE 3.6

Calculate the HLB value of sodium dodecyl sulphate (C12H25SO4– Na+).

Solution The HLB value is calculated from the Eq. (3.10) as described below. HLB = S(hydrophilic group-numbers) – S(group-number per –CH2– group) + 7 Now, from Table 3.7, S (hydrophilic group-numbers) = 38.7

Surfactants and their Properties

Table 3.7 Type of group

HLB group-numbers Group

Group-number

–SO–4

Na+ –COO– K+ –COO– Na+ Sulphonate –N (tertiary amine) Ester (sorbitan ring) Ester (free) –COOH –OH (free) –O– –OH (sorbitan ring) –CH3 –CH2– –CH= –(CH2–CH2–CH2–O–)

Hydrophilic

Lipophilic

87

38.7 21.1 19.1 11 9.4 6.8 2.4 2.1 1.9 1.3 0.5 0.475 0.475 0.475 0.15

S (group-number per –CH2– group) = 12 × 0.475 = 5.7 Thus, HLB = 38.7 – 5.7 + 7 = 40 Therefore, the HLB value of sodium dodecyl sulphate is 40. The HLB values of some surfactants are given in Table 3.8. Table 3.8 Surfactants and their HLB values Surfactant Sodium dodecyl sulphate Potassium oleate Sodium oleate Tween 80 Sorbitan monolaurate Methanol Ethanol n-Propanol n-Butanol Sorbitan monopalmitate Sorbitan monostearate Span 80 Glycerol monostearate Sorbitan tristearate Cetyl alcohol Oleic acid Sorbitan tetrastearate

HLB value 40.0 20.0 18.0 16.5 8.5 8.3 7.9 7.4 7.0 6.6 5.7 5.7 3.7 2.1 1.3 1.0 0.3

The HLB value of a mixture of surfactants can be calculated from the HLB values of the individual surfactants on a weight-prorated basis. It is illustrated in the following example. EXAMPLE 3.7 Calculate the HLB value of a mixture containing 25% (w) potassium oleate and 75% (w) Tween 80.

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Solution From Table 3.8, the HLB values of potassium oleate and Tween 80 are 20 and 16.5 respectively. Therefore, the HLB value of the mixture is HLB = 0.25 × 20 + 0.75 × 16.5 = 17.375

3.3

EMULSIONS AND MICROEMULSIONS

One of the most important uses of surfactants is to stabilise emulsions. Emulsions are dispersions of one liquid into another. They are usually prepared by mixing the two liquids under high shear or by sonification. The dispersed liquid exists in the form of droplets (Figure 3.13). The size of the droplets may vary from a few micrometres to several hundred micrometres. The surfactant molecules adsorb at the surface of the droplets. When the droplets come very close to one another, a thin liquid film separates them. If this film is unstable, it ruptures easily resulting in coalescence of the droplets. If coalescence occurs rapidly, the emulsion becomes unstable Figure 3.13 Emulsion droplets of paraffin-oil in salty water. and the two liquids separate into clear immiscible phases. The role of surfactant is to stabilise the film so that coalescence is prevented. The surfactant achieves this by two mechanisms. The adsorbed surfactant molecules slow down the hydrodynamic drainage of the film. This drainage process is important for the films of viscous liquids. The second mechanism by which the surfactant molecules prevent the film from rupture is by exerting interfacial repulsive force between the droplets. The properties of the surfactant govern the nature of this repulsion. The ionic surfactants exert electrostatic repulsion whereas the nonionic surfactants exert steric repulsion. These forces will be discussed in Chapter 5. Due to this repulsion between the droplets, they cannot approach each other beyond a certain separation, which varies between 5 and 50 nanometres. However, the film can rupture only when the separation between the droplets (i.e. the thickness of the film) is only a few nanometres. The rupture of the thin film is caused by the van der Waals force. Therefore, if the surfactant molecules can generate sufficient repulsion between the approaching droplets, they are prevented from coalescence. These phenomena are crucial for the stability of emulsions. The properties of the surfactant and its concentration are important parameters for stabilising emulsions. Sometimes the emulsion needs to be destabilised so that the two liquids are separated. Examples are dewatering of crude oil during the removal of salt. Certain chemical additives are used to destabilise the ‘tough’ films made of organic compounds present in petroleum crude (e.g. asphaltenes). It has been discussed in Section 3.2.7 that surfactant micelles can solubilise organic liquids, but the amount of this solubilisation is usually small. Under certain circumstances a large amount of the organic compound can be dissolved in an aqueous solution of surfactant, or vice versa. Such a solution is very transparent and stable. It is known as microemulsion. This term was first used by Schulman (1959). The interfacial tension between the aqueous and organic phases needs to be reduced to near-zero values (~1 × 10–6 N/m, or less) to form a microemulsion. In addition to the surfactant, a cosurfactant (e.g. an alcohol such as hexanol) is usually added to prepare the

Surfactants and their Properties

89

microemulsion. At some temperatures, certain nonionic surfactants can form microemulsion without requiring the cosurfactant. A variation in temperature or addition of some compound can destabilise a microemulsion. The size of the droplets in microemulsions is very small (~1–100 nm). The microemulsions form spontaneously, just by simple mixing. Microemulsions are also viewed as ‘swollen micelles’ and the concept of solubilisation of organic compound by micelles is used to describe them. At low interfacial tension, a large swollen micelle can form, which can take up a considerable amount of organic liquid. The solution remains transparent and stable. At this stage, one can view the organic liquid either as emulsified in the aqueous phase or solubilised: both terminologies are used by the scientists. The term ‘solubilised oil’ is also used in the literature. Microemulsions have several important commercial uses. Water-in-oil microemulsions are used in some dry-cleaning processes. Many floor polishes, cleaners, personal-care products, and pesticide formulations are actually microemulsions. An application of great commercial value is in enhanced recovery of petroleum trapped in porous sandstones. Emulsions and microemulsions will be discussed in detail in Chapter 9.

3.4

FOAMS

Foam is a dispersed system in which gas bubbles are separated by a liquid medium. If we shake an aqueous surfactant solution in presence of a gas, foam is formed. Like the emulsions, foams are stabilised by surfactants. Different surfactants stabilise foams in different ways. In absence of a surfactant, foams hardly have any stability. A combination of salt and ionic surfactant produces stable foams. Foams have been of great practical interest because of their widespread occurrence in everyday life such as food products, detergents, personal-care products, industrial applications and hazard management. Foams are present in almost every part of petroleum production and refining process. In some situations, foams are undesirable (e.g. in distillation and fractionation tower, paper production, industrial water purification, and beverage production). Foams are characterised by welldefined interfaces (Figure 3.14). The same factors which stabilise the emulsion droplets (i.e.

Figure 3.14 Foam (Binks and Horozov, 2006) [reproduced by permission from Cambridge University Press, © 2006].

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hydrodynamic drainage of the thin liquid film, and the interfacial forces) govern the stability of the foams as well. The mechanism of collapse of the foams is also similar to the coalescence of droplets in emulsions. When the foaminess is required to be reduced, certain additives are used. These are known as antifoams. Antifoams prevent the formation of foams. Another class of additives, called defoamers, causes the already-formed foams to collapse. The presence of oil droplets (e.g. silicone oil) and/or hydrophobic solid particles (e.g. hydrophobised silica particles) in the foaming solution can act as antifoam agents. Foams will be discussed in detail in Chapter 9.

SUMMARY This chapter presents an overview of the surfactants and their applications. In the beginning of the chapter, various types of surfactants are discussed with examples. An important property of surfactants is micellisation. Various types of micelles are discussed. The effects of various parameters (e.g. concentrations of surfactant and salt) on the formation of micelles are discussed. A simplistic view of the thermodynamics of micellisation is presented. Formation of vesicles, liposomes and liquid crystals is discussed. Two major uses of surfactants, reduction of interfacial tension and formation of carriers of fine particles and other materials, are illustrated. The Krafft point and cloud point are discussed with examples. A discussion on the hydrophilic–lipophilic balance of the surfactants is presented. The chapter ends with a brief overview of emulsions, microemulsions and foams. Some of these topics will be discussed in more detail in the later chapters.

KEYWORDS Aggregation number Amphipathic Anionic surfactant Antifoam Bancroft’s rule Biosurfactant Cationic surfactant Cloud point Cosurfactant Critical micelle concentration Defoamer Electrolyte effect Emulsion Foam Gemini surfactant Hydrophilic Hydrophilic–lipophilic balance Hydrophobic Interfacial tension Krafft point Liposome Liquid crystal

Lyophilic Lyophobic Marangoni effect Micelle Micellisation Microemulsion Nanoparticle Nonionic surfactant Packing parameter Phase inversion temperature Protein-based surfactant Reverse micelle Solubilised oil Surface Surface tension Surface-active agent Surfactant Swollen micelle Tears of wine Vesicle Zwitterionic surfactant

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91

NOTATION a aCMC amicelle aS B c cCMC H K l lmax m M n N r rc R T v xCMC

optimal cross-sectional area of head-group, m2 activity of surfactant at CMC activity of micelle activity of surfactant second virial coefficient, mol m3 kg–2 concentration, mol/m3 critical micelle concentration, mol/m3 constant, m2 kg–2 mol equilibrium constant length of the hydrophobic group in the micellar core, m maximum value of l, m number of oxyethylene units in nonionic surfactant molecular weight, kg/mol number of carbon atoms of the saturated hydrocarbon chain aggregation number radius of micelle, m radius of the smallest vesicle, m gas constant, J mol–1 K–1 temperature, K volume occupied by the hydrophobic group in the micellar core, m3 mole fraction of surfactant in the solution at CMC

Greek Letters 'G0

t

standard Gibbs free energy change for micelle formation per mole of surfactant, J/mol turbidity of solution, m–1

EXERCISES 1. What do you understand by the term surfactant? How does a surfactant differ from an ordinary solute such as sodium chloride? 2. Explain the classification of surfactants. 3. Mention two examples of cationic and anionic surfactants. 4. What is a nonionic surfactant? How does it differ from the ionic surfactants? 5. Explain the salient features of a zwitterionic surfactant. 6. Explain the main features of a gemini surfactant. What is the reason behind their strong surface activity? 7. What are the advantages of the biosurfactants? 8. Explain what you understand by a micelle. What is micellisation? 9. Explain with examples what you understand by aggregation number. 10. What is critical micelle concentration? 11. Explain the effects of surfactant chain length and concentration of electrolyte on critical micelle concentration.

92 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

Colloid and Interface Science

What are the commonly-observed shapes of the micelles? What factors govern the shape of a micelle? Explain in details. Explain what you understand by vesicle and liposome. What are their uses? Explain what you understand by reverse micelle. How does it differ from a normal micelle? Explain how micelles are used to solubilise oil and prepare nanoparticles. Explain how you would determine the free energy of micellisation. Explain what you understand by Krafft point. On what factors does the Krafft point depend? Explain what you understand by cloud point. How does the cloud point of a surfactant vary with its chain length? What do you understand by liquid crystal? What factors govern the viscosity of a liquid crystalline phase? Explain clearly the difference between an ordinary emulsion and a microemulsion. How is a microemulsion prepared? Discuss three major uses of microemulsions. How are the droplets of an emulsion prevented from coalescence by the surfactants? Explain three important uses of foams. How does an antifoam agent differ from a defoamer? Explain the HLB concept of classification of surfactants. If a surfactant has HLB = 3, what type of emulsion would you expect it to form?

NUMERICAL PROBLEMS 3.1 Arrange the following surfactants in the increasing order of CMC in aqueous solution at 298K: (i) C8H17N+(CH3)3Br– (ii) C10H21N+(CH3)3Br– (iii) C12H25N+(CH3)3Br– (iv) C14H29N+(CH3)3Br– 3.2 Estimate the changes in the slopes of turbidity-concentration and osmotic pressureconcentration plots when the concentration of an aqueous solution of sodium dodecyl sulphate (C12H25SO4Na) reaches the CMC. Take aggregation number = 75. 3.3 Using the Tanford equations, calculate the minimum cross-sectional area for a hydrocarbon chain. 3.4 We have seen in Example 3.2 that the SDS micelles in water are slightly non-spherical. For what value of the aggregation number will the micelles be spherical? 3.5 The aggregation number of the surfactant C10H21N(CH3)3Br has been reported to be 36. Can its micelles be spherical? 3.6 The critical micelle concentration of cetyltrimethylammonium bromide is 1 mol/m3. Estimate the standard Gibbs free energy change due to micellisation. 3.7 For a double-chained lipid, the optimal head-group area is 0.6 nm2. The values of v and l are 1.15 nm3 and 2.2 nm respectively. What would be the radius of the smallest vesicle that can be formed with the lipid? What would be the aggregation number? 3.8 The light scattering data from aqueous solutions of sodium dodecyl sulphate [NaC12H25SO4] are given next (Tartar and Lelong, 1955).

Surfactants and their Properties

c (kg/dm3) 4.9 5.8 7.2 1.0 1.3 1.6 2.0 2.5

× × × × × × × ×

93

H (c  cCMC ) – 10 4 (kmol/kg) W

10–3 10–3 10–3 10–2 10–2 10–2 10–2 10–2

0.97 0.98 1.27 1.75 1.93 2.60 3.31 3.80

where t is the turbidity in excess of that of the solvent, c is the concentration of the surfactant and cCMC is the critical micelle concentration. Calculate the molecular weight of the micelle and the aggregation number from these data. Given: cCMC = 2.3 × 10–3 kg/dm3. 3.9 Calculate the HLB value of n-propanol using the group-numbers given in Table 3.8. 3.10 An optimum emulsion having the necessary stability and the desired size of the droplets can be formulated by using a blend of Tween 80 and Span 80 in 4:1 ratio by weight. Calculate the HLB value of the surfactant blend.

FURTHER READING Books Adamson, A.W. and A.P. Gast, Physical Chemistry of Surfaces, John Wiley, New York, 1997. Binks, B.P. and T.S. Horozov (Eds.), Colloidal Particles at Liquid Interfaces, Cambridge University Press, New York, 2006. Evans, D.F. and H. Wennerström, The Colloidal Domain, Wiley-VCH, New York, 1999. Hartley, G.S., Aqueous Solutions of Paraffin-Chain Salts, Hermann et Cie, Paris, 1936. Holmberg, K. (Ed.), Handbook of Applied Surface and Colloid Chemistry (Vol. 1), John Wiley, Chichester (England), 2002. Hunter, R.J., Foundations of Colloid Science, Oxford University Press, New York, 2005. Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, London, 1997. Kosaric, N. (Ed.), Biosurfactants (Surfactant Science Series, Vol. 48), Marcel Dekker, New York, 1993. Kumar, P. and K.L. Mittal (Eds.), Handbook of Microemulsion Science and Technology, Marcel Dekker, New York, 1999. Myers, D., Surfactant Science and Technology, John Wiley, New Jersey, 2006. Nnanna, I.A. and J. Xia (Eds.), Protein-Based Surfactants (Surfactant Science Series, Vol. 101), Marcel Dekker, New York, 2001. Porter, M.R., Handbook of Surfactants, Chapman and Hall, London, 1994. Rosen, M.J., Surfactants and Interfacial Phenomena, John Wiley, New Jersey, 2004.

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Stokes, R.J. and D.F. Evans, Fundamentals of Interfacial Engineering, Wiley-VCH, New York, 1997. Tanford, C., The Hydrophobic Effect, John Wiley, New York, 1980.

Articles Abbott, N.L., “New Horizons for Surfactant Science in Chemical Engineering”, AIChE J., 47, 2634 (2001). Fontell, K., “Some Aspects on the Cubic Phases in Surfactant and Surfactant-Like Lipid Systems”, Adv. Coll. Int. Sci., 41, 127 (1992). Gu, T. and J. Sjöblom, “Surfactant Structure and its Relation to the Krafft Point, Cloud Point and Micellization: Some Empirical Relationships”, Coll. Surf., 64, 39 (1992). Griffin, W.C., “Classification of Surface Active Agents by HLB”, J. Soc. Cosmet. Chem., 1, 311 (1949). Israelachvili, J.N., D.J. Mitchell, and B.W. Ninham, “Theory of Self-Assembly of Hydrocarbon Amphiphiles into Micelles and Bilayers”, J. Chem. Soc. Faraday Trans. II, 72, 1525 (1976). Kunieda, H., K. Aramaki, T. Izawa, M.H. Kabir, K. Sakamoto, and K. Watanabe, “Dye Method to Identify the Types of Cubic Phases”, J. Oleo Sci., 52, 429 (2003). McBain, J.W., “Mobility of Highly Charged Micelles”, Trans. Faraday Soc., 9, 99 (1913). Schulman, J.H., W. Stoeckenius, and L.M. Prince, “Mechanism of Formation and Structure of Microemulsions by Electron Microscopy”, J. Phys. Chem., 63, 1677 (1959). Tartar, H.V. and A.L.M. Lelong, “Micellar Molecular Weights of Some Paraffin Chain Salts by Light Scattering”, J. Phys. Chem., 59, 1185 (1955).

4

Surface and Interfacial Tension

Pierre-Simon Laplace (1749 –1827)

Pierre-Simon Laplace was a renowned French mathematician and astronomer. He was born in Normandy (France). Laplace studied at the Caen University in the early part of his student life where his interests in mathematics flourished. Later, he studied in Paris and his mentor was the famous mathematician d’Alembert. Laplace was appointed professor of mathematics at École Militaire at the age of 19. The famous works of Laplace were on statistics, stability of the solar system, potential theory, Laplace transformation, surface tension, speed of sound, and many other topics in mathematics. One of Laplace’s illustrious students was Napoleon Bonaparte. Laplace’s greatest work was Traité de Mécanique Céleste published in 1799. In 1806, Laplace became Count of the Empire, and was named a marquis in 1817. Laplace died in Paris.

TOPICS COVERED © © © © © © © © ©

Surface and interfacial tension Correlations for estimation of surface tension Contact angle and wetting Radii of curvature Shapes of surfaces and interfaces Pendant and sessile drops Capillary action Kelvin equation Measurement of surface and interfacial tension 95

© © © © © © ©

Drop-weight method du Noüy ring method Wilhelmy plate method Maximum bubble-pressure method Spinning-drop method Measurement of contact angle Advancing and receding contact angles © Contact angle hysteresis

96

4.1

Colloid and Interface Science

INTRODUCTION

Surface tension and interfacial tension are the two fundamental properties by which gas–liquid and liquid–liquid interfaces are characterised. As mentioned in Chapter 1, the zone between a gaseous phase and a liquid phase or between two liquid phases looks like a surface of zero thickness. The interface acts like a membrane under tension. To investigate the origin of this tension, we will look into the difference between the gaseous and liquid states. In the gaseous state, the molecules are in random motion due to their translational kinetic energy. When the pressure is high, the gas molecules experience attraction between them. In the liquid state, the intermolecular attractive forces are very high due to the dipolar and dispersion interactions. In fact, one of the most striking demonstrations of the effect of intermolecular forces is surface tension. The potential energy of a molecule is decreased when another molecule is brought by the attraction forces from a large distance to a distance very close to it. A molecule in the bulk of the liquid phase is surrounded by a large number of neighbour molecules. Therefore, its potential energy is minimum. However, when the molecule is present in the surface layer of the liquid, it is surrounded by lesser number of molecules in the gaseous region at the top of the surface. This causes the energy of the molecule at the surface to be greater than the energy of a molecule in the bulk phase. Therefore, for the formation of unit area of the surface, energy has to be spent for bringing molecules from the bulk to the surface. Surface energy and surface tension will be discussed in Section 4.2. Interfacial tension will be discussed in Section 4.3. The concept of contact angle will be presented in Section 4.4. The shapes of the surfaces and their relationship with the surface tension will be discussed in Section 4.5. The techniques for measuring surface and interfacial tensions will be described in Section 4.6. The methods for experimental determination of contact angle will be discussed in Section 4.7.

4.2

SURFACE TENSION

Let us consider a liquid in contact with its vapour, as depicted in Figure 4.1. A molecule in the bulk liquid is subjected to attractive forces from all directions by the surrounding molecules. It is practically in a uniform field of force. But for the molecule at the surface of the liquid, the net

Figure 4.1

Illustration of the origin of surface tension.

Surface and Interfacial Tension

97

attraction towards the bulk of the liquid is much greater than the attraction towards the vapour phase, because the molecules in the vapour phase are more widely dispersed. This indicates that the molecules at the surface are pulled inwards. This causes the liquid surfaces to contract to minimum areas, which should be compatible with the total mass of the liquid. The droplets of liquids or gas bubbles assume spherical shape, because for a given volume, the sphere has the least surface area. If the area of the surface is to be extended, one has to bring more molecules from the bulk of the liquid to its surface. This requires expenditure of some energy because work has to be done in bringing the molecules from the bulk against the inward attractive forces. The amount of work done in increasing the area by unity is known as the surface energy. If the molecules of a liquid exert large force of attraction, the inward pull will be large. Therefore, the amount of work done will be large. Therefore, Surface energy = (amount of work done)/(amount of area extended) = (force × distance)/area = N.m/m2 = J/m2 = N/m m2

(4.1)

10–3

For example, the amount of work required to create 1 surface is about 72.8 × J for water. Surface tension is defined as the force at right angle to any line of unit length in the surface. Therefore, Surface tension = force/distance = N/m (4.2) Therefore, it is apparent from Eqs. (4.1) and (4.2) that the units of surface energy and surface tension are identical. Surface energy can be determined by measuring the surface tension. EXAMPLE 4.1 Show that among all three-dimensional bodies with a given surface area, the sphere has the largest volume (or for a given volume, the area will be minimal when the body has spherical shape). Solution Let us solve this problem by employing the calculus of variations using the EulerLagrange equation. For simplicity, let us assume that the surface is symmetric about the x-axis passing through the origin as shown in Figure 4.2. The volume V is given by

Figure 4.2

A volume element of a closed surface of fixed area which is symmetric about the x-axis.

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Colloid and Interface Science

S Ô y 2 dx

V

(i)

This volume is subject to the condition that the area A is fixed. 2S Ô y ds

A

(ii)

where ds is a line element of the surface in the x–y plane. Let us introduce a change of variable such that yds = dw (iii) where w can vary from 0 to

A . Now, from Figure 4.2, we have 2S

(dx )2  (dy)2

(ds)2

(iv)

and therefore, 2

È dx Ø È dy Ø ÉÊ dw ÙÚ  ÉÊ dw ÙÚ

2

È ds Ø ÉÊ dw ÙÚ

2

1 y2

(v)

The volume can be expressed in terms of w from Eq. (i) as a

V

È dx Ø S Ô y2 É Ù dw Ê dw Ú

(vi)

0

where a œ A /(2S ). Now, substituting dx/dw from Eq. (v) to Eq. (vi), we get 12

2 Ë È dy Ø Û S Ô y Ì1  y 2 É Ù Ü Ê dw Ú Ü 0 Ì Í Ý a

V

dw

(vii)

Let us now make the following substitution,

]

y2 / 2

(viii)

Therefore, Eq. (vii) becomes V

12



ÎÑ È d] Ø 2 ÞÑ Û S 2 Ô Ì] Ï1  É Ù ß Ü Ê dw Ú Ñ Ü 0Ì àÝ Í ÑÐ

dw

(ix)

The integrand is a function (f) of z and dz/dw (but not an explicit function of w). Therefore, the Euler–Lagrange equation for the maximum value of V is given by df È d] Ø f É Ù Ê dw Ú d (d] /dw)

c, c = constant

(x)

12

where

f

Ë ÎÑ È d] Ø 2 ÞÑÛ Ì] Ï1  É Ù ßÜ ÌÍ ÐÑ Ê dw Ú àÑ ÜÝ

(xi)

Surface and Interfacial Tension

99

From Eqs. (x) and (xi), we get 12

]

12

Ë È d] Ø 2 Û Ì1  É Ù Ü Ê dw Ú Ü ÍÌ Ý

]

12

2 2 È d] Ø Ë È d] Ø Û  1 Ì ÉÊ ÙÚ É Ù Ü dw ÍÌ Ê dw Ú ÝÜ

1 2

c

(xii)

Equation (xii) simplifies to 12

Ë Û ] Ì 2 Ü Í 1  (d] / dw) Ý

(xiii)

c

Therefore, Ë È d] Ø 2 Û b Ì1  É Ù Ü (xiv) ÌÍ Ê dw Ú ÜÝ Rewriting Equation (xiv) and integrating, we get –2b(1– z/b)1/2 = w + I, I = constant of integration (xv) Putting the limit: z = 0 when w = 0, we get I = –2b. Therefore, Eq. (xv) becomes 2b – 2b(1– z/b)1/2 = w (xvi) Rearranging and squaring, we get (w – 2b)2 = 4b2 (1– z / b) (xvii) Now, z = 0 for w = a. Therefore, from Eq. (xvii), we get, b = a/4. Substituting the value of b in Eq. (xvii), we get (w – a/2)2 = (a2/4)(1– 4z /a) (xviii) Simplifying Eq. (xviii), we get z = w (1 – w/a) (xix) This is the equation for the maximising curve. From Eqs. (v) and (viii), we have ]

È dx Ø ÊÉ dw ÚÙ Now,

dy dw

2

1 È dy Ø  2] ÊÉ dw ÚÙ

dy d] d] dw

1 (2] )1 2

2

È d] Ø ÉÊ dw ÙÚ

(xx) 2

(xxi)

From Eqs. (xx) and (xxi), we get È dx Ø ÊÉ dw ÚÙ

2

1 2]

Ë È d] Ø 2 Û Ì1  É Ù Ü ÌÍ Ê dw Ú ÜÝ

(xxii)

From Eq. (xix), we have d] dw

1  2w a

(xxiii)

Substituting dz/dw from Eq. (xxiii) and z from Eq. (xix) in Eq. (xxii), we get

È dx Ø ÉÊ dw ÙÚ

2

2 a

(xxiv)

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Colloid and Interface Science

Integration of Eq. (xxiv) gives

Ô dx

È 2Ø ÉÊ a ÙÚ

12w

È 2Ø ÉÊ a ÙÚ

12

x

È aØ xÉ Ù Ê 2Ú

12

x

0

Ô dw

(xxv)

w

(xxvi)

0

which gives

From Eqs. (viii) and (xix), we get

y2 2



x2 2

(xxvii)

Equation (xxvii) can be written as 2

12 Ë a È aØ Û Ì x  É Ù Ü  y2 (xxviii) Ê 2Ú Ü 2 ÌÍ Ý Therefore, the surface of revolution is a sphere of radius (a/2)1/2. The maximising volume is (4p/3)(a/2) 3/2. The surface area of this sphere is, A = 2pa. Therefore, we can write the relation between the volume and surface area as

V

4 È aØ S 3 ÉÊ 2 ÙÚ

32

4 È AØ S 3 ÉÊ 4S ÙÚ

32

(xxix)

Therefore, (xxx) A3 36S V 2 In general, for a three-dimensional body, the following inequality, known as isoperimetric inequality [see Osserman (1978)] holds. (xxxi) A3 • 36S V 2 The least surface area for a certain volume will be obtained when the equality condition is satisfied. We have noted at the beginning of this section that the natural tendency of the surface of a liquid is to contract to minimise the surface area. Therefore, if we attempt to increase the area, work will be required. Consider a thin film of liquid, ABCD, contained in a rectangular wire-frame (Figure 4.3). The boundary BC (length = l) is movable. Imagine now that the film is stretched by

Figure 4.3

Illustration of work done in increasing the surface area.

Surface and Interfacial Tension

101

moving the boundary BC by Dx to the new position EF. If g be the surface tension, the force acting on the film is 2g l, because the film has two surfaces. The work done in stretching the film is

W where 'A

2J l ' x J ' A

(4.3)

2l 'x is the change in total area on the two sides of the film. J

Therefore,

W 'A

(4.4)

This depicts the equivalence between surface tension and surface energy. The surface tension of some liquids is presented in Table 4.1. Table 4.1 Substance Acetic acid Acetone Aniline Benzaldehyde Benzene Bromobenzene Bromoform Carbon disulphide Carbon tetrachloride Chlorobenzene Chloroform Cyclohexane Cyclohexanol Dichloromethane Ethyl acetate Ethyl alcohol Ethyl mercaptan Iodobenzene

Surface tension of some liquids at 293 K

Surface tension (mN/m) 27.4 23.3 42.9 40.0 28.9 36.5 41.5 32.3 26.8 33.3 27.2 25.5 32.7 26.5 23.8 22.3 23.2 39.7

Substance Mercury Methyl acetate Methyl alcohol Methyl ethyl ketone m-Xylene n-Butyl alcohol n-Decane n-Heptane n-Hexane Nitrobenzene n-Octane n-Pentane n-Propyl alcohol o-Nitrotoluene o-Xylene p-Xylene Toluene Water

Surface tension (mN/m) 476.0 24.8 22.6 25.1 28.9 24.5 23.9 20.4 18.4 43.4 21.8 16.0 23.8 41.5 30.0 28.3 28.5 72.8

Sugden (1924) proposed the following equation to calculate surface tension from the physical properties of the compound. J

Ë¥( Ul  Uv ) Û Ì Ü M Í Ý

4

(4.5)

where rl and rv are the densities of the liquid and vapour respectively, and M is the molecular weight. ¶ is known as parachor, which means ‘comparative volume’. If we neglect the density of vapour in comparison with the density of the liquid, from Eq. (4.5), we have ¥ vl J 1 4 , where vl is the molar volume of the liquid. Therefore, a comparison of the parachors of different liquids is equivalent to the comparison of their molar volumes under the condition of equal surface tension. ¶ is a weak function of temperature for a variety of liquids over wide ranges of temperature, and generally assumed to be a constant. Additive procedures exist for calculating ¶. Equation (4.5) suggests that surface tension is very sensitive to the value of parachor and the liquid density. The structural

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Colloid and Interface Science

contributions for calculating ¶ are given in Table 4.2. The total value of ¶ for a compound is the summation of the values of the structural units. Table 4.2

Structural contributions for calculating the parachor ¶ × 106*

Structural unit C H CH3 CH2 in –(CH2)n, n < 12 CH2 in –(CH2)n, n > 12 1-Methylethyl 1-Methylpropyl 1-Methylbutyl 2-Methylpropyl 1-Ethylpropyl 1,1-Dimethylethyl 1,1-Dimethylpropyl 1,2-Dimethylpropyl 1,1,2-Trimethylpropyl C 6H 5 –COO– –COOH –OH –NH2 –O– –NO2 –NO3 –CONH2

1.600 2.756 9.869 7.113 7.166 23.704 30.569 37.646 30.818 37.255 30.302 36.899 36.970 43.301 33.716 11.345 13.124 5.299 7.558 3.557 13.159 16.538 16.307

Structural unit R–[–CO–]–R¢, R + R¢ = 2 R–[–CO–]–R¢, R + R¢ = 3 R–[–CO–]–R¢, R + R¢ = 4 R–[–CO–]–R¢, R + R¢ = 5 R–[–CO–]–R¢, R + R¢ = 6 R–[CO–]–R¢, R + R¢ = 7 –CHO O (not mentioned above) N (not mentioned above) S P F Cl Br I Double bond, terminal Double bond in 2,3-position Double bond in 3,4-position Triple bond Three-membered ring Four-membered ring Five-membered ring Six-membered ring

¶ × 106* 9.123 8.714 8.447 8.233 8.056 7.842 11.737 3.557 3.112 8.731 7.202 4.641 9.816 12.092 16.058 3.397 3.148 2.899 7.220 2.134 1.067 0.533 0.142

*The unit of ¶ is in kg1/4 m3 s–1/2 mol–1.

EXAMPLE 4.2 Calculate the surface tension of ethyl acetate at 293 K using the parachor data. Given: rl = 900.63 kg/m3. Solution

Since Ul !! Uv at 293 K, Eq. (4.5) simplifies to J

Ë¥Ul Û Ì M Ü Í Ý

4

To calculate the value of ¶, let us use the values of the structural units given in Table 4.2. For ethyl acetate, CH3(COO)C2H5, we have Total ¶ = {9.869 + 11.345 + (9.869 + 7.113)} × 10–6 = 38.196 × 10–6 kg1/4 m3 s–1/2 mol–1. Therefore, J

Ë 38.196 – 10 6 – 900.63 Û Ì Ü 0.088 ÍÌ ÝÜ

The experimental value is 0.02397 N/m.

4

0.02335 N/m

Surface and Interfacial Tension

103

4.2.1 Effect of Temperature on Surface Tension The surface tension of most liquids decreases with increase in temperature. Since the forces of attraction between the molecules of a liquid decrease with the increase in temperature, the surface tension decreases with increase in temperature. One of the classical equations correlating the surface tension and temperature is the Eötvös–Ramsay–Shields equation, given by the following relationship:

J (vl )2 3

ke (T c T  6)

(4.6)

where vl is the molar volume of the liquid, Tc is the critical temperature and ke is a constant. For nonassociated liquids the value of ke is 2.12, and for associated liquids its value is less than this value. Thus, according to this equation, the surface tension will become zero at a temperature six degrees below the critical temperature, which has been supported experimentally. Theoretically, the value of surface tension becomes zero at the critical temperature since at this temperature the surface of separation between a liquid and its vapour disappears. However, it has been observed that the meniscus disappears a few degrees below the critical temperature for some liquids. The Brock and Bird (1955) correlation relates surface tension to the critical properties of the liquid by the relationship

J

Pc2 3Tc1 3Q(1  Tr )11 9

(4.7)

Ë (T / T ) ln( Pc /101325) Û 7 5.55134 – 10 8 Ì1  b c Ü  1.295 – 10  1 / T T b c Í Ý

(4.8)

where Q is given by, Q

where Tb is the normal boiling point of the liquid, Pc is the critical pressure and Tr is the reduced temperature. The quantity J Pc 2 3Tc1 3 k 1 3 (where k is Boltzmann's constant) is dimensionless. It was suggested by van der Waals that this group may be correlated with the quantity (1 – Tr). This method of estimation of surface tension is also known as corresponding states method. The Brock–Bird method is not suitable for liquids which exhibit strong hydrogen bonding (such as alcohols and acids). Sastri and Rao (1995) presented the following correlations for surface tension of such liquids. For alcohols, they proposed

J

T Ø È 1 É Ù T c 1.282 – 10 4 Pc0.25Tb0.175 É Ù É 1  Tb Ù ÉÊ Tc ÙÚ

0.8

(4.9)

For acids, the proposed correlation is

J

T Ø È 1 É Tc Ù 3.9529 – 10 7 Pc0.5Tb1.5Tc1.85 É Ù É 1  Tb Ù ÉÊ Tc ÙÚ

11 9

(4.10)

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Colloid and Interface Science

For any other type of liquid, the following correlation was suggested by them: 11 9

T Ø È 1 É Ù T c J 4.9964 – 10 7 Pc0.5Tb1.5Tc1.85 É Ù (4.11) T É1 b Ù ÉÊ Ù Tc Ú Note that the correlations given by Eqs. (4.8)–(4.11) have been modified from their original source as per the requirements of the SI units. EXAMPLE 4.3 Estimate the surface tensions of acetic acid, aniline and ethanol at 293 K using Brock–Bird and Sastri–Rao correlations. The required properties of the liquids are given below. Pc = 5.74 × 106 Pa, Tb = 391.1 K Acetic acid: Tc = 591.95 K, Aniline: Tc = 699 K, Pc = 5.35 × 106 Pa, Tb = 457.4 K Ethanol: Tc = 513.92 K, Pc = 6.12 × 106 Pa, Tb = 351.4 K Solution The values of surface tension for these three liquids using the two correlations are estimated as follows: Acetic acid (Brock–Bird correlation): Ë (391.1 / 591.95) ln(5.74 – 10 6 /101325) Û 7 Q = 5.55134 – 10 8 Ì1  Ü  1.295 – 10 1  391.1 / 591.95 ÌÍ ÜÝ

g = (5.74 – 10 6 )2 / 3 (591.95)1/ 3 Q(1  293 / 591.95)11/ 9 Acetic acid (Sastri–Rao correlation):

3.6239 – 10 7

0.0423 N/m

È 1  293 / 591.95 Ø g = 3.9529 – 10 7 (5.74 – 10 6 )0.5 (391.1) 1.5 (591.95)1.85 É Ê 1  391.1 / 591.95 ÙÚ

11/ 9

0.0268 N/m

The experimental value is 0.0274 N/m. The errors are given below. For Brock–Bird correlation:

H

J calc  J expt J expt

– 100

0.0423  0.0274 – 100 0.0274

54.4%

– 100

0.0268  0.0274 – 100 0.0274

2.2%

For Sastri–Rao correlation:

H

J calc  J expt J expt

Aniline (Brock–Bird correlation): Ë (454.4 / 699) ln(5.35 – 10 6 /101325) Û 7 Q = 5.55134 – 10 8 Ì1  Ü  1.295 – 10 1 457.4 / 699  ÍÌ ÝÜ

g = (5.35 – 10 6 )2 / 3 (699)1/ 3 Q(1  293 / 699)11/ 9 Aniline (Sastri–Rao correlation):

3.4289 – 10 7

0.0479 N/m

g = 4.9964 – 10 7 (5.35 – 10 6 )0.5 (457.4) 1.5 (699)1.85 ÈÉ 1  293 / 699 ØÙ Ê 1  457.4 / 699 Ú

11/ 9

0.0408 N/m

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105

The experimental value is 0.0429 N/m. The errors are given below. For Brock–Bird correlation:

H

J calc  J expt J expt

– 100

0.0479  0.0429 – 100 11.7% 0.0429

– 100

0.0408  0.0429 – 100 0.0429

For Sastri–Rao correlation:

H

J calc  J expt J expt

4.9%

Ethanol (Brock–Bird correlation): Ë (351.4 / 513.92) ln(6.12 – 10 6 /101325) Û 7 Q = 5.55134 – 10 8 Ì1  Ü  1.295 – 10 1  351.4 / 513.92 ÌÍ ÜÝ

g = (6.12 – 10 6 )2 / 3 (513.92)1/ 3 Q(1  293 / 513.92)11/ 9 Ethanol (Sastri–Rao correlation):

4.18258 – 10 7

0.0399 N/m

È 1  293 / 513.92 Ø g = 1.282 – 10 4 (6.12 – 10 6 )0.25 (351.4)0.175 É Ê 1  351.4 / 513.92 ÙÚ

0.8

0.0227 N/m

The experimental value is 0.0223 N/m. The errors are given below. For Brock–Bird correlation:

H

J calc  J expt J expt

– 100

0.0399  0.0223 – 100 0.0223

– 100

0.0227  0.0223 – 100 1.8% 0.0223

78.9%

For Sastri–Rao correlation:

H

J calc  J expt J expt

The results from this example indicate that the Brock–Bird correlation gives large errors for associated liquids such as acids and alcohols. For other liquids, it gives fairly accurate results. Therefore, this correlation is not suitable for associated liquids. On the other hand, the Sastri–Rao correlations give good results for both associated and non-associated compounds. A computer program is presented in the Appendix of this chapter that uses these correlations to compute surface tension at a given temperature. The variation of surface tension with temperature for some organic liquids is presented in Figure 4.4. It can be observed from this figure that the Sastri–Rao correlations fit the data very well for propanol, however, the error is large for aniline and pyridine. Estimation of surface tension using the parachor data [Eq. (4.5)] usually gives good results. Its simplicity has made it a popular method for calculating surface tension. Several works have used this equation as the basis to improve its accuracy and incorporate temperature dependence. Escobedo and Mansoori (1996) suggested that ¶ is a function of temperature. They proposed the following temperature-dependence for ¶. ¥ ¥0 f (Tr ),

Tr = T/Tc

(4.12)

where ¶0 is independent of temperature but it depends upon the physical properties of the compound such as its critical temperature, pressure, normal boiling point and molar refraction.

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Colloid and Interface Science

Figure 4.4

4.3

Variation of surface tension with temperature. The lines indicate fit by the Sastri–Rao correlation.

INTERFACIAL TENSION

Let us consider two immiscible liquids in contact with each other as shown in Figure 1.1. The molecules at the surface of both of these liquids experience unbalanced forces of attraction as explained in Figure 4.1. These unbalanced forces at the surface of separation between the two immiscible liquids (i.e. at the interface) give rise to interfacial tension. It can be defined in the same way as the surface tension. Antonoff's rule predicts that the interfacial tension (g AB) between two liquids A and B will be equal to the difference between the respective surface tensions (i.e. g A and g B).

J AB

J A J B

(4.13)

Therefore, it can be anticipated from this rule that the interfacial tension should lie between the surface tensions of the two liquids. This is indeed found to be true for some liquids (such as water and carbon tetrachloride). However, for many liquids, this prediction does not hold (e.g. water and insoluble aliphatic alcohols). Girifalco and Good (1957) incorporated the effects of the free energies of cohesion of the two phases and the free energy of adhesion on interfacial tension. They proposed the following equation for interfacial tension: J AB

J A  J B  2 ) J AJ B

(4.14)

where F is a constant, which is defined as )



a 'GAB

'GAc 'GBc

(4.15)

a where 'GAB is the free energy of adhesion for the interface between the phases A and B, 'GAc is the free energy of cohesion for phase A, and 'GBc is the free energy of cohesion for phase B. For many liquid–liquid systems F lies between 0.5 and 1.2. It has been observed that for non-associated liquids, F lies between 0.5 and 0.8, and for the associated liquids, the value of F is higher.

Surface and Interfacial Tension

107

If the two phases are composed of spherical or nearly-spherical molecules, F can be related to the molar volumes of the liquids by the following equation: )

4(v A vB )1 3

(4.16) 2 Ë (v A )1 3  (vB )1 3 Û Í Ý The values of interfacial tension for several organic liquids with water, and the values of F are presented in Table 4.3. Table 4.3

Interfacial tension of organic liquids with water at 293 K

Organic Liquid Aniline Benzaldehyde Benzene Bromobenzene Carbon disulphide Carbon tetrachloride Chlorobenzene Chloroform Cyclohexane Cyclohexanol Decalin Dichloromethane Ethyl acetate Ethyl bromide Iodobenzene Isoamyl alcohol Isobutyl alcohol Isopentane Mesitylene m-Nitrotoluene m-Xylene n-Amyl alcohol n-Butyl acetate n-Butyl alcohol n-Decane n-Heptane n-Hexane Nitrobenzene n-Octane n-Pentane Octanoic acid o-Nitrotoluene o-Xylene p-Xylene Tetrachloroethylene Toluene

Interfacial tension (mN/m) 5.8 15.5 35.0 38.1 48.4 45.0 37.4 31.6 50.2 3.9 51.4 28.3 6.8 31.2 41.8 4.8 2.0 48.7 38.7 27.7 37.9 4.4 14.5 1.8 51.2 50.2 51.1 25.7 50.8 49.0 8.5 27.2 36.1 37.8 47.5 36.1

F 0.98 0.90 0.72 0.69 0.58 0.61 0.70 0.76 0.55 1.04 0.55 0.80 1.08 0.78 0.66 1.11 1.15 0.59 0.67 0.79 0.69 1.09 0.97 1.13 0.55 0.55 0.55 0.81 0.55 0.58 1.03 0.79 0.71 0.69 0.59 0.71

An improvement of the correlation of Girifalco and Good was proposed by Fowkes (1964). He suggested that in strongly polar or metallic liquids (such as mercury) the nonpolar dispersion forces

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Colloid and Interface Science

must be treated differently from the hydrogen bonding or metallic forces. Fowkes assumed that the interaction between the immiscible liquid phases A and B involve only dispersion forces (see Chapter 5). The interfacial tension was given by the following equation: J AB

J A  J B  2 J Ad J Bd

(4.17)

where g d is the contribution from dispersion forces to the surface tension of the pure liquid. For a nonpolar liquid, g = g d. Therefore, this provides a method to determine the unknown g d. The interfacial tension between the strongly polar (or metallic) liquid and a nonpolar liquid is measured to calculate g d using Eq. (4.17). Nonpolar hydrocarbons can be used for this purpose. EXAMPLE 4.4

d For mercury, J Hg

200 mN/m, and for water, J Hd 2 O

22 mN/m at 293 K. Using

these values, compute the interfacial tension between water and mercury at 293 K. Solution

From Table 4.1, the surface tensions of water and mercury at 293 K are

J H2O = 72.8 mN/m g Hg = 476 mN/m From Eq. (4.17), we have d g = J H2O  J Hg  2 J Hd 2OJ Hg

72.8  476  2 22 – 200

416.1

The experimental value of interfacial tension reported in the literature is 426 mN/m.

4.4

CONTACT ANGLE AND WETTING

If a small drop of liquid is placed on a uniform flat solid surface it will, in general, not spread completely over the surface. However, its edge will make an angle (q) with the solid as shown in Figure 4.5. The angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid is called contact angle for that liquid–solid combination. It strongly depends upon the nature of the liquid and the solid, and can have values between zero and p rad. For Water on a hydrophilic surface, the contact angle is less than p/2 rad [Figure 4.5(a)]. On the other hand, if the solid surface is hydrophobic, the contact angle will be greater that p/2 rad [Figure 4.5(b)]. The contact angle for water and glass is p/10 rad, and the same for mercury and glass is 7p /9 rad. If the liquid is very strongly attracted to the solid surface (e.g. water on a strongly hydrophilic solid) the droplet will almost completely spread out on the surface. In that case, the contact angle will be close to zero. If the surface of the solid is less hydrophilic, the contact angle may be close to p/2 rad. On highly hydrophobic surfaces in contact with water, the contact angle can be as high as 5p /6 rad or even higher. On these surfaces, a water droplet will simply rest on the surface without actually wetting the surface to any significant extent.

Figure 4.5 Contact angle.

Surface and Interfacial Tension

109

Let us consider the thermodynamic equilibrium between three phases: air, liquid and solid as depicted in Figure 4.5(c). Here, gAL, gAS and gLS are the interfacial energies (or tensions) at the air– liquid, air–solid and liquid–solid interfaces respectively. At equilibrium, we have J AS

Therefore,

J LS  J AL cos T

cos T

(4.18)

J AS  J LS J AL

(4.19)

Equation (4.18) is known as Young’s equation or Young–Dupré equation. If g AS > g LS, then q < p/2 rad and if g AS < g LS, then q > p/2 rad. If another liquid is present instead of air, the contact angle is defined in a similar manner. The methods for measuring contact angle are discussed in Section 4.7. Knowledge of contact angle is very important in the study of wetting and adhesion. For example, how well a polymer coating will repel solvents is important to the coating manufacturers. The wetting of the biological fluids of the drug powders is important in pharmaceutical industry. A good insecticide should wet the waxy leaves of the plants to protect them from the insects and disease. Contact angle depends on the roughness of the surface and surface heterogeneities. Semiempirical correction factors are often applied to Young's equation for the real surfaces.

4.5

SHAPE OF THE SURFACES AND INTERFACES

To describe the shape of a curved surface or interface, it is necessary to know the radii of curvature. The spherical and cylindrical bodies are rather simple cases for mathematical treatment. In many interfacial phenomena, however, the shapes are more complicated. In this section, the radii of curvature at a point on a curved surface are discussed. The use of principal radii of curvature in quantifying some interfacial properties is illustrated.

4.5.1 Radius of Curvature Consider the curve AB as shown in Figure 4.6. Let P be a point on this curve. The radius of curvature of AB at P is defined as the radius of the circle which is tangent to the curve at point P (i.e. the osculating circle). If Rc is the radius of curvature at point P, the curvature (H) at this point is defined as (4.20) H 1 / Rc Let (a, b) be the coordinates of the centre of the tangent circle at P.

Figure 4.6

Radius of curvature.

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Colloid and Interface Science

The equation of the circle is

( x  a)2  ( z  b)2

Rc2

(4.21)

Let f be the angle between the normal to the curve and the z-axis. tan I



dz dx

(4.22)

Differentiating Eq. (4.21) with respect to x, we get È x  aØ É Ê z  b ÙÚ

dz dx

(4.23)

From Eqs. (4.22) and (4.23), we get xa zb

tan I

Therefore,

1

cos I

(4.24) 1

1  tan I

12

2

2 Û1 2

Ë È x  aØ Ì1  É Ù Ü ÌÍ Ê z  b Ú ÜÝ

d cos I dz

and

zb Rc

1 Rc

(4.25)

(4.26)

From Figure 4.6 it can be observed that cosf decreases with increase in z. Thus, the curvature H (º 1/Rc) is negative. This is consistent with the convention that a curve which is concave outward has a negative curvature and a curve that is convex outward has a positive curvature. The curvature can be expressed in terms of the derivative of z with respect to x as follows: cos I

Therefore,

d cos I dz

1

1

1  tan I 2

12

d Ë È dz Ø Ì1  É Ù dz ÌÍ Ê dx Ú

(4.27)

12

Ë È dz Ø 2 Û Ì1  É Ù Ü ÌÍ Ê dx Ú ÜÝ

2 Û 1 2

1

È dz Ø ÉÊ ÙÚ dx

Ü ÜÝ

È dz Ø È d z Ø 2É Ù É 2 Ù Ê dx Ú Ê dx Ú

d Ë È dz Ø Ì1  É Ù dx ÌÍ Ê dx Ú

2 Û 1 2

Ü ÜÝ

(4.28)

2

d cos I dz



2 È dz Ø Ë È dz Ø Û 2 É Ù Ì1  É Ù Ü Ê dx Ú Ì Ê dx Ú Ü Í Ý

1 Rc

“

3/2



d 2 z dx 2 Ë È dz Ø 2 Û Ì1  É Ù Ü ÌÍ Ê dx Ú ÜÝ

d 2 z dx 2 Ë È dz Ø 2 Û Ì1  É Ù Ü ÌÍ Ê dx Ú ÜÝ

32

3/2

(4.29)

(4.30)

Surface and Interfacial Tension

111

The ± sign has been used in the above equation to emphasise the fact that it is necessary to adjust the sign depending on the geometry. For the curve AB, the negative sign is appropriate. Let us now consider a curved surface. At each point on a given surface, two radii of curvature (which are denoted by r1 and r2) are required to describe the shape. If we want to determine these radii at any point (say P), the normal to the surface at this point is drawn and a plane is constructed through the surface containing the normal. This will intersect the surface in a plane curve. The radius of curvature of the curve at point P is denoted by r1. An infinite number of such planes can be constructed each of which intersects the surface at P. For each of these planes, a radius of curvature can be obtained. If we construct a second plane through the surface, containing the normal, and perpendicular to the first plane, the second line of intersection and hence the second radius of curvature at point P (i.e. r2) is obtained. These two radii define the curvature at P completely. It can be shown that (1/r1 + 1/r2) (called mean curvature of the surface) is constant, which is independent of the choice of the planes. An infinite set of such pairs of radii is possible. Thus, it is not a useful way to describe the shape. For standardisation, the first plane is rotated around the normal till the radius of curvature in that plane reaches minimum. The other radius of curvature is therefore maximum. These are the principal radii of curvature (denoted by R1 and R2). For practical purpose, we will work with the principal radii of curvature. In many interfacial applications, we come across surfaces of revolution. A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. The resulting surface has azimuthal symmetry. Examples of surfaces of revolution are cone (excluding the base), cylinder (excluding the ends) and sphere. Let us consider one such surface shown in Figure 4.7. This surface was obtained by revolving the curve AB about the z-axis. At any point P on this surface R1 is given by 1 R1

“

Figure 4.7

d 2 z dx 2 Ë È dz Ø 2 Û Ì1  É Ù Ü Ê dx Ú Ü ÍÌ Ý

32

Surface of revolution.

(4.31)

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Colloid and Interface Science

The other principal radius of curvature is PQ which is obtained by extending the normal to the curve AB to intersect the z-axis.

R2

x sin I

PQ

(4.32)

We can express R2 in terms of dz/dx as follows: sin I

Ë 1 Û Ì1  2 Ü Í tan I Ý

1 2

Ë 1  dz dx 2 Û Ì Ü ÌÍ dz dx 2 ÜÝ

1 2

dz dx 12

Ë1  dz dx 2 Û Í Ý

(4.33)

From Eqs. (4.32) and (4.33), we get 1 R2

“

dz dx 12

2 x Ë1  dz dx Û Í Ý

(4.34)

The sign needs to be chosen appropriately.

4.5.2

Young–Laplace Equation

There exists a difference in pressure across a curved surface which is a consequence of surface tension. The pressure is greater on the concave side. The Young–Laplace equation relates the pressure difference to the shape of the surface. This equation is of fundamental importance in the study of surfaces, and can be derived easily. Let us consider a small portion of a curved surface (ABCD) shown in Figure 4.8. The surface has been cut by two planes perpendicular to one another. Each of the planes contains a portion of the arc where it intersects the surface. The lengths of the arcs are x and y. The radii of curvature are shown in the figure. The planes have been chosen in such a manner that these radii are the principal radii of curvature (R1 and R2). Now, the surface is displaced outward to a new position (A¢B¢C¢D¢) by a small distance dz such that the arc-lengths are increased by dx and dy. Therefore, the change in area is

dA

( x  dx )( y  dy)  xy

xdy  ydx  dxdy  xdy  ydx

(4.35)

The term dxdy is very small since both dx and dy are small quantities. The work done to form this additional amount of surface is,

dW

J ( xdy  ydx )

(4.36)

Suppose that pressure–volume work is responsible for the expansion of the surface. If the pressure difference across the surface acting on the area xy through a distance dz is Dp, then the pressure– volume work responsible for the expansion of the surface is

dW

'pdV

'pxydz

(4.37)

'pxy dz

(4.38)

From Eqs. (4.36) and (4.37), we get

J ( x dy  y dx)

Surface and Interfacial Tension

Figure 4.8

113

Displacement of surface ABCD to A¢B¢C¢D¢.

Now, we can observe from Figure 4.8 that x

R1[ and x  dx

( R1  dz )[

(4.39)

Therefore, it follows that x x  dx

R1 R1  dz

(4.40)

R2 R2  dz

(4.41)

In a similar manner, it can be shown that y y  dy

Simplifying Eqs. (4.40) and (4.41), we get dx x

dz dy R1 and y

dz R2

(4.42)

Substituting dx and dy from Eq. (4.42) into Eq. (4.38) and simplifying, we get

'p

È 1 1Ø JÉ  Ù Ê R1 R2 Ú

(4.43)

This is the Young–Laplace equation developed independently by Thomas Young (in 1804) and Pierre Laplace (in 1805) [a nice history of this equation has been presented by Pujado et al. (1972)]. Substituting Eqs. (4.31) and (4.34) into Eq. (4.43), we get a differential equation. The solution of this differential equation relates the shape of an axisymmetric surface to the surface tension. This equation, therefore, suggests the possibility of measuring surface tension from the analysis of the

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Colloid and Interface Science

shape of the surface. The simplified forms of Eq. (4.43) for spherical, cylindrical and planar surfaces are given below. • For a spherical surface, R1 = R2 = Rs, therefore, 'p

2J R s .

• For a cylindrical surface, R1 = Rcy and R2 = ¥, therefore, 'p • For a planar surface, R1 = R2 = ¥, therefore, 'p 0 .

J Rcy .

EXAMPLE 4.5 Calculate the pressure inside a 1 mm diameter droplet of water in air at 293 K and atmospheric pressure. Solution

Here, J

72.8 – 10 3 N/m and Rs = 0.5 × 10–6 m. Therefore, 'p

2 – 72.8 – 10 3 0.5 – 10 6

291.2 – 10 3 Pa

291.2 kPa

The atmospheric pressure is 101.3 kPa. Therefore, the pressure inside the droplet is 291.2 + 101.3 = 392.5 kPa. In the rest of this section, we will apply the Young–Laplace equation to a few well-known surfaces of revolution to determine their shapes. The following example illustrates the shape of a soap film stretched between two parallel rings. EXAMPLE 4.6 Consider a soap film stretched between two parallel circular rings having equal diameter as shown in Figure 4.9. Determine its shape neglecting the effects of gravity.

Figure 4.9

Soap film stretched between two parallel circular rings: (a) photograph (Isenberg, 1992) [reproduced by permission from Dover Publications, Inc., © 1992], and (b) schematic sketch.

Surface and Interfacial Tension

Solution

Since the rings are open on both sides, Dp = 0. Therefore, from Eq. (4.43), at all points 1 1  R1 R2

Now,

115

0

1 R1

d cos I dz

1 R2

sin I x

d cos I dx dx dz

d sin I sin I  dx x

Therefore,

1 1  R1 R2

Thus,

1 d x sin I x dx

 cot I

d cos I dx

cos I

dI dx

d sin I dx

1 d x sin I x dx

0

d x sin I 0 dx Integrating with respect to x, we get x sinf = C, C = constant At z = 0, x = d. Here the normal intersects the z-axis at right angle. Therefore, f = p/2 and sinf = 1. Therefore, C = d. Hence, x sinf = d

\

dz dx Now,

sin I

\

dz dx

 tan I



sin I cos I 12

2  Ë1  G x Û Í Ý

G x , cos I

,

f ³ p/2

G x 12

Ë1  G x 2 Û Í Ý

The above equation can be solved with the boundary condition: at z = 0, x = d z

12 Ë 2 ÞÑ Û x ÎÑÈ x Ø Ì “ G ln   1ß Ü Ì G ÏÑÉÊ G ÙÚ Ñà ÜÝÜ Ð ÍÌ

This is the equation of the catenary. The solution can also be written as x G

È zØ cosh É Ù ÊG Ú

4.5.3 Pendant and Sessile Drops When a drop is formed at the tip of a vertical tube (e.g. a burette), its size is mainly determined by the surface tension (or interfacial tension, if the tip is dipped inside another liquid). The drop slowly

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Colloid and Interface Science

forms at the tip of the tube, grows in size and then a ‘neck’ forms. Thereafter, the drop detaches from the tip. The drop suspended from the tube is known as a pendant drop. It is illustrated in the photograph shown in Figure 4.10(a) for a water drop in air. A similar profile of the drop can be observed when it is suspended from a plate (e.g. droplets formed by the condensation of vapour). If a drop is placed on a flat plate, it rests on the plate as shown in Figure 4.10(b) for a mercury drop. It is known as a sessile drop. The drop will deform from its spherical shape as it settles on the solid. From the profiles of pendant and sessile drops, it is possible to determine the surface and interfacial tension. The sessile drop method is widely used for measuring the contact angle. The contact angle reflects the affinity between the liquid and the solid. If the drop is flat, it indicates a high affinity and the liquid is said to ‘wet’ the solid surface. The contact angle is small in this case. A more rounded drop indicates lesser affinity because the angle at which the drop is attached to the solid surface is larger. The liquid is said to be ‘non-wetting’ to the solid in such a situation. Highly accurate instruments are available which can analyse the profiles of pendant and sessile drops. We will discuss these techniques in detail later in this chapter. In this section, we will consider the profiles of sessile drops.

Figure 4.10

Pendant and sessile drops: (a) photo of a pendant water drop, (b) photo of a sessile mercury drop, and (c) schematic diagram of a sessile drop.

Let us consider a drop sitting on a smooth solid surface inside another liquid. The foregoing treatment will remain the same if the drop sits in air. Surface tension would be used in that case instead of interfacial tension. The drop is deformed by gravity. It is assumed that no other external force acts on the drop. In absence of the gravitational force, the drop would remain spherical. The centre of mass of the drop is forced to be lowered by gravity. However, this causes the surface area to increase (since the spherical shape occupies the least surface area), which is opposed by surface tension force. The equilibrium shape depends upon the balance of the two forces. Let us consider Figure 4.10(c) which represents the profile of a sessile drop. The actual surface may be generated by rotating the profile around the axis of symmetry. The origin of the coordinate system is O, which is located at the apex of the surface. The two radii of curvature at point P are R1 and R2. From Eq. (4.32), we have x

R2 sin I

(4.44)

At the point O, R1 = R2 = Ra. Therefore, at this point ( 'p)O

2J Ra

(4.45)

Surface and Interfacial Tension

117

At the point P ( 'p) P

( 'p)O  'U gz

(4.46)

where Dr is the difference in densities of the two phases. If the drop rests in air, Dr » r (where r is the density of the liquid). Substituting (Dp)O from Eq. (4.45) into Eq. (4.46), we get ( 'p) P

2J / Ra  'U gz

(4.47)

From Young–Laplace equation,

È sin I 1 Ø JÉ  Ù R1 Ú Ê x

2J  'U gz Ra

(4.48)

where R1 is given by Eq. (4.31). This equation can be rearranged in the following form: 1 sin I  R1 / Ra x / Ra

2E

z Ra

(4.49)

where b is given by E

'U gRa2 J

(4.50)

b is a dimensionless number which is known as Bond number. It represents the ratio of the gravity force to the force due to surface tension. A small value ( r2, b is positive and the drop is oblate in shape. In this case, the weight of the drop flattens the surface [e.g. the mercury drop shown in Figure 4.10(b)]. If r1 < r2, b is negative, and the shape is prolate. The buoyant force, in this case, causes the surface to elongate in the vertical direction (e.g. a sessile bubble extended into liquid). In a similar manner, it can be deduced that a pendant drop will be prolate and a pendant bubble will be oblate. EXAMPLE 4.7 Calculate Bond number for a sessile carbon tetrachloride drop inside water at 298 K if Ra = 3 mm. Comment on the shape of the drop from the value of the Bond number. Solution To calculate the Bond number, the values of the densities of carbon tetrachloride (r1) and water (r2), and interfacial tension are required. U1

J E

1600 kg/m3,

U2

1000 kg/m3

45 – 10 3 N/m,

Ra

3 – 10 3 m

1600  1000 – 9.8 – (3 – 10 3 )2 45 – 10 3

1.18

This value of b suggests that the drop will be deformed significantly.

4.5.4 Capillary Rise or Depression The rise of water through a capillary immersed in a vessel filled with water was shown in Figure 1.2. The interfacial energy of the water–liquid interface was lower than the glass–air interface in that case (i.e. J LS  J AS ). The height of the liquid column inside the capillary depends on the radius of the tube, surface tension, density of the liquid and contact angle. Let us consider the capillary

Surface and Interfacial Tension

119

(exaggerated for illustration) shown in Figure 4.12 in which the liquid rises to an equilibrium height h above the air–liquid interface. The radius of the capillary is rc. The liquid continues to rise until the vertical component of the lifting force due to surface tension becomes equal to the weight of the liquid column in the capillary.

Figure 4.12

Capillary rise.

The lifting force is, FL

2S rc J cos T

(4.51)

Weight of the liquid in the capillary is, (4.52) FW S rc2 h U g At equilibrium, the lifting force is equal to the force due to the weight of the liquid. Therefore,

2S rc J cos T

S rc2 h U g

(4.53)

The height of the liquid above the air–liquid interface can be calculated by rearranging Eq. (4.53) h

2J cos T U grc

(4.54)

Equation (4.54) is valid for capillary rise as well as capillary depression. In the former case, q lies between 0 and p/2 rad, and in the latter case, q lies between p/2 and p rad. Therefore, one can calculate the surface tension by measuring h and q. If the liquid completely wets the wall of the capillary, q = 0, and the capillary rise will be given by h 2J U grc . A depression of the same height will be observed if q = p rad. It is apparent that a liquid will rise significantly inside a capillary which has a very small radius (as shown in Figure 1.2). If the radius of the capillary is increased, the height of the liquid column would decrease. Eq. (4.54) is also valid for a two-liquid system. In that case, r in this equation needs to be replaced by the density difference between the two liquids, Dr. The following example illustrates the use of Eq. (4.54). EXAMPLE 4.8 Estimate the height of water inside a capillary tube of 0.5 mm radius. Take: g = 72 mN/m and q = 0. Solution

Since q = 0, using Eq. (4.54), we have

h

2J U grc

2 – 72 – 10 3 1000 – 9.8 – 0.5 – 10 3

0.029 m = 2.9 cm

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Colloid and Interface Science

Therefore, water will rise approximately 2.9 cm inside the tube. It is left as an exercise for the reader to explain why this height is approximate. The capillary rise equation can also be derived using the Young–Laplace equation. The horizontal surface shown in Figure 4.12 can be taken as the reference level at which Dp = 0. The pressure just under the meniscus in the capillary is less than the pressure on the other side of the surface due to the curvature of the surface. Therefore, the pressure in the liquid just under the curved surface is less than the pressure at the reference level. This causes the liquid in the capillary to rise until a compensating hydrostatic pressure is generated by the liquid column inside the capillary. Since the capillary has circular cross-section and its radius (rc) is small, the meniscus can be approximated by a cap of a hemisphere of radius, rc /cos q. Therefore, R1 = R2 = rc /cosq. Thus, from Young–Laplace equation, we have

È 1 1Ø JÉ  Ù Ê R1 R2 Ú

'p

2J cos T rc

(4.55)

The hydrostatic pressure difference is Dp = hrg

(4.56)

From Eqs. (4.55) and (4.56), we can obtain Eq. (4.54). The term 2J U g is known as capillary constant or capillary length. Equation (4.54) suggests that every point on the meniscus is at the same height h from the surface of the liquid reservoir, or in other words, the meniscus is flat! A more accurate derivation should take into account the deviation of the meniscus from sphericity considering the elevation of each point above the flat surface of the liquid. This involves the solution of the general Young– Laplace equation with R1 and R2 given by Eqs. (4.31) and (4.34) respectively.

4.5.5

The Kelvin Equation

The pressure difference associated with the curved surfaces has an important effect on the thermodynamic activity of substances. A very important implication of the curvature of the surface is in the vapour–liquid equilibrium. The Kelvin equation gives quantitative information regarding the effect of surface curvature on vapour pressure. The vapour pressure across a flat surface is psat, the saturated vapour pressure of the liquid at the given temperature. The vapour pressure across a curved surface is, however, different from the saturated vapour pressure. Let us consider the drop shown in Figure 4.13. The Young–Laplace equation predicts a pressure difference across the spherical meniscus given by 'p pl  pv 2J Rd , where Rd is the radius of the drop. At a constant temperature, if Rd varies, then we have dpl  dpv

Figure 4.13

2J d (1 / Rd )

Illustration of vapour–liquid equilibrium across a spherical meniscus.

(4.57)

Surface and Interfacial Tension

121

Since equilibrium is maintained during this process, the changes in the chemical potentials of the liquid and the vapour must be equal. Therefore, d Pl

d Pv

vl dpl

(4.58)

vv dpv

where vl and vv are the molar volumes of the liquid and vapour respectively. If we assume that the vapour behaves as an ideal gas and vl  vv , then from Eqs. (4.57) and (4.58), we can write È 1 Ø 2J d É Ù Ê Rd Ú

RT dpv vl pv

(4.59)

where R is the gas constant. Integrating Eq. (4.59) using the condition that the vapour pressure in the case of the flat interface (i.e. Rd  ‡ ) is the saturated vapour pressure of the liquid (psat) we obtain pv p

sat

Ë 2v J Û exp Ì l Ü Í RRd T Ý

Ë 2 MJ H Û exp Ì Ü Í RT Ul Ý

(4.60)

where M is the molecular weight, r l is the density of the liquid and H is the mean curvature. Equation (4.60) is known as Kelvin equation. The sign of H determines whether pv /psat would be greater or less than unity. For the drop shown in Figure 4.13, the liquid is on the concave side of the meniscus (the radius is measured in the liquid). Therefore, H = 1/Rd. However, for a bubble in a liquid, the liquid is on the convex side of the meniscus. Therefore, H = –1/Rb (where Rb is the radius of the bubble). Thus, the foregoing discussion suggests that the vapour pressure of a drop will be greater than the saturated vapour pressure of the liquid (i.e. pv /psat > 1). On the other hand, the vapour pressure inside a bubble will be less than the saturated vapour pressure (i.e. pv /psat < 1). The effect of curvature of the surface is pronounced for very small drops and bubbles. Example 4.9 illustrates this point. EXAMPLE 4.9 Calculate pv /psat from Kelvin equation for water drops at 298 K having diameters of 1 mm and 1 nm. Solution In the present case, r l = 1000 kg/m3, M = 18 × 10–3 kg/mol, R = 8.314 J mol–1 K–1, g = 72.5 × 10–3 N/m and T = 298 K. (i) For 1 mm diameter drop 1 Rd

H

1 0.5 – 10 6

2 – 10 6 m–1

From Eq. (4.60), we have pv psat

Ë 2 MJ H Û exp Ì Ü Í RT Ul Ý

Ë 2 – 0.018 – 72.5 – 10 3 – 2 – 10 6 Û exp Ì Ü 8.314 – 298 – 1000 ÍÌ ÝÜ

(ii) For 1 nm diameter drop H

1 Rd

1 0.5 – 10 9

2 – 10 9

1.002

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Colloid and Interface Science

From Eq. (4.60), we have pv psat

Ë 2 MJ H Û exp Ì Ü Í RT Ul Ý

Ë 2 – 0.018 – 72.5 – 10 3 – 2 – 10 9 Û exp Ì Ü 8.314 – 298 – 1000 ÌÍ ÜÝ

8.223

From these results it is observed that the effect of surface curvature on vapour pressure is not so significant for 1 mm drops, but it is very significant for 1 nm drops. Kelvin equation is used for determining the pore-volume distribution in porous solids. Consider a pore in the solid having radius Rp which is filled with a liquid (see Figure 4.14). For simplicity, let us assume that the contact angle is zero so that the liquid–vapour meniscus is a hemisphere of radius Rp. The liquid is in the convex side of the meniscus. Therefore, H = –1/Rp. From Kelvin

Ë 2 MJ Û psat exp Ì  Ü , where p(Rp) is the vapour pressure of the liquid trapped ÌÍ R p RT Ul ÜÝ inside the pore of radius Rp. The porous solid has pores of different radii. So, if we place the solid in an environment where the vapour pressure is maintained at p(Rp), ideally the liquid will condense into all pores having radius Rp or less. From the mass of the liquid condensed and its density, the volume of liquid that condensed into the pores having radius less than or equal to Rp can be calculated. Suppose that this volume is V(Rp). Since the value of p(Rp) is known, Rp can be calculated from the Kelvin equation. If the vapour pressure is maintained at another value, the volume of liquid condensed into the pores having radii corresponding to this vapour pressure can be calculated. These data representing the variation of V(Rp) with Rp generate the cumulative pore-volume distribution. The pore-volume distributions have important applications in catalysis Figure 4.14 Condensation of liquid inside a pore. and transport through porous media. equation, p( R p )

4.6

MEASUREMENT OF SURFACE AND INTERFACIAL TENSION

The measurement of surface and interfacial tension is one of the most fundamental works for any study in interface science. In this section, we will discuss some of the widely used techniques for measuring these properties.

4.6.1

The Drop-Weight Method

In this method, a drop is allowed to form slowly at the end of a tube having a fine capillary inside it. Then it is slowly released and collected in a container. Several drops (e.g. 100 drops, at the rate of one drop in about 200 seconds) are collected in the same manner and the weight of the liquid is measured. From this weight, the average weight of a drop is calculated. The classical instrument for carrying out this measurement in the laboratory is Stalagmometer. Now-a-days, computer-controlled instruments can form precise drops. The principle behind the drop-weight method is as follows. As the size of the drop at the tip of the tube grows, its weight goes on increasing. It remains

Surface and Interfacial Tension

123

attached to the tube due to surface tension which acts around the circumference of the tube in the upward direction. When the downward force due to gravity acting on the drop becomes slightly greater than the surface tension force, the drop detaches from the tube. Therefore, Upward force = 2prog (4.61) Downward force = mg (4.62) where ro is the outer radius of the tip of the tube. At the point of detachment, mg = 2prog (4.63) Equation (4.63) is known Tate’s law. The use of Eq. (4.63) requires the measurement of ro. To avoid this, a relative method is usually used. A liquid whose surface tension is known is used as a reference liquid. Highly-purified water or an ultrapure organic liquid is used for this purpose. Therefore, the surface tension is calculated from the following equation: È m Ø ÉÊ m ÙÚ J ref ref

J

(4.64)

where g ref is the surface tension of the reference liquid and the weight of a drop of this liquid is mref. When Eq. (4.63) is used to measure surface tension, corrections need to be applied because only a portion of the drop (the larger portion) falls from the tube. A significant amount of liquid (usually upto 40%) may remain attached to the tip of the tube. A correction factor, F , known as Harkins– Brown factor (Harkins and Brown, 1919), is used to correct the surface tension obtained from Eq. (4.63). J

F mg 2S ro

(4.65)

where F is a function of ro/V1/3 (V = volume of the drop), which has been verified experimentally using different liquids and tips of different radii. It has been found that F does not depend upon the viscosity of the liquid. The variation of F with ro/V1/3 is shown in Figure 4.15. The radius of the tip should be chosen such that F is least sensitive on the variation of ro/V1/3. From Figure 4.15, it is observed that the correction factor can be quite large. Since the value of V can differ from liquid to liquid, F may differ from one liquid to another even when the same tip is used. Therefore, the ratio of correction factors in the relative method may not be unity.

Figure 4.15 Harkins–Brown correction factor for the drop-weight method.

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Colloid and Interface Science

Several precautions must be taken to ensure that the tip is smooth at the end. The liquid should either completely wet the tip or do not wet it at all. In the latter case, the internal radius of the tip should be used as ro. The tip must be very clean. The presence of a very small amount surface-active substance can cause significant amount of error in the measurement. The drop-weight method described above can be used to measure the interfacial tension as well. In this case, the drop of the heavier liquid is formed inside the lighter liquid. EXAMPLE 4.10 Determination of surface tension of water at 298 K by drop-weight method. The volume of a water drop was determined to be 3 × 10–8 m3 for a tube having tip-radius of 0.9 mm. Calculate the surface tension of water from these data. Solution

Here,

ro

V1 3 the surface tension is

J

4.6.2

0.29 . From Figure 4.15, we obtain F = 1.37. Therefore, from Eq. (4.65),

F mg 2S ro

FV U g 2S ro

1.37 – 3 – 10 8 – 1000 – 9.8 2 – S – 9 – 10 4

0.0712 N/m

du Noüy Ring Method

This is one of the most widely-used methods for measuring the surface tension. The method is named after the French physicist who developed it in the late 1800s. The measurement is performed by an instrument known as Tensiometer. This instrument has an accurate micro-balance and a precise mechanism to vertically move the sample liquid in a glass beaker. The modern tensiometer has a computer-controlled arrangement that can move the table holding the liquid very slowly (~100 mm/s). The ring is usually made of an alloy of platinum and iridium with well-defined geometry. The measurement procedure is as follows. The ring hanging from the hook of the balance is first immersed into the Figure 4.16 Illustration of du Noüy ring method. liquid and then carefully pulled up by lowering the sample vessel (Figure 4.16). The micro-balance continuously records the force applied on the ring when it pulls through the air–liquid interface. The surface tension is the maximum force needed to detach the ring from the liquid surface. The detachment force is equal to the surface tension multiplied by the periphery of the surface detached. Therefore, for a ring F = 4pRrg

(4.66)

where Rr is the radius of the ring. The force measured by the balance includes the weight of the ring. In actual practice, the weight of the ring is first recorded before it is immersed in the liquid. Sometimes, a calibration is made with a known weight. Usually the results obtained from Eq. (4.66) are in error. Harkins and Jordan (1930) derived a correction factor f such that the correct surface tension can be obtained from the following equation:

J

È F Ø ÉÊ 4S R ÙÚ f r

(4.67)

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125

The correction factor appears due to the weight of the liquid film immediately beneath the ring, which is also raised when the ring pulls [see Figure 4.16]. The correction factor depends upon the complex shape of the meniscus during the detachment of the ring, density of the liquid, radius of the ring (Rr) and the radius of the wire (rw) with which the ring is made. Huh and Mason (1975) have graphically presented the variation of f with Rr3 V (where V is the volume of the meniscus) and Rr /rw. The correlation given by Zuidema and Waters (1941) for f is given below. 12

f

Ë 0.00363J Expt È r ØÛ  0.04534  1.679 É w Ù Ü 0.725  Ì 2 2 Ê Rr Ú ÜÝ ÌÍ S Rr 'U

(4.68)

In Eq. (4.68), the radii Rr and rw are expressed in cm, Dr is expressed in kg/dm3 and g Expt (the experimentally measured value of surface tension) is expressed in mN/m. The Zuidema–Waters correlation gives accurate results when g Expt < 35 mN/m and Dr > 0.1 kg/dm3. Both Huh–Mason and Zuidema–Waters corrections are used by the tensiometer manufacturers. Some precautions must be taken while applying the ring method. Equations (4.67) and (4.68) assume that the contact angle is zero, i.e. the liquid should completely wet the ring. To ensure this, the platinum–iridium ring is cleaned by burning it in a Bunsen burner. The ring is quite delicate and prone to distortion during use. Such distortions should be avoided and it must be ensured that the ring lies flat on a quiescent surface. When used with the surfactant solutions, the ring must be cleaned thoroughly with pure water since the presence of a small amount of surfactant can cause a significant amount of error in the measurement. If the ring is used with viscous oils such as silicone oil or crude petroleum, it must be cleaned with a good solvent (such as acetone) to dissolve and remove the oil. The advantage of the ring method is that it is rapid, very simple and does not need to be calibrated using solutions of known surface tension. When applied to pure liquids with due precautions, the error can be reduced to ± 0.25%. EXAMPLE 4.11 The interfacial tension for water–nitrobenzene system measured by a du Noüy ring has been found to be 26.1 mN/m. Calculate the correction factor f by Zuidema–Waters equation and then calculate the corrected value of the interfacial tension. Given data for the ring: Rr = 0.9545 cm and rw = 0.0185 cm. Solution

From Eq. (4.68), we get 12

f

Ë 0.00363J Expt È r ØÛ 0.725  Ì  0.04534  1.679 É w Ù Ü 2 2 Ê Rr Ú ÝÜ ÍÌ S Rr 'U

For the water–nitrobenzene system, Dr = 0.2 kg/dm3. Therefore, putting the values of Rr, rw, Dr and g Expt in this equation, we get 12

f

Ë 0.00363 – 26.1 È 0.0185 Ø Û 0.725  Ì 2  0.04534  1.679 – É 2 Ê 0.9545 ÙÚ ÝÜ Í S – (0.9545) – (0.2)

0.981

Therefore, the corrected value of interfacial tension is

J

f J Expt

0.981 – 26.1 25.6 mN/m

Note that this value agrees well with the value reported in Table 4.3 (i.e. 25.7 mN/m).

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4.6.3

Wilhelmy Plate Method

This method is named after the German chemist Ludwig Wilhelmy. It is similar to the du Noüy ring method. However, it is simpler and does not require the correction. In this method, a thin plate (usually made of platinum and iridium) is used. It is dipped into the liquid whose surface tension is to be measured [Figure 4.17]. The vessel containing the liquid is gradually lowered and the force measured by the Figure 4.17 Illustration of Wilhelmy plate method. balance at the point of detachment is noted. The Wilhelmy plate is sometimes used in another way. In this approach, the liquid level is raised until it just touches the hanging plate. The force recorded on the balance is noted. The Wilhelmy equation is J

F P cos T

(4.69)

where P is the wetted perimeter of the plate, and q is the contact angle. The contact angle is reduced to near-zero values (so that the liquid wets the plate completely) by cleaning the plate by burning it in the flame of Bunsen burner before each experiment. If the contact angle is close to zero, Eq. (4.69) simplifies to, g = F/P. As mentioned before, the Wilhelmy plate method does not need any correction because the weight of the film hanging from the plate is negligibly small. This method can be used for measuring interfacial tension also.

Advantages and Disadvantages of du Noüy Ring and Wilhelmy Plate Methods Historically, the ring method has been widely used for measuring both surface and interfacial tension. There exist ASTM standards which require the ring method. Many older generation manually-operated tensiometers use the ring method. The ring method has three main problems associated with it. The first two problems: the need for correction, and tendency to deform during use, have been discussed in Section 4.6.2. The third problem arises in surfactant solutions. The ring is designed for keeping the surface in a non-equilibrium state. When the ring is pulled through the surface [see Figure 4.16], it expands the surface searching for the maximum force in the liquid meniscus. Therefore, the measurement of surface tension is performed on a surface that is in a non-equilibrium state. If the surface tension of a pure liquid is being measured, it does not affect the results. However, for surfactant solutions, the expansion of the surface affects the orientation of the surfactant molecules at the surface, and therefore, the results may be inaccurate. The measured surface tension of surfactant solutions can vary with the speed at which the ring is pulled. This is due to the fact that the surfactant molecules require time to orient properly and adsorb at the surface. This time varies from surfactant to surfactant. This problem can be minimised by applying a very slow speed of pulling the ring. However, the measured surface tension is likely to be different from the equilibrium value. The plate method measures equilibrium surface tension. It does not require the correction for the meniscus. It can be placed right at the surface of the liquid and not moved while the surface tension is measured. Therefore, the surfactants are given sufficiently long time to reach the equilibrium state. For this advantage, the plate method often gives superior accuracy in the measurement.

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For interfacial tension measurements, when the plate is wetted with one liquid, zero contact angle with another liquid is not ensured. Therefore, the plate has to pull a meniscus to create zero contact angle, and then the disadvantages of the ring method appear. In this case, the ring method has some advantage since the ring has a higher wetted perimeter. The force measured with the ring is larger than that measured by the plate. This provides more accuracy for the ring method. However, the interface may still not be at equilibrium, which can be important for the surfactant solutions. One remedy is to use a very slow speed, as mentioned before.

4.6.4 Maximum Bubble Pressure Method The Wilhelmy plate method gives the equilibrium (static) surface tension. Surfactant solutions require a significantly higher amount of time than the pure liquids to achieve this equilibrium. In applications such as foaming, cleaning or coating, the interfaces are formed very quickly. For such applications, the dynamics of rapid formation of interface is important, which depend on the mobility of the surfactant molecules (see Section 6.10). The maximum bubble pressure method (MBPM) is an easyto-use technique for measuring the dynamic surface tension (DST). In the MBPM method, gas bubbles are produced in the sample liquid at an exactly-defined rate of generation. The bubble blown at the end of a capillary is stable, and expands with the increasing pressure of the gas in the bubble. The pressure reaches a maximum when the bubble is hemispherical and its radius is equal to the radius of the capillary. If the maximum pressure is pmax (whose value is recorded by the instrument), and the hydrostatic pressure is po, then the following equation gives the dynamic surface tension (Fainerman et al., 1994). J

( pmax  po ) rc 2

(4.70)

where rc is the inner radius of the capillary. After reaching the maximum, the pressure decreases, and the size of the bubble increases. As the size of the bubble becomes larger than a hemisphere, it becomes unstable because the equilibrium pressure within it decreases as it grows. Such a bubble expands further, escapes and rises through the liquid. The entire cycle of bubble formation, its growth and release is repeated. Some of the stages of bubble growth and the corresponding pressures are shown in Figure 4.18. Figure 4.18 Schematic diagram of the variation of pressure The growth of the bubble and its at various stages of bubble formation in the MBPM method. separation can be divided into two time periods. During the first period, the surfactant molecules adsorb on the surface of the bubble and the surface tension varies accordingly. This period is called surface lifetime. The second period is the time in which the bubble grows rapidly and finally separates from the capillary. This period is known as dead time. Modern instruments use electronic sensors for measuring the pressure and the frequency of bubble formation. These instruments can record surface lifetime as small as 0.001 s (which is important when the

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air-liquid interface is formed very quickly). Therefore, the bubbles can be formed very rapidly by these instruments. The details of the various sophisticated techniques used in these measurements are available in the review of Miller et al. (1994). The dynamic surface tension of aqueous Triton X-100 (a nonionic surfactant) solution is shown in Figure 4.19.

Figure 4.19 Dynamic surface tension of aqueous Triton X-100 solution (1.6 mol/m3).

Flow in capillaries and in porous media is affected by DST. This finds importance in enhanced oil recovery where aqueous foams are often used to increase the sweep-efficiency in carbon dioxide flooding. In bioprocessing systems, DST affects the rate of water-oxygenation by influencing the mass transfer coefficient. DST is also important in metal and textile processing, pulp and paper production, and pharmaceutical formulations. One important application of dynamic surface tension is for lung surfactants. The dynamic tension under constant or pulsating area conditions controls the health and stability of the alveoli. In the formulation of pesticides, if the aqueous spray has a low DST, it can be dispersed into smaller droplets, which will spread more easily on the leaves. For these reasons, surfactants are used as pesticide spraying aids known as adjuvants.

4.6.5 Spinning-Drop Method In this method, a small drop is placed in a denser liquid enclosed in a glass tube which is subjected to rotation at a high angular velocity (say, 1000 rad/s) about its horizontal axis. The method is based upon the principle of gyrostatic equilibrium, which is the state of uniform rotation in which every bit of the fluid inside the tube is at rest with respect to the wall of tube. Gyrostatic equilibrium is achieved at high angular velocities when the gravitational force perpendicular to the axis of rotation is negligible as compared with the centrifugal force. When the tube rotates with high velocity, the drop migrates to the axis of rotation and assumes a cylindrical shape with hemispherical ends. For each angular velocity, the drop comes to an equilibrium shape which is characteristic of that velocity. Some of the shapes of the drop are shown in Figure 4.20 at different angular velocities. The drop cannot elongate indefinitely. When the interfacial tension everywhere balances the centrifugal force that produces the pressure difference across the interface, the elongation ceases. The equilibrium condition exists because the force due to the rotating field tending to elongate the drop is proportional to the fourth power of the radius of the drop whereas the opposing interfacial tension force is proportional to the first power of the radius.

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Figure 4.20

Photographs of a heptane drop in glycerol at different angular velocities (Princen et al., 1967) [reproduced by permission from Elsevier Ltd., © 1967].

If the densities of the heavier and lighter liquids are r1 and r2, angular velocity is w, and the equatorial radius of the cylindrical drop is Re (see Figure 4.21), the following formula given by Vonnegut (1942) can be used to calculate the interfacial tension.

J

Figure 4.21

( U1  U2 )Z 2 Re3 4

(4.71)

Schematic diagram of the drop under rotation.

If the ratio of the length (Ld) and the diameter of the drop exceeds 4, Eq. (4.71) can be used to determine interfacial tension with reasonable accuracy (Manning and Scriven, 1977). The advantage of the spinning drop method is that it can be used to measure very low interfacial tensions (~10–6 mN/m). Such low interfacial tensions are encountered in applications such as microemulsions. The spinning drop method has also been used in systems such as polymer melts, bitumen, crude oil and other organic solvents. In modern spinning drop tensiometers, the interfacial tension range is about 10–6 – 50 mN/m. The angular velocity can be as high as 1500 rad/s. Typical radius of the rotating tube is 0.1–0.2 cm. EXAMPLE 4.12 The interfacial tension between an oil and water is 50 mN/m. The density of oil is 850 kg/m3. The tube that contains an oil drop in water has 1 mm radius. If the angular velocity is 1047 rad/s, calculate the radius of the cylindrical drop. Comment on your results. Solution Here, g = 0.05 N/m, Dr = 150 kg/m3, and w = 1047 rad/s. From Vonnegut equation [Eq. (4.71)], 13

Re

Ë 4J Û Ì 2Ü ÍÌ 'UZ ÝÜ

13

Ë 4 – 0.05 Û Ì 2Ü Í 150 – (1047) Ý

1.067 – 10 3 m = 1.067 mm

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Since the tube has only 1 mm radius, the drop will not come off the wall of the tube. A tube having larger radius should be used to measure the interfacial tension.

4.7

MEASUREMENT OF CONTACT ANGLE

The plate method and the sessile drop method are the two most widely used methods for measuring the contact angle. The Wilhelmy plate method has been described in Section 4.6.3 for measuring surface tension. For the measurement of surface tension, the contact angle should be close to zero. A platinum plate is usually used because it has almost-zero contact angles with most liquids. The same method can be used to measure the contact angle of symmetric plate-shaped solids (i.e. plates whose both sides are identical) if it is dipped in a liquid of known surface tension [as shown in Figure 4.22]. Using Eq. (4.69), the contact angle can be determined. T

Ë F Û cos 1 Ì Ü Í 2J (l  t ) Ý

(4.72)

where l is the horizontal length of the plate (when it is suspended from the balance) and t is its thickness. The wetted perimeter is 2(l + t). The surface tension of the liquid is first measured using the platinum Wilhelmy plate. When the sample advances into the liquid, the contact angle measured from the force recorded by the balance is called the advancing contact angle [Figure 4.22(a)]. The sample is immersed to a certain depth and then the process is reversed. As the sample retreats from the liquid, the contact angle measured is called receding contact angle [Figure 4.22(b)]. The profile of the force per wetted perimeter versus the depth of immersion is shown in Figure 4.22(c). When the sample is above the liquid, the force is set to zero. When the sample touches the surface of the liquid, the liquid climbs up the sample causing a positive force if the contact angle is less than p/2 rad. As the sample is immersed into the liquid, the buoyant force increases causing a decrease in the force recorded by the balance. During this process, the forces are measured for the advancing angle (qa). After reaching the desired depth, the sample is pulled out of the liquid. During this process, forces are measured for the receding angle (qr). The contact angle measured in this way is

Figure 4.22

Measurement of contact angle by Wilhelmy plate method.

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131

called dynamic contact angle (DCA). The dynamic contact angle can be measured at various velocities. Dynamic contact angle measured at low velocities is generally expected to be close to the static (equilibrium) contact angle. The sessile drop method is a frequently-used method for measurement of contact angle. The traditional manually-operated equipment is known as contact goniometer. It is a simple contact angle apparatus. This method uses a protractor to align a tangent along the three-phase (i.e. solid, liquid and vapour) contact-point to determine the contact angle of liquid–solid systems. The error associated with using this instrument can be quite large by the present standards, and it varies depending on the expertise of the user. Modern instruments use digitisation of the drop profile along with numerical integration of the Young–Laplace equation (see Section 4.5.2) to generate the bestfit curve. It is a versatile, repeatable and accurate technique for determining the interfacial tension and contact angle. Several scientists have discussed various aspects of this technique. Some of them can be found in the article by Dingle and Harris (2005). To measure the advancing and receding contact angles by the sessile drop method, drops with advanced and receded edges can be generated. Drops can be made to have advanced edges by addition of liquid, and the receded edges may be produced by allowing sufficient evaporation or by withdrawing liquid from the drop. Alternatively, both advanced and receded edges can be produced when the stage on which the solid is held is tilted to the point of incipient motion. Using an instrument with high-speed image-capture capabilities, the shapes of drops can be analysed. The biggest advantage of the goniometer method is that it can be used to measure contact angle between any liquid and any non-porous solid. If the size of the surface is large, this method is good for studying surface heterogeneity. The droplets can be placed at different locations on the surface and the contact angles can be measured at those locations. Another advantage of this method is that the contact angle can be measured quickly and easily. The method has a few disadvantages as well. For conventional goniometers, determination of the contact angle is dependent upon the subjective placement of two lines: one characterising the edge of the drop and the other characterising the plane of contact between the drop and the solid surface. The reproducibility in the measurement depends upon the criteria used to place these lines. With computerised goniometers, these criteria are based upon the analysis of the image of the drop. Factors such as illumination intensity, focus, contrast, refractive indices of the materials and the reflectance of the solid surface affect the measurement. In practice, the non-reflective surfaces pose the biggest measurement problems because the placement of the solid–liquid contact line becomes somewhat nebulous in such a situation. For small contact angles, the error can be as large as 0.1 rad. The Wilhelmy plate method is easy to use with the modern computer-controlled tensiometers. This method is easier to apply for small contact angles since the measured parameter in this method is force, and the force decreases with increase in contact angle. The Wilhelmy method is free from the subjectivity of the goniometer method because no contact line needs to be set. In the Wilhelmy method, the measured contact angle is the average over the entire wetted length of the sample. This inherent averaging process makes the contact angle measurements more reproducible than the data obtained from goniometer. However, it is not suitable for studying the heterogeneity of the surface simultaneously. The main disadvantage of the Wilhelmy method is the sample preparation. The sample must have well-defined cross-section and its wetted length must be known precisely to apply Eq. (4.72). Another important disadvantage is that the two surfaces of the plate must be identical. Therefore, it may be difficult to study the samples with one side coated (such as the asymmetric membranes). If the two surfaces are different, the results will reflect some average contact angle representative of both the surfaces. Sometimes, the sample is folded or bonded back-to-back so that

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only one type of surface is exposed to the liquid. Such preparations of the sample may prove to be cumbersome at times. Basing on Young–Dupré equation [Eq. (4.18)] one would expect that only a single value of contact angle for a particular solid–liquid–gas system should exist. However, it has been observed in practice that qa and qr can be quite different. The difference between them is called contact angle hysteresis. This phenomenon is known for nearly 100 years. The reasons behind hysteresis have been attributed to surface roughness, impurities on the surface, microscopic chemical heterogeneity, molecular reorientation and the penetration of the liquid molecules into the solid surface [see Miller and Neogi (2008) for a detailed discussion on some of these factors]. However, with the development of sophisticated instruments for testing the solid surface (e.g. atomic force microscopy), it has been observed that not only evidently-rough and heterogeneous surfaces cause hysteresis, but different advancing and receding contact angles are observed even on molecularly-smooth surfaces. On smooth and low-energy surfaces such as Teflon, remarkable hysteresis of water and other liquids has been observed. The advancing and receding contact angles of some liquids on the surface of fluorocarbon FC-732 are presented in Table 4.5. These contact angles were measured by the sessile drop method (Chibowski, 2003). Table 4.5 Contact angle of some liquids on FC-732 Liquid Hexane Decane Tetradecane Ethanol Pentanol Octanol Decanol

qa (rad)

qr (rad)

0.92 1.17 1.29 1.22 1.30 1.37 1.39

0.71 1.08 1.19 0.71 0.85 0.95 1.22

Modern theories suggest that hysteresis of contact angle on smooth solid surfaces for liquids such as alkanes or alcohols is due to the sorption of the liquid by the solid surface. Therefore, contact angle hysteresis is likely to depend upon the molecular size of the liquid and it occurs due to the penetration of the liquid and surface swelling.

SUMMARY This chapter presents the fundamental concepts of surface and interfacial tension, and the experimental techniques to measure them. The relationship between surface tension and surface energy is explained. Surface tension data for several liquids are presented. The variation of surface tension with temperature is discussed. The use of parachor for calculating surface tension is illustrated with example. The uses of Brock–Bird and Sastri–Rao correlations are illustrated with examples. A computer program for estimation of surface tension at a given temperature using these two correlations is presented in the Appendix of this chapter. The procedures for the estimation of interfacial tension using Girifalco–Good and Fowkes equations are discussed. Interfacial tensions between organic liquids and water are presented. Contact angle and wetting phenomena on a solid surface are discussed. The radii of curvature of a surface are discussed and their mathematical expressions in terms of the space coordinates are derived. The Young–Laplace equation is derived and its significance is illustrated with examples. The shapes of pendant and sessile drops are

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discussed. Determination of the profile of a sessile drop using the Bashforth–Adams tables is explained. The theory of capillary action is discussed. The Kelvin equation and its use for calculation of pore-volume distribution in a porous solid are analysed. The next part of this chapter describes the various techniques for measuring surface and interfacial tension, such as drop-weight method, du Noüy ring method, Wilhelmy plate method, maximum bubble pressure method and the spinning-drop method. The correction factors used in the first two methods are discussed. The advantages and disadvantages of the du Noüy ring and Wilhelmy plate methods are discussed. The measurement of dynamic surface tension with the maximum bubble pressure method, and the measurement of very low interfacial tension with the spinning-drop method are explained. The last topic discussed in this chapter is the measurement of contact angle. The Wilhelmy plate method and sessile drop method are discussed. The advancing and receding contact angles, and hysteresis of contact angle are illustrated with examples.

KEYWORDS Advancing contact angle Antonoff's rule Bashforth–Adams equation Bond number Brock–Bird correlation Capillary action Capillary constant Contact angle Contact angle hysteresis Corresponding states Dead time Dispersion force Drop-weight method du Noüy ring method Dynamic contact angle Dynamic surface tension Eötvös equation Fowkes equation Girifalco–Good correlation Goniometer Gyrostatic equilibrium Harkins and Jordan correction Harkins–Brown correction Huh and Mason correction Hydrogen bonding Interfacial tension

Intermolecular force Kelvin equation Maximum bubble pressure method Parachor Pendant drop Pore-volume distribution Principal radius of curvature Radius of curvature Receding contact angle Sastri–Rao correlation Sessile drop Spinning-drop method Stalagmometer Surface energy Surface force Surface lifetime Surface tension Tate's law Tensiometer Vonnegut equation Wetting Wilhelmy plate method Young–Dupré equation Young–Laplace equation Zuidema and Waters correction

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NOTATION a A b f F FL FW g h H k ke l Ld m M p po pl pmax psat p(Rp) pv P Pc ¶ ¶0 Q r1 r2 rc ro rw R R1 R2 Ra Rc Rb Rcy Rd Re Rp Rr

distance along the x-axis, m area, m2 distance along the z-axis, m Harkins–Jordan correction factor force, N lifting force, N force due to gravity, N acceleration due to gravity, m/s2 height, m curvature, m–1 Boltzmann’s constant, J/K constant in Eq. (4.6), J mol–2/3 K–1 length, m length of the elongated drop, m mass of a drop, kg molecular weight, kg/mol pressure, Pa hydrostatic pressure, Pa pressure in the liquid phase, Pa maximum pressure, Pa saturated vapour pressure, Pa vapour pressure of the liquid trapped inside the pore of radius Rp, m vapour pressure, Pa wetted perimeter, m critical pressure, Pa parachor, kg1/4 m3 s–1/2 mol–1 parameter in Eq. (4.12), kg1/4 m3 s–1/2 mol–1 parameter in Eq. (4.7), (J/K)1/3 first radius of curvature, m second radius of curvature, m radius of capillary outer radius of the tip of the tube, m radius of the ring wire, m gas constant, J mol–1 K–1 first principal radius of curvature, m second principal radius of curvature, m radius of curvature at the apex of the drop, m radius of curvature, m radius of bubble, m radius of cylinder, m radius of drop, m equatorial radius of drop, m pore radius, m radius of the ring, m

Surface and Interfacial Tension

Rs t T Tb Tc Tr v

radius of sphere, m thickness, m temperature, K normal boiling point, K critical temperature, K reduced temperature molar volume, m3/mol

vl

molar volume of liquid, m3/mol

vv V V(Rp)

molar volume of vapour, m3/mol volume, m3 volume of liquid that condensed into pores having radius less than or equal to Rp, m3 work, J distance along the x-axis, m distance along the y-axis, m distance along the z-axis, m

W x y z

Greek Letters b g gd d DGa DGc Dp Dr e q qa qr m

Bond number surface or interfacial tension, N/m contribution from dispersion force to interfacial tension, N/m minimum radius of film, m free energy of adhesion, J/mol free energy of cohesion, J/mol pressure difference, Pa density difference, kg/m3 error contact angle, rad advancing contact angle, rad receding contact angle, rad chemical potential, J/mol

[ r rl rv f ) F w

angle, rad density, kg/m3 density of liquid, kg/m3 density of vapour, kg/m3 angle, rad constant in Eq. (4.14) Harkins–Brown correction factor angular velocity, rad/s

135

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EXERCISES 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. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

Explain the origin of surface tension from a molecular point of view. What is the relationship between surface tension and surface energy? Explain why a drop or a bubble assumes spherical shape. What is the surface tension of water at room temperature? How does the surface tension of a liquid vary with temperature? What is parachor? Explain the salient features of Brock–Bird and Sastri–Rao correlations. What is the major drawback of the Brock–Bird correlation? Explain the Girifalco–Good correlation for estimation of interfacial tension. Explain the Fowkes correlation for the estimation of interfacial tension. Explain what you understand by contact angle. How does the contact angle reflect the surface properties of a solid material? Write the Young–Dupré equation for contact angle. Explain radius of curvature and curvature. Explain how the radius of curvature on a curved surface can be determined. Explain the principal radii of curvature. Explain how the pressure difference across a curved surface can be related to the principal radii of curvature. Explain what you understand by a pendant and a sessile drop. What are the forces that act on a pendant and a sessile drop? What is Bond number? Explain the significance of the Bashforth–Adams differential equation. Explain the shapes of sessile and pendant bubbles. What is capillary constant? Explain the role of contact angle in capillary rise and depression. What conclusions can you draw from the Kelvin equation? Explain why the vapour pressure of a drop is greater than the saturated vapour pressure of the liquid. Explain how the Kelvin equation may be used to generate the pore-volume distribution of a porous solid. Explain why nanoparticles sinter easily. Name three experimental methods to measure the surface and interfacial tension. Explain why a correction factor is required in the drop-weight method. What precautions will you take to measure surface tension by drop-weight method? Explain how you can measure interfacial tension by the drop-weight method. Explain how surface tension is measured by the du Noüy ring method. Explain the Wilhelmy plate method for measuring surface tension. Explain why a correction factor is required in the du Noüy ring method, but not in the Wilhelmy plate method. How will you ensure a near-zero contact angle while using the Wilhelmy plate method for measuring surface tension? What is the wetted perimeter of Wilhelmy plate? Discuss the advantages and disadvantages of the du Noüy ring and Wilhelmy plate methods. Explain the important aspects of the maximum bubble pressure method.

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40. Explain how the maximum bubble pressure method can be used to measure dynamic surface tension. 41. Explain the terms surface lifetime and dead time used in the maximum bubble pressure method. 42. Explain how interfacial tension is measured by spinning-drop method. 43. Write the Vonnegut equation and explain its terms. 44. What is gyrostatic equilibrium? 45. Under what conditions will the drop fail to detach from the wall of the tube in the spinningdrop method? 46. Explain how the contact angle of a solid substance can be measured. 47. What are advantages and disadvantages of the Wilhelmy plate method for measuring contact angle? 48. Explain how a goniometer is used to measure contact angle. 49. What are the advantages and disadvantages of goniometer for measuring the contact angle? 50. Explain advancing and receding contact angles. 51. What is contact angle hysteresis? Explain with two examples. 52. Why does the hysteresis in contact angle occur?

NUMERICAL AND ANALYTICAL PROBLEMS 4.1 If the surface tension of water is 72.8 mN/m, calculate the energy required to bring one molecule of water from the bulk to the surface. The area of a water molecule is about 0.1 nm2. 4.2 A water drop of 4 mm radius is split up into 100 tiny drops. Find the increase in surface energy. 4.3 Estimate the surface tensions of ethyl alcohol and nitrobenzene at 298 K using the parachor data. 4.4 Calculate the surface tensions of methyl alcohol and toluene at 293 K using the Brock–Bird and Sastri–Rao correlations. Collect the necessary data from a suitable Handbook. 4.5 Calculate the interfacial tension between water and aniline at 300 K using Girifalco–Good correlation. Compare your results with the experimental value given in Table 4.3. 4.6 Calculate the pressures inside air bubbles having 1, 10 and 100 nm radii in water at 298 K and atmospheric pressure. Comment on your results. 4.7 Two small bubbles, one having 10 mm diameter and the other having 100 mm diameter are connected by a capillary. Can you predict which bubble will collapse? Give reasons behind your prediction. Calculate the pressure difference if the surface tension is 30 mN/m. 4.8 Estimate the capillary pressures for pores having 2 mm average diameter for: (a) water, (b) a surfactant solution having surface tension of 30 mN/m, and (c) an aqueous solution of a surfactant, alcohol and a salt whose surface tension is 1 mN/m. Assume that the liquids wet the solid almost completely in each case. Analyse your results. 4.9 If the contact angles in the three systems given in Problem 4.8 are p/4, p/10 and p/20 rad respectively, recalculate the pressures and comment on your results. 4.10 A liquid does not wet the surface of a glass capillary significantly. Derive the equation for calculating the height to which the liquid would rise inside this capillary. Illustrate your answer with a diagram. 4.11 Calculate the heights up to which water will rise in capillaries having 1 mm and 0.1 mm radii.

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4.12 Write a computer program to solve the Bashforth–Adams differential equation. Using this program, generate the profile of sessile drop for b = 25 as shown in Figure 4.11. Also generate the profiles for b = 10 and b = –0.5. Calculate the maximum value of x/Ra in each case. 4.13 Calculate the vapour pressure of a water drop of 10 nm radius at 298 K. What will be the pressure inside the drop? Comment on your results. Given: saturated vapour pressure of water at 298 K is 3.17 kPa. 4.14 Estimate the vapour pressures of water at 323 K in bubbles having 1 nm, 10 nm and 100 nm radii. Given: saturated vapour pressure of water at 323 K is 12.3 kPa. 4.15 The surface tension of a liquid is to be determined at 293 K employing the drop-weight method using water as the reference liquid. The weight of 100 drops of water and the sample liquid were measured, and these values were 3.393 × 10–3 kg and 1.876 × 10–3 kg respectively. From these data, determine the surface tension of the sample liquid. Comment on the accuracy of the measurement. 4.16 The interfacial tension between water and cyclohexanol was measured by a du Noüy ring. The measured value was 4.2 mN/m (at 298 K). Calculate the correction factor by Zuidema-Waters correlation and determine the corrected interfacial tension applying this factor. The ringparameters are: Rr = 0.9545 cm and rw = 0.0185 cm. Collect the necessary physical properties from a Handbook. 4.17 The force measured by a platinum Wilhelmy plate during the measurement of surface tension of water is 1.595 × 10–3 N. The length and thickness of the plate are 10 mm and 1 mm respectively. Calculate the surface tension of water from these data. 4.18 The volume of a drop released from a glass tip of 0.9 mm radius is 22 × 10–9 m3. The density of the liquid is 876 kg/m3. Calculate the Harkins–Brown correction factor. Calculate the surface tension of the liquid. 4.19 In a spinning-drop tensiometer, the internal diameter of the tube is 1 cm. The densities of the two liquids are 1000 kg/m3 and 1260 kg/m3. The angular velocity is 210 rad/s. If the interfacial tension between the liquids is 20 mN/m, calculate the equatorial radius of the cylindrical drop. 4.20 Calculate the contact angle of a coated plastic with water by the Wilhelmy plate method if the measured force is 1.2 mN. The length and the thickness of the plastic sample are 1 cm and 1 mm respectively.

APPENDIX A computer program (written in C++) is presented here which can be used to calculate the surface tension of a liquid at a given temperature. The user needs to provide critical temperature (K), critical pressure (Pa), boiling point of the liquid (K) and the temperature (K) at which surface tension is to be calculated. The program implements Brock–Bird (1955) and Sastri–Rao (1995) correlations for the calculation. The user also needs to specify whether the liquid is an acid or alcohol for using the appropriate correlation. If the liquid is either an alcohol or an acid, only the Sastri–Rao correlation will be used to compute the surface tension. For other liquids, both correlations will be used.

Surface and Interfacial Tension

139

//Program 4A.1 //Program computes surface tension using //Brock-Bird (1955) and Sastri-Rao (1995) correlations # include # include # include void main () { int choice; double gamma, Pc, Q, T, Tb, Tc; double term; coutPc; coutTc; coutTb; coutT; coutchoice; term = (1-T/Tc)/(1-Tb/Tc); if (choice == 1) { gamma = 1.282e-4*pow(Pc,0.25)*pow(Tb,0.175)*pow(term,0.8); cout 4 nm2). In the gaseous phase, the hydrocarbon portions of the molecules make significant contact with the surface. As the monolayer is compressed, a long plateau arises, which is associated with the transition to a liquid phase. This is often called liquid-expanded (LE) phase. The plateau is predicted by the phase rule, which for the insoluble monolayers is similar to the threedimensional systems. When there are two phases present, a pure monolayer has a single degree of freedom. Therefore, if we fix the temperature, the surface pressure is fixed. In the LE phase, the hydrocarbon chains stand upon the surface in a disorderly manner.

Figure 8.4

Figure 8.5

Schematic diagram of Langmuir monolayer isotherm and the orientation of the molecules in different phases.

Surface pressure versus area-per-molecule isotherm of myristic acid monolayer at 294 K. The compression speed was 0.006 nm2 molecule–1 s–1 [Agrawal and Neuman (1988a)] (adapted by permission from Elsevier Ltd., © 1988).

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When the monolayer is compressed further, the liquid-condensed (LC) phase is formed. This phase, however, is not a liquid. The degree of alignment of the chains is higher than that in the LE phase. There is long-range order in this phase. The plateau is not horizontal, which indicates that the LE–LC transition is not of first-order. At even higher compressions (Am » 0.2 nm2), the LC phase is transformed to a phase which is similar to an ordered two-dimensional solid phase. The area per molecule corresponds closely to the packing of the chains found in the three-dimensional crystals of the surfactant. Knobler (1990) used fluorescence microscopy to study the morphology of Langmuir monolayers. Some of his results for pentadecanoic acid monolayers are presented in Figure 8.6. The pentadecanoic acid contained 1% 4-(hexadecylamino)-7-nitrobenz-2-oxa-1,3-diazole, which acted as the probe. The fluorescent probe was excited with laser and the images of the monolayer were detected with a high-sensitivity camera and recorded on a videotape. Figure 8.6(a) corresponds to the LE + G two-phase region. The image contains dark circular bubbles of gas in the white field representing the LE phase. The contrast between the two phases is due to the difference in density. The amount of gas decreases when the monolayer is compressed, as shown in Figure 8.6(b). At a sharply defined area (Am = 0.36 nm2), a completely-white field appears [Figure 8.6(c)] which indicates that all of the monolayer is in the LE phase. This one-phase region persists as Am is decreased further until dark circular domains of the LC phase appear abruptly [Figure 8.6(d)]. The difference in contrast reflects the low solubility of the probe in the LC phase. The fraction of the LC phase grows with increasing density as shown in Figure 8.6(e). If the concentration of the probe is low, the termination of LE + LC coexistence can be detected by the complete loss of the bright LE regions.

8.3.3 Model of Gas-Phase Monolayer When the monolayer is in the gaseous (G) phase, the number of molecules of surfactant in the monolayer is small. In the limit of low film pressure, the two-dimensional equivalent of the ideal gas law applies, which is given by S s Am

(8.2)

kT

where k is the Boltzmann’s constant and T is temperature. In the gaseous phase, the hydrocarbon tails lie almost flat on the surface (Figure 8.4). If the temperature and chain length are known, the surface pressure can be calculated from Eq. (8.2). At higher surface pressures, deviations from Eq. (8.2) similar to that for a real gas are observed. An equation similar to the van der Waals equation of state for real gases has been proposed to account for the excluded volume and intermolecular attractions (Hiemenz and Rajagopalan, 1997).

È a Ø É S s  2 Ù Am  b Am Ú Ê

kT

(8.3)

The parameters a and b are analogous to the van der Waals constants for real gases. This equation can connect the gaseous and liquid-expanded states in the monolayers. The transition between the two phases is equivalent to the critical point. EXAMPLE 8.1 Calculate the surface pressure in the gaseous phase at 300 K, if the length of the hydrocarbon chain of the surfactant molecule is 1.5 nm.

Monolayers and Thin Liquid Films

Figure 8.6

Solution

Fluorescence microscope images of pentadecanoic acid monolayers at 298 K (Knobler, 1990) [reproduced by permission from The American Association for the Advancement of Science and Professor Charles M. Knobler, © 1990].

Since the monolayer is in gaseous phase, therefore Am  S l 2

Ss

kT Am



S 1.5 – 10 9



2

1.381 – 10 23 – 300 7.07 – 10 18

7.07 – 10 18 m 2

5.86 – 10 4 N/m

277

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Colloid and Interface Science

8.3.4 Surface Potential The surface potential is a very important property of the charged monolayers. The usual practice is to measure it along with the surface pressure isotherm. The technique involves the measurement of the potential between the surface of the liquid and that of a metal probe. A popular technique is the vibrating-plate capacitor method (e.g. KSV-SPOT1 surface potential meter). The Helmholtz formula for the potential difference between two conducting plates separated by a distance d and a charge density s is given by 'V

Vd HH 0

(8.4)

where e is the dielectric constant and e0 is the permittivity of the free space. DV is proportional to the surface concentration, and the proportionality constant is a quantity characteristic of the film. The measured value of DV can be used as an alternative means for determining the concentration of molecules in a film and to ascertain whether a film is homogeneous or not. Fluctuation in the value of DV with position across the film may occur if two phases are present (Adamson and Gast, 1997). For weakly-ionised monolayers, the surface potential can be calculated by using the Grahame equation [Eq. (5.54)]. If the surface is considered as a uniformly-charged homogeneous plane with charge density s and the double layer ions are assumed to be single point charges, the Grahame equation gives

\

Ë Û VD È 2kT Ø 1 ÉÊ e ÙÚ sinh Ì ‡ 12Ü ÍÌ (8RT HH 0 c ) ÝÜ

(8.5)

where k is Boltzmann’s constant, T is temperature, e is electronic charge, a is the degree of dissociation in the monolayer, and c ‡ is the concentration of electrolyte in the subphase. EXAMPLE 8.2

Show that Eq. (8.5) simplifies to the following equation at 293 K. \

Ë 43.5D 0.05sinh 1 Ì Ì Am c ‡ Í

Û Ü Ü Ý

where c ‡ is the bulk concentration of the electrolyte (mol/m3), Am is the area per molecule at the surface (nm2), a is the degree of dissociation in the monolayer, and y is the surface potential expressed in volts. Solution

From Eq. (8.5), we have

Ë Û VD È 2kT Ø 1 y = ÉÊ e ÙÚ sinh Ì ‡ 12Ü ÍÌ (8RT HH 0 c ) ÝÜ e = 78.5, e0 = 8.854 × 10–12 C2 J–1 m–1, k = 1.381 × 10–23 J/K e = 1.602 × 10–19 C, R = 8.314 J mol–1 K–1, T = 293 K 2kT 2 – 1.381 – 10 23 – 293 = e 1.602 – 10 19

0.05

Monolayers and Thin Liquid Films

s =

1.602 – 10 19 9 2

Am – (1 – 10 )

0.16 Am

(8RT HH 0 c ‡ )1/ 2 = (8 – 8.314 – 293 – 78.5 – 8.854 – 10 12 – c ‡ )1/ 2 Therefore,

279

VD ‡ 1/ 2

(8RT HH 0 c )

0.16D Am

43.5D

0.00368 c ‡

Am c ‡

0.00368 c ‡

Thus, the simplified form of Eq. (8.5) is given by \

8.3.5

Ë 43.5D 0.05sinh 1 Ì Ì Am c ‡ Í

Û Ü Ü Ý

Monolayers at Liquid–Liquid Interfaces

Many studies have been made at the air–water interface due to the simplicity involved in the experiments. However, biological systems are approximated in a better way by the oil–water interface. Therefore, the films of proteins, lipids and steroids have been studied at oil–water interfaces. The protein layers are more expanded at water–oil interface than air–water interface. Davies (1954) has studied the monolayers of haemoglobin, serum albumin, gliadin and synthetic polypeptide polymers at water–petroleum ether interface. He observed that the molecules forming the monolayer were forced into the oil phase upon compression. Brooks and Pethica (1964) have developed a technique for compressing the monolayer at water–oil interface. They have used a hydrophobic Wilhelmy plate for measuring the interfacial tension. Barton et al. (1988) have studied stearic acid monolayers at water–mercury interface. They used grazing incidence X-ray diffraction method to study the monolayer. A modified design of the KSV Langmuir trough for studying monolayers at liquid–liquid interface has been presented by Galet et al. (1999).

8.4

LANGMUIR–BLODGETT FILMS

In 1920, Langmuir introduced the technique for transferring a floating monolayer to a solid surface by slowly raising the hydrophilic solid through the liquid surface. In 1934, Katharine Blodgett announced the discovery that sequential transfer of monolayer could be accomplished to build up multilayer films. These structures are now universally referred to as Langmuir–Blodgett (LB) films. In addition to the vertical deposition mode, Langmuir and Schaefer (1938) suggested a horizontal deposition method by which the floating monolayer can be transferred to a hydrophobic solid surface by allowing the horizontal solid surface to touch the monolayer. Only one monolayer can be deposited by this method. The Langmuir film-balance can be used for building-up highly organised multilayers. This is accomplished by successively dipping a solid substrate up and down through the monolayer while simultaneously keeping the surface pressure constant by a computer-controlled feedback system between the electrobalance measuring the surface pressure and the barrier-moving mechanism. In this way, multilayer structures of hundreds of layers can be produced. The deposition process is schematically shown in Figure 8.7. Traditionally, the deposition is carried out in the solid phase. The

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surface pressure is then high so that the monolayer does not fall apart during the transfer to the solid substrate. This also ensures the build-up of homogeneous multilayers. The value of surface pressure that gives the best results depends on the nature of the monolayer. It is usually established empirically. However, monolayers can seldom be successfully deposited at ps < 10 mN/m. At ps > 40 mN/m, monolayers can collapse and the film-rigidity can pose problems. When the solid substrate is hydrophilic (e.g. glass or SiO2), the first layer is deposited by raising the solid substrate from the liquid subphase through the monolayer. On the other hand, if the solid substrate is hydrophobic (e.g. silanised SiO2), the first layer is deposited by lowering the substrate into the subphase through the monolayer.

Figure 8.7

Formation of Langmuir–Blodgett film.

The quantity and the quality of the deposited monolayer on a solid substrate is measured by the transfer ratio (t). It is defined as (Peng et al., 2001) Decrease in area of Langmuir monolayer (8.6) Area of the transferred film on the solid substrate If all the area lost from the floating monolayer is a result of deposition (rather than loss through evaporation, dissolution or collapse), then t = 1, which indicates successful deposition, but it may not produce a well-ordered film. During vertical dipping to deposit monolayers on hydrophilic solid surfaces, both the film and a thin water layer are actually applied to the solid surface. Under certain conditions, the substrate is visibly wet immediately after the transfer. This layer of water may be expelled by either drainage or evaporation, leaving the monolayer on the solid substrate. The rate at which the water layer is removed is known as the speed of the deposition. Langmuir (1938) introduced the term zipper angle to refer to the angle formed by the water meniscus against the solid W

Monolayers and Thin Liquid Films

281

plate as it is withdrawn. If the plate emerges wet, the zipper angle is zero. A large zipper angle (> p/6 rad) is observed when the monolayer becomes tightly bound to the solid, expelling water rapidly in a zipper-like action. The zipper angle is the largest when the interaction between the monolayer and the solid is large, and the term ‘reactive’ has been used for such depositions. When the deposition of a monolayer takes place with an intervening hydrous layer, the zipper angle is almost zero and such a deposition is known as ‘nonreactive’ deposition. There are three types of LB deposition which are designated as X, Y and Z. When a solid plate is inserted in and out of a monolayer-covered liquid surface, it is often found that once the first layer has been deposited, an additional layer is deposited each time the plate enters or removed from the liquid. This two-way deposition is called Y-type deposition. This is shown in Figure 8.8. The molecules in the deposited film are arranged head-to-head and tail-to-tail as shown in the figure. Under certain conditions, a layer is deposited only as the plate enters the liquid. This is known as X-type deposition in which the molecules are arranged head-to-tail [Figure 8.8(a)]. The deposition is called Z-type if it only takes place as the plate is withdrawn from the liquid [Figure 8.8(c)]. Intermediate structures are sometimes observed for some LB multilayers. They are known as XYtype multilayers. Surface pressure, dipping speed, properties of the solid substrate and the pH of the subphase are some of the important factors that govern the LB films. A detailed discussion on these factors has been given by Binks (1991). The commonest and most easily produced multilayers are those from Y-type deposition. X-ray diffraction measurements have shown that the spacing of the metal ions incorporated in the film during deposition is nearly twice the single-layer thickness, confirming the head-to-head, tail-to-tail arrangement. The Y-type films exhibit high contact angles for water (> p/2 rad) (i.e. they are hydrophobic) which supports the structure shown in Figure 8.8(b) that the outer surface of the films is composed of the hydrophobic chain (Binks, 1991).

Figure 8.8

X-, Y- and Z-types of Langmuir–Blodgett films.

It might be expected from the deposition characteristics that the X-type films prepared by the one-way mechanism would have a different structure to the Y-films. Since deposition occurs only on the in-stroke, the outer surface should be hydrophilic as shown in Figure 8.8(a). However, for some materials (e.g. fatty acids) it has been found that these films are hydrophobic and give contact angles similar to those on Y-films (Langmuir, 1938). In addition, the spacing between metal ions in multilayers is the same whether the deposition is X- or Y-type. Therefore, it is likely that the molecules in these X-films rearrange during or after deposition to give a structure identical to that of the Y-type films. Only a few examples of the Z-type films have been reported. Substituted anthracene derivatives containing short chain carboxylic acid groups and molecules possessing an a-amino carboxylic acid

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Colloid and Interface Science

head-group and two amide groups along the chain have been reported to form Z-type multilayers under suitable conditions. Arachidic and behenic acid also deposit Z-type films onto freshly-cleaved mica surfaces when the subphase contains no added electrolyte at pH < 5 (Binks, 1991). The Langmuir–Blodgett films have potential applications in optical and electronic devices which are similar to the thin films produced by molecular beam epitaxy (MBE) or chemical vapour deposition (CVD) (see Section 11.6). The organic thin films have very bright future prospects because of the richness of chemical functionality. The LB films of phospholipids and proteins can be used for the development of biosensors and biochemical probes. The potential of the LB films for these applications is sensitive to the details of their molecular packing. Also, these applications require that the layers have a defect-free periodic structure. Defects in the LB films have been studied by conventional surface analysis such as X-ray and electron diffraction. These techniques, however, are not sensitive to defects such as the pinholes or tears within the layers. Atomic force microscopy (AFM) has proved to be a near-ideal, non-destructive and high resolution method to investigate the LB film structure and detect defects at length scales from 0.1 nm to 10 mm (Zasadzinski et al., 1994). A variety of LB films such as lipid and protein films, polymer films, and specially-functionalised molecular films have been studied by AFM. The AFM image of a four-layer film of cadmium arachidate is shown in Figure 8.9.

Figure 8.9

8.5

AFM image (8.5 × 8.5 nm2) of a four-layer film of cadmium arachidate (Garnaes et al., 1992) (reproduced by permission from Macmillan Publishers Ltd., © 1992).

SURFACE DIFFUSION

In systems having insoluble or sparingly soluble monolayers, surface diffusion plays an important role (Levich, 1962). Surface diffusivity is important in longitudinal and transverse wave motions, and in the hydrodynamic behaviour of thin liquid films. It is also important in coalescence of drops and bubbles (Ghosh and Juvekar, 2002). Gaines (1966) has suggested that surface diffusion must be included in the studies of the kinetics of chemical reactions occurring in interfacial films. Surface diffusion is also recognised to be very important in lipid bilayers (Fahey, 1977) and cellular membranes (Feng, 1993).

Monolayers and Thin Liquid Films

283

The experimental determination of surface diffusion coefficient involves a considerable amount of complexity. Usually, radiotracer and fluorescence methods are employed for the measurement of surface diffusion coefficient. The values reported in the literature often show orders of magnitude difference from one another. Temperature and humidity, which cause extraneous surface convective flows, play important roles in these variations (Agrawal and Neuman, 1988b).

8.5.1 The Apparatus The apparatus used by Agrawal and Neuman (1988b) for measuring surface diffusion coefficient of myristic acid by radiotracer method is shown in Figure 8.10. Very precise control of temperature and humidity are two important features of this apparatus. The trough was machined from 25.4 mm-thick Lucite. It was made very shallow to minimise subsolution convective currents and was rigidly

Figure 8.10

Schematic diagram of the surface diffusion apparatus: (A) trough, (B) front barrier, (C) Unislide assembly for front barrier, (D) diffusion barrier, (E) motorised translator, (F) Cahn electrobalance, (G) stand to raise Wilhelmy plate, (H) Wilhelmy plate, (I) Geiger-Müller assembly, (J) Unislide assembly for vertical positioning of the Geiger-Müller tube, (K) Gilman assembly for horizontal positioning of the Geiger-Müller tube, (L) mechanical counter, (M) 10-turn Helipot, (N) thermostatted water, (O) stainless steel tubing, (P) Lucite cabinet, (Q) polystyrene insulation, (R) plywood, and (S) access port (Agrawal and Neuman, 1988b) [reproduced by permission from Elsevier Ltd., © 1988].

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Colloid and Interface Science

mounted to a 25.4 mm thick aluminium tooling plate for greater strength and dimensional stability. The overall inside dimensions of the trough were 78 cm × 15.2 cm × 0.4 cm. The edges of the trough were only 3 mm wide to minimise the leakage of the film. The two channels of the trough were 38 cm × 5.6 cm and were connected to each other by reservoirs. The trough was made hydrophobic by coating the interior and edges with polyfluorocarbon tape. All tape-seams were sealed with purified paraffin wax. The use of compression barriers to adjust the surface pressure of the monolayer was avoided because any leakage through the barrier would cause a profile shift and, this would cause difficulties in interpreting the experimental data. Instead, two rigid dividing barriers made from stainless steel were used to divide the trough into two equal parts. The barriers were covered with polyfluorocarbon tape. The front barrier (B) was connected to a vertical Unislide assembly and could be raised or lowered slowly. This barrier was used for equalising the surface pressure across the back diffusion barrier (D). This second dividing barrier (i.e. the diffusion barrier) was used in the rear channel to separate the radioactive and nonradioactive films prior to the start of the diffusion measurement. The diffusion barrier was connected to a precision motorised translator which was capable of travelling at low speeds (~1 mm/s) without significant vibration. The surface pressures of the radioactive and nonradioactive myristic acid monolayers during spreading were independently measured by two Cahn electrobalances using the Wilhelmy plate technique. Platinum plates (20 mm × 15 mm × 0.1 mm) were sandblasted and electrolytically platinised to obtain good wettability. The b-radiation emanating from the 14C-labelled surface films was measured by a Geiger-Müller tube and decade scaler. The end window was covered with a stainless steel mask having a narrow rectangular aperture (35 mm × 1.5 mm). Care was taken to ensure that the aperture was perpendicular to the trough axis. The Geiger-Müller tube was connected to two mechanical slide assemblies providing horizontal and vertical movement. Accurate measurements of the position of the Geiger-Müller tube along the trough and its height above the film-covered subsolution surface were made possible by the use of a mechanical counter and a 10turn Helipot geared to the horizontal and vertical slide assemblies respectively. The output of the Helipot was connected to a digital multimeter. A Lucite cabinet (94 cm × 50 cm × 24 cm) enclosed the trough. It was heavily insulated with high-density polystyrene and epoxy-coated plywood except for the top of the cabinet which was insulated with polystyrene. The humidity level inside the cabinet was adjusted close to saturation (RH ³ 97%) by placing many Pyrex beakers containing wetted filter paper into the cabinet prior to its closing. Calibrated glass probe thermistors were used to monitor the temperature inside the cabinet. Its fluctuation was controlled within ±0.02 K by circulating thermostatted water through stainless steel tubing attached to the inner cabinet walls. The surface diffusion apparatus was supported by a massive concrete table.

8.5.2

Measurement Procedure

The sweeping barriers, trough and the Wilhelmy plates were cleaned thoroughly. In a typical diffusion experiment, the subsolution surface was divided into two equal parts by the dividing barriers. The trough was filled with the subsolution (10 mol/m3 HCl solution) to a depth of 6.6 mm. The surface of the subsolution was repeatedly cleaned by the barriers. The same surface pressure was maintained on both sides of the diffusion barrier when it was removed from the surface so that there was no initial surface pressure gradient. The diffusion barrier was lowered in such a way as to ensure good contact with the trough edges in order to avoid any film leakage. Similarly, the front barrier was carefully aligned and placed firmly on the trough edges. The Wilhelmy plates were positioned

Monolayers and Thin Liquid Films

285

in the subsolution and the cabinet was sealed. After waiting for about one day to ensure that the gas phase was essentially saturated with water vapour, the nonradioactive and radioactive (14C-labelled) myristic acids were separately spread from hexane solutions on the opposite sides of the dividing barriers from micro-pipettes through small access ports. This procedure minimised any temperature and humidity fluctuation in the equilibrated system during the spreading process. The films were deposited to within ±1 mN/m of the desired pressure. The difference in surface pressure between both films was never more than ±0.5 mN/m. About twenty minutes were allowed for the evaporation of hexane. The surface pressure on both sides of the diffusion barrier was then equalised by slowly raising the front barrier. Typically, the nonradioactive myristic acid was spread on the left side of the diffusion path and the radioactive myristic acid was spread on the right side. The Geiger-Müller tube was then brought into position 3 mm above the nonradioactive film, and the surface activity (background count) was determined at the farthest left position of the diffusion path. Next, the diffusion barrier was very slowly raised by the motorised translator. The detachment of the film-covered subsolution from the diffusion barrier was visually observed from a small peephole strategically located in the insulated cabinet. The lifting of the diffusion barrier was most critical to obtain smooth, error-free surface concentration profiles. After raising the diffusion barrier, the Geiger-Müller tube was raised 75 mm above the film-covered surface and then moved slowly towards the right side of the diffusion path where the radioactive myristic acid was spread. The Geiger-Müller tube was then lowered to 3 mm above the film to monitor the diffusion process. Beginning at about 100 mm from the diffusion barrier, counts were taken for 60 s at 5 mm intervals along the trough moving towards the nonradioactive film. The counting was started half-an-hour after lifting the diffusion barrier. Slow movement of the Geiger-Müller tube is very important to prevent surface convection due to air currents. Thus, the Geiger-Müller tube was advanced slowly (~80 µm/s) between the counting periods until a constant background rate was obtained over the nonradioactive film. After completion of the first profile, the Geiger-Müller tube was raised 75 mm above the surface and moved slowly to its resting place over the nonradioactive film when the radiation counter was not in use. This procedure was repeated for the subsequent profiles and in each case, the background count was measured at the earlier position of the diffusion path prior to the start of a new profile.

8.5.3 Determination of Ds Let us denote the surface concentration of the 14C myristic acid molecules by G and their initial concentration by G0. The ratio G/G0 is plotted against position along the trough at various times. The ratio of the surface concentrations is equal to the ratio of the monolayer activities after subtraction of the background count. Any contribution to the counting rate from desorbed molecules is very small since the rate of film-loss is very small. The surface diffusion coefficient can be calculated from the unsteady-state surface concentration profiles using the model ˜* ˜t

Ds

˜2 * ˜x

2

 vx

˜*  ko * ˜x

(8.7)

where Ds is the surface diffusion coefficient, vx is the surface velocity in the x-direction and ko is the overall film-loss coefficient (the derivation of this equation is given in the Appendix). This differential equation is numerically solved with no-flux boundary conditions and experimentallydetermined initial conditions to obtain G(x, t)/G0. The best fit of the computed profile to the experimental data is determined using a least squares method in which vx and ko are the fitted

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Colloid and Interface Science

parameters. The experimentally-determined initial (zero-time) profile, vx and ko are used to generate subsequent profiles. In another method, each subsequent profile is generated using the previous experimental profile instead of only the initial profile. In this approach, the effect of loss of film molecules from the surface on the surface concentration is minimised. When Ds depends upon surface concentration, it is important that the values of Am for the two profiles be close to one another. A similar procedure but with vx = 0 can also be used to determine Ds. In this method, the centre of the generated profile is shifted to match that of the experimental data to obtain the best fit. The values of Ds determined by these methods should be essentially the same considering the experimental error involved. Some of the profiles depicting the variation of G/G0 with x are shown in Figure 8.11. The lines represent the best-fit profiles to the data. The ideal initial profile would be a step change. In practice, however, even very careful lifting of the diffusion barrier causes some mixing of the radioactive and nonradioactive films. The desorption of film molecules from the surface is the primary cause behind the decrease of G/G0 from its initial value of unity in the subsequent profiles. The surface diffusion coefficient strongly depends upon the molecular packing. To illustrate, Agrawal and Neuman (1988b) found that the value of Ds was 4 × 10–9 m2/s for Am = 0.435 nm2, which was nearly 100 times larger than that for Am = 0.387 nm2. At still higher values of molecular packing (e.g. Am = 0.333 nm2), Ds was less than 10–11 m2/s. The data on surface diffusion coefficient reported in the literature clearly indicate that it depends on the state of monolayer (e.g. G, LE or LC). Various limitations of the radiotracer technique which result in ambiguous values of Ds have been discussed by Agrawal and Neuman (1988b). Some of the values of Ds are presented in Table 8.1.

Figure 8.11

8.6

Surface concentration profiles for myristic acid at 293 K (Agrawal and Neuman, 1988b) (adapted by permission from Elsevier Ltd., © 1988).

THIN LIQUID FILMS

Thin liquid films are important in the stability of emulsions and foams. In addition, they are involved in many biological systems. For example, a bimolecular lipid leaflet with protein adsorbed on it constitutes the basic structural element of the biological cell membrane. Consequently, hydrocarbon

Monolayers and Thin Liquid Films

Table 8.1 Technique

Monolayer material

Radiotracer

Myristic acid

Values of surface diffusion coefficient

Stearic acid Cholesterol Fluorescence

287

PDA in oleic acid1 NBD-egg-PE in DPPC2

Monolayer state

Am (nm2)

Ds × 1011 (m2/s)

LE LE/LC LC G/LC LC G/LC LC

0.30 – 0.37 0.22 0.20 0.33 – 0.75 0.23 0.40 – 0.80 0.38

100 – 300 80 1 188 – 284 161 20 0.6 – 5

LE LE LC

0.47 0.83 0.45

9 – 17 4.7 0.002

1PDA

= 12-(1-pyrene)dodecanoic acid = N-4-nitrobenzo-2-oxa-1,3-diazole egg phosphatidylethanolamine DPPC = L-a-dipalmitoylphosphatidylcholine

2NBD-egg-PE

films composed of phospholipids and proteins have been used as model structures for investigating biological transport processes in cells (Matijevi ü, 1971). Thin liquid films have been extensively studied to investigate the van der Waals, electrostatic double layer and solvation forces (Sheludko, 1967). Thin liquid films have been a subject of interest since the work of Robert Hooke (in 1672) in which he reported ‘holes’ in soap films. Later, the works of Newton and Gibbs have shown that these so called ‘holes’ are, in fact, regions of small thickness in the film where the interference between light reflected from the upper and lower film-surfaces leads to nearly-complete extinction of the reflected light. Several aspects of thin liquid films have been studied. These can be grouped into the following categories: geometrical and optical properties, thermodynamics, hydrodynamics and stability. In this section, we will discuss the hydrodynamic and stability aspects of thin liquid films.

8.6.1 Critical Film Thickness The natural question that arises concerning the thin liquid films is, “how thin is the film and how does it rupture?” We have noted in Chapter 5 that some of the interfacial forces are active when the separation between two interfaces is less than 100 nm. Therefore, these forces can play a significant part on the stability of a liquid film only when the thickness of the film becomes 100 nm or less (Vrij and Overbeek, 1968). However, the energy required to rupture a film (e.g. that stabilises foams and emulsions) of 100 nm thickness is very high. The energy of activation required to form a hole in a thin film is of the order of g h2, where g is the interfacial tension and h is the thickness of the film. The film is not expected to rupture until its thickness is so low that the activation energy for hole formation is a small multiple of kT (where k is Boltzmann’s constant and T is temperature). EXAMPLE 8.3 Calculate the approximate energy required to form a hole in an aqueous thin film if the thickness of the film is 2 nm and the interfacial tension is 10 mN/m. Compare this energy with the thermal energy at the room temperature. Solution

The energy involved in thermal fluctuations is kT. Now, kT = 1.381 × 10–23 × 298 = 4.115 × 10–21 J

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Colloid and Interface Science

The energy of activation for hole formation is g h 2 = 10 × 10–3 × (2 × 10–9) 2 = 40 × 10–21 J Therefore, the activation energy necessary for hole formation is nearly ten times the thermal energy. Surface tension tends to make the surface of the film smooth. However, thermal motions cause a certain amount of roughness. This roughness can be observed experimentally, because a surface that is not completely flat not only reflects but also scatters some of the incident light falling on it. Mathematically, the profile of the surface can be represented as a sum of Fourier waves. For thin films, the shapes of fluctuations in the two surfaces become correlated because of the molecular interactions in the film. To illustrate, the van der Waals force tends to drive the molecules from the thinner to the thicker parts of the film. Consider the thin liquid film shown in Figure 8.12 in which the thickness increases from left to the right. The molecule located in the middle part of the film is attracted more towards the right by the molecules residing in the hatched portions of the film. Otherwise, the attraction on this molecule from the molecules in the left part of the film is exactly balanced by the molecules in the region enclosed by the dotted lines (i.e. the difference between the entire film on the right side of the molecule and the hatched portion). This, in fact, is the mirror image of the part of the film located on the left side of the molecule. This attraction becomes important when the film is so thin that the range of the attractive force is comparable with the thickness of the film. Therefore, the van der Waals force favours ‘corrugation’ in the film which causes disproportionation of the film into thinner and thicker parts. Interfacial repulsive forces such as electrostatic double layer force and polymeric steric force (which are active when surfactant molecules adsorb on the surfaces of the film) favour thicker films. They can play an important role in the stability of the film.

Figure 8.12

Illustration of how a molecule located in the thinner part of a liquid film is attracted towards the thicker part of the film by van der Waals force.

For some deformations, the increase in surface Gibbs energy is more than compensated by the decrease in the van der Waals energy. Such deformations grow spontaneously in amplitude and lead to the rupture of the film. In some cases, rupture does not occur and a stable ‘black film’ forms if the repulsive force is strong enough. We will discuss the black films in Section 8.6.7. When the film is ‘thick’ (e.g. 1 µm), its thinning is mainly governed by viscous flow caused by the pressure gradient

Monolayers and Thin Liquid Films

289

in the film. When the thickness of the film reduces to such a small value that the rate of growth of fluctuations is faster than the thinning by viscous flow, the film can rupture. This film thickness is known as critical film thickness (hc). The value of critical film thickness is less than 50 nm for many emulsion films (Jeelani and Hartland, 1994). Therefore, the total time for the rupture of the film is composed of two parts: the first part corresponds to the viscous drainage of the film and the remaining part corresponds to the rapidly-growing fluctuations leading to the rupture. The second part is of much shorter duration than the first part. Vrij and Overbeek (1968) have given the following equation for the critical film thickness: È a f AH2 Ø 0.267 É Ù Ê 6SJ 'p Ú

hc

17

(8.8)

where af is the area of the film, AH is the Hamaker constant, g is the interfacial tension and Dp is the excess pressure in the film. Now, consider a drop resting on a solid flat plate placed in a liquid which is immiscible with the liquid constituting the drop (Figure 8.13). In this case, the concerned film lies between the drop and the flat surface. The force acting on it is the gravitational force. The radius of the film is given by (Princen, 1963)

Figure 8.13

Thin liquid film beneath a drop pressed to a flat undeformable interface by gravity. The film has been assumed to be flat.

Rf

È 2 'U g Ø Rd2 É Ê 3J ÚÙ

12

(8.9)

where Rd is the radius of the drop, Dr is the difference in density between the liquid constituting the drop and the film liquid, and g is the acceleration due to gravity. The area of the film af is, therefore, given by af

2S Rd4 'U g 3J

(8.10)

If the drop is undeformable but the interface is deformable as shown in Figure 8.14(a), the radius of the film is same as that given by Eq. (8.9). The deformability of the drop can be determined by

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Colloid and Interface Science

calculating the Bond number (see Chapter 4). If both drop and the interface are deformable [Figure 8.14(b)], the radius of the film is given by Rf

È 'U g Ø 2 Rd2 É Ê 3J ÙÚ

12

(8.11)

However, the area of the film can deviate significantly from the quantity S R 2f because of the curvature of the interface. Equations (8.9) and (8.11) were derived from a balance of gravitational and capillary forces. They give reasonably accurate results for very small drops.

Figure 8.14

Models for film radius: (a) drop is undeformable but bulk interface is deformable, and (b) both drop and bulk interface are deformable.

The excess pressure in the film, Dp, is given by Fg/af, where Fg is the gravitational force given by 4 4 S Rd3 ( U d  Uc ) g S Rd3 'U g (8.12) 3 3 where rd is the density of the drop-liquid and rc is the density of the liquid constituting the continuous phase. For a small drop or bubble at a deformable interface, the excess pressure is g /Rd, in which case the film area is given by FgRd/g. Therefore, by calculating af and Dp from the preceding equations, and with knowledge of interfacial tension and Hamaker constant (Section 5.2.3), we can calculate the critical film thickness from Eq. (8.8). Fg

EXAMPLE 8.4 Calculate the critical thickness of the aqueous film, when a 1 mm radius carbon tetrachloride drop rests on a flat water–CCl4 interface at 298 K. Given: density of carbon tetrachloride = 1600 kg/m3 and interfacial tension = 45 mN/m. Solution The system is shown in Figure 8.15. The Hamaker constant can be calculated from the Lifshitz theory [Eq. (5.18)] as shown below. Dielectric constant of CCl4 = 2.24 Dielectric constant of water = 78.5 Refractive index of CCl4 = 1.460 Refractive index of water = 1.333 Mean absorption frequency = 2.85 × 1015 s–1

Monolayers and Thin Liquid Films

291

Figure 8.15 Carbon tetrachloride drop resting at a flat water–carbon tetrachloride interface.

From Program 5A.1 (Appendix 5A), we get, AH = 6.84 × 10–21 J The gravitation force Fg is given by Fg

4 S Rd3 ' U g 3

4 S – (1 – 10 3 )3 – 600 – 9.8 3

2.46 – 10 5 N

The Bond number is (see Section 4.5.3) 'U gRd2 J

600 – 9.8 – (1 – 10 3 )2 0.045

0.13

Since the value of Bond number is much lower than unity, the deformation of the drop would be small. Therefore, we can use Eq. (8.10) to calculate the approximate area of the film. af

2S Rd4 ' U g 3J

3 4 È 2S Ø (1 – 10 ) – 600 – 9.8 ÉÊ 3 ÙÚ 45 – 10 3

'p

Fg

2.46 – 10 5

af

2.74 – 10 7

2.74 – 10 7 m 2

89.8 Pa

From Eq. (8.8), the critical film thickness is given by

hc

È a f AH2 Ø 0.267 É Ù Ê 6SJ ' pÚ

1/ 7

1/ 7

Ë 2.74 – 10 7 – (6.84 – 10 21 )2 Û 0.267 Ì Ü 6S – 45 – 10 3 – 89.8 ÍÌ ÝÜ

28.8 – 10 9 m

Therefore, the critical film thickness is 28.8 nm. Experimental works have confirmed good accuracy of Eq. (8.8) in predicting the critical film thickness in emulsions and foams. Experiments on rupture of thin liquid films have shown that a distribution of the values of hc is always observed, as depicted in Figure 8.16 for chlorobenzene films of different radii (Scheludko and Manev, 1968). In this figure, N is the total number of ruptured films and DN is the number of ruptures at thicknesses between (h – Dh/2) and (h + Dh/2).

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Colloid and Interface Science

Figure 8.16 Distribution curves of critical film thickness for chlorobenzene films (Scheludko and Manev, 1968) [adapted by permission from The Royal Society of Chemistry, © 1968].

8.6.2

Hydrostatics of Thin Liquid Films

Let us consider a thin liquid film formed between two emulsion drops as shown in Figure 8.17 (half of the film is shown in the figure). The film is assumed to be plane-parallel. The liquid in the film exists in contact with the bulk liquid in the Plateau border. The film-liquid would exist in hydrostatic equilibrium with the surrounding liquid if an external force directed perpendicular to the surfaces, known as disjoining pressure (P), balances the internal force (Derjaguin, 1955). The origin of disjoining pressure is van der Waals, electrostatic double layer and steric forces acting between the surfaces of the thin film.

Figure 8.17 The disjoining pressure model of a thin liquid film.

The drop-phase pressure is pd, bulk liquid-phase pressure is pl, interfacial tension is g and the radius of the drop is Rd. In the meniscus region, we have

Monolayers and Thin Liquid Films

pd  pl

2J Rd

293 (8.13)

Since pd is larger than pl, the equilibrium at the film surface is ensured by the action of the disjoining pressure as P = pd – pl (8.14) The following relationship also holds at equilibrium: pd – pf = P (8.15) The disjoining pressure is a function of the film thickness. The disjoining pressure manifests as a normal surface excess force Fs as shown in Figure 8.17. It is taken to be positive when it acts to disjoin (i.e. separate) the film surfaces. Fs can be calculated using the DLVO theory (Section 5.4). The hydrostatic equilibrium model of thin liquid film described above is known as the disjoining pressure model.

8.6.3

Drainage of Liquid Films

Thinning of soap films by the drainage of liquid has been extensively studied since the time of Newton. The qualitative observations made by Newton and Gibbs revealed that the walls of soap bubbles grow thinner in time and pass through thicknesses of the order of the wavelength of the visible light. Different colours appear due to the interference of light reflected from the two surfaces of the film. At the advanced stages of the thinning process, thin black spots are formed, which are sometimes very unstable. The interest in the thinning process of films has grown considerably due to their importance in the understanding of coalescence of drops and bubbles, and hence the stability of emulsions and foams. Mysels et al. (1959) were the first to report in detail the different types of drainage process, concentrating on vertical films formed by withdrawal of glass frames from pools of surfactant solutions. They observed mobile films which drained in minutes and showed turbulent motions along the edges, and rigid films which took hours to drain and showed little or no motion. They proposed that the rapid drainage and turbulence observed for the mobile films were the result of ‘marginal regeneration’. In this phenomenon, a thick film flowed into the Plateau borders near the legs of the frame at some elevations owing to the greater suction force exerted on it by the low pressure in the borders. Simultaneously, a thin film was pulled out of the borders at other elevations to maintain constant surface area for the overall film. Curvature of the interface plays an important role in the drainage process. For example, the film formed between two droplets in an emulsion is actually not flat. The thickness of the film and curvature of the interfaces change with time. When two emulsion droplets approach each other a dimple forms as shown in Figure 8.18(a). Due to the trapped liquid in the central region of the film, the film is thicker in the central part as compared to the peripheral region. The interfaces deform during the drainage of the film and a narrow region is developed near the rim of the film, which is known as the barrier ring (because it causes constriction to the outward flow of the film-liquid, which slows down the drainage process). The shape of the film changes with time as the film thins. The presence of surfactants is believed to influence the film-drainage process. The shape of the film can be studied by interferometry and video microscopy. The various stages of thinning of a film formed between a drop and an initially-flat water–organic interface are shown in Figures 8.18(b) and 8.18(c).The drainage can be symmetric as well as asymmetric, as shown in

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Colloid and Interface Science

Figure 8.18 (a) Schematic diagram of a dimpled thin film formed between a drop and an initially-flat bulk interface, (b) symmetric drainage of the film with time for a 3 mm diameter anisole drop at water–anisole interface in presence of 3 × 10–6 kg/m3 SDS and 10 mol/m3 KCl, and (c) asymmetric drainage of the film for 3 mm diameter anisole drop at water–anisole interface in presence of 2 × 10 –5 kg/m3 SDS and 10 mol/m3 KCl (Hodgson and Woods, 1969) [adapted by permission from Elsevier Ltd., © 1969].

Monolayers and Thin Liquid Films

295

these figures. The dimple sometimes flattens with time and finally a near-flat thin film can be obtained. The theoretical analyses of the thinning process often make simplifying assumptions such as, axisymmetric interfaces (Slattery, 1990) and flat film (Edwards et al., 1991). Asymmetric drainage in foam films has been analysed by Joye et al. (1994).

8.6.4

The Lubrication Flow

When two drops move towards each other along the line of their centres, the hydrodynamic force opposing their motion increases tremendously as the distance between the drops decreases (a similar phenomenon occurs when a drop moves towards a flat interface). As the drops approach, the resulting radially-outward flow in the narrow gap separating the two drops exerts a tangential shear stress on the drop surfaces. This causes a tangential motion of the drop surfaces and drives a flow inside the drops. A sketch of the streamline-patterns is shown in Figure 8.19. This flow has a significant effect on the force that resists the approach of the drops. The flow in the narrow gap and the flow within the drops are discussed in this section.

Figure 8.19 Sketches of streamlines when (a) two drops approach each other, and (b) a drop is in the vicinity of a stationary fluid interface (Rushton and Davies, 1978) [adapted by permission from Elsevier Ltd., © 1978].

Let us consider two spherical drops of radii Rd1 and Rd2 moving towards one another along their line of centres under Stokes-flow conditions, as shown in Figure 8.20(a). The viscosity of the continuous phase is m and that of the drop-liquid is lm. The relative velocity is U = v1 – v2. The closest separation between the drop surfaces is h0. Since we have assumed axisymmetric profile, h0 is the located at the centre of the film. The force that resists the relative motion of the drops and slows down the film-thinning process is dominated by a small region located near the axis of symmetry. This region is known as the lubrication region. Our objective is to derive the velocity profile in this region.

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Colloid and Interface Science

Figure 8.20

(a) Schematic diagram of two drops approaching each other with velocities v1 and v2, and (b) illustration of the near-contact region.

The radial velocity profile u(r, z) can be decomposed into two parts as (Davis et al., 1989) u(r, z) = ut(r) + up(r, z) (8.16) where ut(r) is the tangential velocity and up(r, z) is the velocity for the parabolic portion of the flow driven by the local pressure gradient, which is zero at the drop surfaces (i.e. at z1 and z2). up is given by

up

1 ˜p ( z  z1 )( z  z2 ) 2 P ˜r

(8.17)

The tangential stress exerted by the fluid in the gap on the drop surfaces is ft

P

˜u ˜z z

z1

P

˜u ˜z z

z2



h ˜p 2 ˜r

(8.18)

A mass balance on the fluid flowing out of the film gives the following equation: z2

Sr U 2

2S r Ô u(r , z)dz

(8.19)

z1

Substituting u(r, z) from Eq. (8.16) into Eq. (8.19), carrying out the integration and substituting h = z2 – z1, we get

S r 2U

È h 2 ft Ø 2S r É hut  6 P ÙÚ Ê

(8.20)

Monolayers and Thin Liquid Films

297

The magnitude of the hydrodynamic force that resists the relative motion of the drops is given by ‡

2S Ô p(r )r dr

Fl

(8.21)

0

Since h0 is small compared to the radius of the drops, h0/Rd > up and the interfaces are fully mobile. Therefore, the parameter x characterises interfacial mobility. The effects of surfactants are not considered in this mobility. When surfactants are adsorbed at the interface and an interfacial tension gradient (see Section 7.3) is developed, the interfacial mobility can be significantly different from that considered here. When the viscosity of the drop-liquid is much higher than the viscosity of the continuous phase, the drops behave like rigid spheres. In this case, l >> ( Rd / h0 )1 2 . The thinning of the film is dominated by the parabolic portion of the flow up. The tangential flow, ut is small. As x ® 0, the solution (known as rigid-sphere solution) is given by (Davis et al., 1989) ut = 0

(8.27)

f t = 3mrU/h2

(8.28)

p 3 P RdU / h 2

(8.29)

Fl

and

6 SP R d2 U / h0

(8.30)

If the viscosity of the drop-fluid is comparable to the viscosity of the continuous phase or smaller, 12 then l 1, flocculation is much more rapid than coalescence, and thus coalescence is the rate-controlling step. Several modifications of this coalescence model have been suggested (Borwankar et al., 1992). For example, the rate of increase of the number of drops in an aggregate was assumed to be proportional to n0. However, Eq. (9.19) was obtained from the Smoluchowski's theory which only accounts for the flocculation of drops (i.e. coalescence does not occur). In an emulsion, the size of an aggregate increases due to the addition of individual primary droplets as well as other aggregates, which is accounted for in Eq. (9.19). However, these incoming aggregates have themselves undergone coalescence, and this equation does not account for the coalescence that has occurred in the incoming aggregates. As a result, the theory described above overestimates the rate of increase in the aggregate size. Thus, Eq. (9.24) does not represent the correct physical picture of the flocculation–coalescence process, and it cannot predict the change of the rate-controlling step during the lifetime of the emulsions. Borwankar et al. (1992) adopted a similar approach, but with a somewhat different formulation. Instead of taking a balance on each aggregate, they took an overall balance on all droplets in the emulsion. For linear aggregates, the total number of films in the emulsion is given by

n f nv

(m  1)nv

(9.25)

Instead of Eq. (9.22), the differential equation for n is given by dn K „(m  1)nv (9.26) dt where K¢ is a proportionality constant analogous to K in Eq. (9.22). The overall balance is expressed by Eq. (9.23). Note that m can be expressed through n using Eq. (9.23). The variation of n with time is given by the solution of Eq. (9.26) using Eqs. (9.17), (9.18) and (9.23). The series solution has been given by Borwankar et al. (1992). 

Emulsions, Microemulsions and Foams

n

327

Ë n0D exp ^K „ /(D n0 )` Û  ln(1  D n0 t )  ^K „ /(D n0 )`^(1  D n0 t )  1`  Ü Ì K„ Ì Ü 3 Ì K „ /(D n ) 2 Ü K „ /( D n ) ` ^ ` K „ exp(  K „t ) Ì ^ 0 0 (1  D n0 t )2  1  (1  D n0 t )3  1  "  Ü (9.27) Ü 2 – 2! 3 – 3! D exp ^K „ /(D n0 )` Ì Ì Ü n Ì ^K „ /(D n0 )` Ü n (1  D n0 t )  1  " Ì Ü n – n! ÌÍ ÜÝ

^

`

^

`

^

`

This model contains two parameters, the flocculation-rate constant a and the coalescence-rate constant K¢. The former depends on the droplet interactions as well as on the diffusion coefficient (hence, on the viscosity of the medium), whereas the latter is dependent on the lifetimes of the thin liquid films trapped between the drops. Although the modification of Borwankar et al. (1992) can be considered as an improvement over the van den Tempel equation, there is a common drawback underlying both the theories, which is caused by the use of Smoluchowski equation for calculating nt: coalescence changes nt and correspondingly nv. Danov et al. (1994) have presented a kinetic model in which flocculation and coalescence are interrelated and take place simultaneously. The flocculation scheme shown in Figure 9.4 and coalescence scheme shown in Figure 9.5 are interdependent because the aggregate (after some of the drops have coalesced) is further involved in the flocculation scheme. They have presented a kinetic scheme accounting for the possibilities of flocculation and coalescence in and between the aggregates. They have assigned a rate constant to each elementary act and developed differential equations describing the emulsion system, which are similar to parallel chemical reactions in some respects.

Mechanism of Coalescence of Drops Because coalescence is the most important factor behind the destabilisation of emulsions, it has been studied extensively during the past five decades. Several hundreds of research articles exist on the theoretical and experimental aspects of the mechanism of coalescence. Two experimental techniques are generally used to investigate the mechanism of coalescence: coalescence of one drop at a flat liquid–liquid interface (i.e. drop–interface coalescence) and coalescence between two drops (i.e. binary coalescence). Most of the works are based on the first approach. This method has the advantage that it is very easy to perform experimentally. Detailed reviews of these works have been presented by Slattery (1990) and Edwards et al. (1991). In recent years, some works have extensively studied binary coalescence (Kumar et al., 2006; Mitra and Ghosh, 2007; Giribabu and Ghosh, 2007). From Figure 9.2, it is apparent that the second approach has greater relevance to the stability of emulsions because the droplets in an emulsion grow bigger by binary coalescence. However, in the later stages, when the oil and water phases begin to separate out, the large drops coalesce at the flat oil–water interface. The experimental method of studying drop–interface coalescence is as follows. Two immiscible liquids (e.g. water and toluene) are taken in a coalescence cell made of glass. A drop of water (or organic liquid) is released through a precise burette (or syringe) such that it strikes the flat interface between water and the organic liquid with its terminal velocity. After striking the interface, the drop undergoes an up-and-down oscillatory motion and finally rests on the interface for a certain amount of time until it coalesces with its parent-phase. This period of time is known as rest time or coalescence time. The time is measured by a video camera, and the size of the drop is measured by

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Colloid and Interface Science

image analysis. In such a coalescence experiment, usually 100 drops are studied. The reason for studying a large number of drops lies in the stochastic nature of the coalescence process which is explained later. Binary coalescence of water drops in lighter organic liquids can be studied in the coalescence set-up developed by Mitra and Ghosh (2007). Long coalescence times can be studied in this set-up. A battery of coalescence cells is usually used. The drops are formed using a specially-fabricated glass burette having a delicate long capillary at its end. The tip of the burette is dipped into the organic phase and the aqueous drop is slowly formed at the end of the tip. The drop is equilibrated with its surrounding liquid for one minute. Then it is released so that it falls with its terminal velocity inside the coalescence cell placed beneath it. After the first drop settles inside the cell, the second drop is released. The second drop, after some initial movement due to inertia, rests on the first drop leaning against the inclined side of the cell (see the photograph in Figure 9.6). The coalescence time is measured by a digital video camera. In these coalescence experiments also, usually 100 pairs of drops are studied and a cumulative distribution of coalescence time is developed.

Figure 9.6

Photograph of 3 mm diameter water drops resting inside coalescence cells made of Teflon placed in toluene (the drops were colourised for illustration) (Mitra and Ghosh, 2007) (adapted by permission from Taylor and Francis Ltd., © 2007).

Two theories of coalescence of drops have been proposed: the film-drainage theory and the stochastic theory. The film-drainage theory attributes the coalescence time to the time required for the drainage of the thin liquid film trapped between the drop and the interface (or between two drops). The lubrication model of hydrodynamic drainage of liquid films has been described in Section 8.6. When the thickness of the film becomes small (< 100 nm), its stability is influenced by the interfacial forces (e.g. electrostatic double layer and steric forces). After the thickness of the film reaches the critical value (see Section 8.6.1), it ruptures rapidly by the action of van der Waals attraction. Charles and Mason (1960) proposed two models for coalescence time. The parallel-disc model gives the following equation for coalescence time (tc):

Emulsions, Microemulsions and Foams

tc

K È 'U ga5 Ø Ë 1 1 Û Ì 2  2Ü É Ù 2 4Ê J Ú ÌÍ hc hi ÜÝ

329

(9.28)

where h is the viscosity of the liquid constituting the continuous phase (i.e. the film), g is acceleration due to gravity, Dr is the difference in density between the two liquids, g is interfacial tension, hi is the initial film thickness and hc is the critical film thickness. The above equation was derived based on the assumption that a liquid drop of radius a approaches a flat undeformable interface by gravity and deforms by its own weight. The other model is known as spherical-planar model. It considers the drop as a sphere approaching an unbounded plane. The coalescence time is given by Èh Ø 9K ln É i Ù 2 a ' U g Ê hc Ú

tc

(9.29)

In deriving Eqs. (9.28) and (9.29), the electrostatic double layer repulsion and the van der Waals forces of attraction were ignored. Furthermore, the theory assumes that the interfaces are rigid. The values of coalescence time predicted by Eqs. (9.28) and (9.29) differ considerably, as explained in Example 9.2. EXAMPLE 9.2 Calculate the coalescence time of a 2 mm diameter drop of water at a flat toluene– water interface using Eqs. (9.28) and (9.29). Given: density of toluene = 870 kg/m3, viscosity of toluene = 0.6 mPa s and interfacial tension = 36 mN/m. Given: initial thickness of the film = 1 mm and the critical thickness of the film = 10 nm. Solution Putting r = 1000 kg/m3 for water and g = 9.8 m/s2 in Eq. (9.28), we get tc =

1 Û K È 'U ga5 Ø Ë 1  Ì Ü É Ù 4 Ê J 2 Ú ÍÌ hc2 hi2 ÝÜ

È 0.6 – 10 3 Ø Ë (1000  870) – 9.8 – (1 – 10 3 )5 Û Ë Û 1 1  = É Ì ÜÌ Ü 1475 s Ù  3 2  9 2  6 2 4 (36 – 10 ) (1 – 10 ) Ý Ê Ú ÌÍ ÜÝ Í (10 – 10 ) From Eq. (9.29), we get

tc

Èh Ø 9K ln É i Ù 2 a'U g Ê hc Ú

È 1 – 10 6 Ø ln É Ù 2 – 1 – 10 3 – (1000  870) – 9.8 Ê 10 – 10 9 Ú 9 – 0.6 – 10 3

9.8 – 10 3 s

Therefore, the values of coalescence time obtained from these two models differ widely. The model for coalescence time proposed by Jeelani and Hartland (1994) incorporates the effects of circulation within the adjacent phases of the film. It also considers the interfacial tension gradient when surfactant molecules are adsorbed at the interfaces. The coalescence time can be calculated from the following equation:

tc

9KE 2 R 4f 64a3 'U ghc2

(9.30)

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Colloid and Interface Science

where b 2 is given by, 4 E2

3 È 3K a Ø Ë È S R f Ø È ˜J Ø Û Ì1  É 1 É ÙÉ Ù Ü Ê K d hi ÙÚ Ì Ê 2 fhi Ú Ê ˜r Ú fi Ü Í Ý

(9.31)

where Rf is the radius of the film, hd is the viscosity of the liquid constituting the drop and (¶g /¶r)fi is the initial value of the interfacial tension gradient at the periphery of the film. It is difficult to predict the value of this parameter. Jeelani and Hartland (1994) calculated the values of (¶g /¶r)fi from Eqs. (9.30) and (9.31) using the experimental values of coalescence time. It is evident that this term is zero in absence of surfactant (see Section 8.6.5). Slattery (1990) has presented a set of seven film-drainage models for calculating coalescence time from the physical properties of the system. These equations were derived by a linear stability analysis. The equations for coalescence time are presented in Table 9.1. These models were developed based on varied assumptions, some of which are as follows. Table 9.1 Models for coalescence time which consider van der Waals attraction between the surfaces Equation for coalescence time

tc1

1.07

tc2

K a3.4 ( 'U g)0.6

0.705

tc3

Based on the work of

J 1.2 B0.4 K a3.4 ( 'U g)0.6

Chen et al. (1984)

J 1.2 B0.4

1.046

K a 4.5 'U g J 1.5 B0.5 Mackay and Mason (1963)

tc6

K a4.5'U g

tc4

0.37

tc5

5.202

0.79

J 1.5B0.5 K a1.75 J

0.75 0.25

B

Hodgson and Woods (1969)

K a 4.06 ( 'U g)0.84 J 1.38 B0.46 Slattery (1990)

tc7

0.44

K a4.06 ( 'U g)0.84 J 1.38 B0.46

The two interfaces bounding the draining liquid film were assumed to be axisymmetric. The deformation of the flat interface was assumed to be small. The drop was assumed to be small such that the Bond number was small ( 0.83), the foam cells take on a variety of polyhedral shapes. Foams not only contain gas, liquid and surfactant, but can contain dispersed oil droplets and nano-sized solid particles as well.

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Figure 9.19 Structural aspects of foam lamella [Figure (b) is reproduced from Schramm (2005) by permission from Wiley-VCH Verlag GmbH, © 2005].

9.4.1 Preparation of Foams and Measurement of their Stability The easiest way to prepare foam in the laboratory is to mix a gas and a liquid together in a container and then shake. A foam can be formed in a liquid if bubbles of gas are injected faster than the rate at which liquid between the bubbles can drain away. Although the bubbles coalesce as soon as the liquid between them has drained away, a temporary dispersion can form. An example is the foam formed when bubbles are vigorously blown into a viscous oil. It should be remembered that like the emulsions, foams are not thermodynamically stable: eventually they all collapse. However, in carefully-controlled environments, it has been possible to make surfactant-stabilised static bubbles with lifetimes of months and years! The general methods for generation of foam are as follows (Schramm, 2005): (i) Bubbling gas into a liquid or solution (ii) Causing a stream of liquid to fall onto a pool of liquid or solution engulfing air bubbles (iii) Suddenly reducing the pressure on a solution of dissolved gas, causing rapid nucleation and growth of gas bubbles within the solution (iv) Turbulent mixing with a stirrer such that air is whipped into a liquid (v) Co-injecting liquid and gas into a mechanical foam generator which uses pressure drop, turbulence and/or tortuous flow methods to cause bubble pinch-off and subdivision. Most foams are prepared with the aid of a foaming agent. The foaming agent may comprise one or more of surfactants, macromolecules and finely divided solids. It is needed to reduce the surface tension and thereby facilitate formation of a large amount of interfacial area with the minimal requirement of mechanical energy to create it. The surfactant provides a protective film at the surface of the bubbles which prevents their coalescence. Stability against coalescence can be further enhanced by the inclusion of agents that increase the viscosity and retard evaporation, e.g. addition of glycerine to a foaming solution makes the foam more stable. Several scientists have suggested that high surface viscosity stabilises foams. Micro-foams (also known as colloidal gas aphrons) can be prepared by dispersing gas into a surfactant solution under high shear. Under the appropriate conditions of turbulent wave break-up, it is possible to create a dispersion of very small gas bubbles, each surrounded by a film of surfactant molecules. Under ambient conditions, the bubble diameters are typically in the range of 50 to

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300 mm. Sebba (1971) has presented a simple method for preparing micro-foams in the laboratory by entrainment of gas bubbles in a dilute surfactant solution. The gas volume fraction in a foam is expressed by foam quality. In three-phase systems, when the foam contains solid particles as well, the term slurry quality is used to give the volume fraction of gas and solid, i.e. (Vg  Vs ) /(Vg  Vs  Vl ), where Vg, Vs and Vl denote the volumes of gas, solid and liquid phases respectively. The stability of foam is usually tested by the following methods: (i) The lifetime (i.e. coalescence time) of single bubbles (ii) The steady-state foam volume under given conditions of gas flow, shaking or shearing (iii) The rate of collapse of a static column of foam (known as Ross–Miles test) Detailed experimental and theoretical studies have been made on the first method [see Chaudhari and Hofmann (1994); Giribabu and Ghosh (2007); Giribabu et al. (2008)]. This method is similar to the study of coalescence of drops discussed in Section 9.2.4. Similar to that observed for the drops, stochastic distributions of coalescence time are always observed in these experiments. The stability of the bubbles in different foam systems can be compared by the mean values of the distributions. It can also be compared from the parameters of the stochastic model described in Section 9.2.4 [see Ghosh (2009)]. In the second method, foam is generated by flowing gas through a porous orifice into a test solution. The steady-state foam volume maintained under constant gas flow rate into the column is then measured. This technique is frequently used to assess the stability of evanescent foams. In the static foam test, foam is generated by filling a pipette with a given volume of a foaming solution. Then the solution is allowed to fall a specified distance into a separate volume of the same solution that is contained in a receiver vessel. Foam is produced as the solution from the pipette falls on the solution in the receiver. The volume of foam that is produced immediately upon draining of the pipette is termed initial foam volume. This volume is measured. The decay in foam volume after some time is also measured. From these data, the rate of collapse of the foam column can be calculated. In either the dynamic or the static foam tests, one should bear in mind the many changes in a foam that may occur with time, such as gas diffusion and change in bubble size distribution. The rate of drainage of foam is often expressed in terms of foam number. A foam is formed in a vessel and thereafter the remaining foam volume is measured as a function of time. The foam number is the volume of bulk liquid that has separated after a specified interval, expressed as a percentage of the original volume of liquid foamed.

9.4.2 Structure of Foams Lord Kelvin proved in his work (circa 1887) on the division of space with minimum partitional area that tetrakaidecahedron is the ideal cell in a dry, polyhedral and monodisperse foam. This polyhedron has eight doubly-curved hexagonal faces and six quadrilateral faces with bowed edges [see Figure 9.20(a)]. The curved surfaces are a requirement of the minimisation of surface energy. This structure also satisfies the Plateau’s conditions mentioned in Section 9.4.1. Kelvin’s minimal tetrakaidecahedron is actually a slightly distorted plane-faced isotropic or orthic tetrakaidecahedron, which is obtained by truncating the six corners of a regular octahedron each to such a depth as to reduce its eight original (equilateral triangular) faces to equilateral equiangular hexagons. The orthic tetrakaidecahedron itself is an unsatisfactory foam cell because it does not satisfy Plateau’s conditions. In the minimal tetrakaidecahedron, the corners of the orthic polyhedron are maintained.

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The quadrilateral faces remain planar but acquire bowed-out noncircular edges, each having a total turning angle of ~p/9 rad. The corners of each nonplanar, wavy hexagon are still in one plane, while the hexagon contains three straight lines, i.e. its three long diagonals.

Figure 9.20 The transition from Kelvin’s tetrakaidecahedron (termed a -tetrakaidecahedron), (a), to b -tetrakaidecahedron, (c), through a polyhedron, (b) (Williams, 1968) (reproduced by permission from The American Association for the Advancement of Science, © 1968).

However, the statistical distribution of polygon faces on packed soap bubbles differs markedly from Kelvin’s tetrakaidecahedron, and the bubbles show a predominance of the pentagonal faces. Interestingly, studies of metal crystallites and vegetable cells also show similar distributions. It may be due to the small deviations from monodispersity or the disturbing effect of the container walls. The stringent requirements for the occurrence of the minimal tetrakaidecahedron structure are: every part of the boundary of the group must be either infinitely distant from the place considered, or be so adjusted as not to interfere with the homogeneousness of the interior distribution of cells. Princen and Levinson (1987) suggested a possible reason why the Kelvin’s tetrakaidecahedron, in spite of satisfying the mathematically-correct conditions of space-filling and minimum partitional area, fails to match the structure of foams. According to them, upon drainage of the continuous phase from between the initially spherical bubbles in face-centred cubic packing, the system may get trapped in an intermediate, less-ordered structure that, although at a local surface area minimum, may be energetically separated from the Kelvin’s structure by a high barrier. Evidence for this may be found in the great difficulty one encounters in trying to build a 15-bubble cluster that has a Kelvin’s tetrakaidecahedron at its centre. Ross and Prest (1986), and Princen and Levinson (1987) have calculated the relative surface area and isoperimetric coefficient for several polyhedra which are shown in Table 9.2. The relative surface area is defined as A/A0, where A is the surface area of the polyhedron and A0 is the surface area of the sphere having the same volume as the polyhedron. The isoperimetric coefficient is defined as, Table 9.2 Relative surface area and isoperimetric coefficient for various bodies Body Sphere Icosahedron Kelvin’s tetrakaidecahedron Orthic tetrakaidecahedron Rhombic dodecahedron Pentagonal dodecahedron Octahedron Cube Tetrahedron

A/A0

x

1.0000 1.0646 1.0970 1.0990 1.1053 1.0984 1.1826 1.2407 1.4900

1.0000 0.8288 0.7575 0.7534 0.7405 0.7547 0.6046 0.5236 0.3023

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[ 36S V 2 / A3 , where V is the volume of the polyhedron. These two quantities are related by the following equation: A A0

È 1Ø ÉÊ [ ÙÚ

13

(9.67)

Williams (1968) has pointed out that topological transformations of vertices and sides permit a number of other space-filling polyhedra to be derived from Kelvin’s tetrakaidecahedron. Such derived polyhedra have fewer elements of symmetry than their parent polyhedron, but one of them can be singled out that closely matches the naturally occurring distributions of faces. This tetrakaidecahedron is designated as b -tetrakaidecahedron, shown in Figure 9.20(c). It can be mechanically derived from the Kelvin’s a-tetrakaidecahedron by taking any edge common to two hexagons plus the edges that meet at each end of this edge [Figure 9.20(a)], rotating them p /2 rad and reconnecting them. The resultant polyhedron [Figure 9.20(b)] with four quadrilateral, four pentagonal and six hexagonal faces will also pack to fill space. The same operation is then performed with the same group of edges on the opposite side of the polyhedron. The b -tetrakaidecahedron retains the same average number of sides per face (5.143), faces (14), vertices (24) and edges (36) as the Kelvin’s tetrakaidecahedron. However, it has two quadrilateral, eight pentagonal and four hexagonal faces, which reproduces the predominance of the pentagonal faces that is observed in all polyhedral aggregates, which is so lacking in Kelvin’s tetrakaidecahedron. Williams’ b -tetrakaidecahedra pack together as a body-centred tetragonal lattice. It is not isometric, therefore, if it was metastably composed of soap films, would rearrange spontaneously to an assembly of Kelvin’s polyhedra. This would occur because the b -tetrakaidecahedron has a smaller isoperimetric quotient than Kelvin’s. Williams (1968) reported that the value of A/A0 would be 4% higher than that of a-tetrakaidecahedron to enclose the same volume. This feature makes it impossible to produce with soap films a cluster of 15 equal-sized bubbles having an enclosed b-tetrakaidecahedron. Hence, Williams’ minimal b -tetrakaidecahedron, with its good agreement with the distribution of faces found in foams and other natural packings, and its agreement with the spacefilling requirement, does not meet the condition of minimum surface area that the surface tension imposes. Therefore, from the discussion presented in this section, it appears that no single polyhedral cell can meet all the requirements. It should be noted that many scientists have used foam structures different from tetrakaidecahedron in various models (e.g. drainage of foams). Many of them have used the pentagonal dodecahedral shape of the bubbles: a symmetric cluster of thirteen equal bubbles contains a minimal pentagonal dodecahedron at its center (which is known as the Dewar cluster). This structure, however, does not fill space without voids.

9.4.3 Foam Drainage Because the gas bubbles are polyhedral in shape, the liquid in foam can be divided into films and Plateau borders. From Figure 9.21, it is observed that at the Plateau borders, the gas–liquid interface is quite curved. This generates a low pressure region in the Plateau area (as predicted by the Young– Laplace equation, discussed in Section 4.5.2). Since the interface is flat along the thin film region, a higher pressure exists there. This pressure difference forces liquid to flow towards the Plateau borders, causing thinning of the films, and motion in the foam. Haas and Johnson (1967) assigned separate roles to films and Plateau borders during the drainage process. They showed that flow

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through the films due to gravity was negligible. Instead, the films drain into the adjacent Plateau borders due to the curvature effect as shown in Figure 9.21. The Plateau borders, in turn, form a network, through which the liquid flows due to gravity. This mechanism of foam drainage has been accepted by many scientists, however, the details and complexity vary.

Figure 9.21

Pressure difference across the curved surfaces in foam lamellae and flow of liquid towards the Plateau borders.

Different shapes of the Plateau border have been assumed: Haas and Johnson (1967) assumed the Plateau border to be circular, whereas Desai and Kumar (1982) assumed it to be triangular. Various assumptions have also been made on the rigidity of the walls of the Plateau border. Haas and Johnson (1967), and Hartland and Barber (1974) assumed them to be rigid. On the other hand, Desai and Kumar (1982) have treated them as partially mobile, and the mobility depends upon the surface viscosity. They divided the Plateau border into two categories: the nearly-horizontal and the nearlyvertical Plateau borders. Different roles were assigned to these two types. It was assumed that the films drained into all the Plateau borders equally. The horizontal Plateau borders receive the liquid from the films and drain it into the vertical Plateau borders. The vertical Plateau borders receive liquid from the films, from the horizontal Plateau borders as well as the vertical Plateau borders above, and discharge it into the vertical Plateau borders below them. Gururaj et al. (1995) assumed the b -tetrakaidecahedron structure of the foam bubbles and developed a network model of static foam drainage. The role of surface elasticity on foam stability has been studied in several works (Malysa et al., 1981; Huang et al., 1986). To quantify foam stability, Malysa et al. (1981) used a parameter called retention time, which was defined as the slope of the linear part of the plot of gas flow rate versus the gas volume contained in the system. It characterises the frothability of the flotation frothers. They found that the retention time varied almost linearly with the Marangoni elasticity (see Section 7.4). Figure 9.22 depicts their results for n-octanol and n-octanoic acid solutions. Most of the works on foam drainage have not included the effects of surface forces on the drainage process. As noted by Narsimhan and Ruckenstein (1986), the disjoining pressures arising due to these forces can strongly influence the process of film thinning when the thickness of the film reduces below 100 nm. The critical thickness of rupture is dependent on these forces (see Section 8.6.1). Therefore, the models for foam drainage need to couple the hydrodynamics with the instability of the thin liquid films to predict the conditions for foam collapse.

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Figure 9.22

9.4.4

Variation of retention time with Marangoni dilatational elasticity (Malysa et al., 1981) (adapted by permission from Elsevier Ltd., © 1981).

Applications of Foams

Foams are very commonplace in materials such as foods, shaving creams, fire-fighting materials and detergents. Foams, as froths, are intimately involved in many mineral separation processes. Foams are widely used at many stages of petroleum recovery and processing such as oil-well drilling, reservoir injection and process-plant foams. In enhanced oil recovery, a gas is injected in the form of a foam. Suitable foams can be formulated with air, nitrogen, natural gas, carbon dioxide or steam. In a thermal process, when a steam foam contacts residual crude oil, it tends to condense and create water-in-oil emulsions. In a non-thermal process, the foam may emulsify the oil and form an oil-inwater emulsion. This emulsion is then drawn up into the foam structure. Micro-foams have some interesting potential applications in soil remediation and reservoir oil recovery.

SUMMARY This chapter presents various aspects of emulsions, microemulsions and foams, which constitute a major part of colloid and interface science. Various applications of these colloidal systems are discussed in the Introduction section (and in detail later). Section 9.2 discusses the types of emulsions and the methods for preparing them. The stability of emulsions is discussed. The Ostwald ripening is discussed using a cell model. The flocculation and coalescence of drops is discussed in the light of Smoluchowski’s theory and some of the modifications of this theory are also explained. The mechanism of coalescence of drops is discussed using the film-drainage and stochastic theories of coalescence. Coalescence equipment, especially the electrostatic coalescers are discussed briefly. Next, the phase inversion of emulsions is discussed. Section 9.3 discusses the properties and applications of microemulsions. First, a brief historical overview is presented. Winsor’s classification of the microemulsion systems is presented. The Winsor ratio and its significance are explained. Typical phase diagrams for the Winsor Type I, II and III systems are presented. The liquid crystalline phases in microemulsion are discussed with a typical system. Next, the thermodynamic stability of microemulsions is discussed. A model based on the

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dilution effect was presented and discussed in detail. An equation for the equilibrium droplet radius is derived. Another model of microemulsion based on the interface curvature energy is discussed. Next, the rheology of microemulsions is discussed. The modification of Einstein’s equation of viscosity for colloidal dispersions is discussed along with some other correlations. Section 9.4 discusses foams. The types of foams, foaming agents, antifoamers and defoamers are discussed. The Plateau’s conditions are explained. Some methods of preparation of foams are discussed and the techniques used to measure their stability are explained. The structure of foams is discussed next. Various aspects of Kelvin’s minimal tetrakaidecahedron structure are discussed as well as the modified structure proposed by Williams. The possible reasons for the non-occurrence of Kelvin’s tetrakaidecahedron in foams are discussed. The various drainage mechanisms of foams are discussed thereafter. The chapter ends with a discussion on the various applications of foams.

KEYWORDS Amphiphile Antifoamer Barrier ring Bending free energy Bending modulus Bicontinuous structure Binary coalescence Bubble coalescence Cell model Characteristic diffusion time Cloud point Coalescence Coalescence equipment Coalescence model Colloid mill Colloidal gas aphron Creaming Cumulative distribution coalescence time Defoamer Dimensionless coalescence threshold Drop–interface coalescence Emulsification equipment Emulsion Emulsion stability Equilibrium droplet size Film-drainage theory Flocculation Foam Foam drainage Foaming agent Foam number Foam quality

Foam structure Gaussian curvature modulus Haze point High-pressure homogeniser Interfacial force Kinetic stability Kugelschaum Lamellae Lamellar phase Lifshitz–Slyozov theory Marangoni elasticity Mechanism of coalescence Membrane emulsification Micro foam Microemulsion Microemulsion rheology Middle phase Normalised standard deviation Ostwald ripening Pentagonal dodecahedron structure Phase inversion Phase inversion temperature Plateau border Plateau’s condition Polyederschaum Principal curvature Pseudoemulsion film Rate of ripening Retention time Rotor–stator dispersing machine

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Slurry quality Sonication Spontaneous curvature Steiner angle Stochastic theory of coalescence Surface diffusion

Surface excess Tetrakaidecahedron structure Thermodynamic stability Toothed-disc dispersing machine Winsor ratio Winsor systems

NOTATION a ac a A B c c¥ cs C1, C2 D DG Ec fi fr F(tR) g hc hi hr H H0 k K K¢ Ks m m M n n0 na nf no np nt nv p

drop radius, m critical radius, m number average radius, m area, m2 modified Hamaker constant, J m concentration, mol/m3 bulk solubility of the dispersed phase in the continuous phase, kg/m3 concentration of surfactant, m–3 (or mol/m3) constants in equation (9.66) bulk diffusivity, m2/s surface diffusivity of the surfactant molecules, m2/s curvature free energy, J number frequency of the drops with radius ai repulsive force generated by one mole of surfactant at the barrier ring, N/mol cumulative probability distribution of coalescence time acceleration due to gravity, m/s2 critical film thickness, m initial film thickness, m separation between the drop and the flat interface at the barrier ring, m principal curvature, m–1 spontaneous curvature, m–1 Boltzmann’s constant, J/K coalescence rate constant in Eq. (9.22), s–1 coalescence rate constant in Eq. (9.26), s–1 adsorption constant, m total number of molecules of a species per unit volume of microemulsion, m–3 average number of drops in an aggregate molecular weight, kg/mol total number of drops in the emulsion initial number of drops average number of primary drops in an aggregate number of films in an aggregate number of molecules of oil per unit volume of microemulsion, m–3 total number of drops at time t number of primary drops from the initial set remaining at time t number of aggregates at time t pressure, Pa

Emulsions, Microemulsions and Foams

PG r rb R Rb Rd Rˆ d Rf Rw SG t tc t T vi V Vd Vt wb x

363

dimensionless coalescence threshold radial position, m diffusion boundary radius, m gas constant, J mol–1 K–1 radius of barrier ring, m radius of microemulsion droplet, m radius of stable microemulsion droplet, m radius of film, m Winsor ratio normalised standard deviation time, s coalescence time, s characteristic diffusion time, s temperature, K volume of a molecule of species i, m3 volume, m3 volume of the dispersed phase, m3 volume of emulsion, m3 width of the barrier ring, m mole fraction

Greek Letters a b g G Gm

* * d DF DGf DSf Dr e h hd hr hs q kb kG li

flocculation rate constant, s–1 parameter in Eq. (9.30) interfacial tension, N/m surface excess concentration of the surfactant, mol/m2 minimum value of the surfactant concentration at the barrier ring required to prevent the drop from coalescence, mol/m2 mean value of the distribution of surface excess, mol/m2 surface excess in microemulsion system, m–2 half of the average separation between the drops, m free energy of formation of microemulsion, J/m3 free energy of formation, J entropy of formation, J/K difference in density between two liquids, kg/m3 interaction energy per unit area, J/m2 viscosity of the liquid constituting the continuous phase, Pa s viscosity of the dispersed phase, Pa s relative viscosity viscosity of the solvent, Pa s fraction in Eq. (9.6) bending modulus, J Gaussian curvature modulus, J zeroes of the Bessel function of first kind and order one

364 m n x r s sG t tR j f¥ fd fg fm fs fv c y w

Colloid and Interface Science

chemical potential, J fraction of the total amount of surfactant at the interface which remained at the barrier ring after the displacement of surfactant molecules to the barrier ring isoperimetric coefficient density, kg/m3 surface charge density, C/m2 standard deviation in the distribution of G, mol/m2 tension of the microemulsion droplet surface, N/m dimensionless coalescence time activity coefficient solubility of the dispersed phase liquid in the continuous phase, expressed as a volume fraction volume fraction of the dispersed phase gas volume fraction in foams volume fraction of the dispersed phase liquid in the continuous phase volume fraction of the dispersed phase liquid at the surface of the drop volume fraction of the droplets volume fraction of the dispersed material in a microemulsion surface potential, V rate of Ostwald ripening, m3/s

EXERCISES 1. Give three important applications each of emulsions, microemulsions and foams. 2. Explain how oil-in-water and water-in-oil emulsions are related to the HLB of the surfactants. 3. Mention three methods of preparation of emulsions. Explain their advantages and disadvantages. 4. Explain how an emulsion can be prepared by membrane. 5. Mention the processes by which an emulsion is destabilised. 6. What is Ostwald ripening? When and why is it important? 7. Explain the salient features of Lifshitz–Slyozov theory. 8. What are the assumptions behind Smoluchowski’s theory of flocculation? 9. Explain the drawbacks of the van den Tempel equation of flocculation and coalescence, and how these were rectified in the model of Borwankar et al. (1992). 10. What are the major theories of coalescence of drops? 11. Explain the terms drop–interface coalescence and binary coalescence. 12. Explain the mechanism of coalescence as per the film-drainage theory. What are the main drawbacks of this theory? 13. Explain the mechanism of coalescence as per the stochastic theory. 14. Explain the difference between the parallel-disc and spherical-planar models of coalescence. 15. Explain why the coalescence times obtained from the film-drainage models of Slattery (1990) differ from one another. 16. What is interfacial tension gradient? How does it influence coalescence time? 17. Explain how the effect of van der Waals attraction was incorporated in the film-drainage models presented by Slattery (1990).

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18. What is barrier ring? Why is it called so? 19. What is characteristic diffusion time? How can it be calculated? 20. What is dimensionless coalescence threshold? Explain why at least some drops will never coalescence if the value of this quantity is less than unity. 21. On what parameters does the dimensionless coalescence threshold depend? 22. What is the role of surface diffusion on coalescence? 23. What is normalised standard deviation? 24. What is your opinion on the applicability of Eq. (9.32) to the coalescence of bubbles? 25. Explain three major methods by which an emulsion is broken. 26. What is electrostatic coalescence? For what type of emulsions is it used? 27. Explain the importance of the type of electric field and frequency in electrostatic coalescence. 28. Read the article by Waterman (1965) and explain the mechanism of electrostatic coalescence. 29. Explain why the conventional electrostatic coalescers are large in size. 30. What do you understand by phase inversion? How does it occur? 31. What is phase inversion temperature? 32. Explain how highly stable emulsions containing very fine droplets can be prepared by the phase inversion method. 33. Explain the effect of temperature on an emulsion stabilised by a nonionic surfactant. 34. What is Pickering emulsion? 35. Write in 200 words a note on the history of microemulsions. 36. What is the main difference between an emulsion and a microemulsion? 37. Explain the properties of Winsor Type I – IV microemulsion systems. 38. Explain how the solubilisation of organic liquid in a Type I microemulsion and solubilisation of water in a Type II microemulsion can be increased. 39. Define Winsor ratio and explain its significance. 40. Draw typical ternary phase diagrams for Winsor Type I, II and III systems and explain the main features of these diagrams. 41. Explain briefly the factors which impart thermodynamic stability to microemulsions. What is equilibrium droplet radius? How would you calculate it? 42. What is dilution effect? Explain its significance in microemulsion stability. 43. Explain the significance of interfacial bending free energy in stabilising a microemulsion. 44. Discuss why determination of rheological properties of a microemulsion is important. Are the microemulsions Newtonian or non-Newtonian fluids? 45. Explain the difference between polyederschaum and kugelschaum foams. What is micro-foam? 46. Explain the terms foaming agent, antifoam agent and defoamer. 47. How does oil destabilise foams? 48. Explain how the Taylor’s equation modifies Einstein’s equation for calculating the viscosity of a microemulsion. Under what condition does this equation reduce to Einstein’s equation? 49. Explain the significance of Plateau border. 50. What are Plateau’s conditions? 51. Discuss three methods of preparation of foams. 52. Discuss the methods used for determining the stability of foams. 53. Explain the structure of foams.

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54. How can you obtain the b -tetrakaidecahedron structure from Kelvin’s minimal tetrakaidecahedron? 55. Explain why the Kelvin’s tetrakaidecahedron structure is rarely observed in foams. 56. What is isoperimetric coefficient? What are its values for sphere and Kelvin’s tetrakaidecahedron? 57. What is Steiner angle? 58. Explain the models of foam drainage. What is the role of Plateau borders in foam drainage? 59. What is retention time? How is it related to the surface dilatational elasticity?

NUMERICAL AND ANALYTICAL PROBLEMS 9.1 Calculate the rate of Ostwald ripening for an oil-in-water emulsion at 298 K if the oil has the following properties: density = 830 kg/m3, average molecular weight = 103 kg/mol, diffusivity = 1.3 × 10–9 m2/s and solubility in water = 1.5 kg/m3. The interfacial tension is 5 mN/m. 9.2 Read the article by Charles and Mason (1960) carefully and derive Eqs. (9.28) and (9.29). Write down the assumptions behind the development of these equations. 9.3 Calculate the coalescence time of a 1.7 mm radius carbon tetrachloride drop at water–carbon tetrachloride interface at room temperature using the parallel-disc and spherical-planar models. The initial film thickness is 1 mm and the critical film thickness is 25 nm. Given: density of carbon tetrachloride = 1600 kg/m3 and interfacial tension = 45 mN/m. 9.4 Calculate the coalescence time of a 2.3 mm radius water drop at heptane–water interface at 298 K, using the model of Jeelani and Hartland (1994). Given: density of water = 998 kg/m3, density of heptane = 684 kg/m3, viscosity of water = 1 mPa s, viscosity of heptane = 0.4 mPa s, interfacial tension = 50.3 mN/m and Hamaker constant = 2 × 10–21 J. Take initial film thickness = 1 mm. Calculate critical film thickness and area of the film, using the equations given in Section 8.6.1. 9.5 Calculate the coalescence time of a 3 mm diameter air-bubble at a flat water–air interface in an aqueous solution of sodium dodecyl sulphate using the seven film-drainage models presented in Table 9.1. The surface tension is 62.1 mN/m. Take B = 1 × 10–28 J m. Explain your results. 9.6 Show that when the radius of the droplet in microemulsion approaches zero (i.e. infinite dilution limit), x1 — Rd , where x1 is the mole fraction of surfactant after the formation of the droplets of radius Rd and adsorption of surfactant and cosurfactant at the surface of the droplets. 9.7 For a microemulsion system involving a nonionic surfactant, the interfacial tension is 40 mN/m, volume fraction of the droplets is 0.02 and the concentration of surfactant is 6 × 1019 molecules per cm3 of the microemulsion. Calculate the equilibrium radius of the microemulsion droplet from these data. 9.8 The interfacial bending modulus is 5.35 × 10–21 J, the Gaussian curvature modulus is 4.1 × 10–21 J and the bending force (i.e. kbH0) is 8.23 × 10–13 N in a microemulsion. Calculate the radius of microemulsion droplet that minimises the curvature free energy.

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9.9 In a coalescence experiment involving 2 mm diameter air-bubbles in an aqueous surfactant solution, the coalescence times obtained in 15 observations are given below. Obs. no.

tc (s)

Obs. no.

tc (s)

Obs. no.

tc (s)

1 2 3 4 5

5.7 8.1 17.1 10.5 15.7

6 7 8 9 10

2.8 12.3 4.1 0.6 13.1

11 12 13 14 15

15.0 0.4 7.8 13.5 9.1

Develop the cumulative distribution of coalescence time using these data. Fit the stochastic model to the distribution and obtain the parameters of the model. Show your results graphically. Given: DG = 1 × 10–10 m2/s and g = 66 mN/m. 9.10 Mitra and Ghosh (2007) have reported (binary) coalescence time distributions of water drops in xylene in presence of the surfactant, sodium dodecyl benzene sulphonate. The coalescence time increased with increase in surfactant concentration. The variations of bubble radius and interfacial tension with surfactant concentration are shown below. Surfactant concentration, cs (mol/m3)

Radius (mm)

g (mN/m)

0.09 0.14

2.4 2.3

20.56 17.22

The increase of the surface excess concentration G with the surfactant concentration cs is expressed by the equation, *

È 63.2cs Ø ¹ Calculate the relative variation of PG 1.64 – 10 6 É Ê 1  63.2cs ÙÚ

with surfactant concentration from these data. Assume that the quantity, wbfrn, does not vary with surfactant concentration. 9.11 For the microemulsion system, (tetradecyldimethylamine oxide + 7.5 mol % SDS)-decanewater, the variation of relative viscosity with volume fraction (of surfactant plus decane) at 298 K is given below (Gradzielski et al., 1992). c

hr

c

hr

0.017 0.029 0.057 0.086 0.114

1.09 1.16 1.34 1.59 1.88

0.142 0.212 0.283 0.354

2.20 4.03 9.65 35.10

Fit the equation proposed by Thomas (1965) to these data and determine the constants. Present your results graphically. 9.12 Calculate the values of A/A0 and isoperimetric coefficient for a cube. Verify your results from the values given in Table 9.2.

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FURTHER READING Books Edwards, D.A., H. Brenner, and D.T. Wasan, Interfacial Transport Processes and Rheology, Butterworth-Heinemann, Boston, 1991. Kumar, P. and K.L. Mittal (Eds.), Handbook of Microemulsion Science and Technology, Marcel Dekker, New York, 1999. Schramm, L.E., Emulsions, Foams, and Suspensions, Wiley-VCH, Weinheim, 2005. Sjöblom, J. (Ed.), Emulsions and Emulsion Stability (Surfactant Science Series, Vol. 132), Marcel Dekker, New York, 2006. ———, Encyclopedic Handbook of Emulsion Technology, Marcel Dekker, New York, 2001. Slattery, J.C., Interfacial Transport Phenomena, Springer-Verlag, New York, 1990. Winsor, P.A., Solvent Properties of Amphiphilic Compounds, Butterworth, London, 1954.

Articles Almgren (Jr.), F.J. and J.E. Taylor, “The Geometry of Soap Films and Soap Bubbles”, Scientific American, 235, 82 (1976). Attwood, D., L.R.J. Currie, and P.H. Elworthy, “Studies of Solubilized Micellar Solutions III: The Viscosity of Solutions Formed with Nonionic Surfactants”, J. Coll. Int. Sci., 46, 261 (1974). Bailes, P.J. and S.K.L. Larkai, “Electrostatic Separation of Liquid Dispersions”, UK Patent 217 1031A (1986). Bellocq, A.M., J. Biais, B. Clin, P. Lalanne, and B. Lemanceau, “Study of Dynamical and Structural Properties of Microemulsions by Chemical Physics Methods”, J. Coll. Int. Sci., 70, 524 (1979). Borwankar, R.P., L.A. Lobo, and D.T. Wasan, “Emulsion Stability — Kinetics of Flocculation and Coalescence”, Coll. Surf., 69, 135 (1992). Charles, G.E. and S.G. Mason, “The Coalescence of Liquid Drops with Flat Liquid/Liquid Interfaces”, J. Coll. Sci., 15, 236 (1960). Chaudhari, R.V. and H. Hofmann, “Coalescence of Gas Bubbles in Liquids”, Rev. Chem. Eng., 10, 131 (1994). Chen, J.-D., P.S. Hahn, and J.C. Slattery, “Coalescence Time for a Small Drop or Bubble at a Fluid– Fluid Interface”, AIChE J., 30, 622 (1984). Danov, K.D., I.B. Ivanov, Th.D. Gurkov, and R.P. Borwankar, “Kinetic Model for the Simultaneous Processes of Flocculation and Coalescence in Emulsion Systems”, J. Coll. Int. Sci., 167, 8 (1994). de Bruyn, P.L., J.Th.G. Overbeek, and G.J. Verhoeckx, “On Understanding Microemulsions, III: Phase Equilibria in Systems Composed of Water, Sodium Chloride, Cyclohexane, SDS, and n-Pentanol”, J. Coll. Int. Sci., 127, 244 (1989). Desai, D. and R. Kumar, “Flow Through a Plateau Border of Cellular Foam”, Chem. Eng. Sci., 37, 1361 (1982). Doi, M. and S.F. Edwards, “Dynamics of Rod-Like Macromolecules in Concentrated Solution: Part 2”, J. Chem. Soc., Faraday Trans. II, 74, 918 (1978).

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Engels, T., T. Förster, and W. von Rybinski, “The Influence of Coemulsifier Type on the Stability of Oil-in-Water Emulsions”, Coll. Surf. (A), 99, 141 (1995). Enomoto, Y., K. Kawasaki, and M. Tokutama, “Computer Modelling of Ostwald Ripening”, Acta Metall., 35, 907 (1987). Eow, J.S. and M. Ghadiri, “Electrostatic Enhancement of Coalescence of Water Droplets in Oil: A Review of the Technology”, Chem. Eng. J., 85, 357 (2002). Ghosh, P., “A Comparative Study of the Film-Drainage Models for Coalescence of Drops and Bubbles at Flat Interface”, Chem. Eng. Tech., 27, 1200 (2004). Ghosh, P., “Coalescence of Bubbles in Liquid”, Bubble Sci. Eng. Tech., 1, 75 (2009). Ghosh, P. and V.A. Juvekar, “Analysis of the Drop Rest Phenomenon”, Chem. Eng. Res. Des., 80, 715 (2002). Giorno, L., N. Li, and E. Drioli, “Preparation of Oil-in-Water Emulsions using Polyamide 10 kDa Hollow Fiber Membrane”, J. Membr. Sci., 217, 173 (2003). Giribabu, K. and P. Ghosh, “Adsorption of Nonionic Surfactants at Fluid–Fluid Interfaces: Importance in the Coalescence of Bubbles and Drops”, Chem. Eng. Sci., 62, 3057 (2007). Giribabu, K., M.L.N. Reddy, and P. Ghosh, “Coalescence of Air Bubbles in Surfactant Solutions: Role of Salts Containing Mono-, Di-, and Trivalent Ions”, Chem. Eng. Commun., 195, 336 (2008). Gradzielski, M. and H. Hoffmann, “Structural Investigations of Charged O/W Microemulsion Droplets”, Adv. Coll. Int. Sci., 42, 149 (1992). Gururaj, M., R. Kumar, and K.S. Gandhi, “A Network Model of Static Foam Drainage”, Langmuir, 11, 1381 (1995). Haas, P.A. and H.F. Johnson, “A Model and Experimental Results for Drainage of Solution between Foam Bubbles”, Ind. Eng. Chem. Fundam., 6, 225 (1967). Hartland, S. and A.D. Barber, “Model for a Cellular Foam”, Trans. Inst. Chem. Engrs., 52, 43 (1974). Helfrich, W., “Elastic Properties of Lipid Bilayers: Theory and Possible Experiments”, Z. Naturforsch, 28c, 693 (1973). Hoar, T.P. and J.H. Schulman, “Transparent Water-in-Oil Dispersions: The Oleopathic Hydromicelle”, Nature, 152, 102 (1943). Hodgson, T.D. and D.R. Woods, “The Effect of Surfactants on the Coalescence of a Drop at an Interface II”, J. Coll. Int. Sci., 30, 429 (1969). Huang, D.D., A. Nikolov, and D.T. Wasan, “Foams: Basic Properties with Application to Porous Media”, Langmuir, 2, 672 (1986). Jeelani, S.A.K. and S. Hartland, “Effect of Interfacial Mobility on Thin Film Drainage”, J. Coll. Int. Sci., 164, 296 (1994). Joscelyne, S.M. and G. Tragardh, “Membrane Emulsification — A Literature Review”, J. Membr. Sci., 169, 107 (2000). Kabalnov, A.S. and E.D. Shchukin, “Ostwald Ripening Theory: Applications to Fluorocarbon Emulsion Stability”, Adv. Coll. Int. Sci, 38, 69 (1992). Karbstein, H. and H. Schubert, “Developments in the Continuous Mechanical Production of Oil-inWater Macro-emulsions”, Chem. Eng. Proc., 34, 205 (1995). Koczo, K., L.A. Lobo, and D.T. Wasan, “Effect of Oil on Foam Stability: Aqueous Foams Stabilized by Emulsions”, J. Coll. Int. Sci., 150, 492 (1992).

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Kumar, M.K., T. Mitra, and P. Ghosh, “Adsorption of Ionic Surfactants at Liquid–Liquid Interface in the Presence of Salt: Application in Binary Coalescence of Drops”, Ind. Eng. Chem. Res., 45, 7135 (2006). Lifshitz, I.M. and V.V. Slyozov, “The Kinetics of Precipitation from Supersaturated Solid Solutions”, J. Phys. Chem. Solids, 19, 35 (1961). Mackay, G.D.M. and S.G. Mason, “The Gravity Approach and Coalescence of Fluid Drops at Liquid Interfaces”, Can. J. Chem. Eng., 41, 203 (1963). Malysa, K., K. Lunkenheimer, R. Miller, and C. Hartenstein, “Surface Elasticity and Frothability of n-Octanol and n-Octanoic Acid Solutions”, Coll. Surf., 3, 329 (1981). Mitra, T. and P. Ghosh, “Binary Coalescence of Water Drops in Organic Media in Presence of Ionic Surfactants and Salts”, J. Disp. Sci. Tech., 28, 785 (2007). Narsimhan, G. and E. Ruckenstein, “Hydrodynamics, Enrichment, and Collapse in Foams”, Langmuir, 2, 230 (1986). Princen, H.M., “Shape of a Fluid Drop at a Liquid–Liquid Interface”, J. Coll. Sci., 18, 178 (1963). Princen, H.M. and P. Levinson, “The Surface Area of Kelvin’s Minimal Tetrakaidecahedron: The Ideal Foam Cell (?)”, J. Coll. Int. Sci., 120, 172 (1987). Ross, S. and H.F. Prest, “On the Morphology of Bubble Clusters and Polyhedral Foams”, Coll. Surf., 21, 179 (1986). Ruckenstein, E., “The Origin of Thermodynamic Stability of Microemulsions”, Chem. Phys. Lett., 57, 517 (1978a). ———, “On the Thermodynamic Stability of Microemulsions”, J. Coll. Int. Sci., 66, 369 (1978b). Ruckenstein, E. and J.C. Chi, “Stability of Microemulsions”, J. Chem. Soc., Faraday Trans. II, 71, 1960 (1975). Schulman, J.H., W. Stoeckenius, and L.M. Prince, “Mechanism of Formation and Structure of Micro Emulsions by Electron Microscopy”, J. Phys. Chem., 63, 1677 (1959). Scriven, L.E., “Equilibrium Bicontinuous Structure”, Nature, 263, 123, 1976. Sebba, F., “Microfoams—An Unexploited Colloid System”, J. Coll. Int. Sci., 35, 643 (1971). Shinoda, K., “The Correlation between the Dissolution State of Nonionic Surfactant and the Type of Dispersion Stabilized with the Surfactant”, J. Coll. Int. Sci., 24, 4 (1967). Shinoda, K. and B. Lindman, “Organized Surfactant Systems: Microemulsions”, Langmuir, 3, 135 (1987). Shinoda, K. and H. Saito, “The Stability of O/W Type Emulsions as Functions of Temperature and the HLB of Emulsifiers: The Emulsification by PIT-Method”, J. Coll. Int. Sci., 30, 258 (1969). Smoluchowski, M.V., “Versuch einer Mathematischen Theorie der Koagulationskinetik Kolloider Losungen”, Z. Phys. Chem., 92, 129 (1917). Taylor, G.I., “The Viscosity of a Fluid Containing Small Drops of another Fluid”, Proc. Roy. Soc. Lond. (Ser. A), 138, 41 (1932). Taylor, P., “Ostwald Ripening in Emulsions”, Adv. Coll. Int. Sci., 75, 107 (1998). Taylor, S.E., “Investigations into the Electrical and Coalescence Behaviour of Water-in-Crude Oil Emulsions in High Voltage Gradients”, Coll. Surf., 29, 29 (1988). Thomas, D.G., “Transport Characteristics of Suspension VIII: A Note on the Viscosity of Newtonian Suspensions of Uniform Spherical Particles”, J. Coll. Sci., 20, 267 (1965).

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Waterman, L.C., “Electrical Coalescers”, Chem. Eng. Progr., 61, 51 (1965). Williams, R.E., “Space-Filling Polyhedron: Its Relation to Aggregates of Soap Bubbles, Plant Cells, and Metal Crystallites”, Science, 161, 276 (1968). Winsor, P.A., “Hydrotropy, Solubilisation and Related Emulsification Processes”, Trans. Faraday Soc., 44, 376 (1948). ———, “Binary and Multicomponent Solutions of Aliphatic Compounds: Solubilization and the Formation, Structure, and Theoretical Significance of Liquid Crystalline Solutions”, Chem. Rev., 68, 1 (1968). Yarranton, H.W. and J.H. Masliyah, “Numerical Simulation of Ostwald Ripening in Emulsions”, J. Coll. Int. Sci., 196, 157 (1997).

10

Biological Interfaces

Pierre-Gilles de Gennes (1932 – 2007)

P.G. de Gennes was born in Paris (France) in 1932. He majored from the École Normale in 1955. From 1955 to 1959, he was a research engineer at the Atomic Energy Centre (Saclay) where he worked on neutron scattering and magnetism. During 1959, he was a postdoctoral visitor at Berkeley, and then served for about two years in the French Navy. In 1961, he started research on supraconductors, and in 1968, he switched to liquid crystals. In 1971, he became Professor at the Collège de France, and was a participant of STRASACOL (a joint action of Strasbourg, Saclay and Collège de France) on polymer physics. From 1980, he became interested in interfacial phenomena, in particular the dynamics of wetting. He received the Nobel Prize in Physics in 1991 for discovering the fact that methods developed for studying order phenomena in simple systems can be generalised to more complex forms of matter, in particular to liquid crystals and polymers. In his later years, de Gennes worked at the Institut Curie (Paris) on cellular adhesion and brain function. de Gennes died in Orsay (France).

TOPICS COVERED © © © © © ©

Protein adsorption at the interfaces Biological membranes Interfacial forces Cell adhesion Pulmonary surfactants Biomaterials

10.1 INTRODUCTION Biological systems are complex supramolecular assemblies. An important feature of these systems is that they are constructed from self-assembled, hierarchically ordered microstructures. They facilitate rapid and specific physiologic responses in multicellular systems. The cell-membrane proteins and 372

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lipids are associated with polysaccharides in a complex network which organises the structure of the membrane. The various modes of self-organisation in biological systems are dictated by the principles of thermodynamics. The alternation of system properties and transition from one type of morphology to another type occurs by subtle changes in the balance of intermolecular forces. Light and electron microscopy have revealed an astonishing complexity of the spatial organisation of biological systems. There are two basic factors behind this complexity: (i) these systems contain a large number of different chemical species, and (ii) they represent structures which involve complex patterns of dynamic processes. However, some insight into their spatial organisation can be gained if one considers simple model systems. Lipid bilayers or membranes probably represent the simplest models of this kind. Various physical and chemical interactions control the bio-interfacial phenomena such as immobilisation of enzymes in biocatalysis, biofouling, biofilms, phagocytosis, membrane fusion, and diffusion and transport in biomembranes. The adsorption or adhesion to the surface of a biocomponent often triggers a change in its physico-chemical properties, which affects its biological functioning. Therefore, the presence of interfaces plays a crucial role in biomedicine, food processing, environmental sciences and many other biotechnological applications. Biosurfactants and some elementary properties of vesicles and liposomes have been discussed in Chapter 3. The enzymatic lipolysis reactions will be discussed in Chapter 12. In this chapter, the adsorption of proteins at solid–liquid and gas–liquid interfaces will be discussed. The interfacial phenomena involved in biological membranes will be discussed. The interfacial forces that play very important roles in biological systems will be explained. The adhesion of cells will be discussed. The pulmonary surfactants and the mechanism by which they function will be explained. Finally, the interfacial phenomena involved in biomineralisation will be discussed.

10.2 PROTEIN ADSORPTION AT INTERFACES Protein adsorption at solid surfaces is a widespread phenomenon. Whenever a protein-containing aqueous solution is exposed to a solid surface, protein molecules spontaneously accumulate at the solid–water interface. This has important applications in many disciplines ranging from food science to biomedical science. When a protein molecule adsorbs at an interface, it changes its environment, which is often accompanied by structural rearrangements. Proteins are biopolymers. Therefore, their adsorption is similar in some respects to the simpler molecules such as water-soluble flexible polyelectrolytes [see Cohen Stuart et al. (1991)]. The adsorption of proteins at gas–liquid and liquid–liquid interfaces is important in biological applications such as pulmonary surfactants, food foams and emulsions. Proteins have dual hydrophobic–hydrophilic nature which causes them to adsorb at the fluid–fluid interfaces. Protein molecules tend to adopt globular conformations in aqueous solution in which non-polar groups are congregated in the centres and polar groups concentrate at the periphery. In this way, the free energy of the system is minimised by reducing the interaction between the non-polar groups and the water molecules. Sometimes, however, due to the steric constraints, the periphery of the molecule contains an appreciable number of non-polar groups which interact with water. When this type of a molecule reaches water–non-polar liquid interface, it can adopt a conformation in which the polar groups predominantly interact with the aqueous phase while the non-polar groups can escape. Because of elimination of an area of interface of high free energy (i.e. 72.5 mJ/m2), a considerable lowering of free energy of the system occurs. This is the major driving force for the adsorption of proteins at air– water interface.

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In this section, we will discuss some aspects of adsorption of proteins at solid–liquid and fluid– fluid interfaces.

10.2.1

Adsorption of Proteins at Solid–Liquid Interfaces

Proteins are copolymers containing approximately twenty different amino acid monomers. The amino acids are linked to each other in a polypeptide chain, as illustrated in Figure 10.1. Two of the three bonds in the peptide unit are free to rotate, whereas the C–N bond is fixed because of its partial double bond character. The side groups R and R¢ are positioned in the trans configuration, so that rotation around the bonds in the main chain is minimally hindered. The side groups vary greatly in hydrophobicity. Furthermore, some of the side groups are positively charged and others are negatively charged.

Figure 10.1

Structure of a peptide unit in a polypeptide chain. Rotation is possible around two of the three bonds. The middle C–N bond is fixed in the trans configuration (Cohen Stuart et al., 1991) (reproduced by permission from Elsevier Ltd., © 1991).

Protein molecules that are highly solvated and flexible have a disordered coil-like structure. The natural function of many of these proteins is nutritional. Examples of this type of proteins are glutens in wheat grains and caseins in milk. Fibrous proteins have a very regular structure such as helices and pleated sheets. These proteins are usually insoluble in water. They are mainly found in muscles and connective tissues. The greatest proportion of the protein species contains different structural elements, i.e. a-helices, b-pleated sheets and parts that are unordered and folded together into a compact dense globule. These proteins are known as globular proteins. An almost countless number of different kinds of globular proteins exists, and each kind has its specific biological function that is related to its own characteristic three-dimensional structure (e.g. enzymes and antibodies). The adsorption of globular proteins is important in practical applications such as in biomedical engineering, biosensors, immobilised enzymes in bioreactors, immunological diagnostic tests, drug targeting and drug delivery system, stabilising agents in foodstuffs, pharmaceutics and cosmetics. The compact three-dimensional structure of globular proteins in aqueous environment is determined by various types of interaction occurring inside the protein molecule as well as between the protein molecule and its aqueous environment such as hydrophobic interaction, Coulomb interaction, van der Waals interaction and hydrogen bonding. The main structure-determining factors

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are: (i) dehydration of the hydrophobic side groups, which favours the formation of a compact structure, and (ii) the tendency to maximise the conformational entropy of the protein molecule that counteracts the formation of internal structures including a-helices and b-sheets. The hydrogen bonds formed between the peptide units in a-helices and b-sheets largely reduce the rotational freedom of the bonds in the polypeptide chain, which involves increase of the Gibbs free energy by a few hundreds of kJ/mol at the room temperature (Baszkin and Norde, 2000). These and some other minor contributions are the reason behind the low thermodynamic stability of the native globular protein structure (Cohen Stuart et al., 1991). The Gibbs free energy of stabilisation of the native structure (taking the unfolded structure as the reference) is so low that for purely thermodynamic reasons, conformations can be easily changed by alteration of the external conditions. The introduction of an adsorbent is one of them. The interaction between proteins and surfaces is primarily governed by the factors such as (i) changes in the hydration of the adsorbent surface and the protein molecule, (ii) Coulomb interaction between the protein and the adsorbent which results in a redistribution of charged groups, and (iii) structural rearrangements in the adsorbing protein molecules. The contribution of each of these interactions to the overall adsorption process depends on the nature of the system. Structural rearrangements do not contribute significantly to the adsorption process for the proteins which have a strong internal coherence (‘hard’ proteins). These proteins adsorb on hydrophobic surfaces, but they adsorb on hydrophilic surfaces only if they are electrostatically attracted. Proteins of much lower structural stability (‘soft’ proteins) adsorb even under the seemingly unfavourable conditions on a hydrophilic, electrostatically-repelling surface. With these proteins, a large driving force for adsorption results from the structural rearrangements (Norde and Anusiem, 1992). The various steps of adsorption and desorption of a protein molecule on a solid surface are shown in Figure 10.2. The steps are: (1) transport towards the surface, (2) deposition at the surface, (3) relaxation of the adsorbed molecule, (4) detachment from the surface, and (5) transport away from the surface. These processes are indicated by the above-mentioned numbers in the figure. The asterisks indicate the degree of relaxation of the adsorbed molecule. The possible restructuring of the desorbed protein molecule has been shown by the dotted lines.

Figure 10.2 Protein adsorption process (Norde, 1992) (reproduced by permission from Elsevier Ltd., © 1992).

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Let us consider a simple model for protein transport and adsorption. For the transport of protein molecules towards the adsorbent surface driven by a concentration gradient, the flux can be expressed as J

km (cb  cs )

(10.1)

where cb and cs are the protein concentrations in the bulk solution and at the surface respectively, and km is a transport coefficient that depends on the hydrodynamic conditions. Deposition of protein at the surface of the adsorbent can be considered as a first-order process. The rate is given by

r

kd (cs  ceq )

(10.2)

where ceq is the concentration in bulk solution corresponding to the equilibrium value for the adsorbed amount. When r = J, from Eqs. (10.1) and (10.2), we obtain cs

km cb  kd ceq km  kd

(10.3)

Substituting cs from Eq. (10.3) into either Eq. (10.1) or Eq. (10.2), we get r

cb  ceq 1 / km  1 / kd

(10.4)

If the concentration of protein on the surface is G (expressed in mol/m2), then its build-up with time can be expressed by Eq. (10.4) as d* dt

cb  ceq 1 / km  1 / kd

(10.5)

The quantity ceq is a function of G, which can be determined from the adsorption isotherm. An explicit expression for dG/dt can be obtained employing this isotherm. The value of dG/dt reflects the interaction of the protein molecules with the surface. Protein adsorption isotherms depict the amount of protein adsorbed (G) versus the protein concentration in the solution (ceq) (as in the case of adsorption of surfactants discussed in Chapter 6). The equilibrium adsorption experiments for the determination of G versus ceq profiles are as follows (Arai and Norde, 1990). Protein solutions (prepared in buffer media) of different concentrations are added to the dispersions of the adsorbent material in the same buffer solutions in polycarbonate tubes. The tubes are gently rotated for about one day (which is sufficient in most cases for attaining the equilibrium). The solution is filtered and centrifuged, and the protein concentration in the filtrate is determined by liquid chromatography. The amount of protein adsorbed is calculated from the mass balance. The adsorption isotherms for negatively charged bovine serum albumin (BSA) on negatively charged hydrophilic surfaces of haematite and silica are shown in Figure 10.3. As we have discussed before, the driving force for the adsorption of the BSA molecules under these conditions probably stems from structural rearrangements leading to higher conformational entropy in the molecule. The initial parts of the isotherms indicate that the affinity between BSA and the adsorbent surfaces is not very high. Once the protein molecule has attached to the surface, it relaxes towards its equilibrium structure, which is different from its native structure in the solution, as illustrated in Step 3 in Figure 10.2. Structural relaxation implies optimisation of protein–surface interaction, and it normally involves a certain amount of spreading of the protein molecule on the surface of the

Biological Interfaces

Figure 10.3

377

Adsorption isotherms for bovine serum albumin on haematite and silica at 295 K and pH = 7 (Norde and Anusiem, 1992) (adapted by permission from Elsevier Ltd., © 1992).

adsorbent developing a larger number of protein-surface contacts. As a consequence, the adsorbed layer becomes structurally heterogeneous, i.e. the molecules which arrived at an early stage of the adsorption process find sufficient area available for spreading, whereas this is not the case for the molecules which arrive when the surface is already partially covered with protein (Pitt et al., 1986). The hysteresis in protein adsorption–desorption curves is often significant. The G(ceq) profiles for adsorption and desorption rarely coincide. The deviation between adsorption and desorption remains even when the observation time is extended to several days and is, therefore, much longer than the relaxation time of the protein at the surface. The occurrence of such hysteresis indicates that at a given protein concentration two metastable states exist, one on the adsorption isotherm and the other on the desorption isotherm. They are separated by a free energy barrier that prevents transition from one state to the other. Since adsorption and desorption represent two metastable states, it is likely that a physical change takes place in the system between adsorption and desorption. Because the proteins do not easily desorb from the surface upon dilution, it is virtually impossible to study the desorbed molecules. However, it is possible to exchange adsorbed proteins with other surfaceactive compounds. From such exchange studies, it has been found that permanent structural changes indeed take place (Chan and Brash, 1981).

10.2.2

Adsorption of Proteins at Fluid–Fluid Interfaces

Study of adsorption of protein at air–water interface provides information about the fundamental aspects of the adsorption process. A fluid–fluid interface has the advantages that (i) its area can be precisely measured, allowing parameters such as area per molecule to be monitored (e.g. using a Langmuir trough), (ii) impurities can be readily detected and removed, unlike the situation with solid interfaces, and (iii) the change in surface free energy can be obtained by a simple measurement of surface tension. Proteins have the advantage that, being under genetic control, they are homogeneous in size and composition, unlike most synthetic polymer preparations. In addition, they have low diffusion coefficients which make them suitable for studying the role of diffusion in adsorption. Proteins need to be freed from other surface-active compounds. The latter, because of their higher diffusion coefficients, tend to adsorb more rapidly than the proteins. Their presence, even at trace concentrations can, therefore, affect the results.

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After creation of a fresh air–water interface, providing there is no energy barrier to adsorption, all surface-active molecules that are near the interface adsorb. The rate of adsorption is given by the rate of diffusion of molecules from the bulk solution (at a concentration cb) to the sublayer (see Section 6.10). The amount of protein adsorbed at the interface within time t after creation of fresh surface in absence of any convection or desorption is given by (MacRitchie, 1963) *

È Dpt Ø 2cb É Ê S ÙÚ

12

(10.6)

where cb is the bulk concentration of the protein and Dp is its diffusion coefficient. However, if the adsorption process is reversible, then this simple relation can be expected to hold only in the initial stages and, in general, back-diffusion has to be considered (Ward and Tordai, 1946). EXAMPLE 10.1 Calculate the times required for a surface film of bovine serum albumin to reach a surface pressure of 0.1 mN/m if the bulk concentrations are 0.01 and 0.03 kg/m3. The diffusivity of BSA is 6 × 10–11 m2/s. The surface concentration corresponding to the given surface pressure is 7 × 10–7 kg/m2. Solution

For cb = 0.01 kg/m3 È 6 – 10 11 t Ø 2 – 0.01 – É Ù S Ê Ú

7 – 10 7

\ For cb = 0.03 kg/m3

12

t = 64.1 s È 6 – 10 11 t Ø 2 – 0.03 – É Ù S Ê Ú

7 – 10 7

12

\

t = 7.1 s Thus, the time decreases with increase in bulk concentration. The agreement between Eq. (10.6) and experimental results is good at low surface pressures. At high surface pressures, they deviate, which indicates that adsorption can no longer be explained by diffusion alone. All molecules that reach the sublayer are not adsorbed and an energy barrier exists. If the effects associated with electrical charge are minimal, the rate of adsorption of proteins into monolayers depends only on the surface pressure. For a molecule to adsorb at an interface from solution, it must do work against the surface pressure in order to create a hole of area A into which it can move in. The amount of work is given by Ô S s dA , where A is the area required, and this area is expected to correspond closely to the area occupied by the molecule at the interface. If the adsorption step is such that it occurs without any significant change in p s, then this work is simply equal to p sA. The rate of adsorption is given by r

È S AØ K exp É  s Ù Ê kT Ú

(10.7)

where K is a constant representing the rate of arrival of molecules to the interface. Taking logarithms on both sides of Eq. (10.7), we get

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Ss A (10.8) kT One way of testing the plausibility of the interfacial pressure barrier mechanism is to plot the logarithm of the rate of adsorption of protein against the surface pressure. This should produce a linear plot. The area A can be evaluated from the slope of the plot and this may be compared with the expected area of the molecule. Values of A calculated in this manner for five proteins are presented in Table 10.1. ln (r )

Table 10.1

ln (K ) 

Values of A for different proteins calculated from rate processes where ln(r) is a linear function of ps (Baszkin and Norde, 2000)

Protein Lysozyme Ovalbumin Human albumin Human g -globulin Myosin

Molecular weight (kg/mol)

Concentration (kg/m3)

A (nm2)

15 40 67 160 600

0.01 0.03 0.02 0.01 0.03

1.00 1.75 1.00 1.30 1.45

From the data given in Table 10.1 it is apparent that the values of A are very much smaller than the areas of the whole molecules, and correspond to areas of segments of about 6–10 amino acid residues at the surface. Therefore, these data reflect that adsorption is an activated process in which the transition state involves penetration of only a small portion of the protein molecule into the surface, as illustrated in Figure 10.4. According to the model presented in Figure 10.4, adsorption resembles a nucleation process. For the penetration of a small portion of the protein molecule into the surface, two free energy terms of opposite sign are involved. The positive term is associated with the losses of degrees of translational and rotational freedom required for maintaining the molecule in the adsorbed state. On the other hand, the negative free energy term is related to the replacement of an area of air–water interface of high energy by an area of lower free energy formed between nonpolar protein side-chains and air. When the energy of the molecule is sufficient to penetrate a critical

Figure 10.4

Schematic illustration of transition state for adsorption of a protein molecule. The change of free energy of the system with increase in the area of penetration of the molecule in the interface is also shown. The maximum in D G corresponds to the critical area for adsorption (MacRitchie, 1993) (reproduced by permission from Elsevier Ltd., © 1993).

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area of interface, the latter term becomes equal to the former. For areas greater than this, adsorption proceeds spontaneously with an accompanying lowering of the free energy of the system. Because of the lowering of free energy, which is possible as a result of replacing the high energy interface by the protein molecules and allowing the non-polar amino acid residues to escape from the aqueous phase, the protein molecule unfolds to form an extended conformation at the interface. The areas given in Table 10.1 actually represent statistical averages considering all the possible orientations of the globular molecule as it approaches the interface. For example, if the molecular surface that strikes the interface has a paucity of non-polar side chains, a much higher critical area will be needed for the transition state to be reached. Values of A for different proteins under the same conditions might, therefore, be a measure of the surface hydrophobicity, i.e. those with smaller values of A have higher surface hydrophobicity. Values of A can also be affected by the presence of electrical charge. Langmuir and Schaeffer (1939) observed that proteins are highly soluble in aqueous solution, but are extremely difficult to desorb. The protein molecules can reduce their free energy by displacing segments from the interface without the need for the entire molecule to be desorbed. We have discussed before that the activated complex for adsorption involves a segment of about six to ten amino acid residues. This should be the activated state for desorption because once the area occupied by a molecule at the interface is reduced to this value, it will not be stable any more. A protein molecule can reach this state either by compression of the surface or by fluctuations in free energy about the equilibrium conformation. At low surface pressures, the probability of a fluctuation enabling the conformation to reach the transition state conformation is very low. As the surface pressure is increased, the equilibrium between the adsorbed (trains) and expelled (loops and tails) segments shifts towards the expelled segments, and the probability of a fluctuation attaining the transition state increases. At high surface pressures (ps > 20 mN/m) desorption becomes measurable. A schematic illustration of attainment of the transition state is presented in Figure 10.5. Desorption at these surface pressures has been observed in practice. The manner in which the protein molecule folds during the desorption step is of great interest to biologists who like to know about the manner in which protein molecules fold after their synthesis. Using a film balance to follow desorption along with an optical technique for monitoring the conformation of monolayers may be a useful approach to study this phenomenon under controlled conditions.

Figure 10.5

Schematic illustration of increase in probability of a fluctuation attaining the transition state for desorption with increase in surface pressure (MacRitchie, 1993) (reproduced by permission from Elsevier Ltd., © 1993).

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10.3 BIOLOGICAL MEMBRANES One of the theories of origin of life expounds the idea that membranes localised material into natural bioreactors by limiting and controlling diffusion into and from the surroundings. All biological membranes are composed of proteins and lipids. The protein/lipid mass ratio in membranes varies greatly from ~0.23 for myelin sheets to 3.2 in the mitochondrial inner membranes and in many bacteria (Baszkin and Norde, 2000). In most biological membranes, the quantity of protein equals or exceeds that of lipids, but many membrane proteins have most of their mass outside the membrane bilayer. The lipids and proteins act in cohorts as functional and thermodynamic units. Membranes represent the main structural component for the complex architecture of biological systems. The human brain, for example consists of a complex network of membranes with a total surface area of 103–104 m2. They are typically fluid-like. Biophysicists have constructed a number of model membranes that are expected to capture some of the essential features of their biological counterparts [see Bretscher (1973)]. The fluid-mosaic model of membrane structure is depicted in Figure 10.6. In this model, the lipid bilayer is seen as a continuum in which the protein molecules are either partially embedded or associated with the bilayer surface. Proteins are also attached to the lipids by formation of covalent bonds such as by protein acylation, isoprenylation and binding via glycerosyl phosphatidylinositol. This is known as protein lipidation. In the membranes of most eukaryotic organisms, four major classes of lipid are found: glycerophospholipids, glyceroglycolipids, sphingolipids and sterols. Most bacteria (prokaryotes) lack sterols and sphingolipids. The lipid composition of bacterial membranes is similar to that of mitochondrial and chloroplast membranes. This supports the close relationship between these organelles and bacteria. Mammalian membranes were the last to develop on earth. They have most diverse lipid composition. Among the lipid classes encountered in membranes, phospholipids form the largest group. They can make up 40–80% of mammalian and up to 40% of plant membrane mass.

Figure 10.6

The fluid-mosaic model of cell membranes (Singer and Nicolson, 1972) (adapted by permission from The American Association for the Advancement of Science, © 1972).

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Phospholipids, glycolipids and sterols build the matrix of each biological membrane which is typically organised as a lipid bilayer. We have discussed in Chapter 3 that lipids, which cannot pack into micellar structures either due to their small head-group area or due to their bulk hydrocarbon chains, form bilayers. For bilayer-forming lipids the value of the packing parameter [ºv/(al)] must lie close to unity. For micelle-forming lipids, the value of packing parameter lies in the range of 0.33 to 0.5. Such a value can be obtained if the hydrocarbon volume v is approximately twice that of the micelle-forming lipids. Therefore, lipids with two hydrocarbon tails are likely to form bilayers. The doubling of the chains also affects other aggregate properties. It increases the hydrophobicity of the lipids which drastically lowers their critical micelle concentration (CMC). For example, the CMC of the bilayer-forming lipids usually lies between 1 × 10–6 mol/dm3 and 1 × 10–10 mol/dm3 (Israelachvili, 1997). The residence time of the molecules within the aggregates is also increased tremendously. Micelles are usually highly dynamic and there is continuous exchange of monomers between the bulk solution and the aggregate. The diffusive exchange involves an energy barrier that has to be surmounted before a molecule can escape from a micelle or a bilayer to the solution. The residence time t can be expressed as (Israelachvili, 1997), W

55W 0 / cCMC

(10.9)

where cCMC is expressed in and t0 represents a characteristic collision time. The following example illustrates the difference in residence times in micelles and lipid bilayers. mol/dm3

EXAMPLE 10.2 The critical micelle concentration for a single-tailed surfactant is 1 × 10–3 mol/ dm3 and the same for a double-tailed bilayer-forming lipid is 1 × 10–10 mol/dm3. The corresponding values of collision time of the molecules are 1 × 10–9 s and 1 × 10–7 s respectively (Israelachvili, 1997). Calculate the residence times of the molecules within the micelle and the bilayer. Solution

The residence times can be calculated from Eq. (10.9). For the micelle

W

55 – 1 – 10 9 /1 – 10 3

5.5 – 10 5 s

W

55 – 1 – 10 7 /1 – 10 10

5.5 – 10 4 s

For the bilayer,

Therefore, the above results show that the residence time of the double-tailed lipid molecules within the bilayer is much longer than the same for the single-tailed surfactant in their micelle. In aqueous solution, lipid bilayers typically form closed surfaces, i.e. vesicles (see Chapter 3). Multilamellar vesicles consisting of several phospholipid bilayers (i.e. liposomes) were first observed by electron microscopy by Bangham and Horne (1964). This finding showed that among all constituents of biomembranes, the lipid component by itself could produce cell-like organisation upon simple dispersion in water. This has provided a tool to the biochemists to mimic the structure and functions of biomembranes (Bangham, 1993; Rosoff, 1996). The unilamellar lipid vesicles having size in the range of 1 to 10 mm exhibit a large variety of different shapes, some of which are illustrated in Figure 10.7. The shape of vesicles is determined primarily by the bending elasticity and curvature. Let us consider a two-dimensional surface of a membrane. At each point on the surface one can define two principal curvatures H1 and H2. The configurational energy is given by (Lipowsky, 1991) Ec

ËN b

vÔ dA ÌÍ 2

Û ( H1  H 2  H0 ) 2  N G H1 H 2 Ü Ý

(10.10)

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where the integral extends over the whole membrane surface, dA represents the intrinsic area element, k b represents the bending modulus, H0 represents the spontaneous curvature (see Section 10.5) and k G represents the Gaussian curvature modulus. The definitions of the moduli have been presented in the Appendix. The values of k b and k G are extensively tabulated in the literature [see Marsh (2006)]. Typical value of k b is 0.5 × 10–19 J. If both sides of the membrane are identical, H2 1 / Rv . i.e. the membrane is symmetric, then H0 = 0. For a spherical vesicle of radius Rv, H1 Inserting these quantities in Eq. (10.10) with H0 = 0, we get Ec

4S (2N b  N G )

(10.11)

Let us consider a membrane segment of length Lm. If this segment is planar, it has no bending energy but its boundary has edge energy se ( ~1 – 10 10 J/m for phospholipid bilayers) that arises from the partial contact between the water and the hydrocarbon chains of the lipid. The total edge energy is proportional to Lm. However, if the segment forms a sphere, it has no edge energy but the bending energy is non-zero, which is given by 4S (2N b  N G ) . This energy does not depend on Lm. Therefore, for a large value of Lm, the membrane can lower its energy by forming a closed surface. This mechanism of vesicle formation via the hydrophobic effect is quite general, and operates in biological systems which usually contain a large number of such structures. The micrographs in Figure 10.7 depict the shape transformations of a single vesicle induced by a change in temperature. In thermal equilibrium, the vesicle should attain the shape that corresponds to the minimum bending energy.

Figure 10.7

Shape transformations of free vesicles induced by a change in temperature: (a, c) expulsion of a small vesicle from a larger one (budding), and (b, d) inverse budding (endocytosis) via the transformation from a discocyte to a stomatocyte. The shapes are axisymmetric with respect to the broken line (Lipowsky, 1991) (reproduced by permission from Macmillan Publishers Ltd., © 1991).

In recent years, many vesicles have been discovered which can be as large as 200 mm and can take up various shapes. These giant vesicles can assume an amazing variety of shapes according to the nature of the lipid molecules and the physical environment. For example, small changes in temperature and osmotic pressure can induce budding (see Figure 10.8) and vesiculation. These are the two prominent shape transformations of vesicles. The term ‘budding’ is often used to describe the multistep process in which a single spherical (or prolate) vesicle undergoes a sequence of shape changes resulting in the formation of a distinct daughter vesicle linked to the parent via a neck as shown in the figure. ‘Vesiculation’ distinguishes the singular limit at which the radius of the neck becomes microscopic. The mathematical models of these transitions have been presented by Miao et al. (1994).

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Figure 10.8

Budding of a giant vesicle observed with phase contrast microscopy. The prolate vesicle expels a small satellite as the preferred curvature of its membrane is increased. The bud is still connected to the parent vesicle (radius = 5 mm) with a neck (Döbereiner, 2000) (reproduced by permission from Elsevier Ltd., © 2000).

Membranes undergo thermally excited shape fluctuations to increase their configurational entropy. The character of its shape fluctuations depends on the internal state of the membrane, which can be fluid, polymerised or hexatic. As discussed before, biological membranes often contain twodimensional protein networks, such as the spectrin network which is a part of the plasma membrane of red blood cells. The time-scales for breaking and reassembling the molecular connections of these networks are usually large compared to the time-scales involved in the shape fluctuations. At a low temperature or high lateral pressure, the bilayers of lipid molecules exhibit solid-like phases with translational short-range ordering of the molecules. The plasma membrane of red blood cells has been studied extensively. It consists of two coupled membranes: a fluid bilayer which contains a mixture of many lipids and proteins, and a twodimensional network of spectrin molecules. The ‘flickering’ of red blood cells is an interesting phenomenon. These fluctuations exhibit a crossover from fluid-like behaviour to solid-like behaviour (Lipowsky and Girardet, 1990).

10.4 INTERFACIAL FORCES The van der Waals, electrostatic double layer, hydrophobic interaction, hydration and steric forces (such as undulation, peristaltic, protrusion and head-group overlap forces) play important roles in biological systems. There are two major aspects regarding the role of these forces. The first aspect involves their role in the formation of stable assemblies in biological systems (such as membranes, cells and cell organelles). The second aspect is the interactions between the separate assemblies which determine their stability as the building blocks of biological tissues. Cell adhesion and cell fusion are direct consequences of membrane interactions. The hydrophobic interaction is a strong attractive interaction between the hydrocarbon molecules in water and is believed to be much stronger than the van der Waals attraction (see Section 5.5.2). It plays a very important role in the formation of assemblies. The interaction between separate assemblies is generated by a complex interplay of the various forces mentioned above. The hydrophilic head-groups of the molecules which constitute the

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assemblies face the water phase and determine to a large extent the two-body interactions. The hydrophobic moieties are shielded by these hydrated (hydrophilic) head-groups from the water phase so that no long-range hydrophobic attraction is expected, and only attractions through the van der Waals force remain. Therefore, van der Waals interaction is either partly or totally responsible for phenomena like cell adhesion, membrane stacking and cell recognition in immunological processes (Marra, 1986a).

10.4.1 van der Waals Force The van der Waals interaction energy between two planar surfaces (per unit area) is given by (Section 5.2)

I



AH 12S D 2

(10.12)

where AH is the non-retarded Hamaker constant and D is the separation between the surfaces. The value of Hamaker constant for biological systems is ~5 – 10 21 J. Due to the retardation effect in the bilayers, the value of AH diminishes with D. Electrolytes present in the biological systems provide additional reduction in the van der Waals interaction. The zero-frequency contribution to the Hamaker constant is reduced due to the ionic screening. It is given by the relation (Mahanty and Ninham, 1976)

AHQ 0 ( D)

AHQ 0 (0) 2N D exp( 2N D) , kD >> 1

(10.13)

where k is the Debye–Hückel parameter. From Eq. (10.13) it can be observed that this term becomes significantly reduced in salt solution with increase in separation. If the salt concentration Q 0 ( D) becomes 10% of is 0.15 mol/dm3 (in which k –1 = 0.8 nm), it can be easily shown that AH Q 0 AH (0) at D = 1.5 nm.

10.4.2 Hydration Force The repulsive hydration force plays a very important role in lipid bilayers. This force is responsible for the lack of strong adhesion or aggregation of bilayers and vesicles composed of uncharged lipids (e.g. lecithin). It is believed that hydration forces arise when water molecules bind strongly to hydrophilic surface groups because of the energy needed to dehydrate these groups as two surfaces approach each other (Israelachvili, 1997). Its origin has been subject to a large amount of debate. According to Hartley (1936), “there is a widespread tendency to use ‘hydration’ in colloid chemistry as a sort of universal explanation of puzzling phenomena. Its inaccessibility to direct experimental determination fortifies this tendency”. Repulsive short-range forces have been measured between bilayer and other amphiphilic surfaces in water. The typical range of these forces is 1–3 nm, and below this separation, they can dominate over the van der Waals and electrostatic double layer forces. These forces do not have a simple electrostatic origin since they can be observed between uncharged bilayers. These forces have been measured accurately between lecithin and other uncharged bilayers in aqueous solutions. The repulsion per unit area is found to decay exponentially with distance D according to the following equation (Israelachvili and Wennerström, 1990): 3h

C exp (  D / O )

(10.14)

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where l is the decay length whose value is ~0.2 nm, close to the size of a water molecule. Earlier, it was believed that this repulsive force is due to water structure. Apart from the term ‘hydration force’, some scientists call it ‘structural’ force. Marra (1985) used the Langmuir–Blodgett technique (Section 8.4) for depositing lipid bilayers onto molecularly smooth mica surfaces. He deposited phospholipids such as dipalmitoyl phosphatidylethanolamine (DPPE) and dilanroyl phosphatidylcholine (DLPC), and the galactolipids monogalactosyl diglyceride (MGDG) and digalactosyl diglyceride (DGDG). The van der Waals and hydration forces between two opposing galactolipid bilayers were measured using the surface force apparatus (SFA) (see Chapter 5). As galactolipids are uncharged, contribution from electrostatic double layer force was absent. The experimental results indicated the presence of strong short-range repulsive hydration force (Figure 10.9). The DGPG bilayers have an energy minimum at D = 1.3 nm, and the MGDG bilayers have a deeper energy minimum at D = 0.6 nm. Below these separations, the bilayer interaction is strongly repulsive due to the short-range hydration interaction. Below 4 nm separation, the distance-dependence of the long-range van der Waals interaction between DGDG bilayers in aqueous solutions obeyed the nonretarded Hamaker equation. The experimental Hamaker constant in pure water was found to be (7.5 ± 1) × 10–21 J, which decreased to (3.1 ± 1) × 10–21 J upon addition of 0.2 mol/dm3 NaCl.

Figure 10.9

Short-range interactions between the DGDG and MGDG bilayers (Marra, 1985) (adapted by permission from Elsevier Ltd., © 1985). The interaction energy is expressed per unit area.

Early theoretical work on hydration force appeared to confirm the existence of an exponential repulsive force arising from the electric polarisation of water molecules by the surfaces (Marcelja and Radic, 1976). The decay length was attributed to a length characteristic of water. However, the decay length was not easy to derive theoretically, and had to be assumed, or fitted, although it seemed conceivable that it could be close to the size of a water molecule. However, the experimental data obtained in 1980s presented quite a complex scenario. Experiments with different surfactant and lipid bilayers in water yielded values of l in Eq. (10.14) which varied between 0.1 nm and 0.6 nm. With such a large variation, l does not appear to correlate with the size of water molecule or with some obvious characteristic property of water. Molecular dynamics simulations did not predict the monotonically decaying force. Instead, with surfaces modelled on lecithin and mica, decaying

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oscillatory forces were obtained (Kjellander and Marcelja, 1985). Some of the important observations which do not support the modified water-structure origin of hydration force are given below (Israelachvili and Wennerström, 1990). (i) Helm et al. (1989) measured forces between partially hydrophobic bilayers. They found attractive hydrophobic forces and repulsive hydration forces existing simultaneously, each one dominating over a different distance regime. If both of these forces arise from the water-structure effect (see Section 5.5.2), it is unlikely that they would exist simultaneously. (ii) Much weaker or no hydration force was observed between highly charged bilayer surfaces even though these are expected to have equally strong or even stronger effects on water molecules than that for the uncharged surfaces (Marra, 1986b). (iii) The parameter C in Eq. (10.14) varies strongly (by one or two orders of magnitude) for surfaces that are chemically very similar (e.g. lecithins and phosphatidylethanolamines in gel and liquid-crystalline states) (Rand and Parsegian, 1989). The NMR measurements of hydration and water structure reveal only minor differences. (iv) Many surfactant bilayers in water do not swell at all when they are in the solid state. However, they swell in the liquid-crystalline (LC) state. There has been no satisfactory explanation to why more solvent structure develops in the less ordered liquid-crystalline state than in the more ordered solid-crystalline state. (v) Another important observation is that the repulsion between bilayers in water usually increases with increase in temperature (Marra and Israelachvili, 1985). If the hydration model is applied, this trend would suggest that the water-structure is enhanced with increase in temperature. However, this is very unlikely because the amphiphilic molecules become less ordered with increase in temperature, and thus one would expect less order in the adjoining water molecules. It has been suggested that the short-range repulsive force between amphiphilic surfaces originate from the entropic (osmotic) repulsion of molecular groups that are thermally excited to protrude from these fluid-like surfaces (Israelachvili and Wennerström, 1990, 1992). These forces are discussed in the following section. The genuine hydration effects play a rather minor role. They mainly determine the hydrated size of the protruding groups.

10.4.3

Steric Forces

The structures such as bilayers and biological membranes are aggregates of weakly held amphiphilic molecules. These structures are thermally mobile. Their shape changes continuously as their molecules twist, turn and bob in and out of the surfaces (Israelachvili, 1997). Four types of entropic forces between amphiphilic surfaces arise when they approach each other. These are the undulation force, the peristaltic force, the protrusion force and the head-group overlap force. The molecular origin of these forces is explained in Figure 10.10 (A–D). None of these forces should exist between hard surfaces such as solid colloid particles. Of the four types of osmotic forces, the undulation force has longer range than the others. It can extend well-beyond 3 nm separation between the surfaces. At separations smaller than 2 nm, the protrusion and head-group overlap forces can dominate the undulation repulsion.

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Figure 10.10

Four types of thermal fluctuation forces between amphiphilic surfaces such as surfactant and lipid bilayers (Israelachvili, 1997) (adapted by permission from Elsevier Ltd., © 1992, and American Chemical Society, © 1992).

Undulation Force The lipid bilayers have thermal undulations whose amplitude increases with increasing temperature and decreasing bilayer bending modulus. For two bilayers at a mean distance D apart under no external tension, the repulsive force per unit area is given by (Helfrich, 1978; Israelachvili and Wennerström, 1992) 3u

3S 2 ( kT )2 64N b D 3

(10.15)

where kb is the bending modulus. The undulation force [Figure 10.10(A)] is essentially an entropic force, which arises from the confinement of thermally excited undulation waves into a smaller region when two surfaces approach each other. The undulation force has been measured and the dependence of Pu on D –3 has been experimentally verified (Safinya et al., 1986). The repulsive undulation force has the similar form as the non-retarded van der Waals force (i.e. µ D –3). However, the undulation force can be drastically reduced or even eliminated when a membrane is in tension since it suppresses the undulations. Mechanical or osmotic stresses can bring about tension, and it can enhance the

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adhesion between stressed membranes or bilayers. The van der Waals force, on the other hand, does not change when the surfaces are subjected to a tensile or compressive stress.

Peristaltic force In addition to the bending fluctuations, bilayers or membranes also undergo peristaltic (or squeezing) fluctuations. The thickness of the membrane fluctuates about the mean thickness as shown in Figure 10.10(B). The peristaltic pressure between two membranes is given by (Israelachvili and Wennerström, 1992), 3p

2(kT ) 2

(10.16)

S 2N a D5

where k a is the area expansion or compressibility modulus, which is associated with the elastic energy of the membrane. The two elastic properties of the membrane k a and k b are different and have different dimensions though they are not necessarily independent of each other.

Protrusion force The surfaces of amphiphilic aggregates are molecularly rough. This hypothesis is supported by the quasi-elastic neutron scattering studies of liquid crystalline dipalmitoyl phosphatidylcholine (DPPC) bilayers (Pfeiffer et al., 1989). Computer simulations of micelles and bilayers have shown that the interfaces are very rough or diffuse. When two amphiphilic surfaces come close enough that their molecular-scale protrusions overlap, a repulsive pressure develops [see Figure 10.10(C)]. This force is analogous to the steric force between surfaces with adsorbed polymer layers (see Section 5.6). In this case, the protruding segments of the approaching surfaces are forced back into the surfaces, whereas for the polymers, the molecules are compressed but remain between the surfaces. The protrusion forces are very important between amphiphilic surfaces interacting in aqueous and highly polar liquids. The protrusion force per unit area (i.e. protrusion pressure) is given by (Israelachvili and Wennerström, 1990)

3 pr

nD (D / O) exp(D / O ) , O [1  (1  D / O) exp(D / O )]

kT / D

(10.17)

where n is the number of protrusion sites per unit area, l is the protrusion decay length and a is an interaction parameter. The value of a for the single-chained and double-chained amphiphiles in water range between 1.5 × 10–11 J/m and 5 × 10–11 J/m at 298 K, which corresponds to decay lengths between 0.08 nm and 0.3 nm (Israelachvili, 1997). The value of n is ~2 × 1018 m–2. When the separation between the surfaces lies in the range from l to 10l, the protrusion pressure varies exponentially with the decay length, similar to that given by Eq. (10.14)

3 pr

2.7 n D exp(D O), l < D < 10l

(10.18)

Equations (10.17) and (10.18) were derived considering only one type of protruding mode. In reality, surface groups generally have several conformational degrees of freedom which leads to the development of additional protrusion modes. Therefore, an exponentially decaying entropic repulsion always exists between fluid amphiphilic surfaces whose decay length depends upon the amphiphile– solvent interaction. The role of hydration is to determine the size of the hydrated protruding head-groups and the interaction between them (see Figure 10.10). Therefore, the hydration effects modulate the thermal forces.

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Head-group Overlap Force The head-groups of many lipids are longer than the chains. They extend into the aqueous phase and repel each other. A flexible head-group is similar to the end-grafted polymer as described in Section 5.6. The mean separation between the head-groups is typically 0.6–0.9 nm and they extend into the solution by a similar distance. Thus, the head-group overlap interactions [Figure 10.10(D)] can be described by the interaction between polymer brushes (i.e. polymer molecules grafted at one end to the surface), which is given by the de Gennes equation [Eq. (5.99)]. For D/2L in Eq. (5.99) in the range 0.2 to 0.9, the repulsive pressure Ps is roughly exponential and given by 3s

100 kT s

3

exp( S D / L ) 100 n3 / 2 kT exp (  D / O ) , l = L/p

where s is the mean distance between the head-groups such that s

(10.19)

1/ n .

EXAMPLE 10.3 The variation of repulsive force with separation between adsorbed monolayers of C18EO40 in water at 298 K is given below (Homola and Robertson, 1976). D (nm) Ps (Pa)

20.9 30.5

17.1 1377.8

12.1 4338.7

11.1 7196.9

8.8 9426.7

7.2 20000

5.9 32000

5.5 39000

Fit the de Gennes equation to the data taking L = 10.5 nm and obtain the value of s . Present your results graphically. Solution The de Gennes equation [Eq. (5.99)] is given by 3s

94 34 kT Ë È 2 L Ø È DØ Û  Ì É Ù ÉÊ 2 L ÙÚ Ü , s 3 ÌÍ Ê D Ú ÝÜ

D  2L

Given: L = 10.5 nm. The variation of Ps with D as per the given data is shown in Figure 10.11. After substituting k = 1.38 × 10–23 J/K and T = 298 K in the above equation, s was obtained by fitting the equation to the experimental data (using the ‘solver’ of Microsoft Excel). The fitted value of s is 12.96 nm.

10.4.4 Electrostatic Double Layer Force The electrostatic double layer force is important in charged lipid bilayers. For example, phosphatidyl glycerol is the major negatively charged lipid in bacterial and plant membranes. It carries a single charged phosphate group. The contribution of electrostatic forces in intrinsically uncharged lipids such as phosphatidylcholine, phosphatidylethanolamine and galactolipids becomes important in electrolyte solutions. The electrostatic double layer force arises from the adsorption of the cations (such as Ca+2 and Mg+2 to the bilayer surface), which gives the bilayers a net surface charge. The electrostatic double layer force is sensitive on the electrolyte concentration, pH of the solution and the surface charge density (see Chapter 5). Marra (1986b) measured the electrostatic double layer repulsive force between the negatively charged bilayers of distearoylphosphatidyl glycerol (DSPG) and dimyristoylphosphatidyl glycerol (DMPG) using the Surface Force Apparatus (SFA). The experimental procedure was as follows. Two thin molecularly smooth mica surfaces were silvered on one side with a 50 nm-thick highly-reflecting

Biological Interfaces

Figure 10.11

391

Force between adsorbed monolayers of C18EO40 in water (Example 10.3).

coating and glued on two cylindrically-curved silica-glass disks (radius of curvature = 1 cm), with the silvered sides down. A bilayer was deposited on each mica surface applying a surface pressure in the Langmuir trough that was found to give the desired transfer ratios of lipids before these glass disks were mounted in the force-measurement apparatus. The bilayers were kept immersed in water throughout since they lose their outer monolayer upon being retracted from water. In order to maintain equilibrium between the lipid molecules in the bilayers and the free lipid monomers in solution, the water in the apparatus was pre-saturated with lipid monomers by adding a lipid crystal to the water the previous day. This precaution is particularly important for fluid state DMPG bilayers because the outer DMPG monolayers desorbed within a few hours unless the presaturation was done. Pre-saturation for a time period of 18 hours was sufficient to carry out reproducible experiments. For the gel state DSPG bilayers, these precautions were less important, probably because of the much lower solubility of DSPG monomers. In the force-measurement apparatus, one of the glass disks was mounted on a rigid support, and the other (which faced the first one) was positioned on a spring with a known spring constant. Now, when the bilayer surfaces were brought to close separation, the surfaces experienced an interaction from each other which could be measured through the deflection of the spring. The separation between the two surfaces was measured by employing an optical technique using fringes-of-equal-chromatic-order (FECO) interferometry. From the position and the shape of the FECO fringes observed in the spectrometer, the distance between the two bilayers could be measured to an accuracy of 0.1–0.2 nm. By measuring the surface force (F) as a function of the surface separation (D), the force profile was obtained. The measured force between two DSPG bilayers at 293 K and pH = 9 at various concentrations of sodium chloride are shown in Figure 10.12. The inter-bilayer force can be accounted for by the electrostatic repulsion down to a bilayer separation of 2 nm, below which the force measurements were not possible. At surface separations below 2 nm, the double layer repulsion became so strong that the supporting curved mica surfaces began to flatten. In calcium chloride solutions, the surface charge reduced due to the binding of Ca+2 ions to the bilayer.

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Figure 10.12

Measured forces between two DSPG bilayers. The lines indicate DLVO profiles assuming fully charged bilayers and a Hamaker constant of 6 × 10–21 J (Marra, 1986b) (adapted by permission from The Biophysical Society, © 1986).

10.4.5 Hydrophobic Forces The attractive hydrophobic interaction between hydrocarbon molecules and surfaces in water is of long range. It is much stronger than the van der Waals attraction (see Section 5.5.2). In unstressed bilayers, the hydrophilic head-groups shield the underlying hydrocarbon groups from the aqueous phase. This effectively masks the hydrophobic interaction between them. However, when the bilayers are stretched, they expand laterally. The increased hydrophobic area exposed to the aqueous phase allows hydrophobic interaction to occur between the bilayers. The direct fusion of bilayers takes place by the hydrophobic interaction (Helm et al., 1989). The bilayers do not have to overcome the repulsive force barrier (e.g. the barrier due to the hydration force) before they can fuse. Once the bilayer surfaces come within ~1 nm of each other, local deformations and molecular rearrangements cause parting of the head-groups on opposite sides. In this way, hydrophobic hydrocarbon regions are exposed or opened up. Since the hydrophobic interaction is of long-range, these hydrophobic regions facing each other become unstable and jump together spontaneously, or break through across the gap and fuse. The force-profiles between two dilauroyl phosphatidylcholine (DLPC) bilayers deposited by the Langmuir–Blodgett method on solid dipalmitoyl phosphatidylethanolamine (DPPE) monolayers are shown in Figure 10.13. The force profile for the DLPC surfaces in water saturated with DLPC shows van der Waals attraction beyond 2.5 nm separation and hydration repulsion below 2.5 nm. There was no fusion even up to force/radius value of 1000 mN/m. The van der Waals attraction caused the bilayers jump into adhesive contact from the point A at D = 4.2 nm. The force profile between partially depleted DLPC monolayers in which the bilayers had thinned to about 85% of the original

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thickness showed the effect of hydrophobic interaction. The depletion of bilayers caused more exposure of the hydrophobic groups and as a consequence, a long-range strong hydrophobic attraction emerged that caused the two surfaces to jump into contact from a greater distance from the point B in Figure 10.13. The bilayers spontaneously fused into one bilayer when the pressure between them reached about 0.3 MPa. For thinner bilayers, the attractive force was even greater in range as well as in magnitude.

Figure 10.13 Force versus distance profiles between two LB-deposited DLPC monolayers (each on a solid DPPE monolayer) (Helm et al., 1989) (adapted by permission from The American Association for the Advancement of Science and Professor Jacob N. Israelachvili, © 1989).

Fusion, therefore, is caused by the hydrophobic attraction between the internal hydrocarbon chains that become exposed to each other across the aqueous phase. The attractive van der Waals force plays a negligible role in fusion, but it enhances bilayer adhesion. As we have discussed before, fusion can occur spontaneously between repelling bilayers without requiring to overcome the repulsive hydration force barrier. Highly localised molecular rearrangements allow this to happen through spontaneous instabilities or breakthrough mechanisms. The hydrophobic interaction acts between the interiors of membranes. The attractive van der Waals forces between the exterior surfaces of membranes should only lead to adhesion. The lipids in free bilayers can undergo deformations more easily than the supported bilayers. In vesicles and membranes, the exposed areas could emerge from inhomogeneous ionic or osmotic stresses, or local packing stresses induced by integral membrane proteins.

10.5

CELL ADHESION

Adhesion of membranes plays an essential part in many biological phenomena. For example, the formation of tissue is based on the mutual adhesion of cell membranes, and on the adhesion of these membranes to a network of macromolecules. Many transport processes involve the binding and unbinding of vesicles to and from the membrane surfaces of cells. An important application of this process is the delivery of drugs to specific cells. Cell adhesion is a fascinating and complex process.

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The adhesion of soft shells such as vesicles or cells to any surface is controlled by various interfacial forces and the mechanical properties of the shells such as bending elasticity, tension of the membrane and area-to-volume ratio of the shells. For studying the electrostatically-controlled adhesion of soft shells to substrates in a systematic way, model systems consisting of partially-charged giant vesicles and oppositely-charged supported membranes are frequently used (Nardi et al., 1997). The elasticity of biomembranes can be described by the classical theory of elasticity of shells. The additional parameters are the tension generated by the thermal undulations and the specific surface energy due to the coupling of the bilayer with the surrounding aqueous phase. These tensions result in partial penetration of water into the semipolar region of the bilayer. An important consequence of the extreme softness of fluid lipid bilayers is the excitation of pronounced bending undulations (which is known as flickering). These undulations are excited by thermal fluctuations (and are equivalent in some ways to Brownian motion). They are strongly overdamped due to the friction caused by the coupling of the undulations to hydrodynamic flows in the aqueous phase. The general theory of elasticity of adhering shells is complex. However, it can be simplified if we consider only weak bending corresponding to weak adhesion. Membrane deformations can be described by the three modes of deformation: lateral extension, bending and shearing. The elastic energy (per unit area) associated with an isotropic lateral extension is given by (Sackmann, 1994)

gex

1 È 'A Ø F 2 ÉÊ A ÙÚ

2

(10.20)

where c is the area compressibility modulus. The strain resulting from a bending deformation is expressed in terms of the local curvature of the membrane, which is characterised by the two principal curvatures H1 and H2. The bending free energy of the closed shell is expressed as 2 1 Nb v dA( H1  H 2  H 0 ) (10.21) v Ô Ô 2 where H0 is the spontaneous curvature which accounts for possible membrane asymmetries. H0 can be visualised in terms of a gradient of lateral pressure p l (z) across the membrane (in the z-direction) generating a bending moment per unit length (see Figure 10.14).

D Gb

Figure 10.14

Illustration of generation of one-dimensional bending moment M by gradient of lateral pressure p l (z) across the bilayer.

Biological Interfaces

Ô z³S l (z)dz

M

395 (10.22)

The integration limits must cover all regions for which the lateral pressure gradient is non-zero. H0

M Nb

(10.23)

H0 is essential for generating small invaginations in cell membranes such as coated pits or caveoli. The shear energy density (expressed per unit area) can be described by gs

1 P ([ 2  [ 2  2) 2

(10.24)

where m is the two dimensional shear elastic constant and x is the ratio of the extended to the original length. The above energy functions account for local deformations of closed shells. For closed stratified shells such as bilayers or composite membranes, one has to consider a global contribution of the bending energy which accounts for the fact that the two opposing shells undergo a net extensional deformation. Typical values of the elastic constants c, kb and m for some biological membranes are presented in Table 10.2 and compared with polyethylene. Table 10.2 Membrane DMPC (La) DMPC + cholesterol (1:1 mixture) DMPC (Lb) Erythrocyte Polyethylene

Elastic constants

c (mJ/m2)

kb (J)

m (J/m2)

140 690 900 1000 5 × 103

25 100 – 5 5 × 103

3 – – 6 × 10–3 300

The adhesion of vesicles is similar to the partial wetting of solids by fluid droplets [see de Gennes (1985)]. The free energy of adhesion can be expressed as (Seifert and Lipowsky, 1990) 2 1 1 (10.25) pv Ô vÔ vÔ dV  2 J vÔ vÔ dA  WAc  2 N b vÔ vÔ dA ( H1  H2  H0 ) The first term in Eq. (10.25) accounts for the osmotic pressure of the shell and the integration extends over the volume V. The second term accounts for the change of the surface energy g due to the changes in area due to adhesion. The third term accounts for the gain in adhesion energy (per unit area) W (³ 0), where Ac is the area of contact. The fourth term accounts for the bending elastic contribution. Minimising the energy function given by Eq. (10.25), the shapes of adhering vesicles, and their state of adhesion as a function of the adhesion energy and degree of deflation can be calculated. Reflection-interference-contrast-microscope (RICM) images of a positively charged vesicle which is electrostatically attracted to a negatively charged substrate at pH = 2.8 and pH = 4.5 are shown in Figure 10.15. The vesicle contained 1 mol% of positively charged dihexadecyldimethyl ammonium bromide (DHDAB), and the supported membrane contained 10 mol% of negatively

D Gadh

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charged lipid octadecyl-[NBD-decyl]-dimethyl ammonium succinic acid. The outer solution of the vesicle contained 5 mol/m3 sodium chloride and 10 mol/m3 sucrose and citric acid. The inner solution contained 40 mol/m3 sucrose and 5 mol/m3 NaCl. At pH 2.8, the vesicle exhibited a small and homogeneous adhesion disk with a large number of diffraction fringes indicating that the contact angle was small and that adhesion was weak. The vesicle showed pronounced flickering (indicating a low tension), which suggests weak adhesion. At pH = 4.5, the adhesion disk decomposed into blisters surrounded by regions of strong adhesion. Flickering was suppressed, indicating that the vesicle was under appreciable tension.

Figure 10.15 RICM images of a positively charged giant vesicle demonstrating adhesion (Nardi et al., 1998) (reproduced by permission from American Physical Society, © 1998).

10.6 PULMONARY SURFACTANT Pulmonary surfactant is a surface-active lipoprotein complex formed by the alveolar cells. The alveoli are the final branchings of the respiratory tree (Figure 10.16). There are about 600 million alveoli in human lungs which produce ~150 m2 surface area. They act as the primary gas-exchange units of the lung. The gas–blood barrier between the alveolar space and the pulmonary capillaries is extremely thin, allowing for rapid gas exchange. To reach the blood, oxygen must diffuse through the alveolar epithelium (a thin interstitial space) and the capillary endothelium. Carbon dioxide follows the reverse course to reach the alveoli. There are two types of alveolar epithelial cells. The type I cells have long cytoplasmic extensions which spread out thinly along the alveolar walls and comprise the thin alveolar epithelium. Type II cells are more compact. They are responsible for producing the surfactant. The surfactant is stored in lamellar bodies in these cells. These lamellar bodies, when secreted into the alveoli, form tubular myelin which is the principal precursor of the Figure 10.16 Pulmonary alveoli.

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surface film that lowers the surface tension (Lalchev et al., 2008). In 1929, Dr. Kurt von Neergaard (a Swiss physiologist) performed some simple but ingenious experiments on lungs obtained from cats. He suggested that the surface-active lipid–protein structures cover the inflatable parts of the lung and lower the surface free energy by approximately two third. Research on pulmonary surfactants gained momentum in 1970s. A major impetus was the death of thousands of premature babies struggling for breath (Wrobel, 2004). The major role of the pulmonary surfactant is to allow the lungs to inflate easily. After expiration, the compressed pulmonary surfactant film at the alveolar air–liquid interface reduces the surface tension to less than l mN/m on lung deflation (Schürch et al., 1992). If the lungs do not secrete the surfactant, the surface tension would be much higher preventing the lungs from inflating normally, as is the case in premature infants suffering from infant respiratory distress syndrome. Pulmonary surfactant thus reduces the work of breathing. The reduction in surface tension also reduces fluid accumulation in the alveolus. Not only the alveoli, but also the bronchi are covered by a surfactant film. Inhaled particles, when deposited in the airways or alveoli, are displaced by surface forces towards the epithelium where they are retained. Along with the ability to effectively lower the surface tension to very low values, lung surfactants must possess several other abilities such as good adsorption from the alveolar hypophase onto the interface, rapid variation of the surface tension under dynamic conditions during compression and decompression of surface layer and effective re-spreading after dynamic compression. Some of the synthetic pulmonary surfactants are Exosurf, Pumactant, KL-4 and Venticute. Some of the animal-derived surfactants are Alveofact, Curosurf, Infasurf and Survanta. Pulmonary surfactant of mammalian lungs consists of a variety of macromolecular complexes comprised of lipids and specific proteins. The lamellar bodies and tubular myelin both contain lipid and protein components of the surfactant. However, the compressed surface film primarily consists of dipalmitoyl phosphatidylcholine (DPPC). Surfactant obtained through bronchiolar lavage contains ~90% lipid, ~10% protein and small amounts of carbohydrate. DPPC accounts for approximately half the lipid in the surfactant and it is mainly responsible for the reduction in surface tension. The proteins associated with the surfactant are designated as SP-A, SP-B and SP-C. The protein SP-A is variably glycosylated with a molecular mass of 28 to 36 kg/mol in the reduced state. Protein SP-B has a molecular mass of 15 kg/mol in the non-reduced state and SP-C has a molecular mass of 3 to 5 kg/mol in the reduced or non-reduced states. The surfactant proteins directly influence the interfacial properties of the surfactant lipids. Rapid adsorption of surfactant phospholipids to the air– liquid interface is thought to be critical for maintaining the morphological integrity of the gasexchange region of the lung. The proteins SP-B and SP-C substantially enhance the rate of formation of a surface film at the interface in vitro, and the rate is enhanced by SP-A. The mechanism of action of the lung surfactant at the air–water interface has been a subject of intense research. The most important questions about the properties of lung surfactants are the state of lipid monolayer and the liquid expanded–liquid condensed (LE–LC) balance, formation of bilayer and multilayer structures at the air–water interface, phase behaviour and molecular dynamics in the surface film. Scanning electron micrographs of a lung filled with saline (with surface tension close to zero) and an air-filled lung which has been rinsed with a detergent (with surface tension ~20 mN/ m) are shown in Figure 10.17. The alveolar septa in the saline-filled lung are not homogeneously expanded, which suggests some structural redundancy. The alveolar texture is highly irregular due to the bulging capillaries. In contrast, the normal air-filled lung shows flat alveolar walls. Rearranged capillaries and septal material are accommodated beneath a smooth surface lining layer, and the alveolar duct is wider.

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Figure 10.17

Scanning electron micrographs of alveolar ducts of (a) saline-filled, and (b) air-filled rabbit lungs. Note the increase in diameter of the ducts with increase in surface tension and the change in alveolar architecture (Bachofen and Schürch, 2001) (reproduced by permission from Elsevier Ltd., © 2001).

The alveolar lining layer is believed to have the following structural properties: (i) it is a continuous duplex layer consisting of an aqueous hypophase covered by a thin surfactant film, (ii) the surfactant film is a monolayer with dipalmitoyl phosphatidylcholine (DPPC) as the most important component, and (iii) the surface film is stable in the lung volume range of normal breathing. There have been numerous studies of the structure of the lining layer. An experimental difficulty often encountered is the preservation and sample-preparation of this delicate structure for microscopy. Recent investigations on the alveolar film structure after fixation with non-aqueous osmium fluorocarbon mixture have demonstrated a continuous film covering the entire alveolar surface, including the pores of Kohn. Figure 10.18 shows that the surfactant film is preserved and continuous. It overlies a thin hypophase above the type I epithelial cells. At higher magnifications, the surface film appears to be multilaminated in which the number of lamellae varies from 2 to 7. Disk-like structures or multilamellar vesicles appear attached or partially integrated into the planar multilayered film. Frequently, it is observed that there is an odd number of lamellae, but at some sites the film facing the air phase appears to be a bilayer or to have an even number of lamellae. If an odd number of lamellae is observed, the film might consist of a monolayer with attached stack of bilayers in which the monolayer is placed on top facing the air phase. The spacing between two adjacent lamellae is ~5 nm.

10.7

BIOMATERIALS

An important area where biological interfaces play a crucial role is biomineralisation. Understanding the structure and dynamics at the interface between biological templates and minerals is one of the most challenging problems in molecular biology. In the last few decades, scientists have been exploring various avenues for the synthesis of biomaterials by directly using available biological constructions in synthetic systems. An important example of this is the use of DNA in materials science and technology. Before the discovery of its structure and role in heredity, DNA was considered as exotic but not very useful. However, DNA is now at the centre stage of biotechnology [see Willner (2002)]. The strong and selective bonds between complementary DNA sequences permit one to tag a material with a unique code. These selective attractions can be used to link colloid particles and assemble particles on surfaces. The small size and information-content of DNA can make it the ultimate assembly module. The DNA-strand can be converted into a metal by selectively coating the strand with a metal sheath by an electrochemical technique (Braun et al., 1998). Such

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Figure 10.18 Transmission electron micrograph from guinea pig lung after fixation with non-aqueous osmium fluorocarbon mixture. The surfactant film (shown by the arrows) is preserved and continuous. It overlies a thin hypophase which is thicker at some sites (AL) above the type I epithelium (Ep). ‘End’ represents the endothelial cells and ‘BM’ represents the basement membrane (Schürch et al., 1998) (reproduced by permission from Elsevier Ltd., © 1998).

materials can be useful in nano-scale electronic devices. The usefulness of DNA is due to the small length scale, which is still not accessible to photolithography (see Section 11.8). The DNA can be assembled into cubic and other shapes with varied topology (Seeman, 1998). Proteins can have an almost limitless variation of their sequence. Therefore, once folded, they are imparted with a unique 3D shape. They are central in biological catalysis (enzymes), transport (motor proteins and transmembrane pores) and structural functions (actin filaments). Several common proteins are routinely used in materials applications, e.g. the use of strong biotin– streptavidin interaction for the immobilisation of a desired tag onto the surface of a material. The inside of a virus capsid can act as a near-perfect chemical reactor for the synthesis of materials. The protein residues on the interior and exterior of a self-assembled virus shell are highly organised. They are amenable to precise and predetermined modifications. The protein coats of virus particles (i.e. virions) commonly comprise hundreds of subunits that self-assemble into a cage for transporting viral nucleic acids. Many virions, moreover, can undergo reversible structural changes that open or close gated pores to allow switchable access to their interior. Douglas and Young (1998) showed how a virion of the cowpea chlorotic mottle virus can be used as a host for the synthesis of materials. They mineralised paratungstate and decavanadate clusters inside this virion, controlled by pH-dependent gating of the virion’s pores. The procedure was as follows. The viral RNA was removed and ultracentrifugation was employed to purify the virus shell. This was followed by selective encapsulation of WO42 inside the cationic cage of the virus at pH > 6.5. The virus capsid swells at pH > 6.5 allowing it to be loaded by the inorganic ions (see Figure 10.19). When the pH was reduced below 6.5, the pores of the virus capsid closed and the confined tungstate anions 10 underwent oligomerisation and mineralisation to form H 2W12O 42 within the viral shell. The composite matter was purified subsequently by centrifugation. The TEM images of paratungstatemineralised virus particles are shown in Figure 10.19. The mineral core was surrounded by the

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protein cage and the internal diameter of the particle was 15 nm. The diversity in size and shape of such virus particles can make this procedure a versatile strategy for materials synthesis and molecular entrapment.

Figure 10.19

Mineraliable virus capsids (Douglas and Young, 1998) (reproduced by permission from Macmillan Publishers Ltd., © 1998).

Apart of viruses, bacteria have also been used to synthesise inorganic materials. They can be used as templates for inorganic colloid particles to prepare semiconductor and magnetic fibre composites. The bacterial cultures are drawn at the air–water interface to produce macroscopic bacterial thread with organised internal superstructure that can be reversibly swollen in aqueous inorganic colloidal suspensions. This allows for the infiltration of particles to form inorganic–organic fibrous composite materials. Semiconducting CdS nanoparticles have been incorporated into the organised superstructure of B. subtilis (Davis et al., 1998). The air-dried materials consisted of a close-packed array of multicellular bacterial filaments of 0.5 mm diameter coated with 30–70 nm thick layers of the aggregated colloid particles. The surface charge on the bacterial filaments was negative which allowed negatively charged colloid particles to infiltrate into the swollen interfilament spaces. In a similar manner, silicate micelles have been imbibed into the voids between the close-packed bacterial filaments and polymerised to form siliceous fibre composites. The macropores were filled with the bacterial filament and the silica channel walls were permeated with ordered mesopores (pores having diameter between 2 nm and 50 nm).

SUMMARY This chapter presents some important interfacial phenomena at the biological interfaces. The chapter begins with an introduction to these phenomena. The adsorption of proteins at solid–liquid, gas– liquid and liquid–liquid interfaces are discussed next. The mechanisms of adsorption and desorption

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401

of protein molecules are discussed. The next topic discussed in the chapter is on biological membranes. Some of the structural aspects of biological membranes, and mimicking the structure and functions of cell membranes by bilayers and vesicles are discussed. The interfacial forces important in biological systems are discussed with examples. Cell adhesion is discussed and the role of interfacial forces and the mechanical properties of the membrane on adhesion are explained. The properties and functions of pulmonary surfactants in the alveoli are discussed. The last topic discussed in this chapter is the role of interfaces in the synthesis of biomaterials.

KEYWORDS Alveoli Bending modulus Bilayer Biological membrane Biomineralisation Budding Cell adhesion Electrostatic double layer force Energy barrier Flickering Fluid mosaic model Gas–liquid interface Gaussian curvature modulus Globular protein Head-group overlap force Hydration force Hydrophobic force Infant respiratory distress syndrome

Interfacial force Liquid–solid interface Osmotic repulsion Packing parameter Peristaltic force Protein adsorption Protrusion force Pulmonary surfactant Steric force Surface pressure Undulation force van der Waals force Vesicle Vesiculation Virus encapsulation

NOTATION a A Ac AH cb cCMC ceq cs C D Dp E Ec

optimal cross-sectional area of head-group, m2 area, m2 area of contact, m2 Hamaker constant, J bulk concentration, mol/m3 critical micelle concentration, mol/m3 concentration in bulk solution corresponding to the equilibrium value for surface concentration, mol/m3 surface concentration, mol/m3 constant in Eq. (10.14), N/m2 separation between two surfaces, m diffusion coefficient, m2/s energy barrier, J configurational energy, J

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F g H HG H H0 J k km kd K l L Lm M MG M n p r Rv s t T v V W

force, N energy per unit area, J/m2 mean curvature, m–1 square root of the quantity representing Gaussian curvature, m–1 principal curvature, m–1 spontaneous curvature, m–1 flux, mol m–2 s–1 Boltzmann’s constant, J/K transport rate coefficient, m/s rate constant for deposition, m/s constant in Eq. (10.7), mol m–2 s–1 length of the hydrophobic group in the micellar core, m thickness of the polymer brush, m length of membrane segment, m mean bending moment (per unit length), N Gaussian bending moment (per unit length), N bending moment (per unit length), N number of protrusion sites per unit area, m–2 pressure, Pa rate, mol m–2 s–1 radius of vesicle, m mean distance between the head-groups, m time, s temperature, K volume occupied by the hydrophobic group in the micellar core, m3 volume, m3 adhesion energy, J/m2

Greek Letters a g G DG DGadh DGb k ka kb kG l m n x ps pl

interaction parameter, J/m surface energy, J/m2 surface concentration, mol/m2 (or kg/m2) change in free energy, J free energy of adhesion, J bending free energy, J Debye-Hückel parameter, m–1 expansion modulus, N/m bending modulus, J tad Gaussian curvature modulus, J decay length, m shear elastic constant, J/m2 frequency, s–1 ratio of the extended length to the original length surface pressure, N/m lateral pressure, N/m

Biological Interfaces

P s se t t0 f c

403

disjoining pressure, Pa Poisson’s ratio edge energy, J/m residence time, s collision time, s interaction energy (per unit area), J/m2 compressibility modulus, J/m2

EXERCISES 1. Explain the salient features of biological interfaces. 2. Give five applications where biological interfaces are important. 3. Explain how the protein molecules undergo structural rearrangement in aqueous solutions to minimise the free energy. 4. What are globular proteins? 5. What factors are important for adsorption of proteins on solid surfaces? 6. Illustrate the various steps of adsorption and desorption of proteins at solid surfaces. 7. Explain how the amount of protein adsorbed on a solid surface depends on the equilibrium concentration of the protein in the solution. 8. If the adsorption of protein at a gas–liquid interface is diffusion-controlled, how does the amount adsorbed on the surface vary with time? 9. Explain the activation-energy-barrier concept of protein adsorption at gas–liquid interface. 10. What is the typical area occupied by a bovine serum albumin molecule at the air–water interface? 11. Illustrate with a diagram how proteins adsorb at gas–liquid interface. 12. Explain the structure of biological membranes according to the fluid-mosaic model. 13. Explain the conditions under which lipids form bilayers. 14. What is the critical micelle concentration at which bilayers are formed? 15. Justify the use of vesicles as models for biological membranes. 16. How will you correlate the configurational energy to the membrane moduli? 17. What are giant vesicles? 18. Explain the budding and vesiculation processes. 19. What is flickering of cells? 20. Explain the role of interfacial forces in biological systems. 21. What are the major interfacial forces that operate at the biological interfaces? 22. How does the van der Waals force depend upon the concentration of electrolyte in the medium? 23. What is the main role of hydration in biological interfaces? 24. What is undulation force? How does the disjoining pressure due to the undulation force depend upon the separation between the surfaces? 25. What is peristaltic force? How does the disjoining pressure due to the peristaltic force depend upon the separation between the surfaces? 26. What is protrusion force? How does the disjoining pressure due to the protrusion force depend upon the separation between the surfaces? 27. What is head-group overlap force? How does it compare with the polymeric steric force discussed in Section 5.6?

404 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

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In what biological systems is the electrostatic double layer force important? Explain why. Explain how the hydrophobic interaction force causes the fusion of bilayers. What factors are responsible for cell adhesion? What is pulmonary surfactant? Why is it important for prematurely-born babies? What cells produce the pulmonary surfactant? How does the pulmonary surfactant help in breathing? Discuss the composition of alveolar surfactant. What is biomineralisation? Explain how DNA is used in biomineralisation. Explain how virus particles can be used for mineralising inorganic salts.

NUMERICAL AND ANALYTICAL PROBLEMS 10.1 For a phospholipid bilayer, the values of bending and Gaussian moduli are 0.5 × 10–19 J and –0.4 × 10–19 J respectively (Marsh, 2006). Calculate the bending free energy if it forms a vesicle. 10.2 Calculate the repulsive force per unit area between bilayers generated by the undulation force at 298 K for separation in the range of 1 nm to 3 nm. Given: bilayer bending modulus = 0.4 × 10–19 J. Present your results graphically. 10.3 The variation of repulsive pressure between fluid state dipalmitoylphosphatidylcholine bilayers in water is given below (Israelachvili and Wennerström, 1992). D (nm) 0.181 0.195 0.214 0.295 0.343 0.402 0.417 0.439 0.491

P (Pa) 2.15 1.54 5.73 6.15 1.96 5.62 9.75 7.82 3.00

× × × × × × × × ×

D (nm)

108

0.521 0.554 0.602 0.661 0.669 0.735 0.953 1.245

108 107 107 107 106 106 106 106

P (Pa) 2.89 2.23 1.72 1.07 7.13 4.41 2.26 4.29

× × × × × × × ×

106 106 106 106 105 105 105 104

Assuming that the force law: 3 C exp (– D / O ) is valid, fit the given data to this force law, and determine C and l. 10.4 The variation of force with separation between adsorbed monolayers of C18EO20 in water at 298 K is given below (Homola and Robertson, 1976). D (nm) Ps (Pa)

15.6 28.5

13.3 1147.3

9.9 3550.1

6.5 11953.2

4.6 25071.2

3.4 38213.1

Fit the de Gennes equation to the data taking L = 7.75 nm and obtain the value of s . Present your results graphically.

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405

10.5 The Helfrich-expression for the free energy of bending of a membrane (or monolayer) of surface area A is given by (Helfrich, 1973) [see Eq. (10.10)]. 'Gb

1 N b A ( H  H 0 )2  N G AHG2 2

For a spherical vesicle, derive the expression for bending energy. 10.6 Estimate the reduction of Hamaker constant at 298 K in a 100 mol/m3 aqueous solution of sodium chloride with separation between the surfaces in the range of 0.5 nm to 2 nm.

APPENDIX Let us define the mean curvature H as H

H1  H 2

(10A.1)

H1 H2

(10A.2)

and define the Gaussian curvature as

HG2

The bending modulus (also known as bending rigidity) kb is defined in terms of the derivative of the mean bending moment M with respect to the mean curvature. For a membrane surface of area A, kb is defined as Nb

È ˜M Ø ÉÊ ˜H ÙÚ A, H

(10A.3) G

It can also be defined in terms of the free energy of bending DGb as 1 È ˜ 2 'Gb Ø A ÉÊ ˜H 2 ÙÚ

Nb

(10A.4) A , HG

Another alternative definition is in terms of the gradient of lateral pressure within the membrane Nb



˜³S l ( z ) zdz ˜H

(10A.5)

where z is the distance from the bilayer mid-plane. The integration covers all regions for which Ñp l (z) ¹ 0. This is essentially from 0 to d for a monolayer and from –d to +d for a bilayer, where d is the thickness of the monolayer including the associated hydration layer. The Gaussian curvature modulus can be defined analogously in terms of a Gaussian bending moment M G as NG

È ˜M G Ø ÉÊ ˜H ÙÚ G

(10A.6) A, H

The spontaneous or intrinsic curvature is defined in terms of the first moment of the pressure profile. It is given by 1 H0 z ³S l ( z )dz (10A.7) Nb Ô

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The first moment does not depend on the choice of the origin for z, because

Ô ³S l (z) dz

0.

The relationship between the moduli kb and kG can be obtained by considering the anisotropy of the membrane stresses which is parameterised for solids by Poisson’s ratio, s. In such an anisotropic coupled system, the principal bending moments M1 and M2 are given by,

and

M1

N b ( H1  V H 2 )

(10A.8)

M2

N b ( H2  V H1 )

(10A.9)

The elastic free energy of bending (ignoring the spontaneous curvature) is given by A ( M1 H1  M 2 H 2 ) 2 Equation (10A.10) can be written as 'Gb

'Gb

1 N b A ( H1  H 2 )2  N b (1  V ) AH1 H 2 2

1 N b A ( H1  H 2 ) 2  N G AH1 H 2 2

(10A.10)

(10A.11)

where, NG

(1  V )N b

(10A.12)

Here s represents the ratio of the fractional linear extension within the membrane plane to the fractional contraction in membrane thickness in response to isotropic membrane tension. The maximum value of s is 0.5. Therefore, from Eq. (10A.12) we observe that kG /kb £ –0.5. The minimum value of s is zero, which corresponds to kG = –kb.

FURTHER READING Books Adamson, A.W. and A.P. Gast, Physical Chemistry of Surfaces, John Wiley, New York, 1997. Baszkin, A. and W. Norde, Physical Chemistry of Biological Interfaces, Marcel Dekker, New York, 2000. Hartley, G.S., Aqueous Solutions of Paraffin-Chain Salts, Hermann et Cie, Paris, 1936. Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, London, 1997. Kosaric, N. (Ed.), Biosurfactants (Surfactant Science Series, Vol. 48), Marcel Dekker, New York, 1993. Mahanty, J. and B.W. Ninham, Dispersion Forces, Academic Press, New York, 1976. Nnanna, I.A. and J. Xia (Eds.), Protein-Based Surfactants (Surfactant Science Series, Vol. 101), Marcel Dekker, New York, 2001. Rosoff, M. (Ed.), Vesicles (Surfactant Science Series, Vol. 62), Marcel Dekker, New York, 1996.

Articles Arai, T. and W. Norde, “The Behavior of some Model Proteins at Solid–Liquid Interfaces. 1. Adsorption from Single Protein Solutions”, Coll. Surf., 51, 1 (1990).

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Bachofen, H. and S. Schürch, “Alveolar Surface Forces and Lung Architecture”, Comp. Biochem. Physiol. (A), 129, 183 (2001). Bangham, A.D., “Liposomes: The Babraham Connection”, Chem. Phys. Lipids, 64, 275 (1993). Bangham, A.D. and R.W. Horne, “Negative Staining of Phospholipids and their Structural Modification by Surface-Active Agents as Observed in the Electron Microscope”, J. Mol. Biol., 8, 660 (1964). Braun, E., Y. Eichen, U. Sivan, and G. Ben-Yoseph, “DNA-Templated Assembly and Electrode Attachment of a Conducting Silver Wire”, Nature, 391, 775 (1998). Bretscher, M.S., “Membrane Structure: Some General Principles”, Science, 181, 622 (1973). Chan, B.M.C. and J.L. Brash, “Conformational Change in Fibrinogen Desorbed from Glass Surface”, J. Coll. Int. Sci., 84, 263 (1981). Cohen Stuart, M.A. et al., “Adsorption of Ions, Polyelectrolytes and Proteins”, Adv. Coll. Int. Sci., 34, 477 (1991). Davis, S.A., S.L. Burkett, N.H. Mendelson, and S. Mann, “Bacterial Templating of Ordered Macrostructures in Silica and Silica–Surfactant Mesophases”, Nature, 385, 420 (1997). Davis, S.A. et al., “Brittle Bacteria: A Biomimetic Approach to the Formation of Fibrous Composite Materials”, Chem. Mater., 10, 2516 (1998). de Gennes, P.G., “Wetting: Static and Dynamics”, Rev. Mod. Phys., 57, 827 (1985). Döbereiner, H.-G., “Properties of Giant Vesicles”, Curr. Opin. Coll. Int. Sci., 5, 256 (2000). Douglas, T. and M. Young, “Host-Guest Encapsulation of Materials by Assembled Virus Protein Cages”, Nature, 393, 152 (1998). Helfrich, W., “Elastic Properties of Lipid Bilayers: Theory and Possible Experiments”, Z. Naturforsch, 28c, 693 (1973). ———, “Steric Interactions of Fluid Membranes in Multilayer Systems”, Z. Naturforsch, 33a, 305 (1978). Helm, C.A., J.N. Israelachvili, and P.M. McGuiggan, “Molecular Mechanisms and Forces Involved in the Adhesion and Fusion of Amphiphilic Bilayers”, Science, 246, 919 (1989). Homola, A. and A.A. Robertson, “A Compression Method for Measuring Forces between Colloidal Particles”, J. Coll. Int. Sci., 54, 286 (1976). Israelachvili, J.N. and H. Wennerström, “Entropic Forces between Amphiphilic Surfaces in Liquids”, J. Phys. Chem., 96, 520 (1992). ———, “Hydration or Steric Forces between Amphiphilic Surfaces?”, Langmuir, 6, 873 (1990). Kjellander, R. and S. Marcelja, “Perturbation of Hydrogen Bonding in Water near Polar Surfaces”, Chem. Phys. Lett., 120, 393 (1985). Lalchev, Z., R. Todorov and D. Exerowa, “Thin Liquid Films as a Model to Study Surfactant Layers on the Alveolar Surface”, Curr. Opin. Coll. Int. Sci., 13, 183 (2008). Langmuir, I. and V.J. Schaeffer, “Properties and Structure of Protein Monolayers”, Chem. Rev., 24, 181 (1939). Lipowsky, R., “The Conformation of Membranes”, Nature, 349, 475 (1991). Lipowsky, R. and M. Girardet, “Shape Fluctuations of Polymerized or Solidlike Membranes”, Phys. Rev. Lett., 65, 2893 (1990).

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MacRitchie, F., “Adsorption of Biopolymers”, Coll. Surf. (A), 16, 159 (1993). MacRitchie, F. and A.E. Alexander, “Kinetics of Adsorption of Proteins at Interfaces. Part I. The Role of Bulk Diffusion in Adsorption”, J. Coll. Sci., 18, 453 (1963). MacRitchie, F. and L. Ter-Minassain-Saraga, “Concentrated Protein Monolayers: Desorption Studies with Radiolabelled Bovine Serum Albumin”, Coll. Surf., 10, 53 (1984). Marcelja, S. and N. Radic, “Repulsion of Interfaces due to Boundary Water”, Chem. Phys. Lett., 42, 129 (1976). Marra, J., “Controlled Deposition of Lipid Monolayers and Bilayers onto Mica and Direct Force Measurements between Galactolipid Bilayers in Aqueous Solutions”, J. Coll. Int. Sci., 107, 446 (1985). ———, “Direct Measurements of Attractive van der Waals and Adhesion Forces between Uncharged Lipid Bilayers in Aqueous Solutions”, J. Coll. Int. Sci., 109, 11 (1986a). ———, “Direct Measurement of the Interaction between Phosphatidylglycerol Bilayers in Aqueous Electrolyte Solutions”, Biophys. J., 50, 815 (1986b). Marra, J. and J.N. Israelachvili, “Direct Measurements of Forces between Phosphatidylcholine and Phosphatidylethanolamine Bilayers in Aqueous Electrolyte Solutions”, Biochemistry, 24, 4608 (1985). Marsh, D., “Elastic Curvature Constants of Lipid Monolayers and Bilayers”, Chem. Phys. Lipids, 144, 146 (2006). Miao, L., U. Seifert, M. Wortis, and H.-G. Döbereiner, “Budding Transitions of Fluid-Bilayer Vesicles: The Effect of Area-Difference Elasticity”, Phys. Rev. (E), 49, 5389 (1994). Nardi, J., R. Bruinsma, and E. Sackmann, “Adhesion-Induced Reorganization of Charged Fluid Membranes”, Phys. Rev. (E), 58, 6340 (1998). Nardi, J., T. Feder, R. Bruinsma, and E. Sackmann, “Electrostatic Adhesion between Fluid Membranes: Phase Separation and Blistering”, Europhys. Lett., 37, 371 (1997). Norde, W., “The Behavior of Proteins at Interfaces, with Special Attention to the Role of the Structure Stability of the Protein Molecule”, Clinical Mater., 11, 85 (1992). Norde, W. and A.C.I. Anusiem, “Adsorption, Desorption and Re-adsorption of Proteins on Solid Surfaces”, Coll. Surf., 66, 73 (1992). Pfeiffer, W., Th. Henkel, E. Sackmann, W. Knoll, and D. Richter, “Local Dynamics of Lipid Bilayers Studied by Incoherent Quasi-Elastic Neutron Scattering”, Europhys. Lett., 8, 201 (1989). Pitt, W.G., K. Park and S.L. Cooper, “Sequential Protein Adsorption and Thrombus Deposition on Polymeric Biomaterials”, J. Coll. Int. Sci., 111, 343 (1986). Rand, R.P. and V.A. Parsegian, “Hydration Forces between Phospholipid Bilayers”, Biochim. Biophys. Acta, 988, 351 (1989). Sackmann, E., “Membrane Bending Energy Concept of Vesicle- and Cell-Shapes and ShapeTransitions”, FEBS Letters, 346, 3 (1994). Safinya, C.R., D. Roux, G.S. Smith, S.K. Sinha, P. Dimon, N.A. Clark, and A.M. Bellocq, “Steric Interactions in a Model Membrane System: A Synchrotron X-Ray Study”, Phys. Rev. Lett., 57, 2718 (1986). Schürch, S., F.H.Y. Green, and H. Bachofen, “Formation and Structure of Surface Films: Captive Bubble Surfactometry”, Biochim. Biophys. Acta, 1408, 180 (1998).

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Schürch, S., M. Lee, and P. Gehr, “Pulmonary Surfactant: Surface Properties and Function of Alveolar and Airway Surfactant”, Pure Appl. Chem., 64, 1745 (1992). Seeman, N.C., “Nucleic Acid Nanostructures and Topology”, Angew. Chem. Int. Ed., 37, 3220 (1998). Seifert, U. and R. Lipowsky, “Adhesion of Vesicles”, Phys. Rev. (A), 42, 4768 (1990). Singer, S.J. and G.L. Nicolson, “The Fluid Mosaic Model of the Structure of Cell Membranes”, Science, 175, 720 (1972). von Neergaard, K., “Neue Auffassungen über einen Grundbegriff der Atemmechanik: Die Retraktionskraft der Lunge, abhängig von der Oberflächenspannung in den Alveolen”, Z. Ges. Exp. Med., 66, 373 (1929). Ward, A.F.H. and L. Tordai, “Time-Dependence of Boundary Tension of Solutions. I: The Role of Diffusion in Time-Effects”, J. Chem. Phys., 14, 453 (1946). Willner, I., “Biomaterials for Sensors, Fuel Cells, and Circuitry”, Science, 298, 2407 (2002). Wrobel, S., “Bubbles, Babies and Biology: The Story of Surfactant”, FASEB J., 18, 1624 (2004).

11

Nanomaterials

Richard Smalley was born in Akron (Ohio, USA). He attended Hope College in Michigan and later studied at the University of Michigan where he received his B. S. degree in chemistry in 1965. After graduation, he joined Shell Chemical Company as a chemist. He received his doctorate from Princeton University in 1973 under the supervision of Elliot R. Bernstein. He did postdoctoral work at the University of Chicago where he was a pioneer in the development of supersonic beam laser spectroscopy. Richard Smalley won the Nobel Prize in Chemistry in 1996 [along with Robert F. Curl (Jr.) and Sir Harold W. Kroto] for their discovery of fullerenes. They had discovered the C60 Buckminsterfullerene in 1985. Later, Smalley focussed research on the tubular variant of the fullerenes and iron-catalysed synthesis of carbon nanotubes. He firmly believed that major new technologies would be developed over Richard Errett Smalley the coming decades from fullerene tubes, fibres and cables. His research-group (1943 – 2005) invented the high-pressure carbon monoxide method of producing large batches of high-quality nanotubes. Smalley established a company, Carbon Nanotechnologies Inc. and also took a leading role to create the Rice Centre for Nanoscale Science and Technology. He was strongly of the opinion that tiny robot-assemblers cannot build nanomaterials atom by atom controlling the subtle chemistry of the process. In his later years, Smalley was outspoken about the need for cheap and clean energy, which he described as the main problem facing humanity in the 21st century. Smalley died in Houston (Texas, USA).

TOPICS COVERED © © © © © © © ©

Bottom-up and top-down techniques Self-assembly Nanoparticles Nanowires, nanorods and nanotubes Thin films Microporous and mesoporous materials Lithographic techniques Toxic effects of nanomaterials 410

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11.1 INTRODUCTION The term nano originated from the Greek nanos which means ‘dwarf’. As we know, it is one billionth of a metre. Therefore, whenever we think about nanoscience or nanotechnology, very small objects come to the mind. Indeed, this branch of science and technology deals with materials having at least one spatial dimension in the size range of 1 to 1000 nm (Ozin and Arsenault, 2006). Some scientists put the upper limit of size to 100 nm (Poole and Owens, 2006), or a few hundred nanometres (Cao, 2006). Richard P. Feynman (Nobel Prize in Physics, 1965) is often credited for introducing the concept of nanotechnology about 50 years ago. In the annual meeting of the American Physical Society at California Institute of Technology on 26 December, 1959 he delivered a famous lecture entitled “There’s Plenty of Room at the Bottom” [see Feynman (1992)]. In this lecture, he talked about writing twenty four volumes of the Encyclopaedia Britannica on the head of a pin, and miniaturising the computer. He also suggested that it would be possible to arrange the atoms the way we want. Therefore, a physicist should be able to synthesise any chemical substance by putting the atoms down where the chemist says. The lecture of Feynman has inspired many scientists in various ways. For example, K. Eric Drexler wrote a book in 1986 entitled “Engines of Creation: The Coming Era of Nanotechnology”. Drexler envisioned a world completely transformed by nanoscale robot assemblers. These assemblers would manipulate and build things atom-by-atom working furiously running a nanofactory. They would be able to build anything with absolute precision and no pollution. However, it is unlikely that such a manufacturing process will be possible from the point of view of basic chemistry because the region where these nanoscale robots will have to operate is too small, and the atoms which are to be manipulated will adhere to the arms of the robots due to the attractive forces (Smalley, 2001; Baum, 2003). Nanomaterials are expected to have a wide range of applications in various fields such as electronics, optical communications and biological systems. These applications are based on factors such as their physical properties, huge surface area and small size, which offers possibilities for manipulation and room for accommodating multiple functionalities. In recent years, major progress has been achieved in molecular electronics. As the physical limits of the conventional silicon chips are being approached, researchers are seeking the next small thing in electronics through chemistry. By making devices from small groups of molecules, researchers may be able to pack computer chips with billions of transistors, more than 10 times as many as the current technology can achieve. Researchers in molecular-electronics think that it is possible to make complex circuitry by utilising DNA’s ability to recognise molecules and self-assemble. They hope to use DNA as a template for crafting metallic wiring, or even to wire circuits with strands of DNA itself. Makers of computer chips are concerned with the wavelength-limits of light. As the wavelength of the light is reduced, smaller features can be printed on the chip. As a general rule of thumb, a given wavelength can make features about half its length. Typically, light of 248 nm wavelength was used in the past in optical lithography devices, and the smallest features that could be made by these devices were about 120 nm. Chipmakers are trying to build devices which use much shorter wavelengths (e.g. 157 nm). Features as small as 80 nm have already been created on silicon wafers. Researchers expect that the technology will be able to turn out features as small as 10 nm (Service, 2001). A very promising and rapidly-growing field of application of nanotechnology is in medicine. One interesting application involves the use of nanoscale devices which may serve as vehicles for delivery of therapeutic agents and act as detectors or guardians against early disease. They would

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possibly repair the metabolic and genetic defects. They would seek out a target within the body such as a cancer cell and perform some functions to fix it. The fixing can be achieved by releasing a drug in the localised area. The potential side effects of general drug therapy can be reduced significantly in this manner. As nanotechnology becomes more sophisticated, gene replacement, tissue regeneration or nanosurgeries are the promising future developments (Haberzettl, 2002). The possible benefits that can be obtained from nanoscience and technology seem to be almost endless. Many of these dreams may be realised in the near future. In the preceding paragraphs, a few examples have been presented to provide a glimpse to the reader about some of the applications of nanotechnology. In the following sections of this chapter, various types of nanomaterials, their structures and methods to prepare them will be discussed. The potential applications of these materials will also be explained.

11.2 APPROACHES FOR THE SYNTHESIS OF NANOMATERIALS There are two approaches to the synthesis of nanomaterials: bottom-up and top-down. In the bottomup approach, molecular components arrange themselves into more complex assemblies atom-byatom, molecule-by-molecule, cluster-by-cluster from the bottom (e.g. growth of a crystal). In the topdown approach, nanoscale devices are created by using larger, externally-controlled devices to direct their assembly. The top-down approach often uses the traditional workshop or microfabrication methods in which externally-controlled tools are used to cut, mill and shape materials into the desired shape and order. Attrition and milling for making nanoparticles are typical top-down processes. Bottom-up approaches, in contrast, arrange molecular components themselves into some useful conformation using the concept of molecular self-assembly. Synthesis of nanoparticles by colloid dispersions is an example of the bottom-up approach. An approach where both these techniques are employed is known as a hybrid approach: lithography is an example in which the growth of thin film is a bottom-up method whereas itching is a top-down method. The bottom-up approach has been well known to the chemists for a long time. This approach plays a very important role in preparing nanomaterials having very small size where the top-down process cannot deal with the very tiny objects. The bottom-up approach generally produces nanostructures with fewer defects as compared to the nanostructures produced by the top-down approach. The main driving force behind the bottomup approach is the reduction in Gibbs free energy. Therefore, the materials produced are close to their equilibrium state. In top-down techniques such as lithography, significant crystallographic defects can be introduced to the processed patterns. For example, nanowires made by lithography are not smooth and can contain a lot of impurities and structural defects on its surface. Since the surface area per unit volume is very large for the nanomaterials, these defects can affect the surface properties, e.g. surface imperfections may cause reduced conductivity and excessive generation of heat would result. In spite of the defects, the top-down approach plays an important role in the synthesis and fabrication of nanomaterials. The present state of nanoscience can be viewed as an amalgamation of bottom-up chemistry and top-down engineering techniques.

11.3 SELF-ASSEMBLY AND STRUCTURE Self-assembly is a fundamental principle which creates structural organisation from the disordered components in a system. The principles of self-assembly was conceptualised long time ago (circa 400 BC). The ancient Greek philosopher Democritus expounded the idea that atoms and voids organised

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in different arrangements constitute all matter. He explained the growth of the universe from the minutest atomistic building blocks to the stars and galaxies. This is perhaps the oldest recorded vision of matter undergoing self-assembly over all scales (Ozin and Arsenault, 2006). The basic principle of nanochemistry lies in the self-assembly of a target structure from the spontaneous organisation of building blocks. The building blocks can be molecules or nano-scale clusters. There are five important factors that need to be taken into consideration for self-assembly: (i) building blocks, scale, shape and surface structure, (ii) attractive and repulsive interactions between the building blocks, (iii) association, dissociation and adaptable motion of the building blocks in the assembly to attain the lowest energy structure, (iv) interactions of the building blocks with the solvents, interfaces and templates, and (v) the dynamics of the building blocks and mass transport. The building blocks are usually not monodisperse (unless they are single atoms or molecules). The polydispersity present in the building blocks in terms of size and shape dictates the achievable degree of structural perfection of the assembly, and the defects in the assembled system. The making of building blocks with a particular surface structure, charge and functionality is a challenging task. The surface properties control the interactions between the building blocks and their interactions with the environment as well. This determines the geometry and the equilibrium separation between the building blocks in a self-assembled system. The aggregation and deaggregation processes, and the corrective movements of the self-assembled structure allow it to attain the most stable form. The driving forces for molecular organisation can be as varied as ionic, covalent, hydrogenbonding or metal-ligand bonding interactions. The chemistry of self-assembly of materials transcends the chemistry of molecular assembly. It is distinct solid-state materials chemistry where the building blocks and their assemblages are unconstrained by scale, and they are not restricted to just chemical bonding forces. At length scales higher than molecular length scales, other forces such as capillary, colloidal, elastic, electric, magnetic and shear forces can all influence the self-assembly of materials (Ozin and Arsenault, 2006). An important feature of self-assembly is hierarchy: the primary building blocks associate into more complex secondary structures, which are integrated into the next size level in the hierarchy. This organisational scheme continues until the highest level in the hierarchy is attained. The self-assembly of molecules and materials can be directed by templates. The template can be constituted of molecules, molecular assemblies or additive materials which serve to fill space, balance charge and direct the formation of a specific structure. For example, mesoporous zeolites are templated by block co-polymers or lyotropic liquid crystals. A template patterned at the nanoscale can direct the assembly process. The structure-directing templates which can make, organise and interconnect the building blocks can be porous hosts, lithographic patterns, and channels in polymer, alumina and silicon membranes. Some of these templates are widely used to make nanowires, nanorods and nanotubes. A well-known templating method is the use of surfactant micelles and liquid crystals. Many microporous and mesoporous inorganic solids have been prepared by this templating mechanism. The preparation of MCM-41 molecular sieve is schematically shown in Figure 11.1. In this illustration, the silicate material occupies the spaces between the hexagonal arrays of cylindrical micelles. When calcined, the organic material burns-off and the hollow cylinders of silicate remain. We have discussed the hierarchy of the self-assembly process. In fact, it is a well-known characteristic of many self-assembling biological structures. Many biominerals are organised from nanoscale to the macroscopic scale to give hierarchical materials that have complex forms such as spirals, spheroids and skeletons with apparent disregard for the rigid geometric symmetry of their inorganic constituents (Mann and Ozin, 1996; Alivisatos, 2000). At each level of the hierarchy, distinctive building rules can be observed. The characteristics of self-assembly in biological materials

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Figure 11.1 Liquid crystal templating mechanism (Kresge et al., 1992) (reproduced by permission from Macmillan Publishers Ltd., © 1992).

are manifested in a variety of curved shapes, surface patterns and hierarchical order (e.g. a seashell). Ball (2001) has presented a nice description of pattern formation in nature such as grains, bodies and waves. The structures of the nanomaterials can be classified by their dimensions. The zero-dimensional nanostructures are nanoparticles. The one-dimensional nanostructures are whiskers, fibres (or fibrils), nanowires and nanorods. In many cases, nanocables and nanotubes are also considered onedimensional structures. Thin films are considered as two-dimensional nanostructures. Colloids bearing complex shapes have three-dimensional nanostructures.

11.4 SYNTHESIS OF NANOPARTICLES The methods of synthesis of nanoparticles are well known for a long time as compared to the other nanomaterials. For the synthesis of nanoparticles, the processing conditions need to be controlled in such a manner that the resulting nanoparticles have the following characteristics: (i) identical size of all particles, (ii) identical shape, (iii) identical chemical composition and crystal structure, and (iv) individually dispersed with no agglomeration. Nanoparticles can be synthesised by both topdown or bottom-up approaches. Two well-known top-down approaches are milling (or attrition) and thermal cycling. Attrition produces nanoparticles of a wide range of diameter ranging from 20 nm to several hundred nanometres. The shape of the particles varies as well. They may contain impurities from the milling medium. The nanoparticles made by this process are usually used in the fabrication of nanocomposites and bulk materials having nanograins where perfections in size and shape, and presence of impurities do not matter significantly. Moreover, some of the defects can get annealed during the sintering process. A bulk material having very small thermal conductivity but a large coefficient of thermal expansion may be subjected to repeated thermal cycling to produce very fine particles. However, this technique is difficult to design and the control of particle size and shape is difficult. The bottom-up methods are more popular than the top-down methods. There are several bottomup methods such as homogeneous and heterogeneous nucleation processes, microemulsion-based synthesis, aerosol synthesis, spray pyrolysis and template-based synthesis. In this section, some of these techniques will be discussed.

11.4.1 Homogeneous Nucleation In the homogeneous nucleation process, a supersaturation of the growth species is created. The supersaturation can be achieved by a reduction in temperature of a saturated solution or by a

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chemical reaction in situ. These methods were discussed in Chapter 1. A solution that is in equilibrium with the solid phase is called saturated with respect to that solid. However, it is easy to prepare a solution containing more dissolved solid than that represented by the saturation condition. Such a solution is known as supersaturated with respect to the dissolved solid. The state of supersaturation is a prerequisite for all crystallisation processes. If c is the solute concentration in the solution and c* is the solubility of the solute at the given temperature, the supersaturation ratio S is defined as c

S

(11.1)

c

The relative supersaturation (s ) is defined as c  c

V

c

S 1

(11.2)

A supersaturated solution possesses a high Gibbs free energy and the overall energy of the system would be reduced by segregating the solute from the solution. The reduction in the Gibbs free energy is the driving force for nucleation as well as growth of the crystals. The change in Gibbs free energy per unit volume of the solid phase can be expressed as 'G



kT È c Ø ln É Ù Êc Ú v



kT ln(S ) v



kT ln(1  V ) v

(11.3)

where k is Boltzmann's constant, T is temperature and v is molecular volume. If s = 0, i.e. there is no supersaturation, 'G is zero and nucleation would not occur. When s > 0, 'G is negative and nucleation occurs spontaneously. Let us consider a spherical nucleus with radius r. The change in Gibbs free energy is given by 'Gv

4 3 S r 'G 3

(11.4)

However, when a new phase is formed, the increase in surface energy is given by

'Gs

4S r 2J

(11.5)

where g is the surface energy per unit area. Therefore, the net change of Gibbs free energy for the formation of the nucleus is given by 'G

4 3 S r 'G  4S r 2J 3

'Gv  'Gs

(11.6)

The supersaturated state of the solution is a prerequisite, but not a sufficient cause for a system to begin to crystallise. A free energy barrier must be overcome for the generation of a new crystalline phase from the solution. A newly-formed nucleus is stable only when its radius exceeds a critical value. A small nucleus dissolves in the solution to reduce the overall free energy. However, when the radius is larger than a critical value rc, it becomes stable and continues to grow. At the critical size r = rc, dDG/dr = 0. Therefore, rc



2J 'G

(11.7)

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The free energy barrier can be calculated by substituting rc from Eq. (11.7) and 'G from Eq. (11.3) into Eq. (11.6), which gives

'Gc

16SJ 3

16SJ 3 v 2

3( 'G)2

3k 2T 2 (ln S)2

(11.8)

Since rc represents the minimum size of a stable spherical nucleus, it sets the limit on how small nanoparticles may be synthesised. The value of critical radius can be reduced by reducing g, and increasing 'G by increasing S. A few other factors such as solvent and additives in solution are important factors for the nucleation process. The nucleation-rate by primary homogeneous nucleation mechanism is given by È 'Gc Ø A exp É  Ê kT ÙÚ

J

Ë 16SJ 3 v 2 Û A exp Ì  3 3 2Ü ÌÍ 3k T (ln S) ÜÝ

(11.9)

where A is the pre-exponential factor. Equation (11.9) predicts that nucleation would occur only at some high values of supersaturation. The range of values of supersaturation ratio S in which the rate of nucleation increases very rapidly is known as the metastable zone for homogeneous nucleation. EXAMPLE 11.1 The interfacial energy for barium sulphate crystals in saturated aqueous solution is 125 mJ/m2. If the critical radius is 1.5 nm, calculate the value of the Gibbs free energy barrier. Solution

From Eq. (11.7), we get 'G



2J rc



2 – 0.125 1.5 – 10

9

166.7 – 10 6 J/m 3

Therefore, from Eq. (11.8), we get

'Gc

16SJ 3

16S – (0.125)3

3( 'G)2

3 – ( 166.7 – 10 6 )2

1.177 – 10 18 J

Nanoparticles with uniform size distribution can be synthesised if all the nuclei are formed at the same time. In this procedure, all the nuclei are likely to have similar size initially. If their subsequent growth is similar, monodisperse particles will be obtained. Therefore, as discussed in Section 1.4.6, it is highly desirable if the nucleation occurs in a very short period of time. To achieve a sharp nucleation, the concentration of the solute is increased abruptly to a very high supersaturation and then quickly brought below the minimum concentration for nucleation. No more new nuclei form below this concentration whereas the existing nuclei continue to grow until the concentration of the growth species reduces to the equilibrium value. The size distribution of the nanoparticles can be altered in the subsequent growth process also, depending on the kinetics of the growth process. The growth of the nuclei depends upon several steps such as (i) generation of the growth species, (ii) diffusion of the growth species from the bulk to the growth surface, (iii) adsorption of the growth species on the growth surface, and (iv) surface-growth through the irreversible incorporation of the growth species on the growth surface. Let us consider the growth of a spherical nucleus. If the growth process is controlled by the diffusion of the growth species from the bulk solution to the particle surface, the rate of growth is given by (Nielsen, 1964).

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v dr D(cb  cs ) n (11.10) dt r where r is the radius of the nucleus, t is time, D is the diffusion coefficient of the growth species, cb is the bulk concentration, cs is the concentration on the surface of the solid particles and vn is the molar volume of the nuclei. If the initial radius of nucleus is r0 and if the concentration in the bulk does not change appreciably with time, we can integrate Eq. (11.10) to obtain the following equation: r2

r02  2 D cb  cs vn t

r02  kd t

(11.11)

It can be shown that for two particles with different initial radius, the radius-difference decreases as the time increases and the particles grow in size (see Problem 11.2). Therefore, uniformly-sized particles are formed by the diffusion-controlled growth. If the diffusion process is rapid, cs » cb, and the growth-rate is controlled by the surface process. There are two mechanisms for the surface incorporation: mononuclear growth and polynuclear growth. In the mononuclear growth, the growth proceeds layer-by-layer. The growth species are incorporated into one layer and proceeds to another layer after the growth of the previous layer is complete. On the other hand, for polynuclear growth, the second layer begins to grow even before the growth of the first layer is complete. It is typically observed when the surface concentration is high. The growth-rate for mononuclear growth is proportional to the surface area, i.e. dr km r 2 dt where km is a proportionality constant. The variation of radius with time is given by 1 r

1  km t r0

(11.12)

(11.13)

It can be shown that the difference in radius between two particles increases with time in this mechanism. Therefore, this mechanism of growth does not favour the synthesis of nanoparticles of similar size. In polynuclear growth mechanism, the rate of growth of particle is independent of particle size or time. dr kp (11.14) dt The radius of the particles varies linearly with time. (11.15) r = r0 + kpt In this mechanism the difference in radius between two particles remains constant regardless of the growth time. Therefore, among the three mechanisms described in this section (i.e. the diffusion-controlled growth, mononuclear surface incorporation and polynuclear surface incorporation), it is evident that the diffusion-controlled growth would favour uniformly-sized nanoparticles. However, it is likely that the growth of nanoparticles involves all three mechanisms (Williams et al., 1985). When the nuclei are small, the mononuclear growth mechanism may dominate. As the nuclei become bigger, the polynuclear mechanism may dominate, and for relatively larger particles, the growth is diffusion controlled. The growth conditions are very important to decide which mechanism would control. For example, if the supply of the growth species is very slow (e.g. due to a slow chemical reaction), the diffusion-controlled mechanism would control the process.

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Metal nanoparticles are used as various types of catalysts, adsorbents, sensors and ferrofluids. They have applications in optical, electronic and magnetic devices. Most of these applications critically depend on the size and shape of the nanoparticles. Therefore, the synthesis of wellcontrolled size and shape of these nanoparticles is important for these applications. The reduction of metal complexes to form metallic colloid dispersions has been discussed in Section 1.4.6. Various precursors, reducing agents and polymeric stabilisers are used in the preparation of metallic colloid dispersions. Some of these are presented in Table 11.1. Table 11.1

Precursors, reducing agents and polymeric stabilisers used in the preparation of metallic nanoparticles Category

Name

Precursor

Metal anode Palladium chloride Potassium tetrachloroplatinate II Silver nitrate Chloroauric acid Rhodium chloride Hydrogen Sodium citrate Citric acid Carbon monoxide Methanol Formaldehyde Hydrogen peroxide Sodium tetrahydroborate Poly(vinylpyrrolidone) Polyvinyl alcohol Sodium polyphosphate Sodium polyacrylate

Reducing agent

Polymeric stabiliser

The size of metallic colloids varies significantly with the type of the reducing agent. A strong reducing agent promotes a fast reduction reaction, and if the reaction is fast, generally small nanoparticles are formed. On the other hand, a weak reducing agent induces a slow reaction and large particles are formed. A strong reducing agent generates an abrupt surge of the concentration of the growth species resulting in a very high supersaturation. Consequently, a very large number of nuclei is formed initially. For a given concentration of the metal precursor, the formation of a large number of nuclei results in small size of the nanoparticles. The role of polymeric stabiliser is to form a monolayer on the surface of the nanoparticles and prevent their aggregation. The polymeric stabiliser is also known as capping material. The monolayer of the polymer, however, can affect the growth process significantly. If the growth sites are occupied by the polymer, the rate of growth of nanoparticles may be reduced. If the polymeric stabiliser completely covers the surface of the growing particle, it may hinder the diffusion of the growth species from the surrounding solution to the surface of the particle. The shape of the nanoparticles can be varied by the use of different amounts of the polymeric stabiliser. The shape and size of platinum nanoparticles have been controlled by changing the ratio of the concentration of the polymer (sodium polyacrylate) to the concentration of the platinum cations (Ahmadi et al., 1996). They observed tetrahedral, cubic, irregular-prismatic, icosahedral and cubo-octahedral shapes of the particles (Figure 11.2).

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Figure 11.2 TEM images of platinum nanoparticles: (a) cubic nanoparticles formed when the initial ratio of concentration of polymer (sodium polyacrylate) to that of the metal cation in the solution was 1:1, and (b) tetrahedral nanoparticles formed when the initial ratio was 5:1. The insets show high-resolution images of the particles (Ahmadi et al., 1996) (reproduced by permission from The American Association for the Advancement of Science, © 1996).

Gold nanoparticles are well known for their distinct colours. They have been used in glasses and enamels as colouring agents. A variety of methods exist for the preparation of gold nanoparticles. One of the most common methods is the reduction of chloroauric acid at 373 K by sodium citrate. To prepare a colloidal dispersion of rhodium, the following reduction reaction, using methanol as the reducing agent, can be carried out in presence of a stabiliser such as polyvinyl alcohol. 2RhCl3  3CH3 OH  2Rh  3HCHO  6HCl

(11.16)

The size of nanoparticles depends on the reaction conditions. Ostwald ripening plays an important role in the size of the nanoparticles. Nanoparticles of platinum and palladium can be prepared by reduction using hydrogen: the salts K2PtCl4 and PdCl2 are hydrolysed to form hydroxides, which are then reduced. PdCl2  Na 2 CO3  2H 2 O  Pd(OH)2  H 2 CO3  2NaCl

(11.17)

Pd(OH)2 + H 2  Pd + 2H 2 O

(11.18)

Non-oxide semiconductor nanoparticles can be synthesised by the pyrolysis of organometallic precursor dissolved in anhydrate solvents at elevated temperatures in airless environment in presence of a polymeric stabiliser. The nanocrystals of CdS, CdSe and CdTe have been prepared in this method. The nanocrystals of GaN have been synthesised by a thermal reaction of Li3N and GaCl3 at 553 K under pressure using benzene as the solvent in argon atmosphere (Xie et al., 1996) GaCl3  Li3 N  GaN  3LiCl

(11.19)

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The yield of GaN was 80% and the size of the particles was 30 nm. The product comprised of mostly hexagonal GaN with a small amount of rocksalt GaN. A well-known method for synthesising oxide nanoparticles is sol–gel processing. The sol–gel process typically consists of hydrolysis and condensation of the precursors. The typical precursors are metal alkoxides, or inorganic and organic salts. The precursors are dissolved in aqueous or organic solvents. Sometimes catalysts are used to promote the hydrolysis and condensation reactions. Hydrolysis:

M(OEt)4 + xH 2 O ½ M(OEt) 4- x (OH) x + xEtOH

Condensation: 2M(OEt) 4- x (OH) x ½ (OEt) 4- x (OH) x -1 MOM(OEt) 4- x (OH) x -1  H 2 O

(11.20) (11.21)

where M represents the metal. The nanoclusters formed by condensation often have organic groups attached to them, which may result due to incomplete hydrolysis. The size of the nanoclusters and the structure of the final product can be tailored by suitably controlling the reactions given in Eqs. (11.20) and (11.21). Colloidal dispersions of metal hydrous oxides consisting of particles of considerable uniformity in size and shape have been synthesised by keeping the salt solutions of the respective metals at elevated temperatures for various periods of time. The particle shape and composition depend most strongly on the pH and on the nature of the anions contained in the aging systems (Matijevi ü, 1977). Ferric oxide nanoparticles were synthesised by aging ferric salt solutions with the corresponding acid at 373 K for one day. Electron micrographs of ferric oxide nanoparticles formed from Fe(NO3)3 + HNO3 and Fe(ClO4)3 + HClO4 solutions are shown in Figure 11.3. The anions influence the surface properties and interfacial energy of the nanoparticles, and hence the growth of the nanoparticles. They can also influence the electrostatic double layer repulsion between the particles and hence their stability.

Figure 11.3 Electron micrographs of iron oxide nanoparticles obtained by aging the solutions at 373 K for one day: (a) Fe(NO3)3 (18 mol/m3) + HNO3 (104 mol/m3), and (b) Fe(CIO4)3 (18 mol/m3) + HClO4 (104 mol/m3) (Matijevic„ , 1977) (reproduced by permission from Elsevier Ltd., © 1977).

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11.4.2 Microemulsion-Based Methods Ultrafine metal nanoparticles of diameter between 5 nm and 50 nm can be prepared by water-in-oil microemulsions (see Chapter 9). The nanodroplets of water are dispersed in the oil phase. The size of the droplets can be varied in the range of 5 to 50 nm by changing the water/surfactant ratio. The surfactant molecules provide the sites for particle nucleation and stabilise the growing particles. Therefore, the microemulsion acts as a microreactor. The reactant metal salts and reducing agents are mostly soluble in water. Therefore, the nucleation of particles proceeds in the water pools of the microemulsion. One microemulsion contains the metal salt and the other microemulsion contains the reducing agent. The nanoparticles are synthesised by mixing the two microemulsions as shown in Figure 11.4. During the collision of the water droplets, interchange of the reactants (i.e. the metal salt and the reducing agent) takes place. The interchange of reactants is very fast so that it occurs during the mixing process itself. The nucleation and growth take place inside the droplets. Interchange of nuclei or particles between the drops is hindered because it would require formation of a big hole during the collision of the droplets

Figure 11.4

(a) Mechanism for the synthesis of metal nanoparticles by the microemulsion approach, and (b) the percolation mechanism in detail (Capek, 2004) (reproduced by permission from Elsevier Ltd., © 2004).

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and would require a large change in the curvature of the surfactant layer around the drops, which is not favoured energetically. Since the inorganic salts have very low solubility in the oil phase, the dynamic exchange of reactants between the droplets through the continuous phase is unfavourable. When the particles attain their final size, the surfactant molecules attach themselves to the surface of the particles and stabilise them. Further growth is also prevented by the adsorbed surfactant layer. Percolation is a very important step in the particle nucleation. It is illustrated in Figure 11.4(b) for a bimolecular reaction: A + B ® C, where A represents the metallic salt (e.g. FeCl3), B represents the reducing agent (e.g. NaBH4) and C represents the metal particle. For the reaction to occur, the reactant A located in the water pool of one droplet must find the reactant B located in the water pool of another droplet. This can occur by two mechanisms: (i) movement of A molecule out of the water pool, migration through the oil phase and entry into the water-pool containing B molecule, or (ii) direct transfer between two water-pools containing molecules of A and B at the time of collision between two droplets. If the collisions are energetic and strongly interactive, the latter mechanism would prevail. Eicke et al. (1976) have shown that the inter-droplet communication is very rapid and it occurs by a transitory dimer species which is formed as a result of the collision between two droplets. Generally, the chemical reaction between the salt and the reducing agent is very fast compared to the communication between the droplets. Therefore, the rate-determining step is the second-order communication step. The rate constant for this step is estimated to be in the range of 106 to 107 dm3 mol–1 s–1 for Aerosol OT–water–heptane microemulsion system (Capek, 2004). The concentrations of the reactants affect the reduction rate. The rates of both nucleation and growth are determined mainly by the probabilities of the collisions between two atoms, between one atom and a nucleus, and between two nuclei. The first type of collision is related to nucleation, and the second and third types are related to the growth process. To prepare nanoparticles of different diameters, the water/surfactant ratio can be varied. Barnickel et al. (1992) have synthesised silver nanoparticles of different size by varying this ratio. The nonionic surfactant dodecyl pentaethyleneglycol ether was used for stabilising the reverse micelles. One water-in-oil microemulsion solubilised AgNO3 and the other microemulsion solubilised NaBH4. As the water/ surfactant ratio (W/S) increased from 0.05 to 0.2, the droplet-size in the microemulsion increased. At the same time, the mean particle diameter increased. These results are shown in Figure 11.5. However, with increase in water/surfactant ratio further, this trend was apparently reversed: the particle-diameter decreased, although some very large particles were formed. The formation of the large aggregates was attributed to secondary growth induced by small temperature fluctuations which might have temporarily destabilised the microemulsion. The most uniform product with the narrowest size distribution was obtained for W/S = 0.2. Extensive literature is available on the synthesis of iron, platinum, cadmium, palladium, silver, copper, nickel and gold nanoparticles by the water-in-oil microemulsion method. Capek (2004) has presented a review of this literature.

11.4.3 Synthesis of Carbon Fullerenes Fullerenes are a family of carbon allotropes, molecules composed entirely of carbon in the form of a hollow sphere, ellipsoid, tube or plane. Fullerenes are similar in structure to graphite, which is composed of stacked sheets of linked hexagonal rings, but may also contain pentagonal or sometimes heptagonal rings. The most familiar carbon fullerene is a molecule with 60 carbon atoms, represented as C60. It was discovered in 1985 by Kroto et al. and named as Buckminsterfullerene. The name was coined after the American architect Richard Buckminster Fuller who was famous for the geodesic

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Figure 11.5 Transmission electron micrographs of silver nanoparticles synthesised in water-in-oil microemulsions. The micrographs depict the effect of water/surfactant ratio (W/S) on the size of silver nanoparticles. The most probable particle diameters are indicated on the micrographs (Barnickel et al., 1992) (reproduced by permission from Elsevier Ltd., © 1992).

domes built by him. This unusual molecule was synthesised by the vaporisation of carbon species from the surface of a solid disk of graphite into a high-density helium flow using a focussed pulse laser. The resulting carbon clusters were expanded in a supersonic molecular beam, photoionised using an excimer laser, and detected by time-of-flight mass spectrometry. The vaporisation chamber is shown in Figure 11.6. The vaporisation laser beam was focussed through the nozzle, and it struck a graphite disk which was rotated slowly. The pulsed nozzle passed high-density helium over this vaporisation zone. The helium carrier gas provided the thermalising collisions necessary to cool, react and cluster the species in the vaporised graphite plasma. Free expansion of this cluster-laden gas at the end of the nozzle formed a supersonic beam which was probed 1.3 m downstream with a time-of-flight mass spectrometer. The C60 molecule has a truncated icosahedral structure formed by replacing each vertex on the seams of a football by a carbon atom, as shown in Figure 11.7. There are 20 hexagonal faces and 12 pentagonal faces in the molecule. The average C–C distance is 0.144 nm, which is very close to that in graphite (0.142 nm). Each carbon atom is trigonally bonded to other carbon atoms, same as that in graphite. Out of the three bonds emanating from each carbon atom, there is one double bond and two single bonds. The hexagonal faces consist of alternating single and double bonds and the pentagonal faces are defined by single bonds. The length of the single bonds is 0.146 nm, which is longer than the average bond length (i.e. 0.144 nm), while the double bonds are shorter, 0.14 nm. The diameter of the C60 molecule is 0.71 nm. The inner cavity is capable of holding a variety of atoms.

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Figure 11.6 Schematic diagram of the pulsed supersonic nozzle used to generate carbon cluster beams (Kroto et al., 1985) (reproduced by permission from Macmillan Publishers Ltd., © 1985).

Figure 11.7 The truncated-icosahedral structure of C60 Buckminsterfullerene (Kroto et al., 1985) (reproduced by permission from Macmillan Publishers Ltd., © 1985).

Other fullerenes with smaller as well as larger number of carbon atoms also exist (Dresselhaus et al., 1996). They are represented as Cn. Kroto (1987) has presented a set of simple, empirical chemical and geodesic rules which relate the stability of carbon cages mainly to the disposition of pentagonal rings, or various directly-fused pentagonal ring configurations. He has shown that the fullerenes with n = 24, 28, 32, 36, 50, 60 and 70 should have enhanced stability relative to the near neighbours. Fullerenes can be synthesised in large scale by the method of Krätschmer et al. (1990). In this method, the starting material is pure graphitic carbon soot with a few per cent of C60 molecules. It is produced by evaporating graphite electrodes in an atmosphere of helium at about one-seventh of atmospheric pressure. The resulting black soot is scrapped from the collecting surfaces inside the

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evaporation chamber and dispersed in a solvent such as benzene, carbon tetrachloride or carbon disulphide. The C60 molecules dissolve to give colour that varies between wine-red and brown depending upon the concentration. The liquid is then separated from the soot and dried using gentle heat, leaving a residue of dark brown to black crystalline material. Another procedure is to heat the soot to 673 K in vacuo or in an inert atmosphere. The C60 molecules sublime out of the soot. The sublime coatings are brown to grey depending upon the thickness. The details of purification methods have been given by Krätschmer et al. (1990).

11.5 SYNTHESIS OF NANOWIRES, NANORODS AND NANOTUBES Synthesis of these one-dimensional nanomaterials can be carried out by various techniques such as: (i) spontaneous growth (e.g. evaporation–condensation, vapour–liquid–solid growth and stressinduced recrystallisation), (ii) template-based synthesis (e.g. electroplating, electrophoretic deposition, colloid dispersion, melt or solution filling, and chemical reaction), (iii) electrospinning, and (iv) lithography. Spontaneous growth usually results in the formation of single-crystal nanowires or nanorods along a preferential direction of crystal growth. It depends on the crystal structure and the surface properties of the nanowire material. For the formation of nanowires or nanorods, anisotropic growth is required, because the crystal needs to grow along a certain orientation faster than the other directions. The defects and impurities on the growth surfaces can cause non-uniform products. Template-based synthesis usually produces polycrystalline or amorphous products. The details of these techniques have been presented by Cao (2006).

11.5.1 Nanowires and Nanorods The nomenclature nanorod and nanowire is somewhat arbitrary. However, the terminology used in the literature suggests that they are distinguished by their diameter. The diameter of the nanorods is larger than nanowires, and the demarcation line is approximately 20 nm (Ozin and Arsenault, 2006). Pan et al. (2001) synthesised long ribbon-like nanostructures (known as nanobelts) of semiconducting oxides of zinc, tin, indium, cadmium and gallium by evaporating the commercial metal oxide powders at high temperatures. The oxide powder was placed at the centre of an alumina tube and the tube was inserted in a horizontal tube furnace where the temperature, pressure and evaporation time were controlled. During evaporation, the products were deposited onto an alumina plate placed at the downstream end of the alumina tube. The deposited product was characterised and analysed by X-ray diffraction, scanning electron microscopy, transmission electron microscopy and energy-dispersive X-ray spectroscopy. The oxide nanobelts were pure, structurally uniform and single crystalline. Most of them were free from defects and dislocations. They had rectangle-like cross-section with typical widths between 30 nm and 300 nm, and length up to a few millimetres. The width-to-thickness ratio varied between 5 and 10. The belt-like morphology is a distinctive and common structural characteristic for the family of semiconducting oxides. The TEM images of ZnO nanobelts are shown in Figure 11.8. By controlling the growth kinetics, left-handed helical nanostructures and nanorings may be formed by rolling up single crystal ZnO nanobelts (Kong and Wang, 2003). This phenomenon is attributed to a consequence of minimising the total energy attributed by spontaneous polarisation and elasticity.

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Figure 11.8 TEM images of several straight and twisted ZnO nanobelts displaying the shape characteristics of the belts (Pan et al., 2001) (reproduced by permission from The American Association for the Advancement of Science, © 2001).

One of the most straightforward and general methods of synthesis of nanowires is by filling a template bearing nanosized cylindrical holes. For example, metal nanowires can be synthesised by the reduction of a metal salt into the cylindrical pores of a thin membrane. If a template containing very homogeneous pores is available, this technique can produce nanowires and nanorods having controlled size. There are many porous templates that fulfil such requirements. One important example is microporous zeolites, which can be filled with other materials. Chemically-synthesised templates such as liquid-crystal-templated mesoporous aluminosilicates, electrochemicallysynthesised templates such as anodically-etched aluminium or silicon, self-organised systems such as microphase-separated block copolymers, and track-etched polymer membranes can be used to make nanowires (Martin, 1994; Thurn-Albrecht et al., 2000). Nanochannel silicon membranes with perfectly-ordered pores have been prepared by various techniques. Such silicon membranes can be used to synthesise modulated-diameter gold nanorods (Ozin and Arsenault, 2006). This method has been adapted to create nanorods having different metal segments by electrodeposition. These modulated-composition nanorods are known as nanobarcodes due to their striped appearance. They may be utilised to tag molecules in analytical chemistry and biology (Nicewarner-Pena et al., 2001). Segment-specific anchoring of selected molecules to a barcode metal nanorod is known as orthogonal self-assembly. Suppose that the nanorod has two segments of different metals such as gold and platinum. An organic molecule that has same affinity for the two metals binds indiscriminately to the Au and Pt segments. However, an organic molecule that has greater affinity for Au displaces the former molecule from the segments constituted by Au. In this way, the two metals on the same nanorod can have two distinct monolayers of chemical species. The ability to anchor functional molecules to selective locations of a nanorod provides a number of interesting opportunities to direct the nanorods to self-assemble into predetermined functional architecture. An application of this method is DNA end-functionalisation of gold nanorods. Thiolated DNA is bound to the exposed ends of gold nanorods encased in an alumina membrane. The membrane is then dissolved in NaOH to create a suspension of gold nanorods end-functionalised with DNA. Rhodamine-labelled complementary DNA is end-coupled to the nanorods. By a similar approach, DNA-side functionalised gold nanorods have been created and directed to self-assemble on soft-

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lithographically patterned gold surface sites, which had been functionalised with the complementary DNA (Mbindyo et al., 2001). This technique of coverage of gold surfaces with gold nanowires by linking them with DNA offers good prospects for the assembly of wire structures with particular connectivity. Nanowires have been organised by spinning them to form functional devices. The method is based on electrospinning. When a solution of a polymer such as polyvinylpyridine (or a polymer sol– gel mixture) passes through a high-voltage metal capillary, a thin charged stream emerges from the orifice. This continuous process generates a population of charged nanofibres that are driven to the ground electrode on a substrate (Li et al., 2004). The dimensions of the electrospun nanowire depend on the solvent viscosity, conductivity, surface tension and the precursor concentration. The collector electrodes can be arranged in various configurations to coerce the nanowires into organised arrays. Using this procedure, single or multilayer architecture of nanowires can be created in a simple and reproducible manner. Hollow nanofibres of inorganic and inorganic-polymer hybrid materials have been prepared by electrospinning (Li and Xia, 2004). The technique involves ejection from the double-capillary spinneret of a continuous coaxial jet comprised of a heavy mineral oil core surrounded by a sheath of ethanol–acetic acid–PVP–Ti(OiPr)4. The composite core-shell nanofibres are collected on an aluminium or silicon substrate and allowed to hydrolyse at room temperature in air. Thereafter, the oily core is extracted with octane to leave behind a collection of nanotubes with the walls made of amorphous TiO2 and PVP. Calcination in air at 773 K oxidises the organic polymer and the amorphous titania is converted into the anatase polymorph. The template-based methods which use zeolites, membranes or nanotubes can control crystal growth, but usually form nonuniform and polycrystalline materials. The vapour–liquid–solid (VLS) growth process is one of the most successful ways of synthesising oriented single crystal semiconductor nanowires with control over their diameter, length and composition. The original idea goes back to the 1960s when vapour-growth techniques were developed to produce crystalline semiconductors from hot gaseous reactants (Wagner and Ellis, 1964). It was discovered that whiskers of the semiconductor would spontaneously grow out of gold particles placed in the reaction chamber. In the VLS method of synthesis of nanowires, a catalyst first melts to form a droplet, becomes supersaturated with the precursors, and the precipitating elements extrude out of the catalyst droplet as a single-crystal nanowire. For example, silicon nanowire has been synthesised by laser-ablation of a Fe–Si target in a buffer gas, which creates a dense vapour that condenses into nanoclusters as the Fe and Si species cool through collisions with the buffer gas (Morales and Lieber, 1998). The equilibrium pseudobinary Fe–Si phase diagram is known. Therefore, the composition and temperature that favours liquid Fe–Si in equilibrium with solid Si can be identified. The furnace temperature is controlled to maintain Fe–Si in liquid state. The growth of Si nanowire begins only when the Fe–Si liquid nanocluster becomes supersaturated with Si. The growth of nanowire proceeds as long as the Fe–Si nanoclusters remain in the liquid state and Si-nutrient is available. The diameter of the nanowire is determined by the diameter of the nanoclusters and the length is controlled by the growth speed. The growth of nanowire terminates when it passes out of the hot zone of the reactor. A sheath of silica coats the surface of the wire and the catalytic Fe–Si nanocluster is attached to the end of the wire (see Figure 11.9). Doping can be made by introducing controlled amounts of PH3 (for n-Si nanowire) or BH3 (for p-Si nanowire) in the gas phase during the growth process. Epitaxy at the interface between the nanocluster and nanowire seems to be responsible for the oriented singlecrystal growth. The VLS of SiH4/H2 gaseous mixture on gold nanocluster catalysts generates a narrow size-distribution of single-crystal silicon nanowires having diameters as low as 3 nm (Wu et al., 2004).

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Figure 11.9 (A) TEM image of the nanowires produced after ablation of a Si0.9Fe0.1 target. The growth conditions were 1473 K and argon at 66.7 kPa flowing at 0.83 standard cm3 per second, and (B) diffraction contrast TEM image of a Si nanowire (Morales and Lieber, 1998) (reproduced by permission from The American Association for the Advancement of Science, © 1998).

The growth of a nanowire occurs as follows. When the silicon species condenses at the surface of the droplet, the droplet becomes supersaturated with silicon. Subsequently, the supersaturated silicon diffuses from the liquid–vapour interface and precipitate at the solid–liquid interface resulting in the growth of silicon. The growth proceeds perpendicular to the solid–liquid interface. The rough liquid surface is composed of ledge, ledge-kink or kink sites. Every site over the entire surface can trap the impinging growth species. The catalyst not only acts as a sink for the growth species in the vapour phase, but also aids in the deposition. Consequently, the growth rate of the nanowires by the VLS method is very high in presence of the catalyst. The solubility depends on the surface energy and the radius of curvature of the surface which is given by the Kelvin equation [Eq. (4.60)]. For the growth of nanowires, if facets are developed during the growth, the rates of lateral and longitudinal growths are determined by the growth of the individual facets. The nanowires have very small radius (i.e. large curvature). Therefore, for the growth of uniform nanowires and to prevent any significant growth on the side surface, the supersaturation should be kept low. If a high supersaturation is maintained, other facets would grow and secondary nucleation may occur on the growth surface, which can terminate the epitaxial growth. Using in situ microscopy in the germanium-gold system, Kodambaka et al. (2007) have shown that nanowire growth can occur below the eutectic temperature with either liquid or solid catalysts at the same temperature. They found that the state of catalyst depended on the growth pressure and the thermal history. It has been suggested that these phenomena may be due to the kinetic enrichment of the eutectic alloy composition. As mentioned earlier, chemical dopants can be incorporated during the growth of the nanowires, and it is possible to ensure that the nanowire is n-doped (having extra conduction electrons) or pdoped (with some electrons removed to leave positively-charged holes). Because a nanowire is elongated and easily polarised electrically, it is attracted towards a high electric field with which it lines up. Therefore, when a voltage is applied between two electrodes, a nearby nanowire suspended

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in liquid is drawn-in to bridge the gap between them. In this way, ordered rows of parallel singlenanowire bridges can be created. Duan et al. (2001) have created a junction by placing a p-doped and an n-doped nanowire across each other. Figure 11.10 shows how a light-emitting diode can be created by this technique (Cobden, 2001).

Figure 11.10

11.5.2

(a) A nanowire is produced from two reagents with the help of a metal catalyst, (b) the nanowire is assembled between two metal electrodes using an electric field gradient, and (c) a p-doped and an n-doped nanowire are crossed to form a nanoscale light-emitting diode. When a current is passed between them, electrons and holes are injected across the junction and recombine to emit light (Cobden, 2001) (reproduced by permission from Macmillan Publishers, © 2001).

Nanotubes

Nanotubes made of materials such as polymers, ceramic materials and carbon have been synthesised. Polymer nanotubes can be made by the electrochemical deposition method within the pores of a nanoporous membrane. The deposition and solidification of polymers inside the template pores starts at the surface and proceeds inwardly (Martin, 1994). The electrostatic attraction between the oppositely charged growing polymer and the sites along the pore walls of the membrane may be responsible for the inward growth. The diffusion of monomer molecules through the pores can become a limiting step and the monomer molecules within the pores can be depleted quickly causing the deposition of polymer inside the pores to stop. Silica nanotubes can be formed from an alumina membrane using cycles of liquid-phase surface reactions (Kovtyukhova et al., 2003). First, the membrane is dipped in SiCl4 which reacts with the surface hydroxyl groups in the membrane pores. The excess reactant is washed out with carbon tetrachloride, and then the membrane is reacted with water. After washing out the water, the steps are repeated. This leads to a build-up of a SiO2 shell on the membrane with the elimination of HCl. The nanotubes formed by this process can be released into solution by dissolution of the alumina membrane in concentrated aqueous base. One of the greatest impacts on nanoscience and nanotechnology has been made by the singleand multiwalled carbon nanotubes (which are abbreviated as SWNT and MWNT respectively). The reason for this is that the carbon nanotubes possess remarkable mechanical, electrical and thermal properties that equal, or even surpass, those of other benchmark materials such as steel, copper, and diamond respectively. Applications on the nanometre and micrometre scale, such as SWNT-based transistors and chemical sensors are progressing rapidly. There are excellent books on the synthesis and physical properties of carbon nanotubes (Saito et al., 1998; Harris, 1999; Tanaka et al., 1999).

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The multiwalled carbon nanotubes consist of several nested coaxial single-walled tubules. Typical outer and inner diameters of multiwalled carbon nanotube are 2–20 nm and 1–3 nm respectively. The typical length is 1–100 mm. The intertubular distance is 0.34 nm. The synthesis of C60 and other fullerenes stimulated intense interest in the tubular variant of the fullerenes. Iijima (1991) prepared multiwalled nanotubes as shown in Figure 11.11. They were grown on the negative end of the carbon electrode used in the d.c. arc-discharge evaporation of carbon in an argon-filled vessel at 13.3 kPa pressure. The apparatus was similar to that used by Krätschmer et al. (1990) for large-scale production of C60. The tubes grew on certain regions of the electrode. These tubes were made of a single curved sheet of graphite connected at its edge. The electrical energy and the energy associated with the dangling bonds in a graphite sheet sealed the rolled-up sheets in the form of nanotubes. Electron microscopy revealed that each needle comprised coaxial tubes of graphitic sheets, ranging in number from 2 up to about 50. On each tube, the carbon-atom hexagons were arranged in a helical fashion about the needle axis. The helical pitch varied from needle-toneedle and from tube-to-tube within a single needle. The diameter of the tubes varied between 4 nm and 30 nm, and their length was up to 1 mm.

Figure 11.11

Electron micrographs of nanotubes of graphitic carbon. The parallel dark lines correspond to the (002) lattice images of graphite. A cross-section of each tubule is illustrated: (a) tube consisting of five graphitic sheets, diameter 6.7 nm, (b) two-sheet tube, diameter 5.5 nm, and (c) seven-sheet tube, diameter 6.5 nm, which has the smallest hollow diameter of 2.2 nm (Iijima, 1991) (reproduced by permission from Macmillan Publishers Ltd., © 1991).

The single-walled nanotubes were synthesised in 1993 by Iijima and Ichihashi, and Bethune et al. Iijima and Ichihashi used two vertical electrodes in the carbon-arc chamber. The anode was a graphitic carbon rod and the cathode was a carbon rod. The latter held a small piece of iron. The chamber was filled with a mixture of methane and argon at low pressure. The carbon discharge arc was generated by

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running a d.c. current of 200 A at 20 V between the electrodes. The iron melted and formed a droplet. The iron vapour cooled and condensed into small particles of iron carbide on the electrode. Iron acted as catalyst in the vapour phase in the formation of the single-walled nanotubes. The nanotubes were found in the soot-like deposits. The diameter of the tubes varied between 0.7 nm and 1.6 nm. Bethune et al. (1993) synthesised single-walled carbon nanotubes by covaporising carbon and cobalt in an arc generator. The tubes had about 1.2 nm diameter. The tubes formed a web-like deposit woven through the fullerene-containing soot, giving it a rubbery texture. Thess et al. (1996) presented an efficient route for the synthesis of arrays of single-walled nanotubes. The process involved condensation of a laser-vaporised carbon–nickel–cobalt mixture at 1473 K. The yield was more than 70%. X-ray diffraction and electron microscopy showed that these single-walled nanotubes were almost uniform in diameter and they self-organised into ‘ropes’ which consisted of 100 to 500 nanotubes in a two-dimensional triangular lattice. The ropes were metallic with a single-rope resistivity of less than 10–6 W m at 300 K. The high-pressure-carbon-monoxide (HiPCO) method developed by Nikolaev et al. (1999) can produce large quantities of carbon nanotubes. The catalyst for the growth of single-walled nanotube forms in situ by thermal decomposition of iron pentacarbonyl in a heated flow of carbon monoxide at pressures in the range of 101.325 kPa to 1013.25 kPa, and temperatures between 1073 K and 1473 K. The yield of SWNT and the diameter of the nanotubes produced by this process can vary over a wide range, which is determined by the condition and geometry of the flow-cell. Nikolaev et al. (1999) produced SWNTs of 0.7 nm diameter. The products of thermal decomposition of Fe(CO)5 reacted to produce iron clusters in the gas phase. These clusters acted as nuclei upon which the SWNTs nucleated and grew. The solid carbon was formed through disproportionation of CO (known as the Boudouard reaction): CO + CO ® C(s) + CO2 (11.22) This reaction occurred catalytically on the surface of the iron particles. The iron particles promoted the formation of the tube's characteristic graphitic carbon lattice. The rate at which the reactant gases were heated determined the amount and quality of the SWNTs produced. The temperature and pressure had important effects on the yield of the nanotubes. TEM image of the single-walled nanotube produced by this method is shown in Figure 11.12. Ericson et al. (2004) produced well-aligned macroscopic fibres of single-walled carbon nanotubes from the concentrated dispersions of SWNTs in 102% sulphuric acid employing a wetspinning technique. Because of the high-temperature stability of the SWNTs, melt-spinning is not an option and wet-spinning is the only viable approach. The main challenge to the production of neat SWNT fibres is dispersing the SWNTs at high enough concentrations suitable for efficient alignment and effective coagulation. However, due to their chemical inertness and strong van der Waals attractions, SWNTs aggregate into ropes with limited solubility in aqueous, organic or acidic media. If a surfactant is used to disperse the SWNTs, there are complications of removing the surfactant from the fibre during coagulation or after processing. In sulphuric acid of concentration greater than 100%, SWNTs form charge-transfer complexes of individual positively charged nanotubes surrounded by a finite number of sulphuric acid anions. At very low concentrations, such charged tube-anion complexes behave as Brownian rods. At concentrations greater than 0.03 weight %, a small amount of dissolved individual tubes coexists with a SWNT spaghetti-phase consisting of seemingly endless swollen ropes of well-aligned positively charged SWNTs intercalated by sulphuric acid anions. The SWNTs in the spaghetti are mobile and at a high enough concentration (> 4 weight %), they coagulate and form ordered domains behaving similarly to nematic liquid crystalline rod-like polymers. The SWNT-acid system is very sensitive to water and if a very small amount of moisture enters the

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Figure 11.12

High-magnification TEM image of single-walled nanotube produced by the HiPCO process (Nikolaev et al., 1999) (reproduced by permission from Elsevier Ltd., © 1999).

system, phase separation occurs and discrete needle-like crystal-solvates precipitate. This ordered SWNT dispersion can be extruded and coagulated in a controlled fashion using conventional fibrespinning techniques to produce continuous lengths of macroscopic neat SWNT fibres. The apparatus used for the spinning process is shown in Figure 11.13. The scanning electron micrographs depicting the aligned macroscopic fibres consisting solely of the single-walled nanotubes are shown in Figure 11.14. The morphology of the fibres was strongly dependent on the coagulation conditions.

Figure 11.13

The spinning process of single-walled carbon nanotubes (SWNTs) in 102% sulphuric acid: (A) the apparatus for mixing and extruding neat fibres, (B) a jet of SWNT dispersion being extruded from a capillary tube, and (C) a 30 m spool of water-coagulated fibre (Ericson et al., 2004) (reproduced by permission from The American Association for the Advancement of Science, © 2004).

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Figure 11.14

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SEM images showing the evolution of purified SWNTs into continuous fibre: (A) SWNTs after the purification process, (B) view inside the purified SWNTs showing the tangled mass of SWNT ropes that are 20–30 nm in diameter, (C) an annealed neat SWNT fibre spun from 8 wt% dispersion in 102% sulphuric acid and coagulated in water, and (D) higher magnification of the neat fibre surface showing that all the ropes have merged into aligned super-ropes that are 200 nm or larger in diameter (Ericson et al., 2004) (reproduced by permission from The American Association for the Advancement of Science, © 2004).

In the theoretical analysis of carbon nanotubes, the major focus has been on single-walled tubes, which are cylindrical in shape with caps at each end, such that the two caps can be joined together to form a fullerene. The cylindrical portions of the tubes consist of a single graphene sheet (a single layer of carbon atoms from a three-dimensional graphite crystal) that is shaped to form the cylinder. It is convenient to describe a carbon nanotube in terms of the chiral vector C h and the chiral angle q as shown in Figure 11.15. The points O and A are crystallographically equivalent on the graphene sheet. These points are connected by the chiral vector C h , which is defined in terms of the basis vectors a1 and a2 of the honeycomb lattice. C h

na1  ma2

(11.23)

where n and m are integers. The chiral angle q is shown in Figure 11.15. OB represents the normal to C h at point O. Depending on the value of the chiral angle, a single-walled carbon nanotube can have three basic geometries: armchair, zigzag and chiral, as illustrated in Figure 11.15. The diameter of a carbon nanotube is given by

dt

C h S

(11.24)

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Figure 11.15 Schematic model for single-walled carbon nanotubes with the tube-axis OB normal to: (a) q = p /6 rad direction (an ‘armchair’ nanotube), (b) q = 0 direction (a ‘zigzag’ nanotube), and (c) 0 < q < p /6 rad (a ‘chiral’ nanotube) (Dresselhaus et al., 1995) (reproduced by permission from Elsevier Ltd., © 1995).

The integers n and m uniquely determine dt and q. The circumference of the nanotube is given by (Dresselhaus et al., 1995) L

C h

a( n2  m 2  nm )1/ 2 , 0 … m … n

(11.25)

where a is the length of unit vector. The carbon nanotubes can either have metallic conductivity or can be semiconducting depending on the tube diameter and the chiral angle. Metallic conduction in carbon nanotubes can be achieved without the introduction of doping or defects. Ballistic electron transport and electrical conductivity without phonon and surface scattering have been observed in metallic carbon nanotubes (Bockrath et al., 1997; Bachtold et al., 2000). Semiconducting nanotubes have a band gap between their conduction and valence bands, which is proportional to 1/dt. It is possible to fabricate a transistor based on a single helical carbon nanotube, which is a step towards molecular electronics (Tans et al., 1998). An AFM image of a single nanotube contacting three platinum electrodes is shown in Figure 11.16. The semiconductor Si-substrate covered with a 300 nm layer of thermally grown SiO2 was used as a back-gate. The metallic variety of tubes had linear current (I) versus voltage (Vbias) curves, and showed no dependence on the gate voltage (Vgate). For the sample shown in Figure 11.16(a), the I versus Vbias curve was slightly nonlinear for Vgate = 0. When Vgate was increased to positive values, a pronounced gap-like nonlinearity developed around Vbias = 0. The curves exhibited a power-law behaviour: I µ (Vbias)a with a lying between 1 and 12. Upon application of a negative Vgate, the I versus Vbias curve became linear with a resistance that saturated around 1 MW. This resistance is mainly due to the contact resistance between the tube and the electrodes. Therefore, the device showed controllable semiconductor-to-metal transition in a onedimensional system. The nonlinearity at room temperature and the asymmetric dependence of the conductance on the gate-voltage polarity indicate that the nanotube is semiconducting. The conductance could be modulated by about six orders of magnitude if Vgate was changed by 10 V.

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Figure 11.16

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(a) Tapping-mode AFM image of an individual carbon nanotube on top of three platinum electrodes, and (b) schematic side-view of the TUBEFET (single carbon nanotube field-effect transistor) device. A single semiconducting nanotube is contacted by two electrodes. The Si substrate, which is covered by a layer of SiO2 acts as a back-gate (Tans et al., 1998) (reproduced by permission from Macmillan Publishers Ltd., © 1998).

Both nanowires and single-walled nanotubes can be used to make molecular field-effect transistors. Advanced electronic devices based on carbon nanotubes and various types of nanowires can have a very important role in next-generation semiconductor architectures. However, at present there is lack of a general fabrication method which has held back the development of these devices for practical applications. A few assembling strategies have been suggested for devices based on these nanomaterials. Lee et al. (2006) used inert surface molecular patterns to direct the adsorption and alignment of nanotubes and nanowires on bare surfaces to form device structures without the use of linker molecules. They have used this method to demonstrate large-scale assembly of nanotube and nanowire-based integrated devices and their applications.

11.6

DEPOSITION OF THIN FILMS

Deposition of thin films has been studied for almost a century. Some of the techniques developed during the past five decades are widely used in industries. The methods for depositing a film can be divided into two categories: (i) vapour-phase deposition (e.g. chemical vapour deposition, evaporation, molecular beam epitaxy, sputtering and atomic layer deposition) and (ii) liquid-based deposition (e.g. electrochemical deposition, chemical solution deposition, Langmuir–Blodgett films and self-assembled monolayers). Development of films of nanoscale thickness involves nucleation and growth on the surface of the substrate. The nucleation step is very important because it governs the crystallinity and microstructure of the film. There are three basic modes of nucleation: (i) island or Volmer–Weber, (ii) layer or Frank–van der Merwe, and (iii) island–layer or Stranski–Krastonov nucleation. These three modes of nucleation are illustrated in Figure 11.17. Island nucleation occurs when the species are bound to each other more strongly than to the substrate. The islands subsequently merge and form a continuous film. Metals on insulators display this type of mechanism. The layer-nucleation is opposite to the island nucleation. The species are bound more strongly to the substrate than to each other. First, a complete monolayer is formed and then the deposition of the second layer begins. The epitaxial growth of single-crystal films is an important example of layer growth. The island–layer nucleation mechanism is a combination of both island and layer nucleations. Evaporation is the simplest method for deposition of thin films. The evaporation system consists of an evaporation source that vaporises the desired material. Both the source and the substrate are placed in the vacuum chamber. The substrate faces the evaporation source. The substrate can be placed and heated as desired. The desired vapour pressure of the source material can be generated by simply heating the source as per requirements of the concentration of the growth species in the

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Figure 11.17

Schematic diagram of Volmer–Weber, Frank–van der Merwe and Stranski–Krastonov mechanisms of film nucleation.

gas phase. Precautions need to be taken for pyrolysis, decomposition and dissociation of the compound being heated. Deposition of thin films by evaporation is carried out at very low pressures (1 × 10–8 – 0.1 Pa). The molecules in the vapour phase do not collide with each other prior to the arrival at the growth surface because the mean free path is very large as compared to the distance between the source and the substrate. The transport of molecules from the source to the growth substrate is straight forward along the line of sight. Therefore, the conformal coverage is relatively poor and it is difficult to obtain a uniform film over a large area. To overcome this difficulty, multiple sources are used and the substrate is rotated. Laser beams have been used to evaporate the material. Pulsed laser beams are used where high power density is required. This process is known as laser ablation. The composition of the vapour can be controlled precisely by using this technique. Laser ablation has been used for depositing metal oxides in superconductor films. Molecular beam epitaxy (MBE) can be considered as a more sophisticated version of the evaporation technique. In MBE, the vacuum is very high such that the pressure inside the reactor is of the order of 1 × 10–8 Pa. At this pressure, the mean free path of the gas molecules far exceeds the distance between the source and the target. Most of the molecular beams are generated by heating the solid source material in effusion cells (known as Knudsen cells). The material is raised to the desired temperature by resistive heating. The molecules strike on the single-crystal substrate resulting in the formation of the desired epitaxial film. The extremely clean environment, slow growth rate (~3 × 10–10 m/s) and independent control of the evaporation of the source material ensures precise formation of the film. Most of the MBE reactors have accessories for real-time structural and chemical characterisation capabilities such as X-ray photoelectric spectroscopy and Auger electron spectroscopy. Sputtering involves use of energetic ions to knock molecules out from a target which acts as one electrode and subsequently deposit them on the substrate that acts as the second electrode. In a typical sputtering chamber, the source and substrate electrodes face each other. An inert gas such as argon at low pressure (~15 Pa) is used as the medium. When a high electric field (~10 kV/cm) is applied to initiate the glow-discharge between the electrodes, free electrons are accelerated by the electric field and ionise the argon atoms. The gas pressure should not be too low, otherwise the electrons will simply strike the anode without having gas-phase collision with the argon atoms. The Ar+ ions, generated in this way, strike the source electrode resulting in the ejection of neutral target atoms. These atoms pass through the discharge and deposit on the substrate electrode. Chemical vapour deposition (CVD) involves reaction of a volatile compound with other gases to produce a nonvolatile solid that deposits on a suitably placed substrate. The CVD process is widely

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used in the manufacture of solid-state microelectronic devices. Both gas-phase and surface chemical reactions are involved in CVD. The major types of chemical reaction are pyrolysis, oxidation, reduction and disproportionation. The CVD process is versatile because for depositing a given film, many different reactants or precursors may be used. From the same precursors and reactants, different films can be obtained by varying the ratio of reactants and deposition conditions. Various types of CVD methods and reactors are used depending on the reactants, reaction conditions and the forms of energy used to activate the reactions. For example, when an organometallic compound is used as the precursor, the process is called metalorganic CVD. When plasma is used to promote the reaction, the process is known as plasma-enhanced CVD. There are modified CVD processes such as low-pressure CVD, laser-enhanced CVD and aerosol-assisted CVD. The reactors used in the CVD processes can be classified into hot-wall and cold-wall types. In hot-wall CVD reactors, heating is accomplished by surrounding the reactor with resistance elements. In cold-wall reactors, the substrates are directly heated inductively by graphite susceptors and the chamber walls are cooled by air or water. The details of CVD reactors have been presented by Sze (1985). Atomic layer deposition (ALD) is a unique method for depositing thin films. It is also known as atomic layer epitaxy, atomic layer growth or atomic layer CVD. Its most distinctive feature is that it has a self-limiting growth. Each time only one molecular layer can grow. Therefore, ALD offers a very good method for depositing films having thickness in the nanometre range. In a typical ALD process, the surface is first activated by chemical reaction, e.g. to deposit a titania film by ALD, the substrate is hydroxylated. When the precursor is introduced in the deposition chamber, the molecules of the precursor react with the surface species and form bonds with the substrate. Since the precursor molecules do not react with each other, a film of only a single molecular thickness can be deposited. In the next step, the monolayer is activated again by surface reaction and a layer of the same or a different precursor is deposited on top of the layer of the previous precursor molecules. A few layers of the same or different precursors can be deposited by this procedure. The choice of appropriate precursors is a very important aspect of the ALD process (Ritala and Leskelä, 1999). A major disadvantage of the ALD method is the slow deposition rate. The typical deposition rate is 0.2 nm per cycle (i.e. less than half a monolayer per cycle) with the theoretical maximum of one monolayer per cycle. Hausmann et al. (2002) have deposited layers of amorphous silicon dioxide and aluminium oxide nanolaminates at rates of 12 nanometres per cycle, which is equivalent to more than 32 monolayers per cycle. Vapours of trimethylaluminium (Me3Al) and tris(t-butoxy)silanol [(ButO3SiOH] were supplied in alternating pulses to the heated surface on which transparent, smooth films of alumina-doped silica grew. To test whether the ALD reactions saturate by a self-limiting mechanism, they deposited films on a silicon wafer in which deep, narrow holes had previously been itched. After depositing four ALD cycles, the wafer was cleaved and cross-sectional scanning electron micrographs were recorded. The micrographs of the cross-sections of the top, middle and bottom parts of a hole are shown in Figure 11.18. The images show uniform, conformal coating indicative of an ideal self-limiting ALD reaction. Electrochemical deposition is a well-established method for thin film deposition. It is a special kind of electrolysis resulting in the deposition of solid material on an electrode. The electrochemical deposition process involves oriented deposition of charged growth species (e.g. cations) through a solution when an external electric field is applied, and reduction of the charged growth species at the deposition surface which also serves as an electrode. Generally, this method is applicable to the electrically-conductive materials such as metals, alloys, semiconductors and conductive polymers. The electrochemical deposition method is widely used for making metallic coatings, which is known as electroplating. A somewhat similar method is electrophoretic deposition. It has been explored for

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Figure 11.18

Cross-sections of holes 7 mm deep and 0.1–0.2 mm in diameter. Magnified images of the (a) top, (b) middle, and (c) bottom parts of a hole coated conformally with a uniform silica film 46 nm thick made by four ALD cycles (Hausmann et al., 2002) (reproduced by permission from The American Association for the Advancement of Science, © 2002).

the deposition of ceramic and organoceramic materials from colloidal dispersions. The material in this case need not be electrically conductive. The colloid particles are stabilised by electrostatic double layer or steric forces.

11.7

SYNTHESIS OF MICROPOROUS AND MESOPOROUS MATERIALS

The natural tendency of solid materials is to have close-packing by which the energy can be minimised. Atoms, ions, molecules, clusters, colloids and polymers pack in a solid body as closely as possible to minimise the energy. However, if opportunity is available, kinetic factors lead to the formation of metastable structures with voids. Although thermodynamically unstable to collapse, these open-framework materials can survive indefinitely under suitable conditions. There are plenty of examples in biology (such as siliceous radiolarian and diatom filigree micro-skeletons) and geology (e.g. naturally occurring zeolites riddled with regular arrays of pores). Porous solids are important scientifically and technologically because they can interact with atoms, ions and molecules not only at their surfaces but throughout the bulk of the material as well. The traditional applications of porous materials are in ion exchange, adsorption and catalysis. The pores of solids are classified according to their size. As per the IUPAC classification of the porous solids, pores having diameter in the range of 2 nm and below are called micropores, those in the range of 2 nm to 50 nm are denoted mesopores, and those above 50 nm are macropores. The distribution of size, shape and volume of the void spaces in porous materials govern their ability to perform the desired function in a particular application. The science and engineering of porous materials deal with the methodology to create uniformity in pore size, shape and volume. To illustrate

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the need for uniformity, let us consider the case of zeolites which separate molecules based on their size. A broad distribution of pore-size would limit the ability of a zeolite to separate molecules of different size. Molecular sieves of pure silica are hydrophobic and can adsorb organic components from water, whereas molecular sieves of aluminosilicate are hydrophilic and can adsorb water from organic solvents. Therefore, control over the uniformity in pore-size and the composition of the material are the two very important factors in microporous and mesoporous materials. The first synthesis of a crystalline microporous material with uniform pores larger than 1 nm was reported by Davis et al. (1988). After four years, synthesis of well-ordered mesoporous materials was reported (Kresge et al., 1992). During this period, interesting reports on inclusion of electro- and photo-active guest molecules in porous host materials appeared in the literature (see Stucky and MacDougall, 1990). The synthesis of the aluminophosphate VPI-5 having pore size 1.2 nm in 1988 commenced the era of extra-large pore crystalline materials. An extra-large pore can be obtained if more than 12 oxygen atoms span the circumference of the pore. In VPI-5, the pores are uniformdiameter channels having circular cross-section. The material has 30% void fraction. Several other similar porous materials have been synthesised most of which are phosphate based (see Figure 11.19). The properties of these materials have been presented by Davis (2002). The phosphate-based porous materials have poor thermal and hydrothermal stability as compared to the silica-based molecular sieves. The instability of these materials is not due to the extra-large rings in their structures, but due to the properties of the structural units such as mixed metal-ion coordination, terminal –OH groups and presence of non-tetrahedral framework species such as OH, H2O or F. Crystalline silicas containing 14-membered rings have been synthesised which are as stable as zeolites containing small rings.

Figure 11.19

Pore characteristics in the aluminophosphates AlPO4-11, AlPO4-5 and VPI-5. The line segments represent oxygen atoms that bridge between two tetrahedral atoms (intersection points) that are in this case either Al+3 or P+5 with alternation to give a composition of AlPO4. Rhomboids indicate the unit cell (Davis, 2002) (reproduced by permission from Macmillan Publishers Ltd., © 2002).

A different approach for the synthesis of microporous solids involves the coordination of metal ions into organic linker moieties resulting in the formation of open framework structures. Several metal–organic frameworks (MOFs) have been synthesised which exhibit stable pores. These materials are unlikely to compete with zeolite and other oxide-based porous materials in hightemperature applications owing to their high cost and lack of stability at high temperatures. However, the MOFs can have application in hydrogen or methane storage (Eddaoudi et al., 2002; Rosi et al., 2003). For example, MOF-5 having composition Zn4O(1,4-benzenedicarboxylate)3 with a cubic three-dimensional extended porous structure adsorbs hydrogen up to 4.5 weight % (17.2 H2 molecules per formula unit) at 78 K and 2 MPa pressure. It can adsorb 1 weight % at room

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temperature and at the same pressure. Topologically similar isoreticular frameworks IRMOF-6 and IRMOF-8 (which have cyclobutylbenzene and naphthalene linkers respectively) have been reported to give nearly double and quadruple the uptake of MOF-5 at room temperature and 1 MPa pressure. The MOF approach may also be suitable for constructing a wide range of chiral porous materials for use in enantioselective adsorption and catalysis (Seo et al., 2000). The single-crystal X-ray structures of MOF-5, IRMOF-6 and IRMOF-8 are presented in Figure 11.20.

Figure 11.20

Single-crystal X-ray structures of MOF-5 (A), IRMOF-6 (B), and IRMOF-8 (C) illustrated for a single cube fragment of their respective cubic three-dimensional extended structure. A cluster of an oxygen-centred Zn4 tetrahedron that is bridged by six carboxylates of an organic linker is located on each of the corners. The large sphere represents the largest sphere that would fit in the cavity without touching the van der Waals atoms of the framework (Rosi et al., 2003) (reproduced by permission from The American Association for the Advancement of Science, © 2003).

Intense research activities ensued after the scientists of Mobil Research Corporation (Kresge et al., 1992) reported synthesis of ordered mesoporous silicas having hexagonal and cubic symmetry with pore size between 2 nm and 10 nm. They used surfactants which acted by the liquid-crystal-templating mechanism depicted in Figure 11.1. Surfactants have been shown to organise silica into a variety of mesoporous forms through the influence of electrostatic, hydrogen-bonding, covalent and van der Waals interactions. The silica-surfactant assembly occurs with condensation of the inorganic species which yields mesoscopically-ordered composites. Generally the mesoporous materials are not crystalline with very few exceptions (Inagaki et al., 2002). Non-silica oxides have been used to synthesise large-pore semicrystalline mesoporous materials (Yang et al., 1998). Since most of the ordered mesoporous materials are not crystalline, they can be synthesised by a variety of methods using element combinations found in amorphous materials, e.g. aerogels, xerogels, organic-inorganic hybrid materials, metals and mixed-metal alloys. Due to such varied methods of synthesis, the degree of order within the material shows tremendous variability. Some possible reasons behind the lack of crystallinity in the mesoporous materials have been discussed by Davis (2002). Surfactant micelles, lyotropic liquid crystals and block copolymers have been used to synthesise mesoporous materials. The block copolymers co-assemble with a wide variety of inorganic precursors to form mesoporous inorganic materials having pores in the range of 5 nm to 35 nm. The material contains the block copolymer template, which can be removed yet the mesoporosity retained by various techniques such as surfactant exchange, thermal treatment in air, plasma etching, ozone treatment and vacuum UV or microwave treatment. Zhao et al. (1998) synthesised ordered hexagonal mesoporous structures of SBA-15 having pores in the above range using poly(alkene oxide) triblock copolymer templates. The copolymer was removed by calcination. TEM images of the calcined mesoporous material are shown in Figure 11.21.

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Figure 11.21

441

Block-copolymer-templated mesoporous silica SBA-15. The average pore diameters are, (A) 6 nm, (B) 8.9 nm, (c) 20 nm, and (D) 26 nm. The thicknesses of the silica walls are: (A) 5.3 nm, (B) 3.1 nm, (C) and (D) 4 nm (Zhao et al., 1998) (reproduced by permission from The Americal Association for the Advancement of Science, © 1998).

Hierarchical microporous and mesoporous structures have also been developed in the past decade. Zeolite particles have been used as the building blocks to construct hierarchical porous structures. Pure crystalline molecular sieves have been produced as free-standing films. Films, fibres and spheres possessing fairly uniform mesoporosity have been fabricated using surfactants as structure-directing agents. The procedure is similar to that employed for the synthesis of ordered mesoporous materials. The first zeolite-based supported and free-standing films were reported in the 1990s. Zeolite thin films are of interest for use as membranes in separation equipment and membrane reactors. They can also act as chemical sensors and hosts for various molecules. Apart from the traditional use of porous materials in catalysis, separation and ion exchange, the discovery of mesoporous materials has opened up many promising areas in which they can be used. For example, microchip devices require insulators with low dielectric constant. The films of ordered mesoporous materials with dielectric constant less than 2.2 can be used as low dielectric constant materials in place of silicon dioxide which has dielectric constant close to 4. Another promising application is magnetic resonance imaging (MRI) for physical diagnosis. This technique relies on administering contrast agents to patients to improve the quality of imaging. The contrast agents contain high-spin metals that bind water molecules and thereby yield proton spin relaxation times that are orders of magnitude faster than those with free water. Gadolinium ions (Gd+3) perform very well as contrast agents. However, they cannot be administered directly owing to their inherent toxicity. They have been contained in zeolites and used in the gastrointestinal tract (Balkus et al., 1992). The zeolites are not toxic when introduced in the gastrointestinal tract. They mitigate the toxicity of the Gd+3 ions by immobilising them but still allow hydration so that they can exhibit the relaxation effect mentioned above. Several other uses of micro and mesoporous materials have been reported in literature. An interesting example is zeolite-dye microlaser. Organic dye molecules acting as guest molecules are incorporated in microporous and mesoporous materials which act as hosts. The pores of crystalline sieves have been used as a confining reaction environment to produce single-walled carbon nanotubes. SWNTs of very small diameter (~0.4 nm) have been prepared using the pores of a crystalline molecular sieve as microreactors in which occluded tripropylamine was carbonised (Wang et al., 2000).

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11.8 LITHOGRAPHIC TECHNIQUES Historically, lithography refers to a method for printing using a plate (or stone) with a smooth surface. Lithography uses oil or fat and gum arabic to divide the smooth surface into hydrophobic regions which accept the ink, and hydrophilic regions which reject it and thus become the background. It was invented by the Austrian actor and playright Alois Senefelder in 1796. A modern version of lithography is offset lithography. In nanotechnology, lithography refers to a class of patterning methods capable of structuring materials on a fine scale. Typically, features smaller than 100 nm are considered nanolithographic. Photolithography is a widely-used technique for mass production of integrated circuits. The process consists of producing a mask carrying the requisite pattern information and subsequently transferring that pattern into a photoactive polymer (known as resist) employing an optical technique. The basic steps of photolithography are shown in Figure 11.22. In the first step, designated as ‘coat’, the resist material is applied as a thin coating over the substrate. It is subsequently exposed to light through a photomask so that only the selected parts are exposed to the light. The ‘development’ step involves creation of a three-dimensional relief image in the resist material which is a replication of the opaque and transparent areas of the mask. Depending on the chemical nature of the resist material, the exposed area may be rendered more soluble in a developing solvent than the unexposed area, or the converse may be true. In the former case, a positive tone of the mask is created, whereas in the latter case, a negative tone is created. In the etching step, the main role of the resist is to cover the underlying substrate and prevent the etchant from attacking it. After etching, the resist is removed (i.e. the ‘strip’ step shown in Figure 11.22). The limit of maximum resolution achievable or the minimum size of the elements is set by the diffraction of light. Although the photolithographic technology is the most commercially-advanced form of nanolithography, other techniques are also used. For example, electron-beam lithography is capable of much higher patterning resolution (sometimes as small as a few nanometres). A beam of electrons is scanned in a patterned fashion across a surface covered with the resist. The development part involves selective removal of either the exposed or non-exposed parts of the resist. Electron-beam lithography is also commercially important, primarily for its use in the manufacture of photomasks. Electron-beam lithography, as it is usually practised, is a form of maskless lithography because no mask is required to generate the final pattern. Instead, the final pattern is created directly from a digital representation on a computer by controlling an electron beam as it scans across a resist-coated substrate. An advantage of electron-beam lithography is that its resolution is not limited by the diffraction factors mentioned before. Electron-beam lithography has the disadvantage of being much slower than photolithography. The soft-lithographic techniques are based on the concept of self-assembly [see Xia and Whitesides (1998)]. The inspiration came from the work of Nuzzo et al. (1987) in which they reported the self-assembly of alkanethiols on gold to form well-ordered monolayers. It is a low-cost microfabrication method which has been proposed to complement photolithography. In those ultrasmall-scale fabrication requirements where radiation of short wavelength is necessary, the photolithographic technique requires very expensive facilities and, therefore, the soft-lithographic technique may provide a viable alternative. This technology is easily accessible to the chemists who are interested in working with reliable, convenient and inexpensive methods for patterning planar or curved surfaces with organic, inorganic, polymer, liquid crystal, ceramic or biological structures. The soft-lithographic techniques have shown good potentials in optics, sensors, microelectromechanical systems and cell biology.

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Figure 11.22 Schematic diagram of the steps involved in the photolithographic process.

An interesting application of soft-lithography is in nanoscale printing. The feature-size of softlithographically printed patterns of self-assembled monolayers can be reduced to sub-100 nm levels by directly writing alkanethiols on gold using the fine tip of an atomic force microscope. This process is known as dip-pen lithography (Piner et al., 1999). It operates by delivering collections of alkanethiol molecules from the tip via a water meniscus formed at the tip–substrate interface. It can be adapted to a wide variety of surface chemistry. Patterns as fine as 10 nm can be written on substrates such as metals, semiconductors and dielectrics. The dip-pen lithographic method has been used to generate covalently-anchored nanoscale patterns of oligonucleotides on metallic as well as insulating surfaces (Demers et al., 2002). Features sizing from several micrometres to less than 100 nm were synthesised. The patterns exhibited the sequence-specific binding properties of DNA. Lee et al. (2002) made nanoarrays of proteins with 100–350 nm features. The protein array was fabricated by initially patterning 16-mercaptohexadecanoic acid on a gold substrate in the form of dots or grids. The areas surrounding these features were passivated by a surfactant. The protein was adsorbed on the patterns by immersing the substrate in a solution containing the protein. Lysozyme (an ellipsoidal protein) assembled on the nanopatterns in a clean manner. The tapping mode AFM image is shown in Figure 11.23. Protein molecules did not bind to the passivated portions at all. A variety of surface-

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mediated biological recognition processes can be studied using these arrays. Thus, such arrays can be very important for proteonics and cell research.

Figure 11.23

11.9

(A) Tapping mode AFM images and height profile of a hexagonal lysozyme nanoarray, and (B) 3D topographic image (contact mode) of a lysozyme nanoarray consisting of a line grid and dots (Lee et al., 2002) (reproduced by permission from The American Association for the Advancement of Science, © 2002).

TOXIC EFFECTS OF NANOMATERIALS

In recent years, concerns have been raised about the toxicity of some of the nanomaterials, especially the nanoparticles. The main safety issues surrounding the use of nanoparticles are regarding their biocompatibility with the cells. A material which is quite harmless at the micrometre-scale may turn out to be harmful when its size is reduced to the nanometre scale. This is mainly due to the large surface/volume ratio of the nanoparticles which leads to large number of surface sites where harmful reactions can occur. The possible undesirable effects of the nanoparticles on cell are: (i) corrosion within the cell membrane or in the proximity of the cell which may release toxic chemicals causing death of the cell, and (ii) ingestion of the nanoparticle through cell membranes or adhesion to the surface of the cell, which may impair the function of the cell. Derfus et al. (2004) studied the effect of CdSe nanoparticles on liver cells. Their results have shown that the CdSe nanoparticles in water release Cd+2 ions and the concentration of the cadmium ions correlate with the toxicity. The exposure to air and UV enhance the deterioration of the CdSe lattice which causes the liberation of free Cd2+ ions. The harmful effect can be significantly reduced by capping with a ligand such as mercaptopropionic acid, wrapping with a bio-macromolecule like bovine serum albumin, or coating with a sheath of semiconductor such as ZnS. These observations provide some guidelines for reduction of release of toxic materials from nanoparticles and their harmless application in the life sciences and medicine.

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SUMMARY This chapter presents an overview of the synthesis, analysis, structure and use of various nanomaterials. The chapter begins with an introduction on nanotechnology with a brief history of its development. The bottom-up and top-down methods of synthesis of nanomaterials are discussed with examples. Their relative advantages and disadvantages are also discussed. The self-assembly of materials is discussed. The classification of nanomaterials based on their structure is explained. The various methods of synthesis of nanoparticles are described with emphasis on the theories of nucleation and growth. The homogeneous nucleation method and water-in-oil microemulsion methods are discussed in detail. The methods of synthesis of carbon fullerenes are discussed. Next, the synthesis of nanowires, nanorods and nanotubes is presented and explained with various examples and applications. The synthesis of carbon nanotubes is discussed in detail. Various techniques for the deposition of thin films on a substrate are discussed. Next, the microporous and mesoporous materials are discussed with examples. The potentials of these materials in various important applications are discussed. Next, the lithographic techniques such as photolithography and soft-lithography are discussed. The chapter ends with a brief discussion on the cytotoxicity of nanoparticles.

KEYWORDS Atomic layer deposition Bottom-up method Boudouard reaction Buckminsterfullerene Carbon nanotube Chemical vapour deposition Chiral angle Chiral vector Critical radius Dip-pen lithography Electron-beam lithography Electrospinning Evaporation Fullerene Hierarchy High-pressure-carbon-monoxide method Homogeneous nucleation Laser ablation Lithography Macropore Macroporous material Mesopore Mesoporous material Microemulsion Micropore Microporous material

Molecular beam epitaxy Multiwalled nanotube Nucleation Nanobarcode Nanomaterial Nanoparticle Nanorod Nanotechnology Nanotube Nanowire Orthogonal self-assembly Photolithography Polymer capping Self-assembly Single-walled nanotube Soft-lithography Sputtering Supersaturation Template Thin film Top-down method Vapour–liquid–solid growth Zeolite

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NOTATION a a1 , a2 A c cb cs c* C h dt D I J k kd km kp L m, n r r0 rc S t T v vn Vbias Vgate

length of unit vector, m unit vectors pre-exponential factor, kg–1 s–1 solute concentration in the solution, mol/m3 bulk concentration, mol/m3 surface concentration, mol/m3 solubility of the solute, mol/m3 chiral vector tube diameter, m diffusion coefficient, m2/s current, amp nucleation rate, kg–1 s–1 Boltzmann's constant, J/K constant in Eq. (11.11), m2/s proportionality constant in Eq. (11.12), m–1 s–1 constant in Eq. (11.14), m/s circumference of nanotube, m integers radius, m initial radius of nucleus, m critical radius, m supersaturation ratio time, s temperature, K molecular volume, m3 molar volume of the nuclei, m3/mol bias voltage, V gate voltage, V

Greek Letters a g DG DGc 'G q s

exponent surface energy per unit area, J/m2 change in Gibbs free energy, J Gibbs free energy barrier, J change in Gibbs free energy per unit volume of the solid phase, J/m3 chiral angle, rad relative supersaturation

EXERCISES 1. Explain five major areas of technology where nanotechnology can play important roles. 2. Explain with examples the bottom-up and top-down techniques. What are their merits and demerits? 3. What is self-assembly? What are the important parametres behind self-assembly?

Nanomaterials

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. 44. 45. 46.

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What is template-directed self-assembly? Explain how mesoporous materials can be synthesised by surfactant-based templates. What is hierarchy in self-assembled nanostructures? Give one example each of zero-, one- and two-dimensional nanomaterials. Describe five methods of synthesis of nanoparticles. What is supersaturation? What is critical radius of nucleus? Explain how the growth of a nucleus takes place. What type of growth-conditions favour uniform-sized nanoparticles? What do you understand by capping material? Where is it used? Explain how you would synthesise palladium nanoparticles. Explain the steps involved in the synthesis of nanoparticles using water-in-oil microemulsions. What is percolation? Explain how the Buckminsterfullerene was synthesised by Kroto et al. (1985). Explain the structure of the C60 molecule and its stability. What are the large-scale synthesis methods of fullerenes? Explain why fullerenes with n = 24, 28, 32, 36, 50, 60 and 70 are more stable as compared to their neighbours. Explain the difference between a nanowire and a nanorod. Mention three methods for the synthesis of nanowires and nanorods. What is orthogonal self-assembly? What is its significance? Explain how nanowires can be synthesised by the VLS method. What is the role of catalyst in the VLS method? Explain the salient features of carbon nanotubes. Explain how carbon nanotubes are synthesised by the HiPCO method. Explain the terms chiral vector and chiral angle. What is their significance? Discuss the use of nanowires and nanotubes in electronic devices. Mention three methods for deposition of thin films on solid substrates. Describe the basic modes of nucleation for the development of thin films on a substrate. Explain how thin films can be deposited by evaporation. What is laser ablation? Explain molecular beam epitaxy. What is chemical vapour deposition? Explain atomic layer deposition. How can the speed of this method be enhanced? Classify the porous materials in terms of the size of their pores. Explain the importance of microporous and mesoporous materials. Explain why the phosphate-based porous materials lack stability. What can be the major uses of the metal-organic frameworks? Explain why most of the mesoporous materials are not crystalline. Give two potential future applications of mesoporous materials. What is lithography? Explain the various steps involved in photolithography. What is soft-lithography? What are its potential applications? What is dip-pen lithography? Explain how protein arrays can be created by this technique. Explain why some nanomaterials can be toxic towards cells. How can this problem be avoided?

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NUMERICAL AND ANALYTICAL PROBLEMS 11.1 The growth of gold nanoparticles with time was measured and the data are as follows. t, ks r, nm

1.0 6.5

1.5 7.5

2.5 8.3

3.0 9.5

4.5 10.8

5.5 12.0

7.0 13.5

It has been suggested that the growth is diffusion controlled. Verify this and determine the initial size of the nucleus. 11.2 If two nanoparticle nuclei have initial radius difference Dri, then show that the radius difference decreases with time if the growth is controlled by diffusion, and increases with time if the growth is mononuclear. 11.3 Calculate the diameter of a single-walled nanotube if n = m = 5 and the length of the unit vector is 0.246 nm.

FURTHER READING Books Ball, P., The Self-Made Tapestry: Pattern Formation in Nature, Oxford Press, Oxford, 2001. Cao, G., Nanostructures and Nanomaterials, Imperial College Press, London, 2006. Dresselhaus, M.S., G. Dresselhaus, and P.C. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, 1996. Drexler, K.E., Engines of Creation: The Coming Era of Nanotechnology, Anchor Books, New York, 1986. Edelstein, A.S. and R.C. Cammarata (Eds.), Nanomaterials: Synthesis, Properties and Applications, Taylor and Francis, New York, 1996. Harris, P.J.F., Carbon Nanotubes and Related Structures, New Materials for the Twenty-First Century, Cambridge University Press, Cambridge, 1999. Nielsen, A.E., Kinetics of Precipitation, Pergamon, Oxford, 1964. Ozin, G.A. and A.C. Arsenault, Nanochemistry, RSC Publishing, Cambridge, 2006. Poole (Jr.), C.P. and F.J. Owens, Introduction to Nanotechnology, Wiley, New Delhi, 2006. Saito, R., G. Dresselhaus, and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College ress, London, 1998. Sze, S.M., Semiconductor Devices: Physics and Technology, Wiley, New York, 1985. Tanaka, K., T. Yamabe, and K. Fukui, The Science and Technology of Carbon Nanotubes, Elsevier, Amsterdam, 1999.

Articles Ahmadi, T.S., Z.L. Wang, T.C. Green, A. Henglein, and M.A. El-Sayed, “Shape-Controlled Synthesis of Colloidal Platinum Nanoparticles”, Science, 272, 1924 (1996). Alivisatos, A.P., “Enhanced: Naturally Aligned Nanocrystals”, Science, 289, 736 (2000).

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Bachtold, A., M.S. Fuhrer, S. Plyasunov, M. Forero, E.H. Anderson, A. Zettl, and P.L. McEuen, “Scanned Probe Microscopy of Electronic Transport in Carbon Nanotubes”, Phys. Rev. Lett., 84, 6082 (2000). Balkus, K.J. (Jr), A.D. Sherry, and S.W. Young, “Zeolite-Enclosed Transition and Rare Earth Metal Ions as Contrast Agents for the Gastrointestinal Tract”, US Patent 5,122,363 (1992). Barnickel, P., A. Wokaun, W. Sager, and H.F. Eicke, “Size Tailoring of Silver Colloids by Reduction in W/O Microemulsions”, J. Coll. Int. Sci., 148, 80 (1992). Baum, R., “Nanotechnology”, Chem. Eng. News, 81, 37 (2003). Bethune, D.S., C.H. Kiang, M.S. de Vries, G. Gorman, R. Savoy, J. Vazquez, and R. Beyers,“CobaltCatalysed Growth of Carbon Nanotubes with Single-Atomic-Layer Walls”, Nature, 363, 605 (1993). Bockrath, M., D.H. Cobden, P.L. McEuen, N.G. Chopra, A. Zettl, A. Thess, and R.E. Smalley, “Single-Electron Transport in Ropes of Carbon Nanotubes”, Science, 275, 1922 (1997). Capek, I., “Preparation of Metal Nanoparticles in Water-in-Oil (W/O) Microemulsions”, Adv. Coll. Int. Sci., 110, 49 (2004). Cobden, D.H., “Nanowires Begin to Shine”, Nature, 409, 32 (2001). Davis, M.E., “Ordered Porous Materials for Emerging Applications”, Nature, 417, 813 (2002). Davis, M.E., C. Saldarriaga, C. Montes, J. Garces, and C. Crowder, “A Molecular Sieve with Eighteen-Membered Rings”, Nature, 331, 698 (1988). Demers, L.M., D.S. Ginger, S.-J. Park, Z. Li, S.-W. Chung, and C.A. Mirkin, “Direct Patterning of Modified Oligonucleotides on Metals and Insulators by Dip-Pen Nanolithography”, Science, 296, 1836 (2002). Derfus, A.M., W.C. Chan, and S.N. Bhatia, “Probing the Cytotoxicity of Semiconductor Quantum Dots”, Nano Lett., 4, 11 (2004). Dresselhaus, M.S., G. Dresselhaus, and R. Saito, “Physics of Carbon Nanotubes”, Carbon, 33, 883 (1995). Duan, X.F., Y. Huang, Y. Cui, J.F. Wang, and C.M. Lieber, “Indium Phosphide Nanowires as Building Blocks for Nanoscale Electronic and Optoelectronic Devices”, Nature, 409, 66 (2001). Eddaoudi, M., J. Kim, N. Rosi, D. Vodak, J. Wachter, M. O’Keeffe, and O.M. Yaghi, “Systematic Design of Pore Size and Functionality in Isoreticular MOFs and their Application in Methane Storage”, Science, 295, 469 (2002). Eicke, H.-F., “Exchange of Solubilized Water and Aqueous Electrolyte Solutions between Micelles in Apolar Media”, J. Coll. Int. Sci., 56, 168 (1976). Ericson, L.M. et al. “Macroscopic, Neat, Single-Walled Carbon Nanotube Fibres”, Science, 305, 1447 (2004). Feynman, R.P., “There’s Plenty of Room at the Bottom”, J. Microelectromech. Sys., 1, 60 (1992). Haberzettl, C.A., “Nanomedicine: Destination or Journey?”, Nanotechnology, 13, R9 (2002). Hausmann, D., J. Becker, S. Wang, and R.G. Gordon, “Rapid Vapour Deposition of Highly Conformal Silica Nanolaminates”, Science, 298, 402 (2002). Iijima, S., “Helical Microtubules of Graphitic Carbon”, Nature, 354, 56 (1991). Iijima, S. and T. Ichihashi, “Single-Shell Carbon Nanotubes of 1-nm Diameter”, Nature, 363, 603 (1993).

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Inagaki, S., S. Guan, T. Ohsuna, and O. Terasaki, “Mesoporous Organic-Silica Hybrid with CrystalLike Pore Walls”, Nature, 416, 304 (2002). Kodambaka, S., Tersoff, J., Reuter, M.C. and Ross, F.M., “Germanium Nanowire Growth below the Eutectic Temperature”, Science, 316, 729 (2007). Kong, X.Y. and Z.L. Wang, “Spontaneous Polarisation-Induced Nanohelixes, Nanosprings, and Nanorings of Piezoelectric Nanobelts”, Nano Lett., 3, 1625 (2003). Kovtyukhova, N.I., T.E. Mallouk, and T.S. Mayer, “Templated Surface Sol–Gel Synthesis of SiO2 Nanotubes and SiO2-Insulated Metal Nanowire”, Adv. Mater., 15, 780 (2003). Krätschmer, W., L.D. Lamb, K. Fostiropoulos, and D.R. Huffman, “Solid C60: A New Form of Carbon”, Nature, 347, 354 (1990). Kresge, C.T., M.E. Leonowicz, W.J. Roth, J.C. Vartuli, and J.S. Beck, “Ordered Mesoporous Molecular Sieves Synthesised by a Liquid-Crystal Template Mechanism”, Nature, 359, 710 (1992). Kroto, H.W., “The Stability of the Fullerenes Cn, with n = 24, 28, 32, 36, 50, 60 and 70”, Nature, 329, 529 (1987). Kroto, H.W., J.R. Heath, S.C. O’Brien, R.F. Curl, and R.E. Smalley, “C60: Buckminsterfullerene”, Nature, 318, 162 (1985). Lee, K.-B., S.-J. Park, C.A. Mirkin, J.C. Smith, and M. Mrksich, “Protein Nanoarrays Generated by Dip-Pen Nanolithography”, Science, 295, 1702 (2002). Lee, M., J. Im, B.Y. Lee, S. Myung, J. Kang, L. Huang, Y.-K. Kwon, and S. Hong, “Linker-Free Directed Assembly of High-Performance Integrated Devices based on Nanotubes and Nanowires”, Nature Nanotech., 1, 66 (2006). Li, D. and Y. Xia, “Direct Fabrication of Composite and Ceramic Hollow Nanofibres by Electrospinning”, Nano Lett., 4, 933 (2004). Li, D., Y. Wang, and Y. Xia, “Electrospinning Nanofibres as Uniaxially Aligned Arrays and Layerby-Layer Stacked Films”, Adv. Mater., 16, 361 (2004). Mann, S. and G.A. Ozin, “Synthesis of Inorganic Materials with Complex Form”, Nature, 382, 313 (1996). Martin, C.R., “Nanomaterials: A Membrane-Based Synthetic Approach”, Science, 266, 1961 (1994). Matijevi, E., “The Role of Chemical Complexing in the Formation and Stability of Colloidal Dispersions”, J. Coll. Int. Sci., 58, 374 (1977). Mbindyo, J.K.N., B.D. Reiss, B.R. Martin, C.D. Keating, M.J. Natan, and T.E. Mallouk, “DNADirected Assembly of Gold Nanowires on Complimentary Surfaces”, Adv. Mater., 13, 249 (2001). Morales, A.M. and C.M. Lieber, “A Laser Ablation Method for the Synthesis of Crystalline Semiconductor Nanowires”, Science, 279, 208 (1998). Nicewarner-Pena, S.R., R.G. Freeman, B.D. Reiss, L. He, D.J. Pena, I.D. Walton, R. Cromer, C.D. Keating, and M.J. Natan, “Submicrometre Metallic Barcodes”, Science, 294, 137 (2001). Nikolaev, P., M.J. Bronikowski, R.K. Bradley, F. Rohmund, D.T. Colbert, K.A. Smith, and R.E. Smalley, “Gas-Phase Catalytic Growth of Single-Walled Carbon Nanotubes from Carbon Monoxide”, Chem. Phys. Lett., 313, 91 (1999).

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Nuzzo, R.G., F.A. Fusco, and D.L. Allara, “Spontaneously Organised Molecular Assemblies. 3. Preparation and Properties of Solution Adsorbed Monolayers of Organic Disulfides on Gold Surfaces”, J. Am. Chem. Soc., 109, 2358 (1987). Pan, Z.W., Z.R. Dai, and Z.L. Wang, “Nanobelts of Semiconducting Oxides”, Science, 291, 1947 (2001). Piner, R.D., J. Zhu, F. Xu, S. Hong, and C.A. Mirkin, “‘Dip-Pen’ Nanolithography”, Science, 283, 661 (1999). Ritala, M. and M. Leskalä, “Atomic Layer Epitaxy–A Valuable Tool for Nanotechnology?”, Nanotechnology, 10, 19 (1999). Rosi, N.L., J. Eckert, M. Eddaoudi, D.T. Vodak, J. Kim, M. O’Keeffe, and O.M. Yaghi, “Hydrogen Storage in Microporous Metal–Organic Frameworks”, Science, 300, 1127 (2003). Seo, J.S., D. Whang, H. Lee, S.I. Jun, J. Oh, Y.J. Jeon, and K. Kim, “A Homochiral Metal–Organic Porous Material for Enantioselective Separation and Catalysis”, Nature, 404, 982 (2000). Service, R.F., “Assembling Nanocircuits from the Bottom Up”, Science, 293, 782 (2001). Smalley, R.E., “Of Chemistry, Love and Nanobots”, Scientific American, 76 (September 2001). Stucky, G.D. and J.E. MacDougall, “Quantum Confinement and Host Guest Chemistry—Probing A New Dimension”, Science, 247, 669 (1990). Tans, S.J., A.R.M. Verschueren, and C. Dekker, “Room-Temperature Transistor Based on a Single Carbon Nanotube”, Nature, 393, 49 (1998). Thess, A., R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y.H. Lee, S.G. Kim, A.G. Rinzler, D.T. Colbert, G.E. Scuseria, D. Tománek, J.E. Fischer, and R.E. Smalley, “Crystalline Ropes of Metallic Carbon Nanotubes”, Science, 273, 483 (1996). Thurn-Albrecht, T., J. Schotter, G.A. Kästle, N. Emley, T. Shibauchi, L. Krusin-Elbaum, K. Guarini, C.T. Black, M.T. Tuominen, and T.P. Russell, “Ultrahigh-Density Nanowire Arrayss Grown in Self-Assembled Diblock Copolymer Templates”, Science, 290, 2126 (2000). Wagner, R.S. and W.C. Ellis, “Vapour–Liquid–Solid Mechanism of Single Crystal Growth”, Appl. Phys. Lett., 4, 89 (1964). Wang, N., Z.K. Tang, G.D. Li, and J.S. Chen, “Single-Walled 4 Å Carbon Nanotube Arrays”, Nature, 408, 50 (2000). Williams, R., P.N. Yocom, and F.S. Stofko, “Preparation and Properties of Spherical Zinc Sulfide Particles”, J. Coll. Int. Sci., 106, 388 (1985). Wu, Y., Y. Cui, L. Huynh, C.J. Barrelet, D.C. Bell, and C.M. Lieber, “Controlled Growth and Structures of Molecular-Scale Silicon Nanowires”, Nano Lett., 4, 433 (2004). Xia, Y. and G.M. Whitesides, “Soft Lithography”, Angew. Chem. Int. Ed., 37, 550 (1998). Xie, Y., Y. Qian, W. Wang, S. Zhang, and Y. Zhang, “A Benzene-Thermal Synthetic Route to Nanocrystalline GaN”, Science, 272, 1926 (1996). Yang, P., Z. Dongyuan, D.I. Margolese, B.F. Chmelka, and G.D. Stucky, “Generalised Syntheses of Large-Pore Mesoporous Metal Oxides with Semicrystalline Frameworks”, Nature, 396, 152 (1998). Zhao, D., J. Feng, Q. Huo, N. Melosh, G.H. Fredrickson, B.F. Chmelka, and G.D. Stucky, “Triblock Copolymer Syntheses of Mesoporous Silica with Periodic 50 to 300 Angstrom Pores”, Science, 279, 548 (1998).

12

Interfacial Reactions

Fritz Haber was born in Breslau (Germany). His father was a well-known chemical merchant. Haber studied chemistry at University of Heidelberg, University of Berlin and Technical School at Charlottenburg between 1886 and 1891. After completing studies at these universities, he decided to do research in chemistry (especially electrochemistry) and chemical technology. His early works were on decomposition and combustion of hydrocarbons. He was the Director of the Institute for Physical and Electrochemistry at Berlin-Dahlem between 1911 and 1933. This institute was named Fritz Haber Institute after his death. Haber did research on varied areas such as electrolysis of solid salts, quinhydrone electrode, steam engines and turbines, Bunsen flame and water gas reaction. Haber was awarded the Nobel Prize in Chemistry in 1918 for his work on nitrogen-fixation from air. Haber is probably most famous for his Fritz Haber method known as Haber Process for the production of ammonia from hydrogen and (1868 – 1934) nitrogen over iron catalyst at a high temperature and pressure. Haber was versatile in his talents and possessed an astonishing knowledge of politics, history, economics, science and industry. He left a lasting impression on the minds of all his associates. Haber died in Basel (Switzerland).

TOPICS COVERED © © © © ©

Reactions at fluid–solid interfaces Reactions at liquid–liquid interfaces Reactions at biological interfaces Micellar reactions Phase transfer catalysis

12.1 INTRODUCTION A large number of chemical reactions takes place at the interfaces. A very important class of such reactions is catalysed by solid catalysts, which are carried out in catalytic reactors. The 452

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polycondensation reactions carried out at the liquid–liquid interfaces have found extensive use in the manufacture of membranes and preparation of micro-capsules. Several very fast or instantaneous chemical reactions take place at gas–liquid or liquid–liquid interfaces. One example is aromatic nitration reactions in which the organic compound (e.g. benzene) reacts with nitronium ion generated in the aqueous phase by reaction between concentrated nitric and sulphuric acids. Most biological reactions take place at the interface. The enzymatic lypolytic reactions are important examples of interfacial catalysis. The water-soluble lipolytic enzymes act at the interfaces of insoluble lipid substrates, where the catalytic reactions are coupled with various interfacial phenomena. The mechanism of enzymatic lipolysis depends upon the mode of organisation of the lipid substrate in the interfacial structures such as monolayers, micelles, liposomes, or emulsions. Many organic reactions are enhanced in solutions of micelles as well as reverse micelles. Micelles have the characteristic ability to solubilise substrates. This solubilisation occurs by spontaneous dissolution of a material (which is often a liquid) by reversible interaction with the micelles present in a solvent to form a thermodynamically-stable isotropic solution. The thermodynamic activity of the solubilised material is reduced in this process. Micelles help to solubilise an otherwise insoluble material (e.g. an organic liquid). The micellar catalysis reaction generally occurs at the micelle–solution interface. However, for reverse micelles in non-aqueous medium, the reaction can occur deep inside the inner core of the micelle. In this chapter, these four types of interfacial reactions, viz. reactions at fluid–solid and liquid– liquid interfaces, biointerfaces and micelle–solution interfaces will be discussed. The theories are quite well-developed for the first two types of reactions. Some of the mathematical models of reactions at these interfaces will be discussed. The reactions at biointerfaces and micelle–solution interfaces will be discussed to provide the reader ideas about the present state of scientific development in these areas. A brief description of the mechanism of phase transfer catalysis will also be presented.

12.2 REACTIONS AT SOLID–FLUID INTERFACES The majority of reactions carried out in chemical industries involve a solid catalyst. In these heterogeneous systems, the reaction occurs at some of the catalytic sites, which are generally located on the external surface of the catalyst pellet and inside the pores of the pellet. The reactants have to be transported to these sites for the reaction to take place. The mass transfer to the catalyst surface may influence the overall rate of conversion by a significant extent. The important aspect to study is the interplay between the true chemical rate of reaction and the physical transport phenomena to obtain the overall conversion rate. Consider a porous catalyst particle around which a fluid is flowing containing the reactants (Figure 12.1). The general case of the reaction at the surface of the catalyst can be described by the steps given below. (i) Transport of the reactants from the bulk fluid to the external surface of the catalyst particle. (ii) Transport of the reactants by a diffusional process through the pores into the catalyst particle. (iii) Adsorption of reactants on the catalyst sites (internal and external). (iv) Chemical reaction in the adsorbed state. (v) Desorption of the adsorbed products.

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(vi) Transport of products by a diffusional process through the pores to the outer surface of the catalyst particle. (vii) Transport of products from the external surface into the main flow. If the particle is non-porous, then the steps (ii) and (vi) do not occur. The steps (i) and (ii) are illustrated in Figure 12.1.

Figure 12.1 Transport of reactant from the bulk to the interior of a catalyst partice. The thickness of the film and pore diameter have been exaggerated for illustration.

The best use of a porous catalyst is made when most of the internal surface area is readily accessible to the reactants. One of the ways this can be achieved is by using a small particle of high porosity. The SEM photograph of a catalyst having high surface area (~1 × 106 m2/kg) is shown in Figure 12.2. If the chemical reaction is very fast in comparison with the mass transport, the reaction occurs in the outer part of the particle. In an extreme situation, only the external surface of the catalyst particle may be used in the reaction and the internal surface does not play any role.

Figure 12.2

SEM photograph of a high surface area vanadium-incorporated MCM-41 catalyst (Gucbilmez et al., 2005) [reproduced by permission from Elsevier Ltd., © 2005].

Interfacial Reactions

455

Agglomeration of the porous catalyst particles gives a pellet containing two void regions: small void spaces within the individual particles, and larger spaces between the particles. Such materials contain bidisperse pore system. The void spaces within the particles are called microporous regions and the void regions between the particles are called macroporous regions. As per the modern classification, pores having diameter less than 2 nm are called micropores and the pores which have diameters greater than 50 nm are called macropores. The pores having diameters between 2 nm and 50 nm are called mesopores. Typical microporous materials are zeolites. Examples of mesoporous solids are silicas and modified layered materials. Mesoporous solids can be prepared by various methods (see Section 11.7). One of the methods is templating by liquid crystals, followed by calcination (Kresge et al., 1992). The high-surface-area catalyst shown in Figure 12.2 is an example of mesoporous catalyst.

12.2.1

Langmuir–Hinshelwood Model

A well-known approach to describe the kinetics of gas–solid reactions was proposed by C.N. Hinshelwood (1940) based on Langmuir's theory of chemisorption (see Chapter 6). Let us consider the situation where one reactant molecule occupies a single site (S) on the surface of the catalyst. The adsorption process is represented by A  S U A¹S

(12.1) (mol/m3)

Let us represent the concentration of reactant A in the gas phase as cA and the adsorbed concentration of A on the catalyst surface as cˆ A (expressed as mol/kg of catalyst). The net rate of adsorption of A is given by ra

ka c A (cˆt  cˆ A )  ka„ cˆ A

(12.2)

where cˆt is the concentration corresponding to a complete monolayer on the catalyst (expressed as mol/kg of catalyst). The first term on the right side of Eq. (12.2) is valid if only the reactant A in the gaseous feed adsorbs on the surface of the catalyst. The quantity cˆt  cˆ A represents the concentration corresponding to the vacant sites, cˆv . The adsorption equilibrium constant for A is given by K aA ka / ka„ . Therefore,

ra

È Ø 1 ka É c A cˆv  A cˆ A Ù Ka Ê Ú

(12.3)

If the rate of adsorption is much faster than the other steps in the overall conversion process, the concentration of A on the catalyst surface will be in equilibrium with the concentration of A in the gas phase. The equilibrium concentration of A is obtained by setting ra = 0, which gives cˆ A,eq

K aA c A cˆv

(12.4)

EXAMPLE 12.1 A feed contains a diatomic gas along with an inert gas which does not adsorb on the catalyst. The diatomic gas dissociates upon adsorption and each atom occupies one adsorption site as shown below. A2  2 S U 2 A ¹ S

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Colloid and Interface Science

Derive the expression for the equilibrium concentration of atomically adsorbed A assuming that the rate of adsorption is much faster than the other steps. Solution

The net rate of adsorption is

ra

È Ø 1 ka É c A2 cˆv2  A cˆ 2A Ù Ka 2 Ú Ê

(i)

Since the inert gas does not adsorb, A2 is the only adsorbable gas. Therefore, site balance gives (ii) cˆv cˆt  cˆ A Substituting cv from Eq. (ii) into Eq. (i) and setting ra = 0, we get c A2 (cˆt  cˆ A,eq )2

1 K aA2

cˆ 2A,eq

(iii)

Simplifying the above equation, we get cˆ A,eq

cˆt K aA2 c A2 1  K aA2 c A2

(iv)

Let us now consider the surface reaction step with the reaction expressed as: A  B U C . The reaction may occur with both A and B adsorbed on the catalyst surface or one of the reactants adsorbed on the catalyst site and the other reactant present in the gas phase (see Figure 12.3). Let us consider the first case, which can be expressed as A¹S  B¹S U C ¹S  S

(12.5)

Now, only those A and B molecules will react which are adsorbed on adjacent sites. Therefore, the rate of forward reaction should be proportional to the concentration of the pairs of adjacent sites occupied by A and B molecules, assuming that the rate of forward reaction is first-order in both A and B on the catalyst surface. The concentration of these pairs is approximately equal to cˆ A cˆB / cˆt (which is valid if the fraction of the surface occupied by the A molecules is small). r1

ks cˆ A cˆB / cˆt

Figure 12.3 Surface reaction involving (i) both adsorbed A and adsorbed B on the adjacent sites, and (ii) adsorbed A and non-adsorbed B.

(12.6)

Interfacial Reactions

457

The rate of the reverse reaction is given by r2

ks„cˆC cˆv / cˆt

(12.7)

Therefore, the net rate of surface reaction is rs

(ks cˆ A cˆB  ks„cˆC cˆv ) / cˆt

ks (cˆ A cˆB  cˆC cˆv / K s ) / cˆt

(12.8)

where K s (œ ks / ks„ ) is the equilibrium constant for the surface reaction. If the surface reaction step is intrinsically fast with respect to the other steps, the process would occur at equilibrium. Setting rs = 0, we obtain

Ks

Ë cˆv cˆC Û Ìˆ ˆ Ü Í c A cB Ýeq

(12.9)

In the second case, only the reactant A is adsorbed and the reactant B is present in the gas phase. We can represent the reaction as

A ¹ S  B (g) U C ¹ S

(12.10)

The net rate of surface reaction for this mechanism is given by rs

ks1cˆ A cB  ks„1cˆC

ks1 (cˆ A cB  cˆC / K s„ )

(12.11)

At equilibrium rs = 0. Therefore

K s„

Ë cˆC Û Ìˆ Ü Í c A cB Ýeq

(12.12)

For the desorption of the product C, the mechanism may be expressed as C ¹S U C  S

(12.13)

The rate of desorption is given by

rd

kd„ cˆC  kd cC cˆv

 kd (cC cˆv  cˆC / K dC )

(12.14)

If the rate of desorption is faster with respect to the adsorption and surface reaction steps, the equilibrium concentration of C is given by cˆC ,eq

K dC cC cˆv

(12.15)

At steady state, the rates of adsorption, surface reaction and desorption are equal. The rate Eqs. (12.3), (12.8) [or (12.11)] and (12.14) contain adsorbed concentrations of the reactants, products and the vacant sites (i.e . cˆ A , cˆB , cˆC and cˆv ). To determine the rates from the concentrations in the gas phase, these quantities need to be eliminated. A simple procedure is to assume that one of the steps controls the rate and the other two steps occur at near-equilibrium conditions. This simplifies the rate expressions and reduces the number of rate and equilibrium constants that must be determined from experiment. This is also valid from a practical point of view since in many cases it has been observed that only one of the steps controls the overall catalytic reaction. Let us consider the reaction scheme presented in Table 12.1 and follow the procedure discussed above.

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Table 12.1 Adsorption, surface reaction and desorption schemes for the reaction: A + B U C Step

Mechanism A  S U A¹S

Adsorption

B  S U B¹S A¹S  B¹S U C ¹S  S

Surface reaction

C ¹S U C  S

Desorption

Consider that surface reaction is the rate-controlling step. Then cˆ A , cˆB and cˆC are the equilibrium values corresponding to the adsorption and desorption steps. From Eq. (12.4), we can write cˆB,eq

K aB cB cˆv

(12.16)

From Eq. (12.8), we get

Rate

ks cˆt

rs

Ë A B Û K dC 2 cC cˆv2 Ü Ì K a K a c A cB cˆv  Ks ÌÍ ÜÝ

(12.17)

We can write a site-balance, in which cˆt is given by

cˆt

cˆv  cˆ A,eq  cˆB,eq  cˆC ,eq

(12.18)

Therefore, from Eqs. (12.4), (12.15), (12.16) and (12.18), we get cˆv

cˆt 1  K aA c A

 K aB cB  K dC cC

(12.19)

Now, substituting the value of cˆv from Eq. (12.19) into Eq. (12.17), we obtain Rate

ks cˆt

K aA K aB c A cB  K dC cC / K s (1  K aA c A  K aB cB  K dC cC )2

(12.20)

Several simplifications of Eq. (12.20) can be made depending on the characteristics of the steps given in Table 12.1. For example, if the adsorption of all the components is weak, the denominator of Eq. (12.20) approaches unity, and the rate equation becomes

Rate

ks cˆt [ K aA K aB c A cB  K dC cC / K s ]

(12.21)

Rate expressions for the situations where either adsorption or desorption is the controlling step can be derived easily following a procedure similar to that described above. EXAMPLE 12.2 Derive the rate expression for the reaction: A  B U C , if the adsorption of the reactant A is the rate-controlling step. Assume that the reaction follows the mechanism described in Table 12.1. Solution The adsorption of reactant B, surface reaction and desorption of C will occur at equilibrium in this case. From Eqs. (12.9), (12.15) and (12.16), we obtain

Interfacial Reactions

cˆ A

459

K dC cC cˆv K s K aB cB

Substituting the value of cˆ A in Eq. (12.3), we get Rate

ra

È Ø 1 ka É c A cˆv  A cˆ A Ù Ka Ê Ú

Ë È KC Ø È c Ø Û ka cˆv Ì c A  É A dB Ù É C Ù Ü Ê K a K a K s Ú Ê cB Ú ÝÜ ÍÌ

From site balance, we get cˆv

\

cˆv

Rate

\

cˆt 

K dC cC cˆv K s K aB cB

 K aB cB cˆv  K dC cC cˆv cˆt

È Ø È cC Ø 1 É  K aB cB  K dC cC BÙÉc Ù Ê Ks Ka Ú Ê B Ú K dC

Ë È KC Ø È c Ø c A  É A dB Ù É C Ù Ì Ê K a K a K s Ú Ê cB Ú Ì ka cˆt Ì C Ì 1  È K d Ø È cC Ø  K B c  K C c a B d C É Ì Ê K K B ÙÚ ÉÊ cB ÙÚ s a Í

Û Ü Ü Ü Ü Ü Ý

If the desorption of product C controls the whole reaction, it can be shown that the rate expression is given by

Rate

Ë Û È KC Ø c A cB  É A dB Ù cC Ì Ü A B È Ka Ka Ks Ø Ì Ê Ka Ka Ks Ú Ü kd cˆt É ÙÌ Ü C A B A B 1 K  K c  K c  K K K c c Ê ÚÌ d a A a B a a s A BÜ Ì Ü Í Ý

(12.22)

The derivation of this equation is left as an exercise for the reader. EXAMPLE 12.3 Vapour-phase dehydration of ethanol: 2C 2 H 5 OH = C 2 H 5 OC2 H 5 + H 2 O. Kabel and Johanson (1962) have studied the application of Langmuir–Hinshelwood model for the aforesaid reaction carried out using a sulphonated copolymer of styrene and divinylbenzene as catalyst. If surface-reaction is the rate-controlling step, the following rate equation is applicable when the reaction occurs between two adjacently-adsorbed ethanol molecules. r0 pA0

È K  KE Ø k  r0 É A 1  KE Ê 1  K E ÙÚ

where r0 is the initial reaction rate, pA 0 is the partial pressure of alcohol in the feed, KA and KE are adsorption constants of alcohol and ether respectively, and k is a constant. If the reaction occurs

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Colloid and Interface Science

between an adsorbed ethanol molecule and a gaseous ethanol molecule, the rate equation is given by r0 p2A0

È r Ø k  KA É 0 Ù Ê pA0 Ú

The initial rate data at 393 K are presented below. r0, mol (kg catalyst)–1 s–1

pA0 (Pa)

r0, mol (kg catalyst)–1 s–1

4.772 – 10 4

1.447 – 10 3

8.886 – 10 4

2.147 – 10 3

5.796 – 10 4

1.672 – 10 3

9.595 – 10 4

2.225 – 10 3

1.013 – 10

2.245 – 10 3

pA0 (Pa)

6.495 – 10

4

1.780 – 10

3

5

Which mechanism among the two fits the data? Solution To test the mechanism 1 (i.e. reaction between two adsorbed ethanol molecules), let us plot r0 pA0 versus r0 . The plot is shown in Figure 12.4(a). The linear fit is quite evident in this figure. To test the mechanism 2 (i.e. the reaction occurs between an adsorbed ethanol molecule and a gaseous ethanol molecule), let us plot r0 / p2A0 versus r0 / p A 0 . The plot is shown in Figure 12.4(b). It is observed that the data points clearly do not fall on a straight line. Therefore, it is concluded that the first mechanism is correct, i.e. two adjacent sites are required for the reaction to take place.

12.2.2

External Transport Processes

We have mentioned at the beginning of Section 12.2 that the first step in a heterogeneous catalytic reaction is the transport of the reactants from the bulk of the fluid to the surface of the catalyst. The concentration difference between the bulk and the surface is the driving force for this transport. The concentration of the reactants at the surface of the catalyst is lower than that in the bulk of the fluid. The difference in concentration of a reactant between the bulk and the surface depends upon several factors such as the velocity pattern of the fluid near the surface, the physical properties of the fluid and the rate of chemical reaction. All these factors are expressed in terms of the mass transfer coefficient and the reaction rate constant. For those reactions in which significant thermal effects are involved, temperature difference between the bulk and the surface is developed. The temperature difference between the bulk and the surface depends upon the nature of the reaction (i.e. whether it is endothermic or exothermic). The effect of temperature difference on mass transfer can be significant. Let us consider a first-order irreversible gaseous reaction: A ® B on a solid catalyst pellet. At steady state, the rate of transport of A from the bulk fluid to the gas–solid interface is equal to the rate of reaction at the catalyst surface. The rate of mass transfer from the bulk to the catalyst surface is given by rp

km am (c bA  c As )

(12.23)

where km is the mass transfer coefficient between the bulk and the solid surface based on a unit external area of the catalyst particle, am is the external surface area of the pellet per unit mass, cAb

Interfacial Reactions

Figure 12.4

461

Verification of mechanism of surface reaction: (a) reaction between adjacently-adsorbed ethanol molecules, and (b) reaction between an adsorbed ethanol molecule and a gaseous ethanol molecule.

is the concentration of A in the bulk gas and cAs is the concentration of A at the surface. Since the reaction is of first-order, the rate is given by rp

kc As

where k is the reaction rate constant based on the unit mass of the catalyst. Therefore, related as c As

km am c bA k  km am

(12.24) cAs

and cAb are

(12.25)

From Eqs. (12.24) and (12.25), we can express the rate as

rp

kkm am b cA k  km am

c bA 1 /(km am )  1 / k

(12.26)

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Colloid and Interface Science

The rate expressed by Eq. (12.26) is an example of global rate because it includes the external mass transfer effects and expresses the rate of reaction in terms of the reactant concentration in the bulk phase. However, it is a very simplified global rate expression because it does not consider the heat transfer and internal mass transfer effects. The two terms in the denominator of Eq. (12.26), viz 1/k and 1/(kmam), represent the reaction and mass transfer resistances respectively. If the rate of reaction is much faster than the rate of mass transfer, cAs approaches zero. On the other hand, if the mass transfer rate is much higher than the rate of reaction, cAs approaches cAb. These two limiting cases are schematically shown in Figure 12.5.

Figure 12.5 Mass transfer from bulk of fluid to the surface of catalyst under reaction control and diffusion control.

The heat evolved by chemical reaction is QR

( 'H )rp

È E Ø s ( 'H )k0 exp É  cA Ê RTs ÙÚ

(12.27)

where k0 is the pre-exponential factor, E is the activation energy and R is the gas constant. In Eq. (12.27), it has been assumed that the reaction rate constant k varies with temperature following the Arrhenius equation. The rate of heat loss from the catalyst surface (expressed per unit mass) is given by QL

ham (Ts  Tb )

(12.28)

where h is the heat transfer coefficient. At steady state, the heat evolved by the reaction QR is equal to the heat lost from the catalyst QL. Therefore, from Eqs. (12.27) and (12.28), we get

Ts  Tb

È E Ø s ( 'H )k0 exp É  cA Ê RTs ÚÙ ham

(12.29)

Interfacial Reactions

463

From Eqs. (12.23) and (12.24), we get

È E Ø s k0 exp É  cA Ê RTs ÚÙ km am

kc As

c bA  c As

km am

(12.30)

If DH, A, E, h, km, am, cAb and Tb are known, we can determine cAs and Ts by solving Eqs. (12.29) and (12.30) simultaneously. EXAMPLE 12.4 The following data are available for a first-order catalytic reaction: A ® B. DH = –150 J/mol, E = 100 J/mol, k0 = 770 m3 (kg catalyst)–1 s–1, h = 20 W m–2 K–1, am = 100 (m2 area)/ (kg catalyst) and km = 10 m3 (m2 area)–1 s–1. The bulk temperature is 350 K and the concentration of A in the bulk is 2 mol/m3. Using these data, calculate the surface temperature and the concentration of A at the surface of the catalyst. Solution

From Eqs. (12.29) and (12.30), we get

È E Ø s ( 'H )k0 exp É  cA Ê RTs ÚÙ ham

Ts  Tb

È E Ø s k0 exp É  cA Ê RTs ÙÚ km am

c bA  c As

(i)

(ii)

Dividing Eq. (i) by (ii), we get Ts  Tb c bA

 c As

( 'H )km h

(iii)

Putting the values of DH, km, h, cAb and Tb in Eq. (iii), we get Ts  350 2  c As

150 – 10 20

Ts

500  75c As

75

which simplifies to (iv)

Putting the values of DH, km, h, cAb and Tb in Eq. (iv), we get

Ts  350

È 100 Ø s cA 150 – 770 – exp É  Ê 8.314Ts ÙÚ 20 – 100

which simplifies to

Ts

350  57.75cAs exp(  12.03 Ts )

(v)

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Colloid and Interface Science

Substituting Ts from Eq. (iv) into Eq. (v), we get

500  75c As

È 12.03 Ø 350  57.75c As exp É  s Ù Ê 500  75c A Ú

(vi)

Simplifying Eq. (vi), we get

È Ø 1 c As  0.77c As exp É  s Ù Ê 41.56  6.23c A Ú

2

(vii)

Solving Eq. (vii) by the Newton–Raphson method, we get

c As

1.14 mol/m3

Therefore, from Eq. (iv), we get

500  75c As

Ts

500  75 – 1.14

414.5 K

12.2.3 Determination of Mass and Heat Transfer Coefficients The mass transfer coefficient between the bulk fluid and the surface of the particle can be obtained from the correlations given in the literature (Fogler, 2008). Many such correlations are expressed based on the j-factor defined as (Smith, 1981)

jD

È km U Ø È am Ø È P Ø ÉÊ G ÙÚ É a Ù ÉÊ U D ÙÚ Ê t Ú

23

(12.31)

where G is the superficial mass velocity based on the cross-sectional area of the empty reactor, r is the density of the fluid, D is the diffusivity of the component being transferred, m is the viscosity of the fluid and at is the total external area of the particles. The ratio am /at allows for the possibility that the effective mass transfer area am may be less than the total external area at. The following relationship has been given by Dwivedi and Upadhyay (1977) for calculating the mass transfer coefficient in fixed beds. jD

0.4548 È d p G Ø H b ÉÊ P ÙÚ

0.407

, d p G P ! 10

(12.32)

where eb represents the void fraction of the bed. The j-factor for heat transfer is defined as (Smith, 1981), jH

È h Ø È am Ø È c p P Ø É ÙÉ ÙÉ Ù Ê c p G Ú Ê at Ú Ê kt Ú

23

(12.33)

where cp is the heat capacity and kt is the thermal conductivity of the fluid. Equation (12.32) can also be used for jH as per the Chilton–Colburn analogy. EXAMPLE 12.5 A gaseous feed of SO2 in air is passed through a packed bed of spherical catalyst pellets (dp = 2 mm) at 300 K. The velocity of the gas is 10 cm/s, diffusivity of SO2 is 1.6 × 10–5 m2/s,

Interfacial Reactions

465

density of the gas is 1.8 kg/m3 and the viscosity is 16 × 10–6 Pa s. The void fraction of the bed is 0.5 and the effective mass transfer area is 60% of the total external area of the pellets. Calculate the mass transfer coefficient. Solution

The Reynolds number is Re

\

d pG

d p vU

2 – 10 3 – 0.1 – 1.8

P

P

16 – 10 6

Hb

0.5

jD

0.4548 È d p G Ø H b ÉÊ P ÙÚ

Now,

P UD

Given,

am at

16 – 10 6 1.8 – 1.6 – 10 5

0.407

22.5

È 0.4548 Ø 0.407 ÉÊ 0.5 ÙÚ (22.5)

0.256

0.56

0.6

Therefore, from Eq. (12.31), we get 0.256

\

km

È km Ø 23 ÉÊ 0.1 ÙÚ (0.6)(0.56)

4.076 km

0.063 m/s

12.2.4 Internal Transport Processes The transport of the reactants from the bulk to the surface of the catalyst has been discussed in Section 12.2.2. The next step is the diffusional transport of the reactant inside the pores of the catalyst as depicted in Figure 12.6. This is a very important step because the majority of the active sites of

Figure 12.6

Transport inside the catalyst pores.

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Colloid and Interface Science

a catalyst can stay inside the pellet. The pores are not straight, but have tortuous structure distributed randomly inside the pellet. Let us consider the first-order reaction: A ® B occurring at steady state. The equimolal counterdiffusion model is applicable here: for every mole of A that diffuses from the surface into the pore, one mole of B diffuses back in the opposite direction. The mass transport occurs by both bulk and Knudsen diffusion. The combined diffusivity Dc is given by

1 1 1  A DAB DK

Dc

(12.34)

where DAB is the bulk diffusivity and DK is the Knudsen diffusivity. Since the pores are not straight and cylindrical but a series of tortuous paths, an effective diffusivity De is defined as follows: De

H p Dc

(12.35)

W

where ep is the porosity of the catalyst pellet and t is the tortuosity factor. If the pellet has a bidisperse pore system (i.e. both macro and microporous regions are present), the effective diffusivity is defined as (Smith, 1981) De

È 1 1 Ø ÉD  mÙ Ê AB DK Ú

1

H m2

Ë H P2 1  3H m Û È 1 1 Ø ÜÉ Ì  PÙ ÌÍ 1  H m ÜÝ Ê DAB DK Ú

1

(12.36)

where em is the void fraction in the macroporous regions and em is the void fraction of the microporous regions. Transport in the pellet is assumed to occur by a combination of diffusion through both the regions. Note that there is no tortuosity factor involved in Eq. (12.36). The relative importance of diffusion and reaction is expressed by the effectiveness factor. It is defined as

K

Actual rate for the whole pellet Rate evaluated at outer surface conditions

(12.37)

If the actual rate is expressed by rp and the rate evaluated at the outer surface conditions is expressed by rs, then for the first-order reaction: A ® B, we have rp

K rs

K kc As

(12.38)

The effectiveness factor can be expressed in terms of the effective diffusivity and reaction rate constant as shown later. The concentration profile of reactant A inside the catalyst pellet is required for this purpose. Let us perform a material balance over the spherical-shell of the pellet of inner radius r and outer radius r + Dr as shown in Figure 12.7. At steady state, the rate of diffusion into the element minus the rate of diffusion out of the element must equal the rate of disappearance by reaction within the element. Therefore, we can write the following balance equation:

dc A Û Ë dc A Û Ë 2 2 Ì 4S r De dr Ü  Ì 4S r De dr Ü Í Ýr Í Ýr 'r

4S r 2 'r U p kc A

(12.39)

Interfacial Reactions

Figure 12.7

467

Shell balance on a catalyst pellet.

where rp is the density of the pellet. Taking the limit Dr ® 0, we obtain the following second-order differential equation:

d 2 cA dr



2

2 dc A k U p c A  r dr De

0

(12.40)

The boundary conditions are: dc A dr

0

(12.41)

At r = Rs: c A

cAs

(12.42)

At r = 0:

The solution of Eq. (12.40) with the boundary conditions (12.41) and (12.42) is given by

cA c As

È 3 )r Ø Rs sinh É Ê Rs ÙÚ r sinh 3)

(12.43)

where F is known as Thiele modulus, which is defined as )

Rs È k U p Ø 3 ÉÊ De ÙÚ

12

(12.44)

Therefore, it gives the relative magnitude of reaction rate and diffusion rate. If the Thiele modulus is small, the reaction is the rate-limiting step. On the other hand, if the Thiele modulus is large, diffusion limits the overall rate of reaction. The rate of reaction for the whole pellet is given by

rp

È dc Ø 4S Rs2 De É A Ù Ê dr Ú r 4 S Rs3 U p 3

Rs

È dc Ø 3De É A Ù Ê dr Ú r Rs U p

Rs

(12.45)

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Colloid and Interface Science

Therefore, the effectiveness factor is given by

K

È dc Ø 3De É A Ù Ê dr Ú r

Rs

Rs U p kc As

(12.46)

From Eqs. (12.43) and (12.46), we get K

1Ë 1 1 Û  Ì ) Í tanh 3) 3) ÜÝ

(12.47)

The variation of effectiveness factor h with Thiele modulus is shown in Figure 12.8. From this figure, it is observed that the effectiveness factor approaches unity for small values of F, which indicates that the rate for the whole pellet is same as the rate that would result if the entire interior surface were exposed to the reactant at concentration cAs, i.e. the entire surface is fully effective up to the centre of the pellet. On the other hand, if the value of h is much less than unity, only that part of the interior surface which lies close to the outer periphery of the pellet is effective, and the central part of the pellet is not utilised. Such a situation can be caused by a large value of the Thiele modulus, which may be due to large size of the pellet, low diffusivity, or high rate of reaction.

Figure 12.8

Variation of effectiveness factor with Thiele modulus for a first-order reaction on a spherical catalyst.

From Figure 12.8 it is observed that h is close to unity if F £ 1/3. If this condition is satisfied, it is possible to ignore the effects of intrapellet mass transfer. This is often called Weisz criterion. Therefore, we can write this condition as È kUp Ø Rs É Ù Ê De Ú

12

…1

(12.48)

Therefore, È kUp Ø Rs2 É Ù …1 Ê De Ú

(12.49)

Interfacial Reactions

469

Since h ® 1 when the above condition is valid, from Eq. (12.38) we can write k  rp / c As

(12.50)

From Eqs. (12.49) and (12.50), we can write

È rp U p Ø Rs2 É s Ù … 1 Ê c A De Ú

(12.51)

EXAMPLE 12.6 A first-order reaction: A ® B is carried out using 2 mm diameter spherical catalyst pellets in a packed bed. The effective diffusivity of A is 5 × 10–6 m2/s, the measured rate of reaction is 0.5 mol (kg catalyst)–1 s–1, the estimated concentration of reactant A on the surface of the catalyst is 10 mol/m3, and the density of the pellet is 1200 kg/m3. Calculate the effectiveness factor. Solution

From Eqs. (12.38) and (12.44), we get

)

Rs È k U p Ø 3 ÉÊ De ÙÚ

12

Rs È rp U p Ø É Ù 3 Ê K c As De Ú

12

È 1 – 10 3 Ø È 0.5 – 1200 Ø É 3 ÙÉ Ù Ê Ú Ê K – 10 – 5 – 10 6 Ú

12

1.155 K

Substituting F in Eq. (12.47), we get

K

Ë Û K Ì K Ü 1  1.155 Ì tanh 3.465 K 3.465 Ü ÌÍ ÜÝ





Solving the above equation, we get

K

0.48

12.3 INTERFACIAL POLYCONDENSATION Interfacial polycondensation involves polymerisation of two monomers which are dissolved in two immiscible phases. Typical monomers are diacid chloride and diamine. The acid chloride is dissolved in an organic solvent such as xylene, carbon tetrachloride, hexane or dichloromethane. The diamine is dissolved in water. The polymerisation reaction takes place at the interface. To illustrate, the reaction between sebacyl chloride and hexamethylene diamine, which produces Nylon 6–10, can be represented as xH2N(CH2)6NH2 + xClCO(CH2)8COCl ® [–NH(CH2)6NHCO(CH2)8CO–]x + 2xHCl

(12.52)

The HCl liberated in the reaction is neutralised by an alkali (e.g. NaHCO3) added to the aqueous phase. The reaction does not take place in the acid phase due to the susceptibility of the acid chloride to hydrolysis. The polycondensation reaction can occur in the aqueous phase for situations where the diacid chloride has substantial solubility in the aqueous phase and has good hydrolytic stability (Morgan, 1965). The rate constant for the reaction between diacyl chlorides and diamines is large, ~106 cm3 mol–1 s–1 or higher (Doraiswamy and Sharma, 1984). A major application of interfacial polymerisation is in the preparation of thin-film-composite reverse osmosis membranes (e.g. FT-30). The structure of a thin-film-composite membrane is shown in Figure 12.9. The topmost thin layer is responsible for the separation of solute. This layer is made

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Colloid and Interface Science

by interfacial polycondensation between tri-mesoyl chloride [C6H3(COCl)3] and m-phenylene diamine [C6H4(NH2)2]. The reaction occurs by first soaking the polysulphone support with aqueous solution of m-phenylene diamine, and then spreading the hexane-solution of tri-mesoyl chloride over it.

Figure 12.9

Thin-film-composite reverse osmosis membrane.

Microencapsulation has been found to be an effective method for encapsulating water-insoluble pesticides and biological species (Yadav et al., 1990). These microcapsules are used for controlled or sustained release of the encapsulated material. The performance of these microcapsules depends on the morphology and surface characteristics of the polymeric shell and their size. Both of these characteristics depend upon the materials and process conditions. If the polymer is highly crosslinked, the permeability of the encapsulated substance will be very low. Another important parameter is the time required for encapsulation, which is necessary for designing the encapsulator. The hydrophobic substance to be encapsulated and a hydrophobic monomer (e.g. diisocyanate) is dispersed in an aqueous medium stabilised by a surfactant. The aqueous phase contains a hydrophilic monomer (e.g. a diamine). The reaction starts at the surface of the drops and a shell of the polymer forms at the surface (see Figure 12.10). At subsequent times, the monomer B has to diffuse through the growing shell before it can react with the monomer A, which is present in the organic phase. The polymer swells as it forms. The monomer in the aqueous phase (B) is usually the limiting reactant. The reaction ceases when B is exhausted. The formation of the microcapsule is depicted in Figure 12.10.

Figure 12.10 Formation of microcapsule by interfacial polycondensation.

Interfacial Reactions

471

Let us consider the interfacial polycondensation between hexamethylene-1,6-diamine and hexamethylene-1,6-diisocyanate to encapsulate butachlor (a pesticide) (Yadav et al., 1990). A few assumptions are made to model the process as follows. All the drops are assumed to have the same radius R0 at t = 0. If the dispersion has drops having a wide range of size, the surface-to-volume mean radius can be used to perform the calculation instead of R0. The dispersion is agitated and, therefore, the rate of external mass transfer is expected to be rapid. Thus, only the diffusion of B through the polymer shell and the reaction at the interface are considered. The diffusion of A within the drop is assumed to be rapid and not rate-limiting. The monomer B can diffuse through the polymer as well as through the micropores of the shell. Therefore, an effective diffusivity (De) is used to describe the diffusion of B. De is assumed to be independent of the thickness of the polymeric shell. The rate of movement of the inner boundary due to the growth of the polymer shell is assumed to be small in comparison with the rate of diffusion of B. The pseudo-steady-state approximation is assumed to be applicable for the transport of B through the polymer wall. The interfacial kinetics is expressed as 1 4S

Ri2

dn A dt

ke c Ae cBe

(12.53)

where nA is the number of moles of A in the drop at any time t, and cA e and cBe are volumetric concentrations at the inner radius of the shell Ri. The diffusion of monomer B through the polymeric shell obeys the following equation under the pseudo-steady-state assumption: De d È 2 dcB Ø É r dr ÙÚ r 2 dr Ê

At r = Rc :

cB

0

(12.54)

F cBs

(12.55)

dcB ke c Ae cBe (12.56) dr where cB is the concentration of monomer B in the polymer shell, cBs is the concentration of monomer B in the continuous phase, c is the partition coefficient for monomer B between the continuous and polymer phases, and Rc is the outer radius of the developing microcapsule. It differs from R0 (the original radius of the drop) because of swelling. De

At r = Ri :

Rc

[ Ri3  D ( R03  Ri3 )]1 3

(12.57)

where a is a swelling factor. The concentrations of A and B are referred to the phase in which they are present. The instantaneous concentration profile of B in the polymeric shell is given by

cB

Ë 4S Ri De 1 1 Ì 3k n  R  r e A i F cBs Ì 1 Ì 4S Ri De 1 Ì 3k n  R  R e A i c Í

Û Ü Ü Ü Ü Ý

(12.58)

The conversion of B is defined as xB

1

cBs cB 0

(12.59)

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Colloid and Interface Science

where cB0 is the concentration of monomer B in the bulk solution at t = 0. If the thickness of the polymeric shell is small relative to the radius of the capsule, the following relation between time and conversion is obtained (Yadav et al., 1990): t



2 3I E 3 È a  x B Ø 3I  (3  2b)(a  1) ln ln(1  xB )  xB   (1  a)(3  ab) ÉÊ a ÙÚ E E (a  1)(3  b) bE

Ë I b2  3(3  ab) Û È bx B Ø Ì Ü ln É 1  3 ÙÚ (12.60) ÌÍ (3  b)(3  ab) ÜÝ Ê

where a, b and f are dimensionless parameters which are defined as follows. Nn A 0 VcB 0

a

(12.61)

VcB0 M A  M B

b

(12.62)

Vd Us 3De UsVd2

I

D R0 M A  M B ke VcBo

2

(12.63)

where Vd is the volume of the dispersed organic phase, N is the number of droplets in the system, V is the volume of the aqueous phase, rs is the density of the polymer formed, MA is the molecular weight of monomer A, MB is the molecular weight of monomer B, and nA 0 is the moles of monomer A in the drop at t = 0. The parameter b is defined as E

9 F De Us Vd2

D R02

M A  M B V 2 cB 0

(12.64)

Let us now consider two limiting situations: (i) kinetic control, and (ii) diffusion control. When the rate of chemical reaction is much smaller than that of diffusion, the kinetic control prevails. The interfacial polycondensation is kinetically controlled if the following criterion is satisfied: I !! [ x B (a  x B )(1  2bx B 3)]

(12.65)

The conversion-time relationship in this case is given by t

R0 F ke cB0

Ë b 1 1 È a  xB Ø È bx Ø Û ln É ln(1  xB )  ln É1  B Ù Ü (12.66)  Ì Ù (3  b)(3  ab) Ê 3 ÚÝ Í (a  1)(3  ab) Ê a Ú (a  1)(3  b)

The encapsulation process is kinetically controlled at t = 0 because no polymer shell exists in the beginning which can offer a diffusional resistance. The process may subsequently become diffusion-controlled after a certain conversion is reached. The condition for diffusion control is I  [ x B (a  x B )(1  2bx B 3)]

(12.67)

The conversion-time relationship in this case is given by t

1 E

0 Ë ÎÑ 1  bx B 3 ÞÑÛ 9 È 3  2b Ø È 1  x B Ø  ln ln Ï Ì 2( x B  x B0 )  É ßÜ É Ù Ù Ê 3  b Ú Ê 1  x B Ú (3  b)b ÑÐ 1  bx B0 3 ÑàÜ ÌÍ Ý

(12.68)

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473

where x B0 is the conversion after which the control shifts from kinetic (at the beginning) to diffusion. Equations (12.66) and (12.68) indicate that the time required for a definite conversion is proportional to R0 for kinetic control. On the other hand, t µ R02 in the case of diffusion control. EXAMPLE 12.7 Consider the interfacial polycondensation between hexamethylene-1,6diisocyanate and hexamethylene-1,6-diamine to form polyurea microcapsules. The amount of diisocyanate present in the organic phase is 4 × 10–4 kg. The organic phase is dispersed in the aqueous phase by an emulsifier which prevents coalescence between the droplets. The volumes of the aqueous and organic phases are 100 cm3 and 80 cm3 respectively. Calculate the time required to achieve 70% conversion after 3.5 × 10–4 kg of diamine is added to the aqueous phase. The dispersion is mechanically agitated and the mean diameter of the droplets is 5 mm. The density of the polymer is 1000 kg/m3, rate constant for the reaction is 1 × 10–6 m4 kmol–1 s–1, partition coefficient of the diamine between the aqueous and polymer phases is 0.5, effective diffusivity of the diamine through the polymer is 1 × 10–13 m2/s, and the swelling factor is 1.6. Solution Let us designate hexamethylene-1,6-diisocyanate as A and hexamethylene-1,6-diamine as B. Therefore, MA = 168 kg/kmol and MB = 116 kg/kmol.

1 – 10 4 m 3 and Vd

V

3.5 – 10 4

cB 0

4 – 10 4 168

2.4 – 10 6 kmol

2.4 – 10 6

a

NnA 0 VcB0

b

VcB 0 ( M A  M B ) Vd Us

I

0.03 kmol/m 3

116 – 1 – 10 4

NnA 0

0.8 – 10 4 m 3

1 – 10 4 – 0.03

0.8

1 – 10 4 – 0.03 – (168  116) 8 – 10 5 – 1000

0.011

3De UsVd2

3 – 1 – 10 13 – 1000 – (8 – 10 5 )2

D R0 ( M A  M B )ke (VcBo )2

1.6 – 2.5 – 10 6 – (168  116) – 1 – 10 5 – (1 – 10 4 – 0.03)2

x B (a  x B )(1  2bx B 3)

2 – 0.011 – 0.7 Ø È 0.7 – (0.8  0.7) – É 1  ÙÚ Ê 3

18.8

0.07

Therefore, we observe that I !! x B (a  x B )(1  2bx B 3), which indicates that the reaction is kinetically controlled. t

R0 F ke cB 0

Ë b 1 1 È a  xB Ø È bx Ø Û ln É ln(1  x B )  ln É1  B Ù Ü  Ì Ù (3  b)(3  ab) Ê 3 ÚÝ Í (a  1)(3  ab) Ê a Ú (a  1)(3  b)

R0

2.5 – 10 6 m

F ke

0.5 – 1 – 10 6

5 – 10 7 m 4 kmol –1 s1

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Colloid and Interface Science

R0 F ke cB 0

2.5 – 10 6 5 – 10 7 – 0.03

166.7 s

For xB = 0.7, we get 1 È a  xB Ø ln ( a  1)(3  ab) ÉÊ a ÙÚ

1 ln(1  x B ) (a  1)(3  b) bx Ø b È ln 1  B Ù (3  b)(3  ab) ÉÊ 3 Ú

1 È 0.8  0.7 Ø ln (0.8  1)(3  0.8 – 0.011) ÉÊ 0.8 ÙÚ

1 ln(1  0.7) (0.8  1)(3  0.011)

3.46

2

0.011 0.011 – 0.7 Ø È ln 1  ÙÚ (3  0.011)(3  0.8 – 0.011) ÉÊ 3

3.11 – 10 6

Therefore,

t

166.7 – (3.46  2  3.11 – 10 6 )

243.4 s

Karode et al. (1997) have summarised the various mechanisms proposed for interfacial polycondensation reactions in the literature. In the general situation, three steps are considered: (i) mass transfer of both monomers from the bulk to the interface, (ii) diffusion of diamine through the polymer film, and (iii) second-order reaction. These steps are illustrated in Figure 12.11. The partition coefficient of B between aqueous phase and polymer film is represented by c, and the same between the polymer film and the organic solvent is represented by co. The superscript ‘ap’ indicates the interface between the aqueous phase and the polymer film, and the superscript ‘r’ indicates the reaction zone.

Figure 12.11 Schematic diagram of interfacial polycondensation.

As mentioned before, the reaction cannot take place in the aqueous phase due to the susceptibility of the acid chloride to hydrolysis. The diamine (B) partitions into the organic phase to initiate the polymerisation reaction. It encounters excess of diacid chloride (A) in the reaction zone of volume aid, where d is the thickness of the reaction zone and ai is the interfacial area. The approximate thickness of the reaction zone is ~10 nm. The reaction produces an oligomeric species

Interfacial Reactions

475

which has terminal amine and acid chloride groups. This immediately reacts with the surrounding excess diacid chloride to form a larger oligomer which is acylated at both the ends. The overall reaction is an irreversible coupling of the acid chloride with the incoming diamine. The polymerisation reaction scheme is summarised in Table 12.2. Table 12.2

Interfacial polycondensation between diamine and diacid chloride

Symbol

Monomer/polymeric species

An B Cn

Cl–(OC–R2–CONH–R1–NH)n–OC–R2–COCl H2N–R1–NH2 H–(NH–R1–NHCO–R2–CO)n–Cl A  Bn  C n 1  HCl, rate Cn 1  Am  An  m 1  HCl, rate

4 k p An B 2 k p C n 1 Am

The rate constant kp is for the reaction between one –NH2 group and one –COCl group. Flory’s equal-reactivity hypothesis is assumed to hold for all the reactions. The product of reaction is HCl. It is assumed to diffuse into the aqueous phase rapidly where it is neutralised by an alkali such as NaHCO3. The growth of polymer film as a function of time for the polymerisation of Nylon 6-10 under unstirred conditions is depicted in Figure 12.12.

Figure 12.12

Growth of Nylon 6–10 film by interfacial polycondensation in methylene chloride (Karode et al., 1997) [adapted by permission from Elsevier Ltd., © 1997].

An important interfacial polycondensation is the reaction between bisphenol-A and phosgene to produce polycarbonate. The reaction scheme is shown in Figure 12.13. In this reaction, methylene chloride is usually used as the solvent. An aqueous solution of sodium hydroxide can be used to accept the HCl formed by the reaction (Wielgosz et al., 1972).

Figure 12.13

Manufacture of polycarbonate by interfacial polycondensation.

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Colloid and Interface Science

12.4 GAS–LIQUID AND LIQUID–LIQUID INTERFACIAL REACTIONS Gas–liquid and liquid–liquid reactions can take place in the bulk phase as well as at the interface. To illustrate, a reactive component in a gas, when brought into contact with a liquid that contains another reactant, absorbs at the gas–liquid interface and diffuses through the bulk of the liquid simultaneously undergoing reaction. If the reaction is fast (e.g. absorption of CO2 in aqueous solutions of amines and alkalis, absorption of NO2 in water, and nitration of toluene by mixed acid at high concentrations of sulphuric acid) or instantaneous (e.g. reaction of H2S with aqueous NaOH solution and absorption of chlorine in aqueous solutions of Na2CO3 or NaOH), the diffusing reactant is completely converted near the interface. If the solubility of one reactant is very low in the other phase, the reaction takes place at the interface. In practice, the reaction occurs in a thin film near the interface. The thickness of the film is approximately 10 nm. The concentration profiles of the reactants for an instantaneous interfacial reaction are shown in Figure 12.14 as per the film theory. Reactant A from one phase diffuses into the other phase containing the reactant B. The reaction takes place at a reaction plane, which is depicted by the vertical line at l. The two reactants A and B react instantaneously on reaching the front.

Figure 12.14 Concentration profiles of the reactants for instantaneous reaction at the interface as per the film theory.

The relative magnitudes of rate of reaction and the rate of physical mass transfer are expressed in terms of Hatta number (named after Siroji Hatta of Tohoku University, Japan). It is defined as 12

Ha

Ë Maximum rate of reaction of A in the film per unit interfacial area Û Ì Ü Í Maximum rate of physical mass transfer per unit interfacial area Ý

(12.69)

If the reaction is of second-order, the Hatta number is defined as (Doraiswamy and Sharma, 1984; Westerterp et al., 2001)

Ha

DA k2 cBbulk kL

(12.70)

Interfacial Reactions

477

where k2 is the reaction rate constant, DA is the diffusivity of reactant A, DB is the diffusivity of the reactant B and kL is the liquid-phase mass transfer coefficient. If Ha >> 1 and Ha !!

reaction is instantaneous. On the other hand, if Ha >> 1 and Ha