Structure-Performance Relationships in Surfactants [2 ed.] 0824740440, 9780824740443, 9780824748258

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STRUCTUREPERFORMANCE RELATIONSHIPS IN SURFACTANTS Second Edition, Revised and Expanded edited by

Kunio Esumi Minoru Ueno Tokyo University of Science Tokyo, Japan

MARCEL

MARCEL DEKKER, INC.

NEW YORK • BASEL

ISBN: 0-8247-4044-0 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright # 2003 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

SURFACTANT SCIENCE SERIES

FOUNDING EDITOR

MARTIN J.SCHICK 1918-1998 SERIES EDITOR

ARTHUR T. HUBBARD Santa Barbara Science Project Santa Barbara, California ADVISORY BOARD

DANIEL BLANKSCHTEIN Department of Chemical Engineering

ERIC W. KALER Department of Chemical Engineering

Massachusetts Institute of Technology Cambridge, Massachusetts

University of Delaware Newark, Delaware

S. KARABORNI Shell International Petroleum Company Limited London, England

CLARENCE MILLER Department of Chemical Engineering Rice University Houston, Texas

LISA B. QUENCER The Dow Chemical Company Midland, Michigan

DON RUBINGH The Procter & Gamble Company Cincinnati, Ohio

JOHN F. SCAMEHORN Institute for Applied Surfactant Research

BEREND SMIT Shell International Oil Products B. V. Amsterdam, The Netherlands

University of Oklahoma Norman, Oklahoma

P. SOMASUNDARAN Henry Krumb School of Mines Columbia University New York, New York

JOHN TEXTER Strider Research Corporation Rochester, New York

1. Nonionic Surfactants, edited by Martin J. Schick (see also Volumes 19, 23, and 60) 2. Solvent Properties of Surfactant Solutions, edited by Kozo Shinoda (see Volume 55) 3. Surfactant Biodegradation, R. D. Swisher (see Volume 18) 4. Cationic Surfactants, edited by Eric Jungermann (see also Volumes 34, 37, and 53) 5. Detergency: Theory and Test Methods (in three parts), edited by W. G. Cutler and R. C. Davis (see also Volume 20) 6. Emulsions and Emulsion Technology (in three parts), edited by Kenneth J. Lissant 7. Anionic Surfactants (in two parts), edited by Warner M. Linfield (see Volume 56) 8. Anionic Surfactants: Chemical Analysis, edited by John Cross 9. Stabilization of Colloidal Dispersions by Polymer Adsorption, Tatsuo Sato and Richard Ruch 10. Anionic Surfactants: Biochemistry, Toxicology, Dermatology, edrted by Christian Gloxhuber (see Volume 43) 11. Anionic Surfactants: Physical Chemistry of Surfactant Action, edited by E. H. Lucassen-Reynders 12. Amphoteric Surfactants, edited by B. R. Bluestein and Clifford L Hilton (see Volume 59) 13. Demulsification: Industrial Applications, Kenneth J. Lissant 14. Surfactants in Textile Processing, Arved Datyner 15. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applications, edited byAyao Kitahara andAkira Watanabe 16. Surfactants in Cosmetics, edited by Martin M. Rieger (see Volume 68) 17. Interfacial Phenomena: Equilibrium and Dynamic Effects, Clarence A. Miller and P. Neogi 18. Surfactant Biodegradation: Second Edition, Revised and Expanded, R. D. Swisher 19. Nonionic Surfactants: Chemical Analysis, edited by John Cross 20. Detergency: Theory and Technology, edited by W. Gale Cutler and Erik Kissa 21. Interfacial Phenomena in Apolar Media, edited by Hans-Friedrich Eicke and Geoffrey D. Parfitt 22. Surfactant Solutions: New Methods of Investigation, edited by Raoul Zana 23. Nonionic Surfactants: Physical Chemistry, edited by Martin J. Schick 24. Microemulsion Systems, edited by Henri L. Rosano and Marc Clausse 25. Biosurfactants and Biotechnology, edited by Nairn Kosaric, W. L. Cairns, and Neil C. C. Gray 26. Surfactants in Emerging Technologies, edited by Milton J. Rosen 27. Reagents in Mineral Technology, edited by P. Somasundaran and Brij M. Moudgil 28. Surfactants in Chemical/Process Engineering, edited by Darsh T. Wasan, Martin E. Ginn, and Dinesh O. Shah 29. Thin Liquid Films, edited by /. B. Ivanov 30. Microemulstons and Related Systems: Formulation, Solvency, and Physical Properties, edited by Maurice Bourrel and Robert S. Schechter 31. Crystallization and Polymorphism of Fats and Fatty Acids, edited by Nissim Garti and Kiyotaka Sato

32. Interfacial Phenomena in Coal Technology, edited by Gregory D. Botsaris and Yuli M Glazman 33. Surfactant-Based Separation Processes, edited by John F. Scamehom and Jeffrey H. Harwell 34. Cationic Surfactants: Organic Chemistry, edited by James M. Richmond 35. Alkylene Oxides and Their Polymers, F. E. Bailey, Jr., and Joseph V. Koleske 36. Interfacial Phenomena in Petroleum Recovery, edrfed by Norman R Morrow 37. Cationic Surfactants: Physical Chemistry, edited by Donn N. Rubingh and Paul M. Holland 38. Kinetics and Catalysis in Microheterogeneous Systems, edited by M. Gratzel and K Katyanasundaram 39. Interfacial Phenomena in Biological Systems, ectted by Max Bender 40. Analysis of Surfactants, Thomas M. Schmiti (see Volume 96) 41. Light Scattering by Liquid Surfaces and Complementary Techniques, edited by Domimque Langevin 42. Polymeric Surfactants, Irja Piirma 43. Anionic Surfactants: Biochemistry, Toxicology, Dermatology. Second Edition, Revised and Expanded, edited by Christian Gloxhuberand Klaus Kunstler 44. Organized Solutions: Surfactants in Science and Technology, edited by Stig E Friberg and Bjom Lindman 45. Defoaming: Theory and Industrial Applications, edited by P. R. Garrett 46. Mixed Surfactant Systems, edited by Keizo Ogino and Masahiko Abe 47. Coagulation and Flocculation: Theory and Applications, edited by Bohuslav Dobias 48. Biosurfactants: Production • Properties • Applications, ecWed by Nairn Kosaric 49. Wettability, edited by John C. Berg 50. Fluorinated Surfactants: Synthesis • Properties • Applications, Erik Kissa 51. Surface and Colloid Chemistry in Advanced Ceramics Processing, edited by Robert J. Pugh and Lennart Bergstrom 52. Technological Applications of Dispersions, edited by Robert B. McKay 53. Cationic Surfactants: Analytical and Biological Evaluation, edited by John Cross and Edward J. Singer 54. Surfactants in Agrochemicals, Tharwat F. Tadros 55. Solubilization in Surfactant Aggregates, edited by Sherril D. Christian and John F. Scamehom 56. Anionic Surfactants: Organic Chemistry, edited by Helmut W. Stache 57. Foams: Theory, Measurements, and Applications, edited by Robert K. Prud'homme and Saad A. Khan 58. The Preparation of Dispersions in Liquids, H. N. Stein 59. Amphoteric Surfactants: Second Edition, edited by Eric G. Lomax 60. Nonionic Surfactants: Polyoxyalkylene Block Copolymers, edited by Vaughn M. Nace 61. Emulsions and Emulsion Stability, edited by Johan Sjdblom 62. Vesicles, edited by Morion Rosoff 63. Applied Surface Thermodynamics, edited by A. W. Neumann and Jan K Spelt 64. Surfactants in Solution, edited byArun K. Chattopadhyay and K. L. Mittat 65. Detergents in the Environment, edited by Milan Johann Schwuger

66. Industrial Applications of Microemulsions, edited by Conxita Solans and Hironobu Kunieda 67. Liquid Detergents, edited by Kuo-Yann Lai 68. Surfactants in Cosmetics: Second Edition, Revised and Expanded, edited by Martin M. Rieger and Linda D. Rhein 69. Enzymes in Detergency, edited by Jan H. van Ee, Onno Misset, and Erik J. Baas 70. Structure-Performance Relationships in Surfactants, edited by Kunio Esumi and Minoru Ueno 71. Powdered Detergents, edited by Michael S. Showell 72. Nonionic Surfactants: Organic Chemistry, edited by Nico M. van Os 73. Anionic Surfactants: Analytical Chemistry, Second Edition, Revised and Expanded, edited by John Cross 74. Novel Surfactants: Preparation, Applications, and Biodegradability, edited by Krister Holmberg 75. Biopolymers at Interfaces, edited by Martin Malmsten 76. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applications, Second Edition, Revised and Expanded, edited by Hiroyuki Ohshima and Kunio Furusawa 77. Polymer-Surfactant Systems, edited by Jan C. T. Kwak 78. Surfaces of Nanoparticles and Porous Materials, edited by James A. Schwarz and Cristian I. Contescu 79. Surface Chemistry and Electrochemistry of Membranes, edited by Torben Smith S0rensen 80. Interfacial Phenomena in Chromatography, edited by Emile Pefferkom 81. Solid-Liquid Dispersions, Bohuslav Dobias, Xueping Qiu, and Wolfgang von Rybinski 82. Handbook of Detergents, editor in chief: Uri Zoller Part A: Properties, edited by Guy Broze 83. Modem Characterization Methods of Surfactant Systems, edited by Bernard P. Binks 84. Dispersions: Characterization, Testing, and Measurement Erik Kissa 85. Interfacial Forces and Fields: Theory and Applications, edited by Jyh-Ping Hsu 86. Silicons Surfactants, edited by Randal M. Hill 87. Surface Characterization Methods: Principles, Techniques, and Applications, edited by Andrew J. Milling 88. Interfacial Dynamics, edited by Nikola Kallay 89. Computational Methods in Surface and Colloid Science, edited by Mafaorzata Bordwko 90. Adsorption on Silica Surfaces, eoVted by Eugene Papirer 91. Nonionic Surfactants: Alkyl Polyglucosides, edited by Dieter Balzer and Harald Luders 92. Fine Particles: Synthesis, Characterization, and Mechanisms of Growth, edited by Tadao Sugimoto 93. Thermal Behavior of Dispersed Systems, ectted by Nissim Garti 94. Surface Characteristics of Fibers and Textiles, edited by Christopher M. Pastore and Paul Kiekens 95. Liquid Interfaces in Chemical, Biological, and Pharmaceutical Applications, edrtedbyA/exanderG. Volkov

96. Analysis of Surfactants: Second Edition, Revised and Expanded, Thomas M. Schmitt 97. Fluorinated Surfactants and Repellents: Second Edition, Revised and Expanded, Erik Kissa 98. Detergency of Specialty Surfactants, edited by Floyd E. Fried// 99. Physical Chemistry of Polyelectrolytes, edited by Tsetska Radeva 100. Reactions and Synthesis in Surfactant Systems, edited by John Texter 101. Protein-Based Surfactants: Synthesis, Physicochemical Properties, and Applications, edited by Ifendu A. Nnanna and Jiding Xia 102. Chemical Properties of Material Surfaces, Marek Kosmulski 103. Oxide Surfaces, edrfed by James A. Wingrave 104. Polymers in Particulate Systems: Properties and Applications, edited by Vincent A. Hackley, P. Somasundaran, and Jennifer A. Lewis 105. Colloid and Surface Properties of Clays and Related Minerals, Rossman F. Giese and Caret J. van Oss 106. Interfacial Electrokinetics and Electrophoresis, edited by Angel V. Delgado 107. Adsorption: Theory, Modeling, and Analysis, edited by Jdzsef T6th 108. Interfacial Applications in Environmental Engineering, edited by Mark A. Keane 109. Adsorption and Aggregation of Surfactants in Solution, edited by K. L Mittal andDineshO. Shah 110. Biopolymers at Interfaces: Second Edition, Revised and Expanded, edited by Martin Malmsten 111. Biomolecular Films: Design, Function, and Applications, edited by James F. Rusting 112. Structure-Performance Relationships in Surfactants: Second Edition, Revised and Expanded, edited by Kunio Esumi and Minoru Ueno

ADDITIONAL VOLUMES IN PREPARATION

Liquid Interfacial Systems: Oscillations and Instability, Rudolph V. Birikh, Vladimir A. Briskman, Manuel G. Velarde, and Jean-Claude Legros Novel Surfactants: Preparation, Applications, and Biodegradability: Second Edition, Revised and Expanded, edited by Krister Holmberg Colloidal Polymers: Preparation and Biomedical Applications, edited by Abdelhamid Elaissari

Preface

Surfactant molecules can self-assemble in water, in oil, and in oil–water or solid–liquid mixtures to give a large variety of colloidal structures. Structure–performance relationships in surfactants are of great importance in nearly all fundamental studies and practical applications of surfactants. Six years ago, the first edition of this book was published as Volume 70 in the Surfactant Science series. Its aim was to examine properties and performance of surfactants at various interfaces, such as air–liquid, liquid–liquid, and solid–liquid. Research on new surfactants has been intense in recent years. Now, greatly expanded interest and additional important work in this field have led us to update the book to reflect current trends. This volume has 18 chapters, which can be classified into three parts: theoretical studies of surfactants (Chapter 1 and 2), physicochemical properties of surfactants at the air–liquid interface and in solutions (Chapters 3 through 14), and surfactant behavior at the solid–liquid interface (Chapters 15 through 18). In Chapter 1, Nagarajan presents the quantitative approach to predicting the aggregation properties of surfactants and surfactant–polymer mixtures. Chapter 2, by Koopal, reviews the thermodynamic models for micellization/adsorption and discusses the self-consistent-field lattice model (SCFA) for association and adsorption of surfactants. Aratono, Villeneuve, and Ikeda’s discussion in Chapter 3 focuses on the surface tension and adsorption behaviour of spontaneously vesicle-forming surfactants. In Chapter 4, Ueno and Asano outline the mixed properties of bile salts and some nonionic surfactants and give examples for application of these systems. In Chapter 5, Ishigami describes the molecular design and characterization of biosurfactants, along with applications of multifunctional structure of biosurfactants. Chapter 6, by Koide and Esumi, deals with the physicochemical properties of ring-structured surfactants, including those of crown ether type, those of polyamine type, cyclodextrin, and iii

iv

Preface

calix[n]arene. Zana’s Chapter 7 discusses the physicochemical properties of dimeric surfactants, such as adsorption at the air–solution and solid–solution interfaces, micelle formation, solubilization, micelle size and shape, rheology, phase behavior, and some applications. In Chapter 8, Yoshino compares the synthesis and properties of two series of double chain–type fluorinated anionic surfactants. One of these is a series of surfactants with two fluorocarbon chains in their molecules; the other is a series of hybrid-type surfactants having both fluorocarbon and hydrocarbon chains in one molecule. Chapter 9, by Yoshimura and Esumi, describes the physicochemical properties of telomer-type surfactants having several hydrophobic groups and several hydrophilic groups; these surfactants often exhibit properties of both polymer-type and conventional surfactants. In Chapter 10, Hoffmann analyzes various types of viscoelastic surfactant systems, describing rheological properties and presenting models for understanding the different flow behaviors based on the different microstructures. Kato’s Chapter 11 presents the micelle structure of nonionic surfactants in dilute, semidilute, and concentrated solutions and discusses the thermodynamic models for micellar solutions and the phase transitions in liquid crystal phases. In Chapter 12, Imae reviews the amphiphilic properties and association behavior of concentric dendrimers and hybrid copolymers. Chapter 13, by Zana, describes how polymer hydrophobicity and the surfactant head group affect polymer–surfactant interactions; the chapter also addresses microstructural aspects, solubilization, and dynamic behaviors of polymer–surfactant aggregrates. In Chapter 14, Uddin, Kunieda, and Solans describe the preparation and properties of highly concentrated cubic phase-based emulsions, as well as the correlation between D-phase emulsification and cubic phase-based emulsions. In Chapter 15, Treiner outlines the adsolubilization and related phenomena at solid–solution interfaces and presents some applications. Esumi’s Chapter 16 focuses on the adsorption of polymers and surfactants from their binary mixtures on oxide surface, also discussing the conformation of polymers adsorbed on particles. Chapter 17, also by Esumi, deals with the dispersion of particles by surfactants as well as the properties of surfactant-adsorbed layers. In Chapter 18, Fujii reviews the AFM techniques for the study of surfactant molecules, especially those relating to the morphology of the surfactant aggregations on solid–liquid interfaces. We would like to thank the authors who participated in this effort, and we are indebted to Anita Lekhwani, Joseph Stubenrauch, Michael Deters, and Elissa Ryan of Marcel Dekker, Inc., for their assistance in preparing this volume. Kunio Esumi Minoru Ueno

Contents

Preface Contributors Part I

iii vii

Theoretical Studies of Surfactants

1. Theory of Micelle Formation: Quantitative Approach to Predicting Micellar Properties from Surfactant Molecular Structure R. Nagarajan 2. Modeling Association and Adsorption of Surfactants Luuk K. Koopal

1 111

Part II Physicochemical Properties of Surfactants at the Air–Liquid Interface and in Solutions 3. Adsorption of Vesicle-Forming Surfactants at the Air–Water Interface Makoto Aratono, Masumi Villeneuve, and Norihiro Ikeda

197

4. Physicochemical Properties of Bile Salts Minoru Ueno and Hiroyuki Asano

227

5. Characterization and Functionalization of Biosurfactants Yutaka Ishigami

285

6. Physicochemical Properties of Ring-Structured Surfactants Yoshifumi Koide and Kunio Esumi

309

7. Dimeric (Gemini) Surfactants Raoul Zana

341

v

vi

Contents

8. Fluorinated Surfactants Having Two Hydrophobic Chains Norio Yoshino 9. Surface-Active Properties of Telomer-Type Surfactants Having Several Hydrocarbon Chains Tomokazu Yoshimura and Kunio Esumi 10. Viscoelastic Surfactant Solutions Heinz Hoffmann 11. Microstructures of Nonionic Surfactant–Water Systems: From Dilute Micellar Solution to Liquid Crystal Phase Tadashi Kato

381

399 433

485

12. Association Behavior of Amphiphilic Dendritic Polymers Toyoko Imae

525

13. Polymer/Surfactant Systems Raoul Zana

547

14. Highly Concentrated Cubic Phase-Based Emulsions Md. Hemayet Uddin, Hironobu Kunieda, and Conxita Solans

599

Part III

Surfactant Behaviors at the Solid–Liquid Interface

15. Adsolubilization and Related Phenomena Claude Treiner 16. Adsorption of Polymer and Surfactant from Their Binary Mixtures on an Oxide Surface Kunio Esumi 17. Dispersion of Particles by Surfactants Kunio Esumi

627

673 709

18. Arrangement of Adsorbed Surfactants on Solid Surfaces by AFM Observation Masatoshi Fujii

749

Index

789

Contributors

Makoto Aratono, Ph.D. Department of Chemistry, Faculty of Sciences, Kyushu University, Fukuoka, Japan Hiroyuki Asano, Ph.D. Research Laboratories, Nippon Menard Cosmetic Company, Ltd., Nagoya, Japan Kunio Esumi, Ph.D. Department of Applied Chemistry, Tokyo University of Science, Tokyo, Japan Masatoshi Fujii, Ph.D. Department of Chemistry, Tokyo Metropolitan University, Tokyo, Japan Heinz Hoffmann, Prof.Dr. Department of Physical Chemistry, University of Bayreuth, Bayreuth, Germany Norihiro Ikeda, Ph.D. Department of Environmental Science, Fukuoka Women’s University, Fukuoka, Japan Toyoko Imae, Ph.D. Research Center for Materials Science, Nagoya University, Nagoya, Japan Yutaka Ishigami, Ph.D. Tokyo Gakugei University and Meisei University, Tokyo, Japan, and Dong-Woo Fine-Chem Co. Ltd., Kyunggi-do, South Korea Tadashi Kato, D.Sc. Department of Chemistry, Tokyo Metropolitan University, Tokyo, Japan vii

viii

Contributors

Yoshifumi Koide, Ph.D.y Department of Applied Chemistry Biochemistry, Kumamoto University, Kumamoto, Japan

and

Luuk K. Koopal, Ph.D. Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Wageningen, The Netherlands Hironobu Kunieda, D.Eng. Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, Japan R. Nagarajan, Ph.D. Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania, U.S.A. Conxita Solans Institute of Chemical and Environmental Research, CSIC, Barcelona, Spain Claude Treiner, Ph.D. Laboratoire Liquides Ioniques et Interfaces Charge´es, Universite´ Pierre et Marie Curie, Paris, France Md. Hemayet Uddin Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, Japan Minoru Ueno, Ph.D. Department of Applied Chemistry and Institute of Colloid and Interface Science, Tokyo University of Science, Tokyo, Japan Masumi Villeneuve Japan

Faculty of Science, Saitama University, Saitama,

Tomokazu Yoshimura, Ph.D. Department of Applied Chemistry, Faculty of Science, Tokyo University of Science, Tokyo, Japan Norio Yoshino, Ph.D. Department of Industrial Chemistry and Institute of Colloid and Interface Science, Tokyo University of Science, Tokyo, Japan Raoul Zana, Ph.D.

y

Deceased

Institut C. Sadron, CNRS, Strasbourg, France

1 Theory of Micelle Formation Quantitative Approach to Predicting Micellar Properties from Surfactant Molecular Structure R. NAGARAJAN The Pennsylvania State University, University Park, Pennsylvania, U.S.A.

I.

INTRODUCTION

The numerous, practical applications of surfactants have their basis in the intrinsic duality of their molecular characteristics, namely, they are composed of a polar headgroup that likes water and a nonpolar tail group that dislikes water. A number of variations are possible in the types of the headgroup and tail group of surfactants. For example, the headgroup can be anionic, cationic, zwitterionic, or nonionic. It can be small and compact in size or an oligomeric chain. The tail group can be a hydrocarbon, fluorocarbon, or a siloxane. It can contain straight chains, branched or ring structures, multiple chains, etc. Surfactant molecules with two headgroups (bola surfactants) are also available. Further, the headgroups and tail groups can be polymeric in character, as in the case of block copolymers. This variety in the molecular structure of surfactants allows for extensive variation in their solution and interfacial properties. It is natural that one would like to discover the link between the molecular structure of the surfactant and its physicochemical action so that surfactants can be synthesized or selected specific to a given practical application. Pioneering contributions to our understanding of the general principles of surfactant self-assembly in solutions have come from the early studies of Tanford [1–3], Shinoda [4], and Mukerjee [5–9]. Utilizing their results, we have focused our effort in the last 25 years, on developing quantitative molecular thermodynamic models to predict the aggregation behavior of surfactants in solutions starting from the surfactant molecular structure 1

2

Nagarajan

and the solution conditions. In our approach, the physicochemical factors controlling self-assembly are first identified by examining all the changes experienced by a singly dispersed surfactant molecule when it becomes part of an aggregate [10–16]. Relatively simple, explicit analytical equations are then formulated to calculate the contribution to the free energy of aggregation associated with each of these factors. Because the chemical structure of the surfactant and the solution conditions are sufficient for estimating the molecular constants appearing in these equations, the free energy expressions can be used to make completely a priori predictions. In this chapter we describe in detail our quantitative approach to predicting the aggregation properties of surfactants from their molecular structures. In Section II we present the general thermodynamic equations that govern the aggregation properties of surfactants in solutions. Many conclusions about the aggregation behavior can be drawn from such analysis without invoking any specific models to describe the aggregates. In Section III we summarize the geometrical relations for various shapes of aggregates including spherical, globular, and rodlike micelles and spherical bilayer vesicles, consistent with molecular packing considerations as had been discussed many years ago by Tartar [17]. These considerations lead us to the concept of the packing parameter proposed by Israelachvili et al. [18] that has been widely cited in the literature. The molecular packing model, as it is currently used, is built on the free energy model of Tanford [1]. We examine its predictive power and then show that the model neglects some tail length-dependent free energy contributions; if these are included, they will lead to significantly different predictions of the aggregation behavior. In Section IV the molecular theory of micelle formation is described. The central feature of the theory is the postulation of explicit equations to calculate the free energy of formation of different types of aggregates invoking phenomenological concepts. We suggest how the molecular constants appearing in these equations can be estimated and describe the computational approach suitable for making predictive calculations. We then demonstrate the predictive power of the molecular theory via illustrative calculations performed on a number of surfactant molecules having a variety of headgroups and tail groups. This molecular theory presented in this section is the point of departure for all subsequent models described in this chapter and also for other aggregation phenomena involving solvent mixtures, solubilization, microemulsions, etc. that have been described elsewhere [19]. In Section V we extend the molecular theory to binary mixtures of surfactants and demonstrate its predictive ability for a variety of ideal and nonideal surfactant mixtures. The model is then applied to surfactants in the presence of nonionic polymers in Section VI. How the nonionic polymer changes the aggregate morphology, in addition to the critical concentrations

Theory of Micelle Formation

3

and aggregate microstructures, is explored in this section. The extension of the free energy model to block copolymers is done in Section VII, where the polymeric nature of the headgroup and the tail group of the molecule is taken into consideration in constructing the free energy expressions. The analogy between low-molecular-weight conventional surfactants and the block copolymer amphiphiles is discussed in this section. The last section presents some conclusions.

II. THERMODYNAMIC PRINCIPLES OF AGGREGATION A. Aggregate Shapes Figure 1 illustrates the shapes of surfactant aggregates formed in dilute solutions. The small micelles are spherical in shape. When large rodlike micelles form, they are visualized as having a cylindrical middle portion and parts of spheres as endcaps. The cylindrical middle and the spherical endcaps are allowed to have different diameters. When micelles cannot pack any more into spheres, and if at the same time the rodlike micelles are not yet favored by equilibrium considerations, then small, nonspherical globular aggregates form. Israelachvili et al. [18] have suggested globular shapes generated via ellipses of revolution for the aggregates in the sphere-to-rod transition region, after examining the local molecular packing requirements for various nonspherical shapes. The average surface area per surfactant molecule of the ellipses of revolution suggested by Israelachvili et al. [18] is practically the same as that of prolate ellipsoids, for aggregation numbers up to 3 times larger than the largest spherical micelles,. Therefore, the average geometrical properties of globular aggregates in the sphere-to-rod transition region can be computed as for prolate ellipsoids. Some surfactants pack into a spherical bilayer structure called a vesicle that encloses an aqueous cavity. In the outer and the inner layers of the vesicle, the surface area (in contact with water) per surfactant molecule, and the number of surfactant molecules need not be equal to one another and the thicknesses of the two layers can also be different. In addition to the variety of shapes, it is also natural to expect that surfactant aggregates of various sizes will exist in solution. The size distribution can be represented in terms of a distribution of the number of surfactant molecules constituting the aggregate (i.e., the aggregation number g).

B. Size Distribution of Aggregates Important results related to the size distribution of aggregates can be obtained from the application of general thermodynamic principles to surfactant solutions. The surfactant solution is a multicomponent system con-

4

Nagarajan

FIG. 1 Schematic representation of surfactant aggregates in dilute aqueous solutions. The structures formed include spherical micelles (a), globular micelles (b), spherocylindrical micelles (c), and spherical bilayer vesicles (d).

sisting of NW water molecules, N1 singly dispersed surfactant molecules, and Ng aggregates of aggregation number g, where g can take all values from 2 to 1. (The subscript W refers to water, 1 to the singly dispersed surfactant, and g to the aggregate containing g surfactant molecules.) All shapes of

Theory of Micelle Formation

5

aggregates are considered. Each of the aggregates of a given shape and size is treated as a distinct chemical component described by a characteristic chemical potential. The total Gibbs free energy of the solution G, expressed in terms of the chemical potentials
i of the various species i, has the form G ¼ Nw
w þ N1
1 þ

g¼1 X

N g
g

ð1Þ

g¼2

The equilibrium condition of a minimum of the free energy leads to
g ¼
1 g

ð2Þ

This equation stipulates that the chemical potential of the singly dispersed surfactant molecule is equal to the chemical potential per molecule of an aggregate of any size and shape. Assuming a dilute surfactant solution, one can write a simple expression for the chemical potential of an aggregate of size g (for all values of g including g ¼ 1) in the form:
g ¼
0g þ kT ln Xg

ð3Þ

where
0g is the standard state chemical potential of the species g and Xg is its mole fraction in solution. The standard states of all the species other than the solvent are taken as those corresponding to infinitely dilute solution conditions. The standard state of the solvent is defined as the pure solvent. Introducing the expression for the chemical potential in the equilibrium relation [(Eq. 2)], one gets the aggregate size distribution equation: ! !
0g
g
01 g

0g g g Xg ¼ X1 exp
¼ X1 exp
ð4Þ kT kT where

0g is the difference in the standard chemical potentials between a surfactant molecule present in an aggregate of size g and a singly dispersed surfactant in water [1]. If an expression for

0g is available, then from Eq. (4), the aggregate size distribution can be calculated. From the aggregate size distribution, all other important and experimentally accessible solution properties can be computed as described below.

C. Calculating CMC, Micelle Size, and Aggregate Polydispersity 1. Critical Micelle Concentration (CMC) The critical micelle concentration can be calculated from the P aggregate P size distribution by constructing a plot of one of the functions X , X , gXg , 1 g P 2 or g Xg (which are proportional to different experimentally measured

6

Nagarajan

properties of the surfactant solution such as surface tension, electrical conductivity, dye solubilization, Plight scattering intensity, etc.) against the total concentration Xtot ð¼ X1 þ gXg Þ of the surfactant in solution [1,20–22]. In all cases, the summation extends from 2 to 1. The CMC can be identified as that value of the total surfactant concentration at which a sharp change in the plotted function (representing a physical property) occurs. The CMC has also been estimated as that value of X1 for which the concentration of the singly dispersed surfactant is equalPto that of the surfactant present in the form of aggregates, namely, X1 ¼ gXg ¼ XCMC ½23. For surfactants in aqueous solutions, the estimates of the CMC obtained by the different methods mentioned above are usually close to one another, though not identical [20].

2. Average Micelle Size From the size distribution one can compute various average sizes of the aggregates based on the definitions P P 2 P 3 gXg g Xg g Xg P P gn ¼ ; gw ¼ ; gz ¼ P 2 ð5Þ Xg gXg g Xg where gn, gw, and gz denote the number average, the weight average, and the z average aggregation numbers, respectively, and the summations extend from 2 to 1, as mentioned earlier. The different average aggregation numbers correspond to those determined by different experimental techniques; for example, gn is obtained via membrane osmometry, gw via static light scattering, and gz via intrinsic viscosity measurements. If the summations in Eq. (5) extend from 1 to 1 (i.e., include the singly dispersed surfactant molecules), then one obtains apparent (as opposed to true) average aggregation numbers gn,app , gw,app, and gz,app.

3. Aggregate Size Polydispersity For nonionic and zwitterionic surfactants, the standard state free energy difference

0g appearing in Eq. (4) is not dependent on the concentration of the singly dispersed surfactant X1 or on the total concentration of the surfactant. Consequently, by taking the derivative of the size distribution relation with respect to X1 , one obtains [24–26]: i X 1 hX @ ln Xg ¼ P gXg @ ln X1 ¼ gn @ ln X1 Xg i X 1 hX 2 g Xg @ ln X1 ¼ gw @ ln X1 @ ln gXg ¼ P ð6Þ gXg hX i X 1 g3 Xg @ ln X1 ¼ gz @ ln X1 @ ln g2 Xg ¼ P 2 g Xg

Theory of Micelle Formation

7

where the average aggregation numbers defined by Eq. (5) have been introduced. Equation (6) shows that the average aggregation numbers gn and gw depend on the concentration ðXtot
X1 Þ of the micellized surfactant as P follows (note that gXg ¼ Xtot
X1 ):
 X gn @ ln gXg @ ln gn ¼ 1
gw
 ð7Þ X gz @ ln gw ¼
1 @ ln gXg gw The variances of the size dispersion  2 ðnÞ and  2 ðwÞ are defined by the relations: P
 ðg
gn Þ2 Xg g P ¼ g2n w
1  2 ðnÞ ¼ gn Xg ð8Þ P
 2 ðg
gw Þ gXg 2 2 gz P  ðwÞ ¼ ¼ gw
1 gw gXg Combining Eqs. (7) and (8), we can relate the variance of the size distribution to the concentration dependence of the average aggregation number as follows:


1 ðnÞ 2 gw @ ln gn @ ln gn P P ¼
1¼ 1

1  gn gn @ ln gXg @ ln gXg ð9Þ
 ðwÞ 2 gz @ ln gw P ¼
1¼ gw gw @ ln gXg Equation (9) states that the average aggregation numbers gn and gw must increase appreciably with increasing concentration of the aggregated surfactant if the micelles are polydispersed ½ðnÞ=gn and ðwÞ=gw are large]; the average aggregation numbers must be virtually independent of the aggregated surfactant concentration if the micelles are narrowly dispersed. For ionic surfactants,

0g is dependent on the ionic strength of the solution (discussed in Section IV.A) and thus on the concentration of the surfactant, making Eq. (6) inapplicable to them. However, it has been shown that Eq. (9) derived for nonionic and zwitterionic surfactants based on general thermodynamic considerations is valid also for ionic surfactants [25,26].

D. Sphere-to-Rod Transition of Micelles Micelles having spherical or globular shapes are usually small and narrowly dispersed. A different micellization behavior is, however, observed when

8

Nagarajan

large rodlike micelles are generated [5–9]. These aggregates are visualized as having a cylindrical middle part with two spherical endcaps as shown in Fig. 1. The standard chemical potential of a rodlike aggregate of size g containing gcap molecules in the two spherical endcaps and (g
gcap ) molecules in the cylindrical middle can be written [5–9,16,18,27] as
0g ¼ ðg
gcap Þ
0cyl þ gcap
0cap

ð10Þ

where
0cyl and
0cap are the standard chemical potentials of the molecules in the two regions of the rodlike aggregate, respectively. Introducing the above relation in the aggregate size distribution [Eq. (4)] yields " ! #g !

0cyl

0cap


0cyl exp
gcap Xg ¼ X1 exp
ð11Þ kT kT where

0cyl and

0cap are the differences in the standard chemical potentials between a surfactant molecule in the cylindrical middle or the endcaps of the spherocylindrical micelle and a singly dispersed surfactant molecule. Equation (11) can be rewritten as " !#

0cyl 1 g Y ¼ X1 exp Xg ¼ Y ; ; K kT ! ð12Þ

0cap


0cyl K ¼ exp gcap kT where K is a measure of the free energy advantage for a molecule present in the cylindrical portion compared to that in the spherical endcaps. The possibility of occurrence of rodlike aggregates at a given concentration X1 of the singly dispersed surfactant molecules is indicated by the proximity of the parameter Y to unity. The average aggregation numbers defined by Eq. (5) can be computed on the basis of the size distribution Eq. (12), by performing the analytical summation of the series functions [5–9,18]:


 Y Y 1 ; gw ¼ gcap þ 1þ gn ¼ gcap þ 1
Y 1
Y Y þ gcap ð1
YÞ ð13Þ The total concentration of surfactant present in the aggregated state can be also calculated analytically [5–9,18] and is given by the expression
 g 
X 1 gcap Y cap Y gXg ¼ 1þ ð14Þ K 1
Y gcap ð1
YÞ

Theory of Micelle Formation

9

Equation (13) shows that for values of Y close to unity, very large aggregates are formed. In the limit of Y close to unity and gcap ð1
YÞ  1, Eqs. (13) and (14) reduce to X


2 1 1 ; gXg ¼ K 1
Y




 Y gn ¼ gcap þ ; 1
Y




 Y gw ¼ gcap þ 2 1
Y ð15Þ

P Noting that gXg ¼ Xtot
X1 , the dependence of the average aggregation numbers on the surfactant concentration is obtained:


 1 ¼ gcap þ ðKðXtot
X1 ÞÞ1=2 1
Y
 2 gw ¼ gcap þ ¼ gcap þ 2ðKðXtot
X1 ÞÞ1=2 1
Y gn ¼ gcap þ

ð16Þ

It is evident from Eq. (16) that the weight average and the number average aggregation numbers must substantially deviate from one another if rodlike micelles form (i.e., gn, gw  gcap ), indicating large polydispersity in the micellar size. The polydispersity index gw/gn is close to 2 at sufficiently large surfactant concentrations. Further, the sphere-to-rod transition parameter K must be in the range of 108 to 1012, if the rodlike micelles are to form at physically realistic surfactant concentrations [5–9,18,27]. The critical micelle concentration is calculated from Eq. (12) with the recognition that Y is close to unity:

X1 ¼ XCMC



0cyl ¼ exp kT

! ð17Þ

The thermodynamic results obtained so far are independent of any specific expression for the standard free energy change

0g associated with aggregation and constitute general theoretical principles governing the aggregation behavior of surfactants. However, to perform quantitative predictive calculations of the aggregation behavior, specific expressions for

0g are needed. In the following section, we will introduce the expression for

0g formulated by Tanford [1–3] on a phenomenological basis and use it to examine the molecular packing model of aggregation that is widely cited in the literature on self-assembly.

10

Nagarajan

III. MOLECULAR PACKING MODEL A. Packing Constraint and Packing Parameter The notion of molecular packing into various aggregate shapes has been recognized in the early work of Tartar [17] and Tanford [1-3], as can be seen, for example, from Figure 9.1 of Tanford’s classic monograph (1). However, only after this concept was explored thoroughly in the work of Israelachvili et al. [18], taking the form of the packing parameter, has it evoked wide appreciation in the literature. We start by recognizing that the hydrophobic domain of a surfactant aggregate contains the surfactant tails. If the density in this domain is equal to that in similar hydrocarbon liquids, the surfactant tails must entirely fill the space in this domain. This implies that, irrespective of the shape of the aggregate, no point within the aggregate can be farther than the distance ‘S from the aggregate–water interface, where ‘S is the extended length of the surfactant tail. Therefore, at least one dimension of the surfactant aggregate should be smaller than or at most equal to 2‘S [1–3,17,18]. The volume of the hydrophobic domain is determined from the number of surfactant molecules g in the aggregate and the volume vS of the surfactant tail. The molecular packing parameter is defined as vS =a‘S , where vS and ‘S are the volume and the length of the surfactant tail and a is the surface area of the hydrophobic core of the aggregate expressed per surfactant molecule constituting the aggregate (hereafter referred to as the area per molecule). As we will see below, the concept of the molecular packing parameter allows a simple and intuitive insight into the self-assembly phenomenon.

B. Geometrical Relations for Aggregates For aggregates of various shapes containing g surfactant molecules, the volume of the hydrophobic domain of the aggregate, Vg , the surface area of contact between the aggregate and water, Ag, and the surface area at a distance  from the aggregate–water interface, Ag , are listed in Table 1. Also given in the table is a packing factor P, defined in terms of the geometrical variables characterizing the aggregate. Note that P is slightly different from the packing parameter vS =a‘S introduced above. The area Ag is employed in the computation of the free energy of electrostatic interactions between surfactant headgroups, while the packing factor P is used in the computation of the free energy of tail deformation, both discussed later in this chapter. From the geometrical relations provided in Table 1, it can be seen that, given any surfactant molecule, the geometrical properties of spherical or

Theory of Micelle Formation

11

globular micelles depend only on the aggregation number g. In the case of rodlike micelles, the geometrical properties are dependent on two variables—the radius of the cylindrical part and the radius of the spherical endcaps. For spherical bilayer vesicles, any three variables, such as the aggregation number g and the thicknesses ti and to of the inner and outer layers of the bilayer, determine the geometrical properties.

C. Packing Parameter and Predicting Aggregate Shapes If we consider a spherical micelle with a core radius RS, made up of g molecules, then the volume of the core Vg ¼ gvS ¼ 4R3S =3 and the surface area of the core Ag ¼ ga ¼ 4R2S ; and from these simple geometrical relations we get RS ¼ 3vS =a. If the micelle core is packed with surfactant tails without any empty space, then the radius RS cannot exceed the extended length ‘S of the tail. Introducing this constraint in the expression for RS, one obtains, 0  vS =a‘s  1=3, for spherical micelles. Similarly for cylindrical or bilayer aggregates made up of g surfactant molecules, the geometrical relations for the volume Vg and the surface area Ag are given in Table 1. These geometrical relations, together with the constraint that at least one dimension of the aggregate (the radius of the sphere or the cylinder, or the half-bilayer thickness) cannot exceed ‘s, lead to the following well-known [18] connection between the molecular packing parameter and the aggregate shape: 0  vS =a‘s  1=3 for sphere; 1=3  vS = a‘s  1=2 for cylinder; and 1=2  vS =a‘S  1 for bilayer. Therefore, if we know the molecular packing parameter, the shape and size of the equilibrium aggregate can be readily identified as shown above. This is the predictive sense in which the molecular packing parameter of Israelachvili et al. [18] has found significant use in the literature. For common surfactants, the ratio vS/‘S is a constant independent of   tail length, equal to 21A 2 for single tail and 42A 2 for double tail (1). Consequently, in the packing parameter vS =a‘S , only the area a reflects the specificity of the surfactant.. The area per molecule a is a thermodynamic quantity obtained from equilibrium considerations of minimum free energy and is not a simple variable connected to the geometrical shape and size of the surfactant headgroup. To estimate a, Israelachvili et al. [18] invoked the model for the standard free energy change on aggregation pioneered by Tanford [1]. In the framework of Tanford’s free energy model, the area a is influenced directly by the surfactant headgroup interactions. Hence, the accepted notion in the surfactant literature is, given a headgroup, a is fixed—which then, determines the packing parameter vS =a‘S —thus, the headgroup controls the equilibrium aggregate structure.

12

Nagarajan

TABLE 1

Geometrical Properties of Surfactant Aggregates

Spherical micelles (radius RS  ‘S Þ: Vg ¼

4R3S ¼ gvS 3

Ag ¼ 4R2S ¼ ga Ag ¼ 4ðRS þ Þ2 ¼ ga P¼

Vg v 1 ¼ S ¼ Ag RS aRS 3

Globular micelles (semiminor axis RS ¼ ‘s , semimajor axis b  3‘S , eccentricity E): Vg ¼

4R2S b ¼ gvS 3 "

# "
2 #1=2 sin
1 E RS ¼ ga; E ¼ 1
Ag ¼ 1þ 1=2 2 b Eð1
E Þ " # "
 #1=2 sin
1 E RS þ  2 2 Ag ¼ 2ðRS þ Þ 1 þ E ¼ 1
¼ ga ; bþ E ð1
E2 Þ1=2
1=3 Vg 3Vg v 1  P  0:406; Req ¼ P¼ ¼ S ; Ag RS aRS 4 3 2R2S

Cylindrical part of rodlike micelles (radius RC  ‘s , length LC Þ: Vg ¼ R2C LC ¼ gvS Ag ¼ 2RC LC ¼ ga; P¼

Ag ¼ 2ðRC þ ÞLC ¼ ga

Vg v 1 ¼ S ¼ Ag RC aRC 2

Endcaps of rodlike micelles (endcap radius RS  ‘S , cylinder radius RC  ‘S Þ: H ¼ RS ½1
f1
ðRC =RS Þ2 g1=2  " # 8R3S 2 2
H ð3RS
HÞ ¼ gvS Vg ¼ 3 3 Ag ¼ ½8R2S
4RS H ¼ ga Ag ¼ ½8ðRS þ Þ2
4ðRS þ ÞðH þ Þ ¼ ga P¼

Vg v ¼ S Ag RS aRS

Theory of Micelle Formation

13

TABLE 1 Continued Spherical vesicles (inner/outer radii Ri , Ro ; inner/outer layer thickness ti , to  ‘s Þ; Vg ¼

4½R3o
R3i  ¼ gvS ; 3

g ¼ go þ gi

Vgo ¼

4½R3o
ðRo
to Þ3  ¼ go vS 3

Vgi ¼

4½ðRi þ ti Þ3
R3i Þ ¼ gi vS 3

Ago ¼ 4R2o ;

Agi ¼ 4R2i

Ago ¼ 4ðRo þ Þ2 ¼ go ao Agi ¼ 4ðRi
Þ2 ¼ gi ai

D. Estimating Equilibrium Area from Tanford’s Free Energy Model In his phenomenological model Tanford suggests that the standard free energy change associated with the transfer of a surfactant molecule from its infinitely dilute state in water to an aggregate of size g (aggregation number) has three contributions: 

      

0g ¼

0g Transfer þ

0g Interface þ

0g Head

ð18Þ

The first term, ð

0g ÞTransfer , is a negative free energy contribution arising from the transfer of the tail from its unfavorable contact with water to the hydrocarbonlike environment of the aggregate core. The transfer free energy contribution depends on the surfactant tail but not on the aggregate shape or size. The second term, ð

0g ÞInterface , provides a positive contribution to account for the fact that the entire surface area of the tail is not removed from water but there is still residual contact with water at the surface of the aggregate core. This is represented as the product of a contact free energy per unit area  (or an interfacial free energy) and the surface area per molecule of the aggregate core, a. The third term, ð

0g ÞHead , provides another positive contribution representing the repulsive interactions between the headgroups that crowd at the aggregate surface. Because the repulsion would increase if the headgroups came close to one another, Tanford proposed an expression with an inverse dependence on a. Thus,

14

Nagarajan

the standard free energy change per molecule on aggregation proposed by Tanford has the form  0  0 ð19Þ

g ¼

g Transfer þa þ =a where  is the headgroup repulsion parameter. Starting from the free energy model of Tanford, the equilibrium aggregation behavior can be examined either by treating the surfactant solution as consisting of aggregates with a distribution of sizes or by treating the aggregate as constituting a pseudophase [1–3,15]. If the aggregate is viewed as a pseudophase, in the sense of small systems thermodynamics, the equilibrium condition corresponds to a minimum in the standard free energy change ð

0g Þ. The minimization can be done with respect to either the aggregation number g or the area per molecule a, since they are dependent on one another through the geometrical relations given in Table 1. In this manner one obtains the equilibrium area per molecule of the aggregate: 1=2 @  0 

g ¼ 0 ) 
2 ¼ 0 ) a ¼ ð20Þ @a  a The critical micelle concentration (CMC, denoted as XCMC in mole fraction units) in the pseudophase approximation is obtained from the relation ! !     1

0g

0g ð21Þ ln XCMC ¼ þ aþ ¼ kT kT a kT kT Transfer

where the area a now stands for the equilibrium estimate given by Eq. (20). In Tanford’s free energy expression [Eq. (19)], the first contribution, the tail transfer free energy, is negative. Hence, this contribution is responsible for the aggregation to occur. It affects only the CMC [as shown by Eq. (21)] but not the equilibrium area a [as shown by Eq. (20)]. Hence it does not affect the size and shape of the aggregate. The second contribution, the free energy of residual contact between the aggregate core and water, is positive and decreases in magnitude as the area a decreases. A decrease in the area a corresponds to an increase in the aggregation number g, for all aggregate shapes, as shown by the geometrical relations in Table 1. Hence, this contribution promotes the growth of the aggregate. The third contribution, the free energy due to headgroup repulsions, is also positive and increases in magnitude if the area a decreases or the aggregation number g increases. Hence, this contribution is responsible for limiting the growth of aggregates to a finite size. Thus, Tanford’s model clearly identifies why aggregates form, why they grow, and why they do not keep growing but remain finite in size.

Theory of Micelle Formation

15

E. Predictive Power of Molecular Packing Model The packing parameter vS /a‘S can be estimated using the equilibrium area a obtained from Eq. (20). One can observe that a will be small and the packing parameter will be large if the headgroup interaction parameter  is small. The area a will increase and the packing parameter will decrease if the interfacial free energy per unit area  decreases. These simple considerations allow one to predict many features of surfactant self-assembly as summarized below. 1. For nonionic surfactants with ethylene oxide units as the headgroup, the headgroup parameter  can be expected to increase in magnitude if the number of ethylene oxide units in the headgroup increases. Therefore, when the number of ethylene oxide units is small,  is small, a is small, vS /a‘S is large, and bilayer aggregates (lamellae) are favored. For larger number of ethylene oxide units,  increases, a increases, vS /a‘S decreases, and cylindrical micelles become possible. When the number of ethylene oxide units is further increased, a becomes very large, vS /a‘S becomes small enough so that spherical micelles will form with their aggregation number g decreasing with increasing ethylene oxide chain length. 2. Comparing nonionic and ionic surfactants, the headgroup interaction parameter  will be smaller for nonionics than for ionics, because one has to also consider ionic repulsions in the latter case. Therefore, a will be smaller and vS /a‘S will be larger for the nonionics compared to the ionics. As a result, nonionic surfactants would form aggregates of larger aggregation number compared to ionic surfactants of the same tail length. 3. For a given surfactant molecule, the headgroup repulsion can be decreased by modifying the solution conditions. For example, adding salt to an ionic surfactant solution decreases ionic repulsions; increasing the temperature for a nonionic surfactant molecule with an ethylene oxide headgroup decreases steric repulsions. Because  decreases, a will decrease and vS /a‘S will increase. Thus, one can achieve a transition from spherical micelles to rodlike micelles, and possibly to bilayer aggregates, by modifying solution conditions that control headgroup repulsions. 4. If single-tail and double-tail surfactant molecules are compared, for the same equilibrium area a, the double-tail molecule will have a packing parameter vS /a‘S twice as large as that of the single-tail molecule. Therefore, the double-tail molecule can self-assemble to form bilayer vesicles while the corresponding single-tail molecule aggregates into only spherical or globular micelles. 5. If the solvent is changed from water to a mixed aqueous-organic solvent, then the interfacial tension parameter  decreases. For a given surfactant, this would lead to an increase in the equilibrium area per mole-

16

Nagarajan

cule a and, hence, a decrease in vS /a‘S . Therefore, on the addition of a polar organic solvent to an aqueous surfactant solution, bilayers will transform into micelles, rodlike micelles into spherical micelles, and spherical micelles into those of smaller aggregation numbers including only small molecular clusters. All the above predictions are in agreement with numerous experiments and are by now well established in the literature. One can thus see the evidence for the predictive power of the molecular packing parameter model and its dramatic simplicity.

F. Neglected Role of the Surfactant Tail From the previous discussion, it is obvious that the equilibrium area a has become closely identified with the headgroup of the surfactant because of its dependence on the headgroup interaction parameter . Indeed, a is often referred to as the ‘‘head-group area’’ in the literature. This has even led to the erroneous identification of a as a simple geometrical area based on the chemical structure of the headgroup in many papers, in contrast to the actuality that a is an equilibrium parameter derived from thermodynamic considerations. For the same surfactant molecule, the area a can assume widely different values depending on the solution conditions like temperature, salt concentration, additives present, etc.; hence, it is meaningless to associate one specific area with a given surfactant. While the role of the surfactant headgroup in controlling self-assembly is appreciated in the literature, in marked contrast, the role of the surfactant tail has been virtually neglected. This is because the ratio vS /‘S appearing in the molecular packing parameter is independent of the chain length for common surfactants and the area a depends only on the headgroup interaction parameter  [Eq. (20)] in the framework of Tanford’s free energy model. However, as discussed below, it is necessary to consider an extension to Tanford’s free energy expression to account for the packing entropy of the surfactant tail in the aggregates. When this is done, the area a becomes dependent on the tail length of the surfactant and also on the aggregate shape. In formulating and evaluating his free energy expression, Tanford had already noted [1] that the transfer free energy of the tail has a magnitude different from the free energy for transferring the corresponding hydrocarbon chain from aqueous solution to a pure hydrocarbon phase. He had correctly surmised that the difference arises from the packing constraints inside the aggregates that are absent in the case of a bulk hydrocarbon liquid phase. He took account of this factor empirically, as a correction to the transfer free energy, independent of the aggregate size and shape. Such an

Theory of Micelle Formation

17

empirical expression for the tail packing free energy, independent of the aggregate size and shape, was employed in our early treatments of micelle and vesicle formation [11–13]. Subsequently, detailed chain packing models to estimate this free energy contribution were developed following different approaches, by Gruen [28– 30], Dill et al. [31–35], Ben-Shaul et al. [36–38], Puvvada and Blankschtein [39–41], and Nagarajan and Ruckenstein (16). The inclusion of this free energy contribution obviously leads to the surfactant tail directly influencing a, and hence the packing parameter, size, and shape of the equilibrium aggregate. This is discussed in detail in Ref. [42]. The molecular theory of micellization outlined in the next section is strongly influenced by Tanford’s work. Various contributions to the free energy of aggregation are identified phenomenologically. Explicit expressions to estimate these free energy contributions are then developed in molecular terms, involving only molecular constants that are readily estimated from knowledge of surfactant molecular structure and solution conditions. This allows a truly predictive approach to the phenomenon of surfactant self-assembly.

IV. THEORY OF MICELLIZATION OF SURFACTANTS A. Free Energy of Micellization 1. Contributions to the Free Energy of Micellization From our previous discussion, it is evident that at the heart of the theory of micellization is the formulation of an expression for the standard free energy difference

0g between a surfactant molecule in an aggregate of size g and one in the singly dispersed state. Once such an expression is available, all solution and aggregation properties can be a priori calculated. Tanford’s phenomenological model [1] has already suggested the essential components of this free energy change. One can decompose

0g into a number of contributions on the basis of phenomenological and molecular considerations [11,13,16]. First, the hydrophobic tail of the surfactant is removed from contact with water and transferred to the aggregate core, which is like a hydrocarbon liquid. Second, the surfactant tail inside the aggregate core is subjected to packing constraints because of the requirements that the polar headgroup should remain at the aggregate–water interface and the micelle core should have a hydrocarbon liquidlike density. Third, the formation of the aggregate is associated with the creation of an interface between its hydrophobic domain and water. Fourth, the surfactant headgroups are brought to the aggregate surface, giving rise to steric repulsions between them. Finally, if the headgroups are ionic or zwitterionic, then electrostatic

18

Nagarajan

repulsions between the headgroups at the aggregate surface also arise. Explicit analytical expressions developed in our earlier studies [11,13,16] are presented in this section for each of these free energy contributions in terms of the molecular characteristics of the surfactant.

2. Transfer of the Surfactant Tail On aggregation, the surfactant tail is transferred from its contact with water to the hydrophobic core of the aggregate. The contribution to the free energy from this transfer process is estimated by considering the aggregate core to be like a liquid hydrocarbon. The fact that the aggregate core differs from a liquid hydrocarbon gives rise to an additional free energy contribution that is evaluated immediately below. The transfer free energy of the surfactant tail is estimated from independent experimental data on the solubility of hydrocarbons in water [43,44]. On this basis, the transfer free energy for the methylene and methyl groups in an aliphatic tail as a function of temperature T (in 8K) is given by [16] ð

0g Þtr 896
36:15
0:0056T (for CH2 Þ ¼ 5:85 ln T þ T kT ð

0g Þtr 4064
44:13 þ 0:02595T (for CH3 Þ ¼ 3:38 ln T þ T kT

ð22Þ

For surfactant tails made up of two hydrocarbon chains, the contribution to the transfer free energy would be smaller than that calculated based on two independent single chains, because of intramolecular interactions. Tanford [1] has estimated that the second chain of a dialkyl molecule contributes a transfer free energy that is only about 60% of an equivalent single chain molecule, and this estimate is used for calculations involving double chain molecules. For fluorocarbons, using the work of Mukerjee and Handa [45], we estimate the transfer free energy to be
6.2kT for the CF3 group and
2:25kT for the CF2 group, at 258C. The temperature dependencies of the transfer free energies for fluorocarbons are presently not available.

3. Packing and Deformation of the Surfactant Tail Within the aggregates, one end of the surfactant tail, which is attached to the polar headgroup, is constrained to remain at the aggregate–water interface. The other end (the terminal methyl group) is free to occupy any position inside the aggregate as long as a uniform density is maintained in the aggregate core. Obviously, the tail must deform locally, that is, stretch and compress nonuniformly along the tail length, in order to satisfy both the packing and the uniform density constraints. The positive free energy contribution resulting from this conformational constraint on the surfactant tail is referred to as the tail deformation free energy. Different methods for

Theory of Micelle Formation

19

estimating this contribution have appeared in the literature [28–41] that involve either complex expressions or extensive numerical calculations. In constructing our theory, we have followed the method suggested for block copolymer microdomains by Semenov [46], and have developed a simple analytical expression by integrating the local tail deformation energy over the volume of the aggregate. In this manner, we obtain [16] the following expression for spherical micelles: ð

0g Þdef 9P 2 ¼ kT 80

!

R2S NL2

! ð23Þ

where P is the packing factor defined in Table 1, RS is the core radius, L is the characteristic segment length for the tail, and N is the number of segments in the tail (N ¼ ‘S =L, where ‘S is the extended length of the tail). As suggested by Dill and Flory [31,32], a segment is assumed to consist of 3.6 methylene groups (hence, L ¼ 0:46 nm). L also represents the spacing between alkane molecules in the liquid state, namely, L2 ð¼ 0:21 nm2 Þ is the cross-sectional area of the polymethylene chain. Equation (23) is employed also for nonspherical globular micelles and the spherical endcaps of rodlike micelles. For infinite cylindrical rods, Eq. (23) is still applicable, but with the changes that the coefficient 9 is replaced by 10, the radius RS is replaced by the radius RC of the cylinder, and the packing factor P ¼ 1=2. For spherical bilayer vesicles, the molecular packing differences between the outer and the inner layers must be accounted for. Consequently, when Eq. (23) is applied to spherical bilayer vesicles, the radius RS is replaced by the half-bilayer thickness to for the molecules in the outer layer, and ti for the molecules in the inner layer, and the coefficient 9 is replaced by 10 and the packing factor P ¼ 1, as for lamellar aggregates. For surfactant tails with two chains, the tail deformation free energy calculated for a single chain should be multiplied by a factor of 2. The segment size L mentioned above for hydrocarbon tails is employed, also for fluorocarbon tails because most calculations involving fluorocarbons have been done for hydrocarbon– fluorocarbon mixed surfactants, and lattice models that allow for two different sizes for sites on the same lattice become mathematically complicated. For pure fluorocarbon surfactants, one can perform predictive calculations taking L ¼ 0:55 nm consistent with the cross-sectional area of about 0.30 nm2 of a fluorocarbon chain.

4. Formation of Aggregate Core–Water Interface The formation of an aggregate generates an interface between the hydrophobic domain consisting of surfactant tails and the surrounding water medium. The free energy of formation of this interface is calculated as the

20

Nagarajan

product of the surface area in contact with water and the macroscopic interfacial tension  agg characteristic of the interface [11,13,16]: ð

0g Þint agg  ¼ ð24Þ ða
ao Þ kT kT Here, a is the surface area of the hydrophobic core per surfactant molecule, and ao is the surface area per molecule shielded from contact with water by the polar headgroup of the surfactant. For spherical bilayer vesicles, the area per molecule differs between the outer and inner layers and the area a in Eq. (24) is replaced with ðAgo þ Agi Þ=g. Expressions for the surface area per molecule corresponding to different aggregate shapes and sizes are provided in Table 1. The area ao depends on the extent to which the polar headgroup shields the cross-sectional area L2 of the surfactant tail (2L2 for a double chain tail). If the headgroup cross-sectional area ap is larger than the tail cross-sectional area, the latter is shielded completely from contact with water and ao ¼ L2 (or 2L2 for double chains) in this case. If ap is smaller than the crosssectional area of the tail, then the headgroup shields only a part of the cross-sectional area of the tail from contact with water, and ao ¼ ap . Thus, ao is equal to the smaller of ap or L2 (2L2 for a double chain). The aggregate core–water interfacial tension  agg is taken equal to the interfacial tension  SW between water (W) and the aliphatic hydrocarbon of the same molecular weight as the surfactant tail (S). The interfacial tension  SW can be calculated in terms of the surface tensions  S of the aliphatic surfactant tail and  W of water via the relation sw ¼ s þ w
2:0 ðs w Þ1=2

ð25Þ

where is a constant with a value of about 0.55 [47,48]. The surface tension  S can be estimated to within 2% accuracy [16] using the relation s ¼ 35:0
325M
2=3
0:098ðT
298Þ

ð26Þ

where M is the molecular weight of the surfactant tail, T is in 8K, and  S is expressed in mN/m. The surface tension of water [49] can be calculated using the expression [16] w ¼ 72:0
0:16ðT
298Þ

ð27Þ

where the surface tension is expressed in mN/m and the temperature in 8K. For fluorocarbon surfactants,  agg is taken equal to the experimentally determined interfacial tension of 56.45 mN/m at 258C between water and perfluorohexane [45]. The temperature dependency of this interfacial tension can be approximated by a constant coefficient of 0.1 mN/m/8K, in the absence of available experimental data.

Theory of Micelle Formation

21

5. Headgroup Steric Interactions On aggregation, the polar headgroups of surfactant molecules are brought to the surface of the aggregate, where they are crowded when compared to the infinitely dilute state of the singly dispersed molecules. This generates steric repulsions among the headgroups. If the headgroups are compact, the steric interactions between them can be estimated as hard-particle interactions, using any of the models available in the literature. The simplest is the van der Waals approach, on the basis of which the contribution from steric repulsion at the micelle surface is calculated as [11,13,16]  ap  ð

0g Þsteric ¼
ln 1
kT a

ð28Þ

where ap is the cross-sectional area of the polar headgroup near the micellar surface. This equation is used for spherical and globular micelles as well as for the cylindrical middle and the spherical endcaps of rodlike micelles. For spherical bilayer vesicles, the steric repulsions at both the outer and the inner surfaces must be considered, taking into account that the area per molecule is different for the two surfaces. Noting that go and gi surfactant molecules are present in the outer and the inner layers, respectively, one can write

 g g ap ap ð

0g Þsteric
i ln 1
ð29Þ ¼
o ln 1
kT g g Ago = go Agi = gi The above approach to calculating the headgroup interactions using Eqs.(28) or (29) is inadequate when the polar headgroups are not compact, as in the case of nonionic surfactants having polyoxyethylene chains as headgroups. An alternate treatment for headgroup interactions in such systems is presented later in this section.

6. Headgroup Dipole Interactions If the headgroup has a permanent dipole moment as in the case of zwitterionic surfactants, the crowding of the aggregate surface by the headgroups also leads to dipole–dipole interactions. The dipoles at the surface of the micelle are oriented normal to the interface and stacked such that the poles of the dipoles are located on parallel surfaces. The dipole–dipole interaction for such an orientation provides a repulsive contribution to the free energy of aggregation. The interaction free energy is estimated by considering that the poles of the dipoles generate an electrical capacitor and the distance between the planes of the capacitor is equal to the distance of charge separation d (or the dipole length) in the zwitterionic headgroup. Consequently, for spherical micelles one gets [11,13,16] the expression

22

Nagarajan

  ð

0g Þdipole 2 e2 RS d ¼ kT "a kT RS þ þ d

ð30Þ

where e is the electronic charge, " the dielectric constant of the solvent, RS the radius of the spherical core, and  the distance from the core surface to the place where the dipole is located. This equation is also employed for globular micelles and the endcaps of rodlike micelles. For the cylindrical part of the rodlike micelles, the capacitor model yields   ð

0g Þdipole 2 e2 RC d ¼ ln 1 þ ð31Þ kT "a kT RC þd þ  where RC is the radius of the cylindrical core of the micelle. For vesicles, considering the outer and inner surfaces, one can write [12,13]     gi 2 e2 Ri ð

0g Þdipole go 2 e2 Ro d d ¼ þ ð32Þ kT g " ao kT Ro þ þ d g " ai kT Ri

d where the areas ao and ai are defined in Table 1. The dielectric constant " is taken to be that of pure water [49] and is calculated using the expression [16] " ¼ 87:74 expð
0:0046ðT
273ÞÞ

ð33Þ

where the temperature T is in 8K.

7. Headgroup Ionic Interactions If the surfactant has an anionic or cationic headgroup, then ionic interactions arise at the micellar surface. The theoretical computation of these interactions is complicated by a number of factors such as the size, shape, and orientation of the charged groups, the dielectric constant in the region where the headgroups are located, the occurrence of Stern layers, discrete charge effects, etc. [1,4,50,51]. The Debye–Hu¨ckel solution to the Poisson– Boltzmann equation is found to overestimate the interaction energy approximately by a factor of 2 [1,15]. An approximate analytical solution to the Poisson–Boltzmann equation derived [52] for spherical and cylindrical micelles is used in the present calculations. This free energy expression has the form 8 8 9 ! 9 ! 0 < < = 2 1=2 = 2 1=2 ð

g Þionic S S 4 S þ 1þ 1þ ¼ 2 ln

1 :2 ; S: ; kT 4 4 8 9 ð34Þ ! < 2 1=2 = 4C 1 1 S ln þ 1þ
; S :2 2 4 where

Theory of Micelle Formation

S¼

23

4 e2 " a kT

ð35Þ

and  is the reciprocal Debye length. The area per molecule a is evaluated at a distance  from the hydrophobic core surface (see Table 1), where the center of the counterion is located. The first two terms on the right-hand side of Eq. (34) constitute the exact solution to the Poisson–Boltzmann equation for a planar geometry, and the last term provides the curvature correction. The curvature-dependent factor C is given by [16] C¼

2 ; RS þ

2 ; Req þ

1 RC þ

ð36Þ

for spheres/spherical endcaps of spherocylinders, globular aggregates (with an equivalent radius Req defined in Table 1), and cylindrical middle part of spherocylinders, respectively. For spherical bilayer vesicles, the electrostatic interactions at both the outer and inner surfaces are taken into account [12,13]. For the molecules in the outer layer, the free energy contribution is calculated with a replaced by ao and C ¼ 2=ðRo þ Þ. For the molecules in the inner layer, a is replaced by ai and C ¼
2=ðRi
Þ. The reciprocal Debye length  is related to the ionic strength of the solution via !1=2 8 n0 e2 ðC þ Cadd Þ NAv n0 ¼ 1 ð37Þ ¼ ; "kT 1000 In the above equation, n0 is the number of counterions in solution per cm3, C1 is the molar concentration of the singly dispersed surfactant molecules, Cadd is the molar concentration of the salt added to the surfactant solution, and NAv is Avogadro’s number. The temperature dependence of the reciprocal Debye length  arises from both the variables T and " present in Eq. (37).

8. Headgroup Interactions for Oligomeric Headgroups For nonionic surfactants with polyoxyethylene chains as headgroups, the calculation of the headgroup interactions using Eq. (28) and (29) for the steric interaction energy becomes less satisfactory since it is difficult to define an area ap characteristic of the oligomeric headgroups (13,16) without ambiguity. For sufficiently large polyoxyethylene chain lengths, it is more appropriate to treat the headgroup as a polymeric chain when estimating the free energy of headgroup interactions. The treatment developed in our earlier work (16) is based on the following conceptual approach. In a singly dispersed surfactant molecule, the polyoxyethylene chain is viewed as an iso-

24

Nagarajan

lated free polymer coil swollen in water. In micelles, the polyoxyethylene chains present in the region surrounding the hydrophobic core (referred to as shell or corona) can be viewed as forming a solution denser in polymer segments compared to the isolated polymer coil. The difference in the two states of polyoxyethylene provides a contribution to the free energy of aggregation, which is computed as the sum of the free energy of mixing of the polymer segments with water and the free energy of polymer chain deformation. As the polyoxyethylene chain length decreases, the use of polymer statistics becomes less satisfactory. Two limiting models of micellar corona are considered. One assumes that the corona has a uniform concentration of polymer segments. The maintenance of such a uniform concentration in the corona is possible for curved aggregates, only if the chains deform nonuniformly along the radial coordinate. This model may be appropriate when the number of ethylene oxide units in the headgroup is small. The second model assumes a radial concentration gradient of chain segments in the corona consistent with the uniform deformation of the chain. This model may be more appropriate for headgroups with large number of ethylene oxide units. For either of the above models, in order to calculate the mixing free energies, one has to choose some polymer solution theory. Because of its simplicity, the mean-field approach of Flory [53], which requires only the polyoxyethylene–water interaction parameter WE for calculating the free energies, is used here. Usually, it is necessary to consider the composition and temperature dependencies of the interaction parameter WE in order to accurately describe the thermodynamic properties of polymer solutions [54– 56]. Even with such dependencies incorporated, it has not been possible to satisfactorily describe the properties of a polyethylene oxide–water system, indicating the inadequacy of the polymer solution theory to quantitatively represent the aqueous polymer solution. Nevertheless, we have carried out calculations based on the Flory model and taking WE to be a constant, with the justification that the model provides at least an approximate accounting of the mixing free energy needed for our purposes and no better model with comparable simplicity is available. For both uniform concentration and nonuniform concentration models, the free energy of the steric interactions is calculated using Eq. (28) or (29), but taking ap equal to L2 to describe the steric repulsions between neighboring surfactant tails at the sharp interface separating the core from the corona.

9. Headgroup Mixing in Corona Region For a polyoxyethylene chain containing EX oxyethylene units, the number of segments NE is given by NE ¼ EX vE =L3 , where vE is the volume of an oxyethylene unit and the characteristic segment volume is retained to be L3.

Theory of Micelle Formation

25

Based on available density data [57] at 258C, vE ¼ 0:063 nm3. The volume VS of the micellar corona, having a thickness D, is calculated from the geometrical relations given in Table 2. In the uniform concentration model, the volume fraction of the polymer segments in the corona is Eg ¼

g Ex vE g NE L3 ¼ VS VS

ð38Þ

In the nonuniform concentration model, the polymer concentration in the corona is determined from the requirement that the polymer chains be uniformly deformed over the thickness D. The radial variation of polymer concentration in the corona is thus found to be (16) ! 3 g NE L ðrÞ ¼ ð39Þ D 4 r2 for spherical aggregates and ! 3 g NE L ðrÞ ¼ 2r LC D

ð40Þ

TABLE 2 Corona Volume of Micelles with Polyoxytheylene Headgroups Spherical micelles: VS Vg ¼ g g Globluar micelles: VS Vg ¼ g g

"


"
1þ

D 1þ RS

D RS

#

3


1

2
1þ

#  D
1 b

Cylindrical part of rodlike micelles: "
#  VS Vg D 2
1 ¼ 1þ g g RC Spherical endcaps of rodlike micelles: " # VS Vg 8ðRS þ DÞ3 2ðH þ DÞ2 ¼
f3ðRS þ DÞ
ðH þ DÞg
1 g g 3gvS 3gvS

26

Nagarajan

for a cylinder of length LC containing g molecules. The polymer concentration ’R at the micellar core–water interface (at r ¼ R) reduces for both spheres and cylinders to ! 3 NE L R ¼ ð41Þ Da where a is the surface area of the micellar core per surfactant molecule. In the corona each polyoxyethylene chain experiences a potential U(r) because of the interactions between the segments of a single molecule and the segments of all the other molecules. This potential is taken in the meanfield approach to be proportional to the total segment density arising from all the molecules of the micelle. The influence of the solvent is also incorporated in this mean potential via the excluded-volume factor [53,58]. The potential is written [16] as UðrÞ ¼ kTðrÞð12
wE Þ

ð42Þ

where ’(r) is the segment density of polyoxyethylene in the corona at the radial position r. For the uniform concentration model, ’(r) is a constant equal to ’Eg. Therefore, the mixing free energy of the headgroup in the corona with respect to that in an isolated, free polymer coil is given by [16] the expression
 ð ð

0g Þmix;E 1 1 RþD UðrÞ 1 2  r  4
ðrÞ dr ¼ ð43Þ ¼ N E Eg wE g L3 R kT 2 kT Here, R ¼ RS for the spherical micelles. This equation is used to calculate the headgroup mixing free energy for globular aggregates and endcaps of rodlike aggregates, as well. For the cylindrical part of the rodlike micelles, with R ¼ RC , one gets [16]
 ð ð

0g Þmix;E 1 1 RþD UðrÞ 1   2r
¼ ð44Þ ðrÞ dr ¼ N L C E Eg wE g L3 R kT 2 kT For the nonuniform concentration model, using the radial concentration profile given by Eqs. (39)–(41), the headgroup mixing free energy in spherical micelles is obtained to be [16]
 ð ð

0g Þmix;E 1 1 RþD UðrÞ 1 1 2  r  4
¼ ðrÞ dr ¼ N E R wE 3 gL R kT 2 ð1 þ D=RÞ kT ð45Þ This equation is also used for the globular aggregates and the endcaps of spherocylinders, with R denoting the radius of the globular aggregate or of

Theory of Micelle Formation

27

the spherical endcaps. For the cylindrical part of the spherocylinders, with R ¼ RC , one gets [16] ð ð

0g Þmix;E 1 1 RþD UðrÞ ¼ ðrÞ 2r LC dr 3 gL R kT kT ð46Þ
 1 R  lnð1 þ D=RÞ ¼ NE R
wE 2 D

10. Headgroup Deformation in Corona Region For the uniform concentration model, the elastic deformation of the segments is nonuniform along the length of the polymer molecule. An expression for this nonuniform deformation free energy of the polyoxyethylene chains in the corona region of the micelle is obtained by employing the approach of Semenov [46] mentioned earlier. One can calculate the free energy contribution arising from the headgroup deformation in the corona from the expression [16] ð

0g Þdef;E 3 L RS D ¼ 2 a Eg RS þD kT

ð47Þ

for spherical and globular micelles and the endcaps of rodlike micelle and from   ð

0g Þdef;E 3 L RC D ¼ ln 1 þ ð48Þ 2 a Eg kT RC for the cylindrical part of the rodlike micelle. In the nonuniform concentration model, the polymer chain has been assumed to deform uniformly along the chain length. Using Flory’s [53] rubber elasticity theory to estimate the deformation free energy of a chain, one obtains " # ð

0g Þdef;E 1 D2 2 NE1=2 L
3 ð49Þ ¼ þ 2 NE L2 D kT which can be employed for all aggregate shapes.

11. Formation of Core–Corona Interface The interfacial free energy contribution ð

0g Þint is calculated using Eq. (24) but recognizing that  agg is now different from  SW since the interface in these aggregates is that between a domain of surfactant tails and a solution of polyoxyethylene segments in water.  agg is calculated using the Prigogine theory [59,60] for the surface tensions of solutions. This involves determining the surface phase composition ’S for a given bulk solution composition

28

Nagarajan

by equating the chemical potentials of the surface and bulk phases. The surface phase composition, in turn, determines the interfacial tension.  agg is thus dependent on the concentration of the polyoxyethylene segments in the micellar corona and the surfactant tail–polyoxyethylene interfacial tension  SE in addition to the surfactant tail–water interfacial tension  SW. In the uniform concentration model, the micellar corona has a bulk concentration ’Eg. Correspondingly, the concentration ’S of polymer segments at the interface is determined [16] by solving the implicit equation " # ðS =Eg Þ1=NE ð
SE Þ 2=3 vS ln ¼ SW kT ð1
S Þ=ð1
Eg Þ 3 1 þ WE ½ð1
Eg Þ
Eg 
WE ½ð1
S Þ
S  4 2 ð50Þ Once ’S is determined, the interfacial tension  agg is calculated [16] from the explicit equation !
 
  1
S NE
1 agg SW 2=3 ðS
Eg Þ vS ¼ ln þ NE kT 1
Eg ð51Þ   1 S2 3 2 þ WE ð Þ
ðEg Þ 2 4 The interfacial tension  SE between polyoxyethylene and surfactant tails is calculated in terms of the surface tension  S of the surfactant tails and the surface tension  E of polyoxyethylene using the relation SE ¼ S þ E
2:0 ðS E Þ1=2

ð52Þ

The constant is taken to be ¼ 0:55 as for the surfactant tail–water system in Eq. (25) because of the polar nature of the polyoxyethylene group [47,48]. The surface tension  S is calculated using Eq. (26). The surface tension of polyoxyethylene with EX oxyethylene units is estimated (in mN/m) on the basis of the information given in Ref. [61] using the equation E ¼ 42:5
19EX
2=3
0:098ðT
298Þ

ð53Þ

where T is in 8K. For the nonuniform concentration model also, the free energy of formation of the core–corona interface is calculated using Eqs. (50) and (51), but with the concentration ’R replacing ’Eg in both equations.

Theory of Micelle Formation

29

B. Computational Approach The equation for the size distribution of aggregates, in conjunction with the geometrical characteristics of the aggregates and the expressions for the different contributions to the free energy of micellization, allow one to calculate the various solution properties of the surfactant system. In Eq. (4) for the aggregate size distribution, ð

0g Þ is the sum of various contributions: ð

0g Þ ¼ð

0g Þtr þð

0g Þdef þð

0g Þint þð

0g Þsteric þð

0g Þdipole þ ð

0g Þionic

ð54Þ

where the contribution ð

0g Þdipole is included if the surfactant is zwitterionic, the contribution ð

0g Þionic is included if the surfactant is ionic, and neither of the two contributions is relevant when the surfactant is nonionic. For surfactants with polyoxyethylene headgroups, one has to include the contributions ð

0g Þmix;E and ð

0g Þdef;E associated with the mixing and deformation of the headgroups in the corona of the aggregate. Explicit equations for calculating each of these contributions have been discussed above. Using them, the various surfactant solution properties are calculated as follows.

1. Calculations Using Complete Size Distribution An obvious and direct approach to calculating the aggregation behavior of surfactants is based on calculating the entire size distribution of aggregates as a function of the independent variables and then performing the necessary summations to obtain the CMC, average aggregate size, and the variance of the size distribution as described in Section II.C. In contrast to a typical experiment where the total surfactant concentration is fixed and the aggregation behavior is determined, in doing the predictive calculations, it is convenient to calculate the size distribution at a specified value of the singly dispersed surfactant concentration X1 and then obtain the total surfactant P concentration by the summation, as Xtot ¼ X1 þ gXg .

2. Calculations Using the Maximum-Term Method The approach based on the calculation of the aggregate size distribution is not complicated but is time-consuming. Instead, a simpler approach based on the maximum-term method [62] is employed in our calculations as described below. The simpler approach is built on the recognition that for spherical or globular micelles and spherical bilayer vesicles, the size dispersion is usually narrow. The concentrations of aggregates other than that corresponding to the maximum in the size distribution are relatively small. Because the average properties of the solution are strongly influenced by the

30

Nagarajan

species present in the largest amount, the number average aggregation number gn can be taken as the value of g for which the number concentration Xg of the aggregates is a maximum (¼ XgM ) , and the weight average aggregation number gw can be taken as the value of g for which the weight concentration gXg of the aggregates is a maximum ½¼ ðgXg ÞM . These average aggregation numbers are very close to one another and are practically the same as those obtained by calculating the size distribution of aggregates [62]. As mentioned before, only the aggregation number g is the independent variable in the case of spherical or globular micelles; the aggregation number g as well as the inner and outer layer thicknesses ti and to are independent variables in the case of spherical vesicles. Further, for polyoxyethylene surfactants, one has to include the corona thickness D as an additional independent variable. One can obtain quantitative estimates of the variance of the size distribution, also by the maximum-term method. Approximating the derivatives in Eq. (9) by differences, we can write
 ðwÞ 2 @ ln g
ln gw ¼ P w ¼ gw @ gXg
lnðgXg ÞM
 ðnÞ 2 @ ln gn
ln gn P ¼ ¼ gn @ ln gXg
lnðgn XgM Þ

ð55Þ

In the above equation, the average aggregation numbers gn and gw are P taken gXg is as those corresponding to the maximum in Xg or (gXg), the sum approximated by the maximum term gn XgM or ðgXg ÞM , and the derivatives have been replaced by the difference terms denoted by the symbol
. One can determine the maximum in Xg (or in gXg) for two slightly different values of X1 and the two sets of values for gn and XgM or gw and (gXg)M can be introduced in Eq. (55) to calculate the variance in the size distribution. The CMC is estimated by plotting any one of the variables [X1 , XgM , ðgXg ÞM , gn;app , gw;app ] as a function of the total surfactant concentration Xtot and determining the surfactant concentration at which the plotted variable displays a sharp transition in values. It can also be calculated as the value of X1 for which the amount of surfactant in the micellized form P is equal to that in the singly dispersed form, namely XCMC ¼ X1 ¼ gXg ¼ gn XgM ¼ ðgXg ÞM . The predicted average aggregation numbers reported in this chapter correspond to those at the CMC, unless otherwise stated.

3. Calculations for Rodlike Micelles In the case of rodlike micelles, the size distribution Xg [Eq. (12)] is monotonic and does not have a maximum. In this case, by minimizing ð

0cyl Þ for

Theory of Micelle Formation

31

an infinitely long cylinder, the equilibrium radius RC of the cylindrical part of the micelle is determined. Given the radius of the cylindrical part, the number of molecules gcap in the spherical endcaps is found to be that value which minimizes ð

0cap Þ. Given gcap, ð

0cyl Þ, and ð

0cap Þ, the sphere-torod transition parameter K is calculated from Eq. (12), the average aggregation numbers at any total surfactant concentration from Eq. (16), and the CMC from Eq. (17). For polyoxyethylene surfactants, one has to include the corona thickness D as an additional independent variable, and the equilibrium value of D is determined for the cylindrical middle part and the spherical endcaps via the minimizations of ð

0cyl Þ and ð

0cap Þ, respectively. The search for the parameter values that maximize the aggregate concentration Xg (or gXg) or minimize the standard free energy differences ð

0cyl Þ and ð

0cap Þ was carried out using the IMSL (International Mathematical and Statistical Library) subroutine ZXMWD. This subroutine is designed to search for the global extremum of a function of many independent variables subject to any specified constraints on the variables. This subroutine has been used for all the calculations described in this chapter.

C. Estimation of Molecular Constants For illustrative purposes, calculations have been carried out for a number of nonionic, zwitterionic, and ionic surfactants. Examples of molecules employed in these calculations are shown in Fig. 2. The molecular constants associated with the surfactant tail are the volume vS and the extended length ‘S of the tail. For the headgroup, one needs the cross-sectional area ap for all types of headgroups, the distance  from the core surface where the counterion is located in the case of ionic headgroups, the dipole length d, and the distance  from the core surface at which the dipole is located, in the case of a zwitterionic headgroup. All these molecular constants can be estimated from the chemical structure of the surfactant molecule. There are no free parameters, and the calculations are therefore completely predictive in nature. Some examples of how the molecular constants are estimated are given below. The molecular constants characterizing the headgroups of surfactants considered in this chapter are listed in Table 3.

1. Estimation of Tail Volume vS The molecular volume vS of the surfactant tail containing nC carbon atoms is calculated from the group contributions of (nC
1) methylene groups and the terminal methyl group vS ¼ vðCH3 Þ þ ðnC
1ÞvðCH2 Þ

ð56Þ

32

Nagarajan

FIG. 2 Chemical structure of surfactant molecules considered in this chapter and the symbols used to refer to them in the text.

Theory of Micelle Formation

33

TABLE 3 Molecular Constants for Surfactant Headgroups Surfactant headgroup -Glucoside Methyl sulfoxide Dimethyl phosphene oxide -Maltoside N-methyl glucamine Sodium sulfate Sodium sulfonate Potassium carboxylate Sodium carboxylate Pyridinium bromide Trimethyl ammonium bromide N-betaine Lecithin

ap (nm2 )

ao (nm2 )

 (nm)

d (nm)

0.40 0.39 0.48 0.43 0.34 0.17 0.17 0.11 0.11 0.34 0.54 0.30 0.45

0.21 0.21 0.21 0.21 0.21 0.17 0.17 0.11 0.11 0.21 0.21 0.21 0.42

— — — — — 0.545 0.385 0.60 0.555 0.22 0.345 0.07 0.65

— — — — — — — — — — — 0.5 0.62

These group molecular volumes, estimated from the density versus temperature data available for aliphatic hydrocarbons, are given [16] by the expressions vðCH3 Þ ¼ 0:0546 þ 1:24  10
4 ðT
298Þ nm3 vðCH2 Þ ¼ 0:0269 þ 1:46  10
5 ðT
298Þ nm3

ð57Þ

where T is in 8K. For double-tailed surfactants, vS is calculated by accounting for both the tails. For the fluorocarbon tails, using the data available [63–65] for 25oC, we estimate that vðCF3 Þ ¼ 1:67vðCH3 Þ and vðCF2 Þ ¼ 1:44vðCH2 Þ. Extensive volumetric data are not yet available to estimate the temperature dependence of the molecular volumes of CF3 and CF2 groups. As an approximate estimation, the ratio between the volumes of the fluorocarbon and the hydrocarbon groups is assumed to be the same at all temperatures.

2. Estimation of Extended Tail Length ‘S

The extended length of the surfactant tail ‘S at 2988K is calculated using a group contribution of 0.1265 nm for the methylene group and 0.2765 nm for the methyl group [1]. Given the small volumetric expansion of the surfactant tail over the range of temperatures of interest, the extended tail length ‘S is considered as temperature-independent. Therefore, the small volumetric expansion of the surfactant tail is accounted for by small increases in the cross-sectional area of the surfactant tail. The extended length of the fluorocarbon chain is estimated using the same group contributions as for hydro-

34

Nagarajan

carbon tails, namely, 0.1265 nm for the CF2 group and 0.2765 nm for the CF3 group.

3. Estimation of Headgroup Area ap The headgroup area ap is calculated as the cross-sectional area of the headgroup near the hydrophobic core-water interface. The glucoside headgroup in -glucosides has a compact ring structure [5] with an approximate diameter of 0.7 nm, and hence, the effective cross-sectional area of the polar headgroup ap is estimated as 0.40 nm2. For sodium alkyl sulfates, the crosssectional area of the polar group ap has been estimated to be 0.17 nm2. For the zwitterionic N-betaine headgroup, ap has been estimated to be 0.30 nm2. The area per molecule a0 of the micellar core that is shielded by the headgroup from having contact with water, is the smaller of ap or L2, as discussed earlier.

4. Estimation of  and d

For ionic surfactants, the molecular constant  depends on the size of the ionic headgroup, the size of the hydrated counterion, and also the proximity of the counterion to the charge on the surfactant ion. Visualizing that the sodium counterion is placed on top of the sulfate anion, we estimate  ¼ 0:545 nm for sodium alkyl sulfates and 0.385 nm for sodium alkyl sulfonates. For alkyl pyridinium bromide the surfactant cation can approach very near the hydrophobic core, and we estimate  = 0.22 nm. For zwitterionic surfactants we need the molecular constant d, which is the distance of separation of the charges on the dipole (or the dipole length), and also the constant , which is the distance from the hydrophobic core surface at which the dipole is located. The computations in this chapter are based on an estimate of d ¼ 0:5 nm and  ¼ 0:07 nm, for N-alkyl betaines, and d ¼ 0:62 nm and  ¼ 0:65 nm, for the lecithin headgroup.

5. Estimation of WE For nonionic surfactants with a polyoxyethylene headgroup, we need the polyoxyethylene–water interaction parameter WE. This can, in principle, be estimated using the thermodynamic properties of polyoxyethylene–water solutions (such as the activity data or the phase behavior data). The activity data [54] represented in the framework of the Flory–Huggins theory indicate that WE is dependent on the composition of the polymer solution (55). The phase behavior exhibits both a lower critical solution temperature and an upper critical solution temperature [54–56] indicating that WE first increases and then decreases with increasing temperature. Further, the headgroup of the surfactant contains a functional group (such as a hydroxyl) that terminates the polymer chain; the presence of this terminal group may also affect the value of WE when compared to the estimate based on high-

Theory of Micelle Formation

35

molecular-weight polyoxyethylenes. The dependence of WE on polymer concentration, temperature, polymer molecular weight, and the terminating functional group is, however, not known. Because water is a good solvent for polyoxyethylene, values for WE smaller than 0.5, namely WE ¼ 0:1 and 0.3, have been chosen for the illustrative calculations.

D. Illustrative Predictions for Surfactants 1. Influence of Free Energy Contributions on Aggregation Behavior Figure 3 presents the calculated free energy contributions ð

0g Þ expressed per molecule of surfactant, for cetyl pyridinium bromide in water, as a function of the aggregation number g. Whereas the magnitude of ð

0g Þ

FIG. 3 Contributions to the standard free energy difference between a surfactant molecule in the micellized state and one in the singly dispersed state calculated as a function of the aggregation number g of the micelle for cetyl pyridinium bromide in water at 258C. Subscripts refer to the following: total (total of all contributions), transfer (transfer free energy of tails), deform (deformation free energy of tails), interface (interfacial free energy), steric (headgroup steric interactions), and ionic (headgroup ionic interactions). Refer to text for detailed discussion.

36

Nagarajan

influences the CMC, the functional dependence of ð

0g Þ on g determines the shape and size of the equilibrium aggregates. Of all the contributions to ð

0g Þ, only the transfer free energy of the surfactant tail is negative. Therefore, it is responsible for the aggregated state of the surfactant being favored over the singly dispersed state. The transfer free energy contribution is a constant independent of the micellar size and hence has no influence on the structural characteristics of the equilibrium aggregate. All the remaining free energy contributions to ð

0g Þ are positive and depend on the aggregate size. It is clear from the geometrical relations for aggregates (see Table 1) that as the aggregation number g increases, the area per molecule a decreases. Consequently, the free energy of formation of the aggregate core–water interface decreases with increasing aggregation number. This free energy is thus responsible for the growth of aggregates to large sizes and is said to provide the positive cooperativity of aggregation. All remaining free energy contributions (namely, the surfactant tail deformation energy, the steric repulsions between the headgroups, the dipole–dipole interactions between zwitterionic headgroups, and the ionic interactions between ionic headgroups) increase with increasing aggregation number. These free energy contributions thus provide the negative cooperativity of aggregation and are responsible for limiting the aggregate growth to finite sizes. All the free energy contributions affect, however, the magnitude of the CMC. The calculated size distributions for cetyl pyridinium bromide are presented in Fig. 4 for two values of the molar concentration C1 ð¼ 55:55X1 Þ of the singly dispersed surfactant. As expected, near the CMC, a small variation in C1 gives rise to a large variation in the total aggregate concentration, Ctot. The average aggregation number is, however, practically the same at these two concentrations. This implies [see Eq. (9)] that the size dispersion of the aggregates is narrow, which can be seen also from the figure. Various size-dependent solution properties calculated using the model are plotted in Fig. 5 as a function of the total concentration of cetyl pyridinium bromide in water. The concentration at which any one of these properties shows a sharp change in behavior can be taken as the CMC. All four properties plotted in the figure yield CMC values that are very close to one another.

2. Influence of Tail Groups and Headgroups on Aggregation Behavior The CMC values predicted at 258C, for nonionic alkyl -glucosides, zwitterionic N-alkyl betaines, and ionic alkyl sodium sulfates are presented in Fig. 6 as a function of the length of the surfactant tail. The CMC decreases with an increase in the chain length of the surfactant, as a consequence of the increase in the magnitude of the tail transfer free energy. The incremental variation in CMC is roughly constant for a given homologous family of surfactants. The

Theory of Micelle Formation

37

FIG. 4 Calculated size distribution of cetyl pyridinium bromide aggregates in water at 258C at two values of the total surfactant concentration Ctot and the singly dispersed surfactant concentration C1. (a) C1 ¼ 0:54 mM and Ctot ¼ 0:68 mM; (b) C1 ¼ 0:57 mM and Ctot ¼ 5:89 mM.

experimentally measured CMCs [66–71] are in reasonable agreement with the predicted values. For a given surfactant tail length, the CMC is smaller for a nonionic surfactant than for an anionic surfactant. This is a consequence of the strong repulsive interaction between ionic headgroups compared to the weaker steric interactions between nonionic headgroups. The predicted average aggregation numbers at the CMC are plotted as a function of the tail length in Fig. 7 for some zwitterionic and anionic surfactants. For the cases considered, the micelles are spherical or globular and are narrowly dispersed in size. For the zwitterionic alkyl N-betaines, the predicted aggregation numbers are in reasonable agreement with the measured values [70,71]. The aggregation number increases with an increase in the chain length of the surfactant tail. The equilibrium area per molecule of the aggregate changes with a change in the tail length, but the range of values assumed by a is small. Consequently, given an equilibrium area per molecule,

38

Nagarajan

FIG. 5 Calculated size-dependent solution properties of cetyl pyridinium bromide in water as a function of the total surfactant concentration (expressed as mole fraction). gw,app is the apparent weight average aggregation number, gn,app is the apparent number P average aggregation number, X1 denotes the monomer concentration, and gXg is the total concentration of the surfactant in the form of aggregates. All concentrations are expressed in mole fraction units.

the aggregation number of a spherical or globular micelle must increase with increasing tail length, as dictated by the geometrical relations. For anionic sodium alkyl sulfates with chain lengths smaller than dodecyl, a large number of experimental data have been reported in the literature [72–75], from which we select for plotting in Fig. 7 those that show the largest deviation from the predicted values. For tetradecyl and hexadecyl chains, aggregation numbers mentioned in Ref. [76] have been plotted, although in the work of Tartar [72], which is cited as the source of these experimental data, no report of these experimental aggregation numbers is found; therefore, the reported aggregation numbers for these two surfactants should be discounted. For alkyl sulfates the predicted aggregation numbers do not show a significant increase

Theory of Micelle Formation

39

FIG. 6 Dependence of the critical micelle concentration on the number of carbon atoms in the surfactant tail for nonionic alkyl -glucosides (triangles), zwitterionic N-alkyl betaines (circles), and anionic sodium alkyl sulfates (squares). The lines represent the theoretical predictions while the points are experimental data. (From Refs. 66–71.)

with increasing length of the surfactant tail and remain practically constant for the longer tail lengths. These results can be understood by noting that the CMC for the surfactant with the smaller tail length is large, and for this reason the ionic strength is larger. This decreases the headgroup ionic repulsion, thus leading to a smaller equilibrium area per molecule in the aggregate. In contrast, for surfactants with longer tail lengths, the CMCs are low, the ionic strengths smaller, and hence, the equilibrium area per molecule in the aggregate larger. This increase in the equilibrium area per molecule with increasing tail length is responsible for the relatively small changes in the aggregation number with increasing tail length, in contrast to the behavior exhibited by the zwitterionic surfactant.

3. Influence of Ionic Strength on Aggregation Behavior The headgroup ionic interactions at the micelle surface are weakened by the addition of salt to the surfactant solution. Figures 8 and 9 present the predicted dependence of the CMC (Figure 8) and the average aggregation number (Figure 9) of sodium dodecyl sulfate, on the amount of added NaCl

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Nagarajan

FIG. 7 Dependence of the weight average aggregation number of micelles at the CMC, on the chain length of the surfactant tail for sodium alkyl sulfates (squares) and N-alkyl betaines (circles). The points are experimental data. (From Refs. 70–76.) The dotted line shows predicted values for sodium alkyl sulfates while the continuous line describes the predictions for N-alkyl betaines. See text for comments about the reported experimental aggregation numbers for sodium alkyl sulfates with C14 and C16 surfactant tails.

electrolyte. For the range of ionic strengths considered, only narrowly dispersed spherical or globular micelles are formed. With decreasing ionic repulsion between the headgroups, the predicted CMC decreases and the average aggregation number of the micelle increases, in satisfactory agreement with experimental measurements (74,75).

4. Influence of Temperature on Aggregation Behavior The temperature dependence of the CMC and the average aggregation number of the micelle have been calculated for a number of surfactants. Figure 10 compares the predicted CMC values with the experimental data [70,71] for the zwitterionic surfactants N-alkyl betaines for three different alkyl chain lengths. In Fig. 11, the predicted CMCs of the anionic sodium dodecyl sulfate have been plotted for two different concentrations of added NaCl electrolyte and compared with experimental data [77]. Calculated and experimental [78] CMCs are presented in Fig. 12 for the homologous family of sodium alkyl sulfonates. In all cases, the predictions show reasonable

FIG. 8 Influence of added NaCl concentration on the CMC of sodium dodecyl sulfate micelles. The lines denote the predicted values while the points are the experimental measurements at 258C. (From Refs. 74 and 75.)

FIG. 9 Influence of added NaCl concentration on the average aggregation number of sodium dodecyl sulfate micelles. The lines denote the predicted values while the points are the experimental measurements at 258C. (From Refs. 74 and 75.) 41

FIG. 10 The dependence of CMC on temperature for N-alkyl betaines having C10, C11 and C12 chains as surfactant tails. The lines are predictions of the present theory while the points are experimental data. (From Refs. 70 and 71.)

FIG. 11 The dependence of CMC on temperature for sodium dodecyl sulfate in water and in a 0.0125M solution of NaCl. The lines are predictions of the present theory while the points are experimental data. (From Ref. 77.) 42

Theory of Micelle Formation

43

FIG. 12 The dependence of CMC on temperature for sodium alkyl sulfonates having C10, C12, and C14 chains as surfactant tails. The lines are predictions of the present theory while the points are experimental data. (From Ref. 78.)

agreement with experiment. One may note that the experiment indicates some increase in the CMC as the temperature is decreased below about 258C, while the predicted values show a monotonic decrease of the CMC with decreasing temperature. It has been suggested [5] that the hydration state of the ionic headgroup in the micelles may be different from that in the singly dispersed state as the temperature is lowered below 258C. A free energy contribution accounting for such an effect is not included in the theory since it cannot be calculated with any precision at the present time.

5. Transition from Spherical to Rodlike Micelles The predicted results for nonionic alkyl -glucosides indicate that large, polydispersed rodlike micelles form. For octyl glucoside the aggregation numbers are still not very large, but for longer chain lengths very large rods form. The predicted weight average aggregation number is used to compute the hydrodynamic radius of the micelles using the expression RH ¼


 3 gw vS 1=3 þ ‘p 4

ð58Þ

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Nagarajan

where ‘p is the length of the polar headgroup. For the ring structure of the -glucoside, ‘p has been estimated [5] to be 0.7 nm. The computed hydrodynamic radius as a function of surfactant concentration is presented in Fig.13. The two predicted lines correspond to two marginally different values of the parameter ap describing the headgroup of the surfactant. One may note from Eq. (12) that the sphere-to-rod transition parameter K [which determines gw as per Eq. (16)] can be dramatically altered by small changes in the free energy difference ð

0cap Þ
ð

0cyl Þ, since gcap, which appears in the definition of K, is quite large. For example, assuming a typical value of 90 for gcap, a small change of 0.05 kT in the free energy difference ð

0cap Þ
ð

0cyl Þ will cause a change in K of e4:5 ¼ 90, which, in turn, can change the predicted value for gw by a factor of about 10. Therefore, the predicted average aggregation numbers are very sensitive even to small changes in

FIG. 13 The dependence of the micellar size (expressed as the hydrodynamic radius) on the concentration of the surfactant for octyl glucoside. Both lines correspond to predicted values but for marginally different values of the molecular con 2 line corresponds to a ¼ 40A while the dotted line stant ap. The continuous p  corresponds to ap ¼ 39A 2 . The squares denote the reported experimental data. (From Ref. 68.) The circles correspond to modified experimental data if the reported hydrodynamic radius had included one layer of water. See text for discussion.

Theory of Micelle Formation

45

the free energy estimates when rodlike micelles form. This is illustrated by the calculations carried out for two slightly different values of the parameter ap, which affects the magnitude of the headgroup steric interaction energy. The predictions are compared with the data provided by dynamic light scattering measurements [68]. It is not clear whether the reported hydrodynamic radii correspond to the dry or hydrated aggregates. Therefore, both the reported hydrodynamic radii and the radii obtained by subtracting the diameter of a water molecule are plotted in Fig. 13. Given the sensitivity of K to the free energy estimates, the agreement between the measured and predicted aggregate sizes is satisfactory. For anionic sodium alkyl sulfates with NaCl as the added electrolyte, an increase in ionic strength beyond that in Fig. 9 is expected to contribute to a transition from globular micelles to large spherocylindrical micelles. As already noted, the ability to predict ln K with deviations of about 4.5 or less from the measurements can be considered satisfactory. The predicted values for the sphere-to-rod transition parameter K for sodium dodecyl sulfate are plotted in Fig. 14 against the added concentration of NaCl as the electrolyte

FIG. 14 The dependence of the sphere-to-rod transition parameter K for sodium dodecyl sulfate on the concentration of added electrolyte NaCl. The points are from light scattering measurements (from Ref. 50) and the line represents the predictions, both at 258C.

46

Nagarajan

and are compared with light scattering measurements [79]. The largest deviation between predicted and experimental value of ‘nK is about 2, at the highest ionic strength. The predicted radius of the cylindrical part of the aggregate is smaller than the fully extended length of the surfactant tail. It increases from 1.45 nm to 1.49 nm as the electrolyte concentration is increased from 0.45 M to 1.25 M at 258C. For this range of ionic strengths, the radius of the endcaps remains unaltered and is equal to the extended length of the surfactant tail. The predicted CMC, as defined by Eq. (17), decreases from 0.38 mM to 0.204 mM over this range of added salt concentration. The temperature dependence of the parameter K has been calculated for sodium dodecyl sulfate for two concentrations of the added NaCl electrolyte. The predicted values of K are plotted in Fig. 15 along with the experimental estimates based on dynamic light scattering measurements [79]. The largest deviation in ‘nK is about 4.5, and the theory predicts a somewhat stronger dependence on the temperature than that observed experimentally. Figure 16 presents the predicted values of K as a function of the added electrolyte concentration for the homologous family of sodium alkyl sulfates

FIG. 15 Influence of temperature on the sphere-to-rod transition parameter K of sodium dodecyl sulfate in solutions containing 0.45M and 0.80M added NaCl electrolyte. The lines denote the predicted values while the points are experimental data. (From Ref. 79.)

Theory of Micelle Formation

47

FIG. 16 The dependence of the sphere-to-rod transition parameter K on the surfactant tail length and on the concentration of added electrolyte NaCl for sodium alkyl sulfates. The lines are predictions obtained from the present theory while the points denote the experimental data (from Ref. 79), both at 308C. The predicted lines are labeled with the surfactant tail lengths, and the corresponding experimental data are indicated by circles, squares, triangles, and diamonds, respectively.

with 10 to 13 carbon atoms in their hydrophobic tails. The figure shows that even the largest deviation in ln K between the predicted and the experimental values [79] is smaller than 2 for C11, C12, and C13 sodium sulfates and is about 4 for the C10 sodium sulfate. Given the sensitivity of the parameter K to the free energy estimates, the ability of the theory to predict K accurately as a function of the ionic strength, temperature, and tail length for sodium alkyl sulfates is remarkable. However, when the calculations are repeated for alkyl sulfates with other counterions such as Li, K, etc., the predicted K values significantly deviate from the measurements [79]. As mentioned before, this is not surprising given the sensitivity of K. In contrast, the prediction of the CMC and micelle size is quite satisfactory in the presence of various counterions, when only spherical or globular micelles form. Therefore, improved accounting of the counterion effects beyond what is considered in the approximate analytical solution to the Poisson–Boltzmann equation is necessary for predicting accurate values of K. Such treatment of counterion effects remains to be developed.

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Nagarajan

6. Formation of Bilayer Vesicles The aggregation behavior of the anionic dialkyl sodium sulfate at an ionic strength of 0.01 M has been calculated. Because of the presence of two tails, these surfactants can form spherical vesicular structures provided the length of the surfactant tail is large enough and the ionic strength is not too small. The predicted aggregation number, the inner and the outer radii, the thicknesses of the inner and the outer layer, the numbers of molecules of surfactant present in the two layers, and the CMC are all listed in Table 4. The inner and the outer layers differ somewhat in thicknesses, with the inner layer more compressed than the outer. The thicknesses are much smaller than the extended lengths of the tails and are also smaller than the radii of spherical and cylindrical micelles formed of single-tailed sodium alkyl sulfates, discussed before. The equilibrium areas per surfactant molecule are substantially different from one another for the molecules in the two layers. The vesicles are not too large and the aggregation number increases with increasing chain length of the surfactant. As one may anticipate, the CMC is small because of the larger magnitude of the tail transfer free energy in case of the dialkyl tail. Calculated results for dialkyl lecithins with phosphotidylcholine headgroup are also presented in Table 4 for alkyl chain lengths varying from 12 to 18. The vesicles formed of lecithins also have inner and outer layers of differing thicknesses. However, the radii of the lecithin vesicles and, hence,

TABLE 4 Ri (nm)

Predicted Structural Properties of Vesicles ti (nm)

to (nm)

ai (nm2)

ao (nm)

gi

go

Xcmc

di-Cn SO4 Na þ 0:01M NaCl 12 4.63 6.68 0.98 14 5.30 7.57 1.09 16 5.83 8.31 1.20 18 6.36 9.04 1.30

1.07 1.08 1.28 1.38

0.5825 0.6065 0.627 0.646

0.771 0.805 0.836 0.865

462 580 684 787

727 893 1041 1185

7:67  10
8 1:07  10
9 1:46  10
11 1:92  10
13

di-Cn -lecithin 12 20.90 22.67 14 20.11 22.09 16 19.61 21.80 18 19.30 21.70

0.92 1.03 1.14 1.25

0.796 0.813 0.828 0.842

0.794 0.82 0.845 0.869

6929 6260 5855 5577

8163 7490 7086 6822

1:58  10
9 2:01  10
11 2:49  10
13 3:04  10
15

nC

Ro (nm)

0.846 0.95 1.05 1.15

Note: i refers to the inner layer of the bilayer and o to the outer layer, R is the radius at the hydrophobic domain–water interface, t is the layer thickness, a is the area per surfactant molecule, and g is the number of surfactant molecules.

Theory of Micelle Formation

49

the aggregation numbers are much larger than those formed of dialkyl sodium sulfates. Consequently, the areas per molecule in the inner and outer layers are closer to one another. The vesicle radius deceases slightly with an increase in the chain length of the surfactant. The differences between the two types of molecules can be traced to the decrease in the headgroup repulsions in the case of the zwitterionic lecithin headgroups compared to that in the case of anionic sodium sulfate headgroups. In all cases listed in Table 4, the aggregates are narrowly dispersed.

7. Influence of Polyoxyethylene Headgroups The predicted CMC values for a surfactant C12Ex with dodecyl hydrocarbon tail and 6 to 53 oxyethylene units are presented in Fig. 17 for both the uniform concentration model and the nonuniform concentration model

FIG. 17 Dependence of the CMC on the length of the polyoxyethylene headgroup of nonionic surfactants C12Ex. The points refer to measured values while the lines denote predictions from the present model, both at 258C. The continuous lines represent the results from the nonuniform concentration model while the dotted lines denote the predictions based on the uniform concentration model. Calculated results are shown for two different values of the polyoxyethylene–water interaction parameter WE . The circles refer to commercial polyoxyethylene glycol ethers (from Refs. 81–84), triangles represent commercial samples where the distribution of oxyethylene chain lengths is reduced by molecular distillation (from Ref. 85), and squares correspond to purified polyoxyethylene methyl ethers (from Ref. 86).

50

Nagarajan

and for WE ¼ 0:1 and 0.3. As mentioned before in Section IV.A.8, the uniform concentration model may be appropriate for small values of Ex and the nonuniform concentration model may be more suitable for larger Ex. The experimental data used for comparison are for polyoxyethylene glycol ethers [80–85] and polyoxyethylene methyl ethers [86]. The considerable scatter in the measured CMCs is a consequence of the heterogeneity of some of the surfactant samples that have been used, the samples containing a range of polyoxyethylene chain lengths distributed around the reported mean value. Figure 18 presents the predicted as well as measured CMC data for a surfactant C16Ex with a hexadecyl hydrocarbon tail and 8 to 63 oxyethylene units. The experimental data are for polyoxyethylene glycol ethers [87–90] and polyoxyethylene methyl ethers [86]. The smaller scatter in the experimental data of Fig. 18 compared to that of Fig. 17, is primarily due to the fewer measurements available for the C16Ex surfactants.

FIG. 18 Dependence of the CMC on the length of the polyoxyethylene headgroup of nonionic surfactants C16Ex. The points refer to measured values while the lines denote predictions from the present model, both at 258C. The continuous lines represent the results from the nonuniform concentration model while the dotted lines denote the predictions based on the uniform concentration model. Calculated results are shown for two different values of the polyoxyethylene-water interaction parameter WE . Circles denote polyoxyethylene glycol ethers (from Refs. 87–90) while the squares represent purified polyoxyethylene methyl ethers (from Ref. 86).

Theory of Micelle Formation

51

The predicted aggregation numbers based on the nonuniform concentration model are plotted in Figs. 19 and 20 for C12Ex and C16Ex surfactants, respectively. Measured aggregation numbers [81,87–90] are also included for comparison. The calculated aggregation numbers are in qualitative agreement with the experimental values. The predicted aggregation numbers are larger when a larger value is taken for WE. The agreement between the predicted and measured aggregation numbers is satisfactory even from a quantitative point of view. The predicted thickness D of the micellar corona region is plotted in Figs. 21 and 22, respectively, for the C12Ex and C16Ex surfactants, on the basis of the non-uniform concentration model. The experimental shell thicknesses are those estimated by Tanford et al. [91] from intrinsic viscosity measurements [87–90]. As expected, the shell thickness D calculated assuming a smaller value for WE ð¼ 0:1, implying a better solvent) is larger than that based on a larger value (=0.3, implying a relatively poorer solvent). The predicted aggregation numbers based on the uniform concentration model are presented in Figs. 23 and 24 for C12Ex and C16Ex surfactants. The

FIG. 19 Influence of the polyoxyethylene headgroup size on the average aggregation number of micelles for surfactants with dodecyl hydrophobic tail. The points are experimental data at 258C (from Refs. 80 and 87-90) and the lines represent the predictions of the present theory. The calculated results are based on the nonuniform concentration model.

FIG. 20 Influence of the polyoxyethylene headgroup size on the average aggregation number of micelles for surfactants with hexadecyl hydrophobic tail. The points are experimental data at 258C (from Refs. 80 and 87–90) and the lines represent the predictions of the present theory. The calculated results are based on the nonuniform concentration model.

FIG. 21 Influence of the polyoxyethylene headgroup size on the shell thickness of the micelles for the C12Ex surfactants. The points are experimental data at 258C (from Ref. 91), and the lines represent the predictions of the present theory. The calculated results are based on the nonuniform concentration model. 52

Theory of Micelle Formation

53

FIG. 22 Influence of the polyoxyethylene headgroup size on the shell thickness of the micelles for the C16Ex surfactants. The points are experimental data at 258C (from Ref. 91), and the lines represent the predictions of the present theory. The calculated results are based on the nonuniform concentration model.

predicted shell thicknesses based on the uniform concentration model are presented in Figs. 25 and 26 for C12Ex and C16Ex surfactants. The predicted dependence of the micelle aggregation numbers on the polyoxyethylene chain length is in reasonable agreement with the experimental data, but the calculated aggregation numbers are smaller than the experimental values. Figures 25 and 26 show that the model predicts much smaller values for the corona thickness D than that estimated from intrinsic viscosity measurements [91]. The temperature dependence of the aggregation behavior of surfactants with polyoxyethylene headgroups is expected to differ from that of ionic and zwitterionic surfactants because of the way the interactions between polyoxyethylene headgroups depend on temperature. For ionic and zwitterionic surfactants, the various contributions to the free energy of micellization display a temperature dependence that leads to a lowering of the aggregation number with increasing temperature. These temperature-dependent effects are, however, overshadowed by the temperature-dependent WE, which governs the interactions between the polyoxyethylene headgroups and water in the micellar shell region. As suggested by the observed phase behavior of the polyoxyethylene–water systems [54–56], the interaction

FIG. 23 Influence of the polyoxyethylene headgroup size on the average aggregation number of micelles. The surfactants contain dodecyl hydrophobic tails. The points are experimental data at 258C (from Refs. 80 and 87–90), and the lines represent the predictions of the present theory. The calculated results are based on the uniform concentration model.

FIG. 24 Influence of the polyoxyethylene headgroup size on the average aggregation number of micelles. The surfactants contain hexadecyl hydrophobic tails. The points are experimental data at 258C (from Refs. 80 amd 87–90), and the lines represent the predictions of the present theory. The calculated results are based on the uniform concentration model. 54

FIG. 25 Influence of the polyoxyethylene headgroup size on the thickness of the micellar shell for the C12Ex surfactants. The points are experimental data at 258C (91) and the lines represent the predictions of the present theory. The calculated results are based on the uniform concentration model.

FIG. 26 Influence of the polyoxyethylene headgroup size on the thickness of the micellar shell for the C16Ex. surfactants. The points are experimental data at 258C (from Ref. 91), and the lines represent the predictions of the present theory. The calculated results are based on the uniform concentration model. 55

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Nagarajan

parameter WE first increases as the temperature is increased (giving rise to the lower critical solution temperature, LCST), passes through a maximum, and then decreases with increasing temperature (giving rise to the upper critical solution temperature, UCST). Thus, for temperatures smaller than the LCST (which is the temperature range for which the aggregation behavior is examined), the interaction parameter WE should increase with increasing temperature. The calculations presented above show that the aggregation numbers increase with increasing WE if none of the other variables is changed (Figs. 19 and 20 or 23 and 24). Consequently, for polyoxyethylene surfactants, the headgroup interactions with water promote aggregate growth with increasing temperature. The remaining free energy contributions favor a decrease in the aggregation number with increasing temperature. It is interesting to note that an increase in WE with increasing temperature promotes both the phase separation of the polymer solution (LCST) and also the growth of the micelles to large aggregation numbers [92,93]. The calculated results show that for nonionic surfactants with polyoxyethylene headgroups, the predictions from the nonuniform concentration model are in somewhat better agreement with experiments than those from the uniform concentration model. However, improved agreement between predicted and experimental micellar properties should await a more satisfactory treatment of polyoxyethylene–water solutions than presently available. Alternately, a modified treatment of headgroup interactions, considering the headgroups to be oligomers rather than polymers, is necessary. It is important that the general validity of any model for nonionic polyoxyethylene surfactants be tested by comparing experimental and predicted results for all the micellar characteristics, namely the CMC, the aggregation number, and the shell thickness over an extended range of polyoxyethylene chain lengths. Although these surfactants are among the most commonly used in practical applications, a comprehensive prediction of their solution properties remain to be developed despite many papers that have appeared in the literature.

V. THEORY OF MICELLIZATION OF SURFACTANT MIXTURES A. Size and Composition Distribution of Micelles The thermodynamic treatment and illustrative calculations presented in this section focus on binary mixtures of surfactants for which many experimental results are available. The approach, however, can be readily extended to a mixture of three or more surfactants. A molecular theory of mixed micelles

Theory of Micelle Formation

57

was first proposed in our earlier work [94], which was later improved by incorporating a better treatment of molecular packing in mixed micelles [95]. This latter model constitutes the basis of this section. We consider a solution of surfactant molecules A and B and denote by g the aggregation number of the mixed micelle containing gA molecules of A and gB molecules of B. At equilibrium, in analogy with Eq. (4), one obtains [94–96] for the aggregate size and composition distribution, the equation ! ! !
0g
gA
01A
gB
01B g

0g g

0g gA gB Xg ¼ X1A X1B exp
; ¼ kT kT kT ð59Þ Here,
0g is the standard chemical potential of the mixed micelle, while
01A and
01B are the standard chemical potentials of the singly dispersed A and B molecules, respectively;

0g is the difference in the standard chemical potentials between gA =g molecules of surfactant A plus gB =g molecules of surfactant B present in an aggregate of size g and the same numbers of molecules present in their singly dispersed states in water; X1A and X1B are the mole fractions of the singly dispersed surfactants A and B, while Xg is the mole fraction of aggregates of size g in the solution. The mole fraction Xg is dependent not only on the size g but also on the composition of the micelle. We define the solvent-free composition of the singly dispersed surfactant mixture, the mixed micelle, and the total surfactant mixture by the relations X1A X ¼ 1A ; X1A þ X1B X1 gA gA ; ¼ ¼ gA þ gB g P X1A þ gA Xg P ¼ ; X1 þ gXg

X1B X ¼ 1B X1A þ X1B X1 gB gB ¼ ¼ gA þ gB g P X1B þ gB Xg P ¼ X1 þ gXg

1A ¼

1B ¼

gA

gB

tA

tB

ð60Þ

In the definition for the total surfactant composition, the summation is over two independent variables, namely, the aggregation number g = 2 to 1 and the micelle composition gA ¼ 0 to 1. One may note that for spherical bilayer vesicles, the compositions of the inner and the outer layers need not be the same. Therefore, we define the composition variables gAi and gBi for the inner layer and gAo and gBo for the outer layer, similar to the definitions given above for the overall composition of the aggregate. From the size and composition distribution one can compute the average sizes of

58

Nagarajan

the aggregates via Eq. (5). The average composition of the mixed micelle can be calculated from gA ¼

X

ðgA =gÞ Xg =

X

Xg ;

gB ¼

X

ðgB =gÞ Xg =

X

Xg

ð61Þ

The geometrical relations for various micellar shapes have been presented in Table 1. In these relations, the tail volume vS is now given by vS ¼ ðgA vSA þ gB vSB Þ, where, vSA and vSB denote the volumes of the hydrophobic tails of surfactants A and B.

B. Free Energy of Formation of Mixed Micelles Expressions for the standard free energy difference between the surfactant molecules A and B present in a mixed micelle and those present in the singly dispersed state in water are obtained by a simple extension of the equations developed in Section IV for single-component surfactant systems. Only the modifications necessary for the treatment of surfactant mixtures compared to the analysis of pure surfactant behavior are described below.

1. Transfer of the Surfactant Tail For a mixed micelle having the composition ðgA ; gB Þ, the transfer free energy per surfactant molecule is given by [94,95] ð

0g Þtr ð

0g Þtr;A ð

0g Þtr;B ¼ gA þ gB kT kT kT

ð62Þ

The transfer free energy contribution can be estimated as described in Section IV.A.2 using the group contributions. The transfer free energy calculated from Eq. (62) does not include contributions arising from the mixing of the A and the B tails inside the micellar core, which are accounted for separately.

2. Deformation of the Surfactant Tail When surfactants A and B have different tail lengths, segments of both molecules may not be simultaneously present everywhere in the micellar core. Let us assume that surfactant A has a longer tail than surfactant B, ‘SA > ‘SB. If the micelle radius R is less than both ‘SA and ‘SB, then even the shorter tail can reach everywhere within the core of the micelle. If ‘SA > RS > ‘SB , then the inner region of the micellar core, of dimension ðRS
‘SB Þ, can be reached only by the A tails. Taking into account the different extent to which the A and the B tails are stretched for the two situations described above, one obtains [95] the expression

Theory of Micelle Formation

" # ð

0g Þdef Q2g R2S ¼ Bg gA þ gB ; kT NA L2 NB L2

59

9P2 Bg ¼ 80

Qg ¼ RS if RS < ‘SA ; ‘SB ; Qg ¼ ‘SB ¼ NB L if ‘SA > RS > ‘SB

! ð63Þ

This equation is used for spherical and globular micelles and for the spherical endcaps of rodlike micelles. NA and NB stand for the number of segments in the tails of surfactants A and B, respectively, and P is the packing factor defined in Table 1. Because the innermost region of the micelle is not accessible to surfactant B, the micelle must contain a sufficient number of A surfactant molecules to completely fill up the inner region. This packing condition is satisfied if the radius RS is less than the composition averaged tail length, RS  ð A ‘SA þ B ‘SB Þ, where A and B are the volume fractions of surfactant tails in the micellar core. In all the calculations reported here, the upper limit of RS is taken to be this composition averaged tail length [95]. For the cylindrical part of the rodlike micelles, the coefficient 9 in Bg is replaced by 10, the radius RS is replaced by the radius RC of the cylindrical core, and the packing factor P ¼ 1=2. For spherical bilayer vesicles, the coefficient 9 in Bg is replaced with 10, the radius RS is replaced by the layer thickness ti for the molecules in the inner layer and by the layer thickness to for the molecules in the outer layer, and P ¼ 1, as for lamellar aggregates. For fluorocarbon chains, as mentioned in Section IV.A, the calculations are performed retaining the definition for the segment length L to be 0.46 nm.

3. Formation of Aggregate Core–Water Interface The free energy associated with the formation of the hydrophobic core– water interface is given for the case of binary mixtures by the expression [94,95]  ð

0g Þint agg  ¼ a
gA aoA
gB aoB kT kT

ð64Þ

Here, aoA and aoB are the areas per molecule of the core surface shielded from contact with water by the polar headgroups of surfactants A and B. Because the interfacial tension against water of various hydrocarbon and fluorocarbon tails of surfactants are close to one another, the aggregate core–water interfacial tension  agg is approximated by the micelle composition averaged value: agg ¼ A AW þ B BW

ð65Þ

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Nagarajan

where  AW and  BW are interfacial tensions between water and the tails of A and B surfactants. For spherical bilayer vesicles, the area per molecule differs for the inner and the outer layers, and a in Eq. (64) is replaced by ðAgo þ Agi Þ=g.

4. Headgroup Steric Interactions Extending the expression used for single surfactant systems to binary surfactant mixtures, one can write [94,95] for mixed micellar aggregates
 gA apA þ gB apB ð

0g Þsteric ¼
ln 1
ð66Þ kT a For spherical bilayer vesicles, we take into account the composition variation between the inner and the outer layers. Equation (29) for a single surfactant is now extended to the form
 gAo apA þ gBo apB ð

0g Þsteric go ¼
ln 1
kT g Ago =go ð67Þ
 gAi apA þ gBi apB gi
ln 1
g Agi =gi

5. Headgroup Dipole Interactions The dipole–dipole interactions for dipoles having a charge separation d and located at a distance  from the hydrophobic domain surface can be computed for spherical and globular micelles and the spherical endcaps of rodlike micelles from [94,95]   ð

0g Þdipole 2 e2 RS d  ¼ ð68Þ kT " adipole kT RS þ þ d g;dipole For the cylindrical part of the rodlike micelles, one can write   ð

0g Þdipole 2 e2 RC d g;dipole ¼ ln 1 þ kT " adipole kT RC þd þ 

ð69Þ

In the above relations, g,dipole is the fraction of surfactant molecules in the aggregate having a dipolar headgroup. If both A and B are zwitterionic surfactants with the same headgroup, then g;dipole ¼ gA þ gB ¼ 1; adipole ¼ a

ð70Þ

If surfactant A is zwitterionic and surfactant B is nonionic or ionic, then a g;dipole ¼ gA ; adipole ¼ ð71Þ g;dipole

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The dipole–dipole interactions may be relevant even when the surfactants do not possess zwitterionic headgroups. Such a situation occurs when the surfactant mixture consists of an anionic and a cationic surfactant. The two oppositely charged surfactants may be visualized as forming ion pairs. Depending on the location of the charges on the two surfactant headgroups, these ion pairs may act as dipoles. The distance of charge separation d in the zwitterionic headgroup, now refers to the distance between the locations of the anionic and the cationic charges, measured normal to the micelle core surface. Thus, for such systems, g;dipole ¼ the smaller of adipole ¼

a

g;dipole

;

ðgA ; gB Þ 2

d ¼ jA
B j

ð72Þ

The factor 2 in the expression for g;dipole accounts for the fact that a dipole is associated with two surfactant molecules, treated as a pair. A and B represent the distance normal to the hydrophobic core surface at which the charges are located on the A and B surfactants. For spherical bilayer vesicles, Eq. (32) is extended to binary surfactant mixtures taking into account the composition variation between the two layers. Noting that the numbers of surfactant molecules in the two layers are go and gi and go,dipole and gi,dipole are the fractions of surfactant molecules in the outer and the inner layers having a dipolar headgroup, one gets   ð

0g Þdipole go 2e2 Ro d ¼  kT g "ao;dipole kT Ro þ  þ d go;dipole ð73Þ   gi 2e2 Ri d þ  g "ai;dipole kT Ri

d gi;dipole The areas per molecule ao,dipole and ai,dipole and the fractions of surfactant molecules that are dipolar go,dipole and gi,dipole, are evaluated by applying Eqs. (70) to (72) for both the inner and the outer layers.

6. Headgroup Ionic Interactions The ionic interactions that arise at the aggregate surface are calculated using Eq. (34), in conjunction with the curvature factor C given by Eq. (36), with the modification that S is now given by S¼

4 e2 " a;ion kT

ð74Þ

If both A and B are ionic surfactants with the same kind of charged headgroups, then

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g;ion ¼ gA þ gB ¼ 1;

a;ion ¼ a ;

 ¼ gA A þ gB B

If A is ionic while B is nonionic or zwitterionic, then a a;ion ¼  ; g;ion ¼ gA ;  ¼ A g;ion If A and B are both ionic but of opposite charge, then a a;ion ¼  ; g;ion ¼ gA
gB ;  ¼ gA A þ gB B g;ion

ð75Þ

ð76Þ

ð77Þ

For spherical bilayer vesicles, the electrostatic interactions at both the outer and the inner surfaces are taken into account. For the molecules in the outer layer, a is replaced with ao , g;ion with go;ion , and the curvature factor C ¼ 2=ðRo þ Þ. For the molecules in the inner layer, a is replaced with ai , g;ion with gi;ion , and the curvature factor C ¼
2=ðRi
Þ. The fraction of charged molecules go;ion and gi;ion is calculated by applying Eqs. (75) to (77) to both the outer and the inner surfaces of the vesicle.

7. Free Energy of Mixing of Surfactant Tails This is the only contribution that is not present in the free energy model for single-component surfactant solutions. This contribution accounts for the entropy and the enthalpy of mixing of the surfactant tails of molecules A and B in the hydrophobic core of the micelle, with respect to the reference states of pure A and pure B micelle cores. Any available solution model can be employed to calculate the entropy and the enthalpy of mixing. Consistent with our use of the Flory–Huggins model in various cases considered before (because of its relative simplicity), we retain the same model here as well: ð

0g Þmix H 2 ¼ gA ln A þ gB ln B þ ½gA vSA ðH A
mix Þ kT H 2 þ gB vSB ðH B
mix Þ =kT

ð78Þ

H where H A and B are the Hildebrand solubility parameters of the tails of surfactants A and B and H mix is the volume fraction averaged solubility parameter of all the components within the micelle core, H mix ¼ H

A H þ . B B A

8. Free Energy Model and Mixture Nonideality The free energy change on the formation of a mixed micelle can be written in terms of the free energies of formation of the two pure component micelles, the ideal entropy of mixing of surfactants inside the mixed micelle, and an excess free energy that is responsible for all nonidealities. This implies that if the free energy of formation of the mixed micelles, excluding the ideal entropy of mixing, is a linear function of the micelle composition, then

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the excess free energy is zero and the mixed micelles are said to behave ideally. Any nonlinear dependence on the micelle composition is thus a signature for the nonideal behavior of the mixed surfactant system. In the framework of the molecular theory outlined in this section, the transfer free energy is a linear function of the micelle composition. All other contributions have a nonlinear dependence on micelle composition because they are dependent nonlinearly on the micelle core radius or the area per molecule. Further, these structural parameters are themselves dependent nonlinearly on the micelle composition if the tail lengths of the surfactants are different from one another. Depending on the quantitative importance of these nonlinear contributions, the mixed surfactant system displays small or large deviations from ideal behavior, as we will discuss below.

C. Predictions for Surfactant Mixtures 1. Estimation of Molecular Constants and Computational Approach The molecular constants needed for the surfactants have been discussed before in Section IV.C, and the constants characterizing various surfactant headgroups are listed in Table 3. The only additional molecular constant needed for calculations involving surfactant mixtures is the Hildebrand solubility parameter for the hydrocarbon and fluorocarbon tails of the surfactants. This can be estimated using a group contribution approach based on the properties of pure components [97–99]. For hydrocarbon tails, the solubility parameters can be estimated in units of MPa1/2 (1 MPa ¼ 1 J/cm3) from the relation H  ¼

0:7 þ 0:471ðnC
1Þ MPa1=2 Þ; vS in nm3 vS

ð79Þ

The solubility parameter for the fluorocarbon tail of the surfactant sodium perfluoro octanoate (SPFO) estimated using the group contribution approach yields 12.3 MPa1/2. Since the solubility parameters estimated in this manner have been found inadequate for the quantitative description of hydrocarbon–fluorocarbon mixture properties, Mukerjee and Handa [45] have estimated group contributions to the solubility parameters by fitting the critical solution temperature of the hydrocarbon–fluorocarbon mixtures. On the basis of these group contributions, one can calculate the solubility parameter of the SPFO tail to be approximately 9.5 MPa1/2. Computed results for hydrocarbon–fluorocarbon surfactant mixtures are presented later, utilizing both of these estimates. The predictive computations have been carried out using the maximumterm method described in Section IV.B.2. For the binary mixtures, the

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concentration of the singly dispersed surfactant mixture X1 (¼ X1A þ X1B ) and the composition of this mixture 1A are used as inputs and the aggregation number g of the equilibrium aggregate and the composition gA of the equilibrium aggregate are obtained by finding the maximum in the aggregate size distribution. For spherical bilayer vesicles, one has to determine the thicknesses of the inner and the outer layers and the compositions of each layer as well. The critical micelle concentration is estimated using the results based on the maximum-term method calculations as described earlier in Section IV.B.

2. Nonionic Hydrocarbon–Nonionic Hydrocarbon Mixtures The calculated aggregation properties of binary mixtures of decyl methyl sulfoxide (designated as C10SO) and decyl dimethyl phosphene oxide (designated as C10PO) at 248C are presented in Figs. 27 and 28. In Fig. 27 the CMC is plotted against the composition of the micelles and the composition of the singly dispersed surfactants. One can practically equate the composition of the singly dispersed surfactant to the composition of the total surfactant, when the total surfactant concentration is equal to the CMC since

FIG. 27 The CMC of C10PO þ C10SO binary mixture as a function of the composition of micelles (dashed line) and that of singly dispersed surfactants (solid line) at 248C. The points are experimental data from Ref. (100).

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FIG. 28 The average aggregation number of C10PO þ C10SO mixed micelles at the CMC as a function of the composition of micelles (solid line) and that of the singly dispersed surfactants (dashed line) at 248C.

the amount of surfactant present as micelles is small. Figure 27 contains also the experimental data obtained by Holland and Rubingh [100] on the basis of surface tension measurements. Figure 28 presents the average aggregation numbers predicted by the theory as a function of the composition of the mixed micelle and that of the monomers. No experimental data are, however, available for comparison. The size of the mixed micelle varies approximately linear with the composition. It has been shown that the CMC of this binary surfactant system can be calculated from the CMC values of the individual surfactants by assuming the mixture to be ideal [100]. In the framework of the molecular theory presented here, nonidealities in this binary mixture can arise even when the two surfactants have somewhat different headgroup cross-sectional areas while possessing identical tails [94,95]. Because there are no volume differences between the hydrophobic tails of the two surfactants, for any aggregation number, the area per molecule of the mixed micelle is independent of the micelle composition. However, the steric interaction between headgroups at any aggregation number is a nonlinear function of the micelle composition. This constitutes a source of mixture nonideality. However, because of the small magnitude of the steric repulsion free energy, the devia-

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tion from ideal mixing is rather small. For this reason, the ideal mixed micelle model can satisfactorily predict the mixture CMC.

3. Ionic Hydrocarbon–Ionic Hydrocarbon Mixtures The predicted CMC of mixtures of two anionic surfactants sodium dodecyl sulfate (SDS) and sodium decyl sulfate (SDeS), which differ from one another in their hydrocarbon tail lengths, is plotted against the composition of the singly dispersed surfactant in Fig. 29. The figure also contains the experimental data obtained by Mysels and Otter [101] based on conductivity measurements, and the data of Shedlowsky et al. [102] based on e.m.f. measurements. The predicted mixed micelle composition as a function of the composition of the singly dispersed surfactants is compared in Fig. 30 with the data obtained by Mysels and Otter [101]. In these binary mixtures, one of the sources of nonideality arises from the volume differences between the hydrophobic tails of the two surfactants. Consequently, at any given aggregation number, the area per molecule of the mixed micelle is a nonlinear function of the micelle composition and, hence, nonlinearity is reflected in all the free energy contributions. Another source of nonideality is the change in

FIG. 29 The CMC of SDS þ SDeS mixtures as a function of the composition of singly dispersed surfactants. (The experimental data shown by circles are from Ref. 101 and those shown by triangles are from Ref. 102.)

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FIG. 30 The composition of SDS þ SDeS mixed micelles as a function of the composition of singly dispersed surfactants. (The experimental data shown by circles are from Ref. 101.)

ionic strength of the solution as the composition is modified. In the absence of any added salt, the ionic strength is determined mainly by the concentration of the singly dispersed surfactants. This concentration changes with the composition of the mixed micelle and thus modifies the ionic interactions at the micelle surface nonlinearly as a function of the micelle composition. Given the importance of the ionic interactions to the free energy of micellization, the nonideality is more perceptible in these binary mixtures. Similar behavior is exhibited by the mixtures of cationic surfactants, dodecyl trimethyl ammonium bromide (DTAB), and decyl trimethyl ammonium bromide (DeTAB). This mixture is similar to the SDS-SDeS mixture as concerns the tail lengths of the two surfactants. However, the trimethyl ammonium bromide headgroup has a larger area ap compared to that of the anionic sulfate headgroup. The predicted CMC as a function of the micelle and the monomer compositions is presented in Fig. 31, which also contains the experimental CMC data obtained by Garcia-Mateos et al. [103] using electrical conductivity measurements. One can observe large CMC changes for the DTAB-DeTAB mixture as the composition is altered, which also influences the ionic strength of the solution. The predicted aggregation numbers do not show much growth beyond the size of the pure component

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FIG. 31 The CMC of DTAB þ DeTAB mixtures as a function of the composition of the micelles (dashed line) and that of singly dispersed surfactants (solid line). (The points are experimental data from Ref. 101.)

micelles. In contrast, for SDS-SDeS mixtures, the mixed micelles are larger than the pure component micelles. This different behavior is a consequence of the larger steric repulsion for these cationic surfactants with a bulky headgroup, when compared to the sulfate headgroups of SDS-SDeS. To explore the effect of different chain lengths of the hydrophobic tails, we have computed the micellization behavior of mixtures of anionic potassium alkanoates, namely, potassium tetradecanoate (KC14)–potassium octanoate (KC8) and potassium decanoate (KC10)–potassium octanoate (KC8) at 258C. The calculated CMC and micelle composition are plotted in Figs. 32 and 33 as a function of the composition of the singly dispersed surfactant, together with the experimental CMC data [104] obtained by Shinoda using dye solubilization measurements. One can observe from Fig. 33 that the less hydrophobic KC8 is almost completely excluded from the micelles in KC8 þ KC14 mixtures because of the much stronger hydrophobicity of KC14. The singly dispersed surfactants contain almost exclusively the less hydrophobic KC8 molecules. Such exclusion of KC8 from the mixed micelles is reduced in case of the KC8 þ KC10 mixtures, where the chain length difference between the surfactants is smaller.

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FIG. 32 The CMC of mixtures of potassium alkanoate (KCn) surfactants, KC8 þ KC10 mixtures, and KC8 þ KC14 mixtures as a function of the composition of singly dispersed surfactants. (The experimental data [circles and triangles] are from Ref. 104.)

4. Ionic Hydrocarbon–Nonionic Hydrocarbon Mixtures The mixture of anionic sodium dodecyl sulfate (SDS) and nonionic decyl methyl sulfoxide (C10SO) at 248C, in the presence of 1-mM Na2CO3, is taken for illustrative calculations. The calculated CMC is plotted in Fig. 34 as a function of the composition of the singly dispersed molecules. The experimental CMC data based on surface tension measurements [100] are also included in the figure. The results show that this binary mixture exhibits considerable nonideality. The CMC of the mixed system is substantially smaller than that anticipated for the ideal mixed micelle. The two surfactants differ somewhat in their hydrophobic tail lengths and in the sizes of the polar headgroups. Similar differences occurred in the case of nonionic–nonionic mixtures considered before, but they did not give rise to significant nonidealities. However, in the present case, one component is ionic while the other is nonionic. Therefore, the area per charge at the micelle surface and the ionic strength are strongly affected by the composition of the micelle. This results in a large variation in the ionic interaction energy at the micelle

FIG. 33 The composition of KC8 þ KC10 mixed micelles and KC8 þ KC14 mixed micelles as a function of the composition of singly dispersed surfactants.

FIG. 34 The CMC of SDS þ decyl methyl sulfoxide (C10SO) mixtures as a function of the composition of singly dispersed surfactants. (The experimental data [circles] are from Ref. 100.) 70

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surface as the micelle composition is changed. This free energy contribution is mostly responsible for the nonideal behavior exhibited by this system. The predictions for binary mixtures of the anionic sodium dodecyl sulfate (SDS) and nonionic -dodecyl maltoside (DM) are now compared with the experimental data obtained by Bucci et al. [105] in Figs. 35 to 37. In this system the two surfactants have identical hydrophobic chains, and hence the nonideality arises from the differences in the size and charge of the two headgroups. The calculated average aggregation numbers and those estimated from neutron scattering measurements [105] are plotted in Fig. 35 with and without added salt. The average aggregation number and the average micelle composition are estimated at a total surfactant concentration of 50 mM and at 258C. Figures 36 and 37 present the CMC and micelle composition as functions of the composition of the singly dispersed surfactants at two concentrations of added electrolyte, NaCl. They show that the nonionic DM molecules are preferentially incorporated in the micelles over most of the composition range. In the absence of any added electrolyte, this

FIG. 35 The average aggregation number of SDS þ dodecyl maltoside (DM) mixed micelles as a function of the total surfactant composition at a total surfactant concentration of 50 mM. (The experimental data are from Ref. 105, where the circles refer to micelle sizes in the absence of any added salt while the triangles correspond to a 0.2M concentration of added NaCl.)

FIG. 36 The CMC of SDS þ DM mixtures as a function of the composition of monomers. The conditions correspond to those in Fig. 35.

FIG. 37 The composition of mixed micelles of SDS þ DM as a function of the composition of monomers. The conditions correspond to those in Fig. 35. 72

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73

preference is stronger since this decreases the positive free energy contribution due to the presence of the anionic SDS in micelles. Obviously, in the presence of NaCl, the electrostatic repulsions between ionic headgroups are reduced, and hence a larger number of SDS molecules is incorporated into the mixed micelles.

5. Anionic Hydrocarbon–Cationic Hydrocarbon Mixtures When present together, the anionic and cationic surfactants are expected to form ion pairs with no net charge; this decreases their aqueous solubility and results in precipitation [106,107]. These surfactant mixtures can also generate mixed micelles or mixed spherical bilayer vesicles in certain concentration and composition ranges. As noted earlier, depending on the location of the charges on the anionic and the cationic surfactants, one can associate a dipole moment with each ion pair. Consequently, these surfactant mixtures can behave partly as ionic single chain molecules and partly as zwitterionic paired chain molecules. We have calculated the aggregation characteristics of binary mixtures of decyl trimethyl ammonium bromide (DeTAB) and sodium decyl sulfate (SDeS). The calculated and experimental [100] CMC values are presented in Fig. 38, while information regarding the micelle

FIG. 38 The CMC of SDeS þ DeTAB mixtures as a function of the composition of singly dispersed surfactants. (The points are experimental data from Ref. 100.)

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composition is provided by Fig. 39. The calculations show that rodlike mixed micelles are formed over most of the composition range. The micelle composition data show that the mixed micelles contain approximately equal numbers of the two types of surfactants over the entire composition range. The small deviation from the micelle composition value of 0.5 arises because of the different sizes of the polar headgroups. Some binary mixtures of anionic and cationic surfactants have been observed to give rise to spherical bilayer vesicles in aqueous solutions [108–111]. In these mixtures, the surfactant tails have appreciably differing lengths. Here, we have calculated the solution behavior of binary mixtures of cationic cetyl trimethyl ammonium bromide (CTAB) and anionic sodium dodecyl sulfate (SDS), in the presence of 1-mM NaBr as electrolyte, to explore the formation of micelles versus vesicles in such mixtures. Figure 40 shows the calculated critical aggregate (micelle or vesicle) concentration as a function of the composition of the aggregate. The lower the critical concentration for the formation of a given kind of aggregate, aggregates of that type are favored at equilibrium. In CTAB-poor systems, for CTAB mole fractions in the aggregates g;CTAB less than 0.2, only micelles are generated. For CTAB compositions in the region 0.2 < g;CTAB < 0.29,

FIG. 39 The average composition of SDeS þ DeTAB mixed micelles as a function of the composition of monomers.

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FIG. 40 The critical micelle (or vesicle) concentration of CTAB þ SDS mixtures as a function of the composition of the aggregates. Squares denote micelles and triangles refer to spherical bilayer vesicles. The lines simply connect the calculated results. The dashed line near the composition region between 0.29 and 0.57 corresponds to a precipitated surfactant phase of lamellar aggregates.

spherical bilayer vesicles are predicted to form. In CTAB-rich systems, again for g;CTAB > 0.57, only micelles are present in solution. In the narrow composition region of 0.565 < g;CTAB < 0.567, vesicles are formed in solution. For a wide range of CTAB composition 0.29 < g;CTAB < 0.565, the calculations indicate the formation of lamellar aggregates rather than spherical bilayer vesicles. This may correspond to the formation of a precipitating surfactant phase. Figure 41 presents the calculated average aggregation numbers of the micellar and vesicular aggregates. The micelles correspond to spherical or globular aggregates. The smallest vesicles formed are not very much larger than the larger micellar aggregates. The vesicles (corresponding to the results shown in the figure) have outer radii in the range of 3 to 14 nm. The inner and outer layers of the bilayer vesicles have differing compositions. In CTAB-poor vesicles the inner layer is enriched somewhat with CTAB, while in CTAB-rich vesicles the inner layer is somewhat depleted of CTAB.

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FIG. 41 The aggregation number of mixed micelles and mixed vesicles formed of CTAB þ SDS mixtures as a function of the composition of the aggregates. Squares denote micelles and triangles refer to vesicles. The lines simply connect the calculated results. The dashed line near the compositions 0.29 and 0.57, which extend vertically, suggest the formation of infinite lamellar aggregates in that region corresponding to the precipitation of a surfactant phase.

6. Anionic Fluorocarbon–Nonionic Hydrocarbon Mixtures The aggregation behavior of mixtures of nonionic alkyl-N-methyl glucamines (MEGA-n) and anionic sodium perfluoro octanoate (SPFO), studied experimentally by Wada et al. (112) was investigated using the present model. In such mixtures the nonideality associated with the mixing of hydrocarbon and fluorocarbon surfactant tails is superimposed on the nonideality associated with the mixing of anionic and nonionic headgroups. For hydrocarbon surfactants with these polar headgroups, the results presented earlier revealed considerable negative deviations from ideality (i.e., the free energy of mixed micelle formation is lower than the composition averaged sum of the free energies of the pure component micelles) [94,95]. For mixtures of hydrocarbon and fluorocarbon tails one can anticipate strong positive deviations from ideality. Thus, for the mixtures under study, both negative and positive deviations from ideality occur, which partially compensate for one another. As a result, these binary mixtures exhibit reduced nonideality. Figures 42 and 43 present

FIG. 42 The CMC of MEGA-8 þ SPFO mixtures as a function of the composition of singly dispersed surfactants. The calculated results are presented for two alternate estimates of the solubility parameter of the SPFO tail. (The experimental data [circles] are from Ref. 112.)

FIG. 43 The average composition of MEGA-8 þ SPFO mixed micelles as a function of the composition of monomers. The conditions correspond to those described in Fig. 42. 77

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the calculated CMC and the composition of the mixed micelles as a function of the composition of the singly dispersed surfactant for MEGA-8SPFO mixtures. Similar results for the MEGA-9-SPFO mixtures are presented in Figs. 44 and 45. The calculations have been performed for two different values of the solubility parameter of SPFO, one determined from the properties of pure fluorocarbons (12.3 MPa1/2), and the other determined from the properties of fluorocarbon–hydrocarbon mixtures (9.5 MPa1/2). Both MEGA-8-SPFO and MEGA-9-SPFO mixtures display the same qualitative behavior. The miscibility between the two surfactants is promoted by the headgroup interactions, which lead to a single kind of mixed micelles in solution. As shown by the data presented in Figs. 43 and 45, there is no demixing of micelles for the MEGA-8-SPFO and MEGA-9SPFO mixtures.

7. Anionic Hydrocarbon–Anionic Fluorocarbon Mixtures The micellization behavior of sodium perfluoro octanoate (SPFO) and sodium decyl sulfate (SDeS) mixtures is plotted in Figs. 46 and 47. The

FIG. 44 The CMC of MEGA-9 þ SPFO mixtures as a function of the composition of singly dispersed surfactants. The calculated results are presented for two alternate estimates of the solubility parameter of the SPFO tail. (The experimental data [circles] are from Ref. 112.)

FIG. 45 The average composition of MEGA-9 þ SPFO mixed micelles as a function of the composition of monomers. The conditions correspond to those described in Fig. 44.

FIG. 46 The CMC of SDeS þ SPFO mixtures as a function of the composition of micelles (dashed line) and that of singly dispersed surfactants (solid line). (The experimental data [circles] are from Ref. 112.) The experimental and calculated results indicate the coexistence of two micelle populations. 79

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FIG. 47 The average aggregation number of SDeS þ SPFO mixed micelles as a function of the composition of micelles. A single type of mixed micelle exists in the SDeS-rich and SPFO-rich mixtures, while for intermediate compositions, two distinct micelle sizes corresponding to the two coexisting micelle populations are indicated.

CMCs are compared with available experimental data [112] in Fig. 46, which also provides the composition of the mixed micelle. There are positive deviations in the CMC in contrast to the negative deviations observed in the binary mixtures examined previously. This positive deviation is a direct consequence of the interactions between the hydrocarbon and fluorocarbon tails. This positive deviation, which was also present in SPFOMEGA-n mixtures, was partially compensated for there by the negative deviations due to headgroup interactions. Such compensating effects are, however, absent in the SPFO-SDeS mixtures since the headgroups of both surfactants are anionic. The calculations show the interesting feature that over a certain composition domain, two types of micelles coexist. One is hydrocarbon-rich and the other one is fluorocarbon-rich, with average compositions gA ¼ 0:32 and 0.79, respectively. The predicted average aggregation numbers of the mixed micelles are plotted in Fig. 47. The

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fluorocarbon- and hydrocarbon-rich micelles that coexist have different aggregation numbers.

VI. MICELLIZATION OF SURFACTANTS IN THE PRESENCE OF POLYMERS The interactions between surfactant molecules and synthetic polymers in aqueous solutions are of importance to many applications in detergents and personal care products, chemical, pharmaceutical, mineral processing, and petroleum industries. In general, the mutual presence of polymer and surfactant molecules alter the rheological properties of solutions, adsorption characteristics at solid–liquid interfaces, stability of colloidal dispersions, the solubilization capacities in water for sparingly soluble molecules, and liquid–liquid interfacial tensions [113–117]. The ability of the surfactant and the polymer molecules to influence the solution and interfacial characteristics is controlled by the state of their occurrence in aqueous solutions, namely whether they form mixed aggregates in solution and, if so, the nature of their microstructures.

A. Polymer–Surfactant Association Structures Various morphologies of polymer-surfactant complexes can be visualized [118–121] depending on the molecular structures of the polymer and the surfactant and on the nature of the interaction forces operative between the solvent, the surfactant, and the polymer. A schematic view of these morphologies is presented in Fig. 48. Structure A denotes only the polymer, implying that no polymer–surfactant association occurs. This would be the case when both the polymer and the surfactant carry the same type of ionic charges. This could also occur when the polymer is relatively rigid and for steric reasons does not interact with ionic or nonionic surfactants. It could also be the situation when both the polymer and the surfactant are uncharged and no obvious attractive interactions, promoting association, exist between them. Structure B denotes a system where the polymer and the surfactant carry opposite electrical charges. Their mutual association is promoted by electrostatic attractions. These cause the creation of a complex with reduced charge and, hence, reduced hydrophilicity. Indeed, this eventually leads to the precipitation of these complexes from solution. Structure C also occurs in systems containing surfactant and polymer possessing opposite charges. In this case, the surfactant promotes intramolecular bridging within a polymer molecule by interacting with multiple sites on one molecule or intermolecular bridging by interacting simultaneously with sites on different polymer molecules.

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FIG. 48 Schematic representation of association structures involving surfactant and polymer molecules. A. Polymer molecule does not interact with surfactants for electrostatic or steric reasons. No surfactant is bound to the polymer. For example, the surfactant and the polymer are both anionic or both cationic. B. The polymer and the surfactant are oppositely charged. Single-surfactant molecules are bound linearly along the length of the polymer molecules. C. The polymer and the surfactant are oppositely charged. A single-surfactant molecule binds at multiple sites on a single-polymer molecule, giving rise to intramolecular bridging. Alternatively, it binds to more than one polymer molecule, allowing intermolecular bridging. D. The polymer is an uncharged random or multiblock copolymer. The surfactant molecules orient themselves at domain boundaries separating the polymer segments of different polarities. E. Polymer is hydrophobically modified. Individual surfactant molecules associate with one or more of the hydrophobic modifiers on a singlepolymer molecule or multiple-polymer molecules. F. Polymer is hydrophobically modified. Clusters of surfactant molecules associate with multiple hydrophobic modifiers on a single-polymer molecule. G. Polymer is hydrophobically modified. Clusters of surfactant molecules associate with each of the hydrophobic modifier on a single-polymer molecule. H. The polymer segments partially penetrate and wrap around the polar headgroup region of the surfactant micelles. A single polymer molecule can associate with one or more surfactant micelles.

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Structure D depicts a situation when the polymer is a random copolymer or multiblock copolymer with relatively short blocks. In this case, the polymer molecule assumes a conformation in solution characterized by segregation of dissimilar segments or blocks of varying polarity. Depending on whether the polymer is a random copolymer or a block copolymer, the segregation in the polymer can take different forms, including the formation of polymeric micelles. In either case, one can imagine the surfactant molecules to locate themselves at the interfaces between the segregated regions. Structures E, F, and G pertain to hydrophobically modified polymers. In this case the size of the hydrophobic modifier, its grafting density along the polymer, and the relative concentrations of the surfactant and the polymer all influence the nature of the association structure. In general, at low surfactant concentrations, structure E may be obtained with single-surfactant molecules or very small surfactant clusters (dimers, trimers, etc.) interacting with one or more hydrophobic modifiers, without causing any conformational changes on the polymer. When the surfactant concentration is increased, somewhat larger surfactant clusters form co-aggregates with multiple hydrophobic modifiers belonging to the same polymer molecule, causing the polymer conformation to change significantly. At larger surfactant concentrations, it is possible to obtain structure G, where surfactant aggregates are formed around each of the hydrophobic modifier. Structure H denotes a complex consisting of the polymer molecule wrapped around surfactant micelles with the polymer segments partially penetrating the polar headgroup region of the micelles and reducing the micelle core–water contact. Such a structure can describe a nonionic polymer interacting with surfactant micelles. Such a structure can also be imagined in the case of an ionic polymer interacting with oppositely charged micelles. For each type of polymer molecule such as those depicted in Fig. 48, including the nonionic polymer, charged polymer, hydrophobically modified polymer, star polymer, random or block copolymer, some of the unique features of the polymer molecule will have to be invoked in developing quantitative models of polymer–surfactant association. Given the widespread use of nonionic polymers, in this paper we explore the association of surfactants with such polymers. The general ideas of molecular aggregation and the formation of competitive association structures discussed in this paper are applicable to the various types of polymers mentioned above, but the details of the modeling would have to be modified in each case to account for the particularities of the polymer molecule.

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B. Modeling Nonionic Polymer–Surfactant Association The model presented below follows our earlier thermodynamic analysis of polymer-micelle association [121]; a somewhat different approach has been presented by Ruckenstein et al. [122]. We have extended the model also to treat the association of nonionic polymers with rodlike micelles and vesicles [123] and various types of microemulsions [124]. We visualize a polymer–surfactant association complex (Structure H in Fig. 48) as consisting of fully formed surfactant aggregates interacting with the polymer segments. In the polymer-bound aggregates, the polymer segments are assumed to adsorb at the aggregate–water interface, shielding a part of the hydrophobic domain of the aggregate from being in contact with water. The physicochemical properties of the polymer molecule determine the area of mutual contact between the polymer molecule and the hydrophobic surface of the aggregate. A characteristic parameter apol is defined to represent this area of mutual contact per surfactant molecule in the aggregate. Although it is not very realistic to anticipate that the polymer molecule can provide a uniform shielding of the aggregate surface from water by the amount apol per surfactant molecule, this area parameter is defined in the mean-field spirit to serve as a quantitative measure of the nature of polymer–aggregate interactions.

C. Free Energy of Micellization in the Presence of Polymer 1. Modification of Free Energy Contributions Due to Polymer The various contributions to the standard free energy of formation of micelles or vesicles have been summarized in Eq. (54), and detailed expressions to calculate each of these contributions are described in Section IV.A. For an ionic surfactant, the free energy of micellization has the form ð

0g Þ ð

0g Þtr ð

0g Þdef ð

0g Þint ð

0g Þsteric ð

0g Þionic þ þ þ þ ¼ kT kT kT kT kT kT ð80Þ where the five terms respectively account for the transfer free energy of the surfactant tail, the deformation free energy of the tail, the free energy of formation of the micelle–water interface, the steric repulsions between the headgroups at the micelle surface, and the electrostatic repulsions between the ionic headgroups. The last term is absent for nonionic surfactants and is replaced by the dipole interaction free energy term for zwitterionic surfactants. The standard free energy of formation of aggregates bound to the nonionic polymer molecule can be written in a similar form, with modifications to account for the presence of polymer segments at the aggregate–

Theory of Micelle Formation

85

water interface. As mentioned above, we assume that the polymer segments shield the hydrophobic domain from water by an area apol per surfactant molecule. This gives rise to three competing contributions to the free energy of aggregation. First, a decrease in the hydrophobic surface area of the aggregate exposed to water occurs. This decreases the positive free energy of formation of the aggregate–water interface and thus favors the formation of polymer-bound aggregates. Second, steric repulsions arise between the polymer segments and the surfactant headgroups at the aggregate surface. This increases the positive free energy of headgroup repulsions and thus disfavors the formation of the polymer-bound aggregates. Finally, the contact area apol of the polymer molecule is removed from water and transferred to the surface of the aggregate, which is concentrated in the surfactant headgroups. This alters the free energy of the polymer and can favor or disfavor the formation of the polymer-bound micelles depending on the type of interactions between the polymer segments and the interfacial region rich in headgroups. Taking these factors into account, one may write ð

0g Þ ð

0g Þtr ð
0g Þdef agg þ þ ða
ao
apol Þ ¼ kT kT kT kT
 ap þ apol ð

0g Þionic
pol apol

ln 1
þ a kT kT

ð81Þ

The first two terms and the fifth term are identical to those appearing in Eq. (80). The modified third term accounts for the enhanced shielding of the micellar core from water provided by the polymer. The modified fourth term accounts for the increase in the steric repulsions due to the presence of the polymer. The sixth term is new and represents the change in the interaction free energy of the polymer molecule. This interaction free energy is written as the product of an interfacial tension and the area of the polymer that is removed from water and brought into contact with the micellar surface. Here,
 pol is the difference between the macroscopic polymer-water interfacial tension and the interfacial tension between the polymer and the aggregate headgroup region. The factor
 pol is obviously affected by both the hydrophobic character of the polymer molecule and its interactions with the surfactant headgroups that are crowded at the interface. One may expect
 pol to be positive and larger for more hydrophobic polymers and for polymers that may have attractive interactions with surfactant headgroups. Given the above free energy equation, the equilibrium aggregation number, the surface area per surfactant molecule of the bound micelle, and the CMC are calculated as described in Section IV.B.2 by finding the maximum in the aggregate size distribution.

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2. Estimation of Molecular Constants Two new molecular constants, apol and
 pol, that depend on the type of the polymer are necessary for predictive calculations. At present, there are no a priori methods available for estimating apol, while
 pol can, in principle, be determined from interfacial tension measurements as suggested by Ruckenstein et al. [122]. Consequently, a reasonable and simple way to estimate both these parameters is by fitting the measured CMC and aggregation number of any one surfactant that forms spherical or globular micelles to the model; the fitted parameters can then be used to make a priori predictions for any other surfactant and for systems forming not only globular micelles but those forming rodlike micelles, vesicles, and microemulsions as well. For the illustrative calculations discussed below, we take apol ¼ 0:20 nm2 and
 pol ¼ 15, 0, or
15 mN/m. The three values for
 pol allow the investigation of the subtle role this contribution plays in governing the formation of polymer-bound aggregates. One may note that
 pol affects only the CMC and not the aggregation number since the polymer interaction free energy contribution is a constant independent of the aggregation number g.

D. Interaction of Polymer with Globular Micelles The equilibrium aggregation number, the area per molecule of the micelle, and the CMC calculated for the anionic surfactant—sodium dodecyl sulfate (SDS)—and the nonionic surfactant—decyl dimethyl phosphene oxide (C10PO)—are given in Table 5. In polymer-free solutions the anionic SDS micelles are small, only marginally larger than the largest spherical micelle possible, and the CMC is approximately 7.85 mM. In the presence of the polymer, the CMC is substantially decreased (2.45 mM, 1.2 mM, and 5.0 mM for
 pol ¼ 0, 15, and
15 mN/m, respectively), corresponding to a lower free energy for the formation of polymer-bound micelles. Therefore, polymer-bound globular micelles will be favored over free micelles in this case. The aggregation number of the polymer-bound micelles are smaller than those of the free micelles. Results for the nonionic surfactant C10PO are also given in Table 5. In the absence of the polymer, small ellipsoidal micelles (slightly larger than the largest spheres that are geometrically allowed) are formed and the CMC is approximately 0.386 mM. In the presence of the polymer, the CMC is significantly decreased (0.18 and 0.088 mM) if
 pol ¼ 0 or 15 mN/m. Therefore, micelles that are bound to the polymer will be favored over free micelles. The aggregation number of the polymer-bound spherical micelles are substantially smaller compared to that of the free micelles. The CMC of 0.368 mM calculated for
 pol ¼
15 mN/m is closer to but still smaller than that in the polymer-free solution,

0.386 0.18 7.85 2.45 0.243 0.128 0.313 0.057 4:26  10
3 2:61  10
3

C10 PO C10 PO þ P* C12 SO4 Na C12 SO4 Na þ P C12 PO C12 PO þ P C12 SO4 Na þ 0:8M NaCl C12 SO4 Na þ 0:8M NaCl þ P di-C12 SO4 Na þ 0:01M NaCl di-C12 SO4 Na þ P þ 0:01M NaCl

0.584 0.789 0.629 0.781 0.582 0.795 0.477 0.561 0.583 0.771 1.065

a (nm2 ) 52 20 56 29 — 28 — 88 462 727 60

g

—

— — — — 5:6  109 4:5  102 6:0  104 6:1  104 —

K

Globular

Globular Sphere Sphere Sphere Rod Sphere Rod Globular Vesicle**

Shape

*P refers to polymer-bound aggregates, the corresponding CMCs listed are for
pol ¼ 0 mN/m. These CMCs should be multiplied by 0.49 if
pol ¼ 15 or by 2.043 if
pol ¼
15 mN/m. **The two values listed for both a and g correspond to the inner and the outer layers of vesicle.

CMC (mM)

Surfactant

TABLE 5 Polymer Association with Surfactant Aggregates at 258C

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suggesting that polymer-bound micelles will be formed in this case as well. But a further decrease in
 pol (say,
16.5 mN/m) will result in the CMC for polymer-bound micelles being larger than the CMC for the polymer-free micelles. Therefore, a preference for free micelles over bound micelles would be predicted. Evidently, subtle free energy differences associated with the hydrophobicity of the polymer and its interactions with the surfactant headgroups can tilt the equilibrium favoring free micelles over bound micelles, or vice versa. The contrasting behavior of anionic and nonionic surfactants can be interpreted in the framework of the present thermodynamic model. If the polymer associates with the micelle, the presence of the polymer segments increases the equilibrium area per surfactant molecule. For nonionic surfactants, this incremental increase in the equilibrium area a is roughly comparable to the polymer-micelle contact area apol (see Table 5). Correspondingly, the negative increment in the free energy of shielding of the micellar core from water by the polymer segments is close in magnitude to the positive increment in the free energy of steric repulsions at the aggregate surface. If only these two effects are considered, then the CMC values for the formation of bound and free micelles are always very near one another, with the free micelles generally favored over the bound micelles. The tail deformation free energy decreases for the polymer-bound micelles because of their reduced size, thus favoring the bound micelles over the free micelles. The polymer interaction free energy provides a positive or negative contribution to the formation of the polymer-bound aggregates, depending on the polymer hydrophobicity and the polymer–surfactant headgroup interactions. This factor decreases further the CMC when
pol > 0 and tilts the balance more in favor of bound micelles. On the other hand, when
 pol < 0 and has a sufficiently large magnitude, this factor inhibits the occurrence of binding. The calculated results for the anionic surfactants are different from the above behavior of nonionic surfactants. In the case of anionic surfactants, the increase in the equilibrium area per molecule of the aggregate due to the presence of the polymer is generally smaller than the polymer–aggregate contact area apol. This is because of the importance of the electrostatic headgroup interaction term in determining the equilibrium area of the aggregate. Therefore, the negative increment in the free energy of shielding of the aggregate core from water by the polymer segments more than offsets the positive increment in the free energy of steric repulsions at the micellar surface. More importantly, there is a negative increment in the free energy of electrostatic interactions at the aggregate surface because the equilibrium area in the presence of the polymer is somewhat larger. These free energy contributions dominate over the polymer interaction free energy with the

Theory of Micelle Formation

89

result that the polymer-bound anionic micelles are always favored over free micelles. It has been reported that cationic surfactants do not bind with a nonionic polymer such as polyoxyethylene [117–121]. This differing behavior compared to anionic surfactants must thus come because
 pol < 0 and has a sufficiently large magnitude than that considered, for example, here. Such a situation can arise because of the weakly cationic nature of the ether oxygens in polyoxyethylene [118–121] and may lead to a larger CMC if polymer-bound aggregates are to form; hence binding does not occur in these cases.

E. Interaction of Polymer with Rodlike Micelles The general thermodynamic analysis of rodlike micelles was discussed in Section II.D. Given Eqs. (80) and (81) for the free energy of micellization, the sphere-to-rod transition parameter K and the CMC can be calculated as described in Section IV.B. We again note that the radius of the cylindrical middle part of the micelle and that of the spherical endcaps are allowed to differ from one another and are determined by the equilibrium conditions. The calculated results are included in Table 5 for the surfactant solution containing the anionic SDS in the presence of 0.8M NaCl. In the absence of the polymer, large polydispersed rodlike micelles are formed in solution as shown by the large value for K. When the polymer molecules are present, the magnitude of K is dramatically reduced. Consequently, only small globular micelles are formed. The CMC corresponding to the formation of the polymer-bound globular micelles (0.057, 0.028, and 0.116 mM for
 pol ¼ 0, 15, and
15 mN/m, respectively) is lower than that for the formation of polymer-free rodlike micelles (0.313 mM). Therefore, the equilibrium favors the formation of the smaller ellipsoidal micelles in the presence of the polymer. Thus, a rod-to-globule transition is induced by the addition of the polymer. In the case of the nonionic surfactant dodecyl dimethyl phosphene oxide (designated as C12PO), large rodlike micelles are formed in polymer-free solutions as reflected in the large value for K and at a CMC of 0.243 mM. In the presence of the nonionic polymer, K is dramatically decreased and the allowed aggregate is a small ellipsoid. The calculations show that if
 pol ¼ 0 or 15 mN/m, the CMC is 0.128 or 0.063 mM in the presence of the polymer. This implies that equilibrium will favor the formation of small polymerbound ellipsoidal micelles. In contrast, if
 pol ¼
15 mN/m, the CMC corresponding to the polymer-bound micelles is larger (0.262 mM) than the CMC corresponding to the polymer-free micelles. In such a case, the binding with polymer does not occur and polymer-free rodlike micelles coexist with free polymer molecules in solution. One can also visualize

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situations when
 pol assumes values such that the CMCs corresponding to both polymer-free and polymer-bound micelles are comparable. This condition will lead to the coexistence of both large rodlike and small polymer-bound ellipsoidal micelles.

F. Interaction of Polymer with Vesicles The free energy of aggregation is still given by Eqs. (80) and (81), with the understanding that the headgroup interaction free energies and the free energy of formation of the aggregate–water interface should be written for both inner and outer layers of the bilayer vesicle, the areas per molecule at these two layers can differ from one another, and the layers can have different thicknesses. The polymer molecules are assumed to be present at both inner and outer surfaces of the spherical bilayer vesicle. Calculations have been carried out for didodecyl sodium sulfate (designated as di-C12 SO4Na), and the results are included in Table 5. In the absence of the polymer, calculations show (see also Table 4) that di-C12 SO4Na forms small spherical bilayer vesicles. The inner and the outer radii of the hydrophobic shell are 4.63 nm and 6.68 nm, respectively. The volume of the aqueous core per surfactant molecule is approximately 0.350 nm3, whereas the volume of the polar headgroup of di-C12 SO4Na is less than 0.100 nm3. When the polymer molecule is present, the formation of the vesicular structure is not predicted. Calculations to examine the formation of closed aggregates have been carried out for the aggregate geometries described in Table 1. The results show that small globular micelles are favored (see Table 5). Thus, the addition of polymer to the solution containing vesicles can lead to a disruption of the vesicle structure and the generation of smaller closed aggregates.

VII. MICELLIZATION OF BLOCK COPOLYMERS A. Aggregation of Block Copolymers in Selective Solvents Block copolymer molecules exhibit self-assembly behavior similar to conventional low molecular-weight surfactants. The different blocks of the AB copolymer can have differing affinities for the solvent: the A block being solvophobic and the B block being solvophilic. The two blocks thus resemble the tail and the head of a conventional low-molecular-weight surfactant. As a result, the block copolymer forms micelles in a variety of solvents that are selective to one block and nonselective to the other block (Fig. 49). In these micelles, the solvophobic blocks constitute the core and the solvophilic blocks constitute the corona or the shell region of the micelle. For BAB triblock copolymers, the A block has to bend back and form a loop so as to

Theory of Micelle Formation

91

FIG. 49 Schematic representation of a micelle formed of AB diblock copolymers in a solvent selective for the B block. The thick lines denote the solvophobic blocks constituting the core of the micelle while the thin lines denote the solvophilic blocks forming the corona region of the micelle. A significant amount of solvent is present in the corona region

ensure that the two B blocks are in the corona; similarly, in the case of an ABA copolymer, the B block has to form a loop and bend back so that the two A blocks can be inside the core. The micellization tendency of block copolymers may be viewed as more general compared to that of low-molecular-weight surfactants in the sense that aggregation of block copolymers is achievable in almost any solvent by selecting the appropriate block copolymer, whereas for low-molecularweight surfactants, water constitutes the best solvent for promoting aggregation [125–127]. Further, in the case of block copolymers, because of the large size of the solvophobic block, the CMC is often too small to be measurable when compared to the CMCs of low-molecular-weight surfactants. The block copolymer aggregates can assume a variety of shapes such as spheres, cylinders, and bilayers including vesicles. Theoretical treatments of block copolymer micelles have been pioneered by de Gennes [128], Leibler et al. [129], Noolandi and Hong [130], and

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Whitmore and Noolandi [131]. de Gennes [128] has analyzed the formation of a diblock copolymer micelle in selective solvents by minimizing the free energy per molecule of an isolated micelle with respect to the aggregation number or core radius. The micelle core was assumed fully segregated and devoid of any solvent. The free energy of formation of the core–corona interface and the elastic free energy of stretching of the core blocks control the micellization behavior. Leibler et al. [129] have treated the problem of micelle formation of a symmetric diblock copolymer in a homopolymer solvent. In their study and in de Gennes’ work, the interface is taken to be sharp. Noolandi and Hong [130] and Whitmore and Noolandi [131] have formulated mean-field models taking into account the possibility of a diffuse interface between the core and corona regions. The ideas of chain deformation and corona mixing effects discussed in these studies have been applied to the treatment of micelles formed of nonionic surfactants with polyethylene oxide headgroups in Section IV. A. In this section the thermodynamic treatment of aggregation is presented for AB diblock and BAB triblock copolymers, and one can observe the close similarity between the model for low-molecular weight surfactants and that for the high-molecular-weight block copolymers.

B. Aggregate Shapes and Geometrical Relations Before developing a free energy model, we will first define the geometrical relations connecting aggregate shapes to the molecular size properties of the block copolymer. The shape of the aggregate and the assumption of incompressibility lead to the geometrical relations summarized in Table 6 for different morphologies. We denote the molecular volumes of the A and the B segments and the solvent by vA, vB, and vW, respectively. The characteristic lengths of the A and the B segments are denoted by LA (¼ v1=3 A ) ). The variables N and N refer to the number of segments and LB (¼ v1=3 A B B of block A and block B for the AB diblock as well as the BAB triblock TABLE 6 Geometrical Properties of Spherical, Cylindrical, and Lamellar Block Copolymer Aggregates Property Vc Vs g a ’B

Sphere

Cylinder

Lamella

4R3 =3 Vc ½ð1 þ D=RÞ3
1 Vc ’A =ðNA vA Þ 3NA vA =ðR’A Þ ðNB vB =NA vA Þ’A ðVc =Vs Þ

R2 Vc ½ð1 þ D=RÞ2
1 Vc ’A =ðNA vA Þ 2NA vA =ðR’A Þ ðNB vB =NA vA Þ’A ðVc =Vs Þ

2R Vc ½ð1 þ D=RÞ
1 Vc ’A =ðNA vA Þ NA vA =ðR’A Þ ðNB vB =NA vA Þ’A ðVc =Vs Þ

Theory of Micelle Formation

93

copolymers, implying that the BAB triblock copolymer has two terminal blocks of size NB/2 attached to a middle block of size NA. We use the variable R to denote the hydrophobic core dimension (radius for sphere or cylinder and half-bilayer thickness for lamella), D for the corona thickness, and a for the surface area of the aggregate core per constituent block copolymer molecule. The number of molecules g, the micelle core volume VC, and the corona volume VS all refer to the total quantities in the case of spherical aggregates, quantities per unit length in the case of cylindrical aggregates, and quantities per unit area in the case of lamellae. The core volume VC is calculated as the volumes of the A blocks, VC ¼ gNA vA . The concentrations of segments are assumed to be uniform in the core as well as in the corona, with ’A standing for the volume fraction of the A segments in the core (’A = 1), and ’B for the volume fraction of the B segments in the corona. If any two structural variables are specified, all remaining geometrical variables can be calculated through the relations given in Table 6. For convenience, R and D are chosen as the independent variables.

C. Free Energy of Micellization of Block Copolymers An expression for

0g is formulated by considering all the physicochemical changes accompanying the transfer of a singly dispersed copolymer molecule from the infinitely dilute solution state to an isolated micelle, also in the infinitely dilute solution state. First, the transfer of the singly dispersed copolymer to the micellar core is associated with changes in the state of dilution and in the state of deformation of the A block. Second, the B block of the singly dispersed copolymer is transferred to the corona region of the micelle, and this transfer process also involves changes in the states of dilution and deformation of the B block. Third, the formation of the micelle localizes the copolymer such that the A block is confined to the core while the B block is confined to the corona. Fourth, the formation of the micelle is associated with the generation of an interface between the micelle core made up of A blocks and the micelle corona consisting of solvent W and B blocks. Further, in the case of a BAB triblock copolymer, folding or loop formation of the A block occurs, ensuring that the B blocks at the two ends are in the aqueous domain while the folded A block is within the hydrophobic core of the micelle. This provides an additional free energy contribution. The overall free energy of micellization can be obtained as the sum of the above individual contributions: ð

0g Þ ¼ ð

0g ÞA;dil þ ð

0g ÞA;def þ ð

0g ÞB;dil þ ð

0g ÞB;def þ ð

0g Þloc þ ð

0g Þint þ ð

0g Þloop ð82Þ

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Nagarajan

One may observe that many of the above free energy contributions are analogous to those considered in Section IV.A for conventional surfactants. The A block dilution contribution is equivalent to the tail transfer free energy. The A block deformation is identical to the tail deformation energy. The B block dilution and deformation contributions are equivalent to headgroup repulsions. The interfacial free energy is the same for both kinds of surfactants. Only the localization and looping free energy contributions are additions that had not been considered before. These two contributions are found to be less important relative to the other contributions. In essence, the free energy model for the block copolymer remains essentially the same as that for all self-assembling systems treated in this chapter. Expressions for each of the contributions appearing in Eq. (82) are formulated below.

1. Change in State of Dilution of Block A In the singly dispersed state of the copolymer molecule in solvent, the A block is in a collapsed state, minimizing its interactions with the solvent. The region consisting of the collapsed A block with some solvent entrapped in it is viewed as a spherical globule, whose diameter 2R1A is equal to the endto-end distance of block A in the solvent. The volume of this spherical region is denoted by V1A . The chain expansion parameter A describes the swelling of the polymer block A by the solvent W.
1=3 4 R31A 6 A ¼ ; 2 R1A ¼ A NA1=2 LA ; NA
1=6 
1=3 V1A ¼ A1  3 ð83Þ where ’A1 (¼ NA vA =V1A ) is the volume fraction of A segments within the monomolecular globule. The first equality in Eq. (83) follows from geometry, while the second equality is based on the definition of the chain expansion parameter A, taking (NA1=2 LA ) as the unperturbed end-to-end distance of block A. The third equality is obtained by combining the first two in conjunction with the definition for ’A1. Applying the suggestion of de Gennes [58], the volume fraction ’A1 is calculated from the condition of osmotic equilibrium between the monomolecular globule treated as a distinct phase and the solvent surrounding it. lnð1
A1 Þ þ A1 þ AW 2A1 ¼ 0

ð84Þ

In Eq. (84), AW is the Flory interaction parameter between the pure A polymer and solvent. In the micelle, the A block is confined to the core region where it is like a pure liquid. The difference in the dilution of block A from its singly dispersed state to the micellized state makes a free energy contribution given by the relation

Theory of Micelle Formation

  ð

0g ÞA;dil v 1
A1 v ¼
NA A lnð1
A1 Þ þ A ð1
A1 ÞAW kT vW A1 vW ! 1=2 2  L 6NA
AW A kT A

95

ð85Þ

The first two terms account for the entropic and enthalpic changes associated with the removal of A block from its infinitely dilute state in the solvent to a pure A state. These terms are written in the framework of the Flory [53] expression for an isolated polymer molecule. The last term accounts for the fact that the interface of the globule of the singly dispersed A block disappears on micellization. This term is written as the product of the interfacial tension ( AW) between pure A and solvent W, the surface area of the globule (4R21A2 ), and the factor ’A1 (volume fraction of the polymer A in the globule) to account for the reduction in the contact area between the block A and solvent W caused by the presence of some solvent molecules inside the monomolecular globule. If the interfacial tension  AW is not available from direct measurements, it can be estimated using the relation AW ¼ ðAW =6Þ1=2 ðkT =L2 Þ, where L ¼ v1=3 W . Such a relation is usually employed for the calculation of polymer–polymer interfacial tensions [132,133].

2. Change in State of Deformation of Block A In the singly dispersed state of the copolymer, the conformation of the A block is characterized by the chain expansion parameter A, which is the ratio between the actual end-to-end distance and the unperturbed end-toend distance of the polymer block. The free energy of this deformation is written using the Flory [53] expression derived for an isolated polymer molecule. Within the micelle the A block is stretched nonuniformly, with the chain ends occupying a distribution of positions within the core while ensuring that the core has a uniform concentration. The free energy contribution allowing for nonuniform chain deformation is calculated using the analysis of chain packing pioneered by Semenov [46] and discussed earlier in detail in Section IV.A. In the case of a BAB triblock copolymer, the A block deformation is calculated by considering the folded A block of size NA to be equivalent to two A blocks of size NA/2. On this basis, one obtains " #
!  2 ð

0g ÞA;def p 2 3 2 R 3 ð
1Þ
ln A ¼ q ð86Þ
2 A kT 80 ðNA =qÞ L2A where q ¼ 1 for an AB diblock copolymer and q ¼ 2 for a BAB triblock copolymer having a middle hydrophobic block. The parameter p is dependent on aggregate shape and has the value of 3 for spherical micelles, 5 for

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cylinders, and 10 for lamellae (see Section IV.A). In Eq. (86), the first term represents the A block deformation free energy in the micelle while the second term corresponds to the deformation free energy in the singly dispersed copolymer.

3. Change in State of Dilution of Block B In the singly dispersed state of the copolymer, the polymer block B is swollen with the solvent. As mentioned before, NB denotes the size of the B block for the AB diblock copolymer, while for a symmetric BAB triblock copolymer, the end blocks are of equal size NB/2. We consider this swollen B block to be a sphere whose diameter 2R1B is equal to the end-to-end distance of isolated block B in the solvent. The volume of this spherical region is denoted by V1B , while ’B1 ½¼ ðNB =qÞvB =V1B  is the volume fraction of B segments within the monomolecular globule; V1B ¼

4 R31B ; 3

2 R1B ¼ B ðNB =q Þ1=2 LB

ð87Þ

The second equality in Eq. (87) is based on the definition of the chain expansion parameter B, which can be estimated using the expression developed by Flory (53). In the Flory expression for B, Stockmayer (134) has suggested decreasing the numerical coefficient by approximately a factor of 2 to ensure consistency with the results obtained from perturbation theories of excluded volume. Consequently, one can estimate B as the solution of 5B
3B ¼ 0:88ð1=2
BW ÞðNB =q Þ1=2

ð88Þ

where BW is the Flory interaction parameter between the B block and water. In the micelle, the B blocks are present in the corona region of volume VS. This region is assumed to be uniform in concentration with ’B (¼ gNB vB =VS ) being the volume fraction of the B segments in the corona. The free energy of the corona region can be written using the Flory [53] expression for a network swollen by the solvent. Therefore, the difference in the states of dilution of the B block on micellization provides the following free energy contribution:   ð

0g ÞB;dil v 1
B v ¼ NB B lnð1
B Þ þ B ð1
B ÞBW kT vW B vW ð89Þ   v 1
B1 v
NB B lnð1
B1 Þ þ B ð1
B1 ÞBW vW B1 vW The first two terms in Eq. (89) describe the entropic and enthalpic contributions to the free energy of swelling of the B block by the solvent in the

Theory of Micelle Formation

97

corona region of the micelle, while the last two terms refer to the corresponding contributions in the singly dispersed copolymer molecule.

4. Change in State of Deformation of Block B In the singly dispersed state the B block has a chain conformation characterized by the chain expansion parameter B. In the micelle the B block is stretched nonuniformly over the micelle corona so as to ensure that the concentration in the corona region is uniform. Semenov [46] has shown that the estimate for the chain deformation energy assuming that the termini of all B blocks lie at the distance D from the core surface is not very different from that calculated assuming a distribution of chain termini at various positions within the corona. This has already been applied to the case of surfactants with oligomeric ethylene oxide headgroups in Section IV. We use that expression here also:     ð

0g ÞB;def 3 LB R 3 2 3 ¼ q P
q ðB
1Þ
ln B 2 ða=qÞ B 2 kT

ð90Þ

where a is the surface area per molecule of the micelle core, q ¼ 1 for AB diblock and 2 for BAB triblock, as mentioned before, and P is a shapedependent function given by P ¼ ðD=RÞ=½1 þ ðD=RÞ for spheres, P ¼ ln½1 þ ðD=RÞ for cylinders, and P ¼ ðD=RÞ for lamellae. The first term in Eq. (90) represents the free energy of deformation of the B block in the micellar corona, while the second term denotes the corresponding free energy in the singly dispersed copolymer molecule.

5. Localization of the Copolymer Molecule On micellization, the copolymer becomes localized in the sense that the joint linking blocks A and B in the copolymer are constrained to remain in the interfacial region rather than occupying all the positions available in the entire volume of the micelle. The entropic reduction associated with localization is modeled using the concept of configurational volume restriction [135]. Thus, the localization free energy is calculated on the basis of the ratio between the volume available to the A–B joint in the interfacial shell of the micelle (surrounding the core and having a thickness LB) and the total volume of the micelle:   ð

0g Þloc d LB ¼
q ln kT Rð1 þ D=R Þd

ð91Þ

Here, d refers to the dimensionality of aggregate growth and is 3 for spherical micelles, 2 for cylinders, and 1 for lamellae.

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6. Formation of Micellar Core–Solvent Interface When a micelle forms, an interface is generated between the core region consisting of the A block and the corona region consisting of the B block and the solvent W. The free energy of formation of this interface is estimated as the product of the surface area of the micellar core and an interfacial tension characteristic of this interface. The appropriate interfacial tension is that between a pure liquid of block A in the micelle core and a solution of block B and solvent W in the micellar corona. Because the corona region is often very dilute in block B, the interfacial tension can be approximated as that between the solvent W and the A block in the micelle core. Denoting the polymer A–solvent W interfacial tension by  AW, the free energy of generation of the micellar core–solvent interface is calculated from ð

0g Þint agg ¼ a; kT kT

agg ¼ AW

ð92Þ

7. Backfolding or Looping in Triblock Copolymer The backfolding of the middle block in a BAB triblock copolymer contributes an entropic term to the free energy of solubilization. This contribution is absent for the case of a diblock copolymer. Jacobson and Stockmayer [136] show that the reduction in entropy for the condition that the ends of a linear chain of N segments are to lie in the same plane or on one side of a plane is proportional to ln N. Therefore, the assumption that the backfolding of the middle block in the micelle follows the same functional form is made. Hence, the backfolding entropy makes the following contribution in the case of a BAB copolymer: ð

0g Þloop 3 ¼ lnðNA Þ 2 kT

ð93Þ

Here is an excluded volume parameter that is equal to unity when the excluded-volume effects are negligible and larger than unity when these effects become important. In our calculations, is taken to be unity. The difference in the estimate for in the case of ABA and BAB triblock copolymers may be important for explaining any observed differences between these two kinds of triblock copolymers having the same overall molecular weight and composition.

D. Predicted Aggregation Behavior of Block Copolymers Illustrative calculations have been carried out for the diblock copolymers, polystyrene-polyisoprene (PS-PI) and polyethylene oxide-polypropylene

Theory of Micelle Formation

99

oxide (PEO-PPO, denoted EXPY), and for the triblock copolymer PEOPPO-PEO (denoted EXPYEX). One may note that for PS-PI block copolymers, the micellization is examined in n-heptane, which is a selective solvent for the PI blocks and is a nonsolvent for the PS block. For the other two block copolymers, the calculations have been done in water as the solvent with PEO being the hydrophilic and PPO the hydrophobic blocks.

1. Estimation of Model Parameters To facilitate quantitative calculations, the values of molecular constants appearing in the free energy expressions are needed. The molecular volumes (vA and vB) of the repeating units appearing in the different polymer blocks are 0.1612 nm3 for styrene, 0.0943 nm3 for butadiene, 0.01257 nm3 for isoprene, 0.0965 nm3 for propyleneoxide, and 0.0646 nm3 for ethyleneoxide [135]. The molecular volume of water is 0.030 nm3 and of n-heptane is 0.2447 nm3. Given the molecular weight M of the copolymer and the block composition (MA/M), the number of repeating units NA and NB in blocks A and B can be calculated, knowing the segmental molecular weights. The characteristic length L is calculated knowing the molecular volume of the solvent. For polyisoprene–heptane the solvent is practically a theta solvent at 258C and, correspondingly, the interaction parameters BW is taken to be 0.5 [130]. For polystyrene–heptane the solvent is a very poor solvent. The interaction parameter at 258C is taken to be AW ¼ 1:9 [130]. The corresponding interfacial tension between pure polystyrene and n-heptane is estimated from to be 5.92 mN/m [135]. For polyethyleneoxide–water and polypropyleneoxide–water systems, the interaction parameters were estimated from the experimental activity data available in the literature [54]. These data display a concentration dependence for the interaction parameters. Because the corona region is a dilute solution of polyethyleneoxide in water, the value for BW is taken from the dilute concentration region of the activity data. Correspondingly, BW ¼ 0.21. Because the core region is made of pure polypropyleneoxide, the value for AW is taken from the concentrated region of the activity data. For this condition AW ¼ 2.1. The corresponding value for the interfacial tension between water and polypropyleneoxide is calculated to be 25.9 mN/m. The estimation of these molecular constants are discussed in detail in Ref. [135].

2. Influence of Free Energy Contributions The role of various free energy contributions in influencing micelle formation can be understood from their dependence on the aggregation number. The formation of micelles in preference to the singly dispersed state of the copolymer occurs because of the large negative free energy contribution

100

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arising from a change in the state of dilution of the solvent incompatible A block. This free energy contribution is a constant independent of the size of the micelle and hence does not govern the aggregation number of micelle. The contribution favorable to the growth of the aggregates is provided by the interfacial energy. The geometrical relations dictate that the surface area per molecule of the micelle decrease with an increase in the aggregation number. Consequently, the positive interfacial free energy between the micellar core and the solvent decreases with increasing aggregation number of the micelle, and thus this contribution promotes the growth of the micelle. The changes in the state of deformation of the A and the B blocks and the change in the state of dilution of the B block provide positive free energy contributions that increase with increasing aggregation number of the micelle. Therefore, these factors are responsible for limiting the growth of the micelle. The free energy of localization and the free energy of backfolding or looping (in the case of BAB triblock copolymers) are practically independent of g and thus have little influence over the determination of the equilibrium aggregation number. The net free energy of formation of the micelle per molecule is negative and shows a minimum at the equilibrium aggregation number.

3. Predicted Micellization Behavior For the PS-PI copolymer, the calculated results have been compared against the measurements of Bahadur et al. [137] in Table 7. The tabulated experimental data have been derived from photon correlation spectroscopy and viscosity measurements, assuming that the micelle core is completely devoid of any solvent. The values for g, R, and D estimated from the measurements in this manner are compared against the model prediction and show reasonable agreement. Experimental observations using electron microscopy indi-

TABLE 7 Micellization Behavior of Polystyrene— Polyisoprene in n-Heptane at 258C 

M 29,000 36,000 39,000 49,000 53,000

MA =M 0.31 0.45 0.49 0.59 0.62

*Source: Ref. 137.

g R (A ) D=R (Experiments)* 94 119 141 194 240

1.77 1.31 1.13 0.96 0.92

248 278 386 666 1113



R ðA Þ D=R g (Predictions) 85 128 144 193 211

1.77 1.24 1.12 0.87 0.80

187 352 422 672 778

Theory of Micelle Formation

101

cated that the micelles are practically monodispersed [137], which is also in agreement with the predictions. Computed results for diblock copolymers of PEO-PPO are summarized in Table 8. The core radius, the corona thickness, the micelle aggregation number, and the calculated CMC (expressed as volume fraction CMC in water) are all listed. The calculations have been carried out keeping the size MA of the PPO block constant, the size MB of the PEO block constant, the composition of the block copolymer MA/M constant, and the overall molecular weight of the polymer M constant. In general, the PPO block plays a dominant role in determining the dimension of the core radius R, and hence the aggregation number g of the micelle, while the influence of the PEO

TABLE 8 Micellization Behavior of PEO–PPO Diblock Copolymers in Water M

MA =M



R ðA Þ

D=R

g


ln CMC

Results for constant 4,750 0.79 8,250 0.45 12,500 0.30 14,000 0.27 15,750 0.24

MA ¼ 3750 113 0.38 89 1.56 80 2.72 78 3.09 76 3.49

976 475 342 318 296

74.1 67.2 63.9 63.2 62.4

Results for constant 10.750 0.19 14,250 0.39 17,250 0.49 20.750 0.58 28,750 0.70 43,750 0.80

MB ¼ 8750 52 3.86 105 2.17 144 1.65 186 1.32 275 0.92 424 0.60

173 528 884 1354 2610 5491

38.8 85.3 116.7 148.3 208.8 298.1

Results for constant 2,000 0.30 7,000 0.30 20,000 0.30 35,000 0.30 50,000 0.30

MA =M 29 58 103 140 169

101 234 461 656 818

17.0 42.6 88.1 127.6 160.8

Results for constant 12,500 0.75 12,500 0.63 12,500 0.50 12,500 0.40 12,500 0.20

M ¼ 12,500 189 0.58 152 0.99 122 1.50 100 2.02 59 3.79

1808 1143 736 512 208

132.4 114.2 95.7 80.3 46.0

¼ 0:30 1.79 2.39 3.03 3.43 3.71

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block on R and g is comparatively smaller. The PEO block dominantly controls the corona dimension D, while the PPO block has only a marginal influence over it. In all cases the CMC is negligibly small because of the large size of the hydrophobic part of the block copolymer. The relative importance of the PEO block (the headgroup) and the PPO block (the tail group) is quite different from that of conventional surfactant micelles. As seen in Section IV, the headgroup has a much more significant effect on the micellar properties for small surfactant molecules. This is because the headgroup interaction free energy is much more significant compared to the tail deformation free energy in the case of small surfactant molecules. Hence, the effect of the headgroup is dominant. In contrast, the tail deformation free energy can be comparable, if not larger, than the headgroup interaction energy in the case of block copolymers, depending on the relative sizes of the two blocks. Therefore, one sees the dominant role of the hydrophobic group in controlling the micelle size in the case of block copolymers.

4. Aggregate Shape Transitions To investigate how the size and composition of the block copolymer affect the transition from one aggregate shape to another in the case of block copolymers, we have computed the aggregation properties of the family of Pluronic triblock copolymers, EXPYEX. The calculated results are summarized in Table 9. The aggregate shape that yields the smallest free energy of aggregation is taken to be the equilibrium shape. Also given are the dimensions of the core (R) and the corona (D), and the aggregation number (g) in the case of spherical aggregates. The numerical values within brackets are some available experimental data (see details in Ref. [138]). The calculations show that lamellar aggregates are favored when the ratio of PEO to PPO is small, whereas spherical aggregates are favored when the PEO to PPO ratio is large. Typically, for block copolymers containing 40 or more wt % PEO, only spherical aggregates form at 258C. For block copolymers containing 30 wt % PEO, cylindrical aggregates are possible. Block copolymers containing 20 or less wt % PEO generate lamellae. This is analogous to the behavior of small-molecular-weight surfactants where the reduction in headgroup repulsions can cause a transition from spherical micelles to rodlike micelles and then to bilayers. In this sense the formation of block copolymer micelles is entirely analogous to the formation of conventional surfactant micelles. One important difference that should be noted is connected to the tail transfer free energy contribution, which is negative and is the driving force for aggregation of surfactants and block copolymers. In

0.75 1.45 3.41 3.07 2.13 4.22 3.63 2.5 2.67 3.86 5.1 4.39 2.83 4.21 3.75

E6 P35 E6 E9 P32 E9 E13 P30 E13 E19 P29 E19 E77 P29 E77 E19 P43 E19 E26 P40 E26 E104 P39 E104 E118 P45 E118 E17 P60 E17 E27 P61 E27 E37 P56 E37 E133 P50 E133 E20 P70 E20 E100 P64 E100 (2.5)

(3.7)

(2.5)

(3.8 to 4.6)

1.92 1.39 1.66 (3.7 to 4.4) 2.21 5.1 (5.3) 2.25 2.83 (3.6) 6.39 7.03 2.15 2.98 3.73 7.66 (15.0) 2.44 7.02

— — 57 43 15 75 53 17 18 — 94 65 20 — 35 (15 to 45, 30)

(13)

(57, 37 to 78)

(22)

(39 to 70)

g for Spheres

L62 L63 L64 L65 F68 P84 P85 F88 F98 P103 P104 P105 F108 P123 F127

D (nm)

Structure

Name

R (nm)

Aggregation Behavior of PEO–PPO–PEO Triblock Pluronic Copolymers in Water

TABLE 9

Lamellae Lamellae Sphere Sphere Sphere Sphere Sphere Sphere Sphere Cylinder Sphere Sphere Sphere Cylinder Sphere

Shape

Theory of Micelle Formation 103

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the case of conventional surfactants the negative free energy associated with the transfer of the tail is significant when water is the solvent and diminishes in magnitude when water is replaced by other polar organic molecules such as alcohols, glycerol, or formamide [139,140]. Therefore, the aggregation tendency is diminished when water is replaced by other solvents in the case of small-molecular-weight surfactants. In contrast, for block copolymers, the core-forming block can be incompatible with a wide variety of solvents, and therefore the tail transfer free energy contribution remains negative at appreciably large magnitude. Hence, significant aggregation of block copolymers is possible in a number of solvents. The aggregation phenomenon where entropy changes (hydrophobic effect) are important as in the case of conventional surfactants is replaced by an aggregation process where enthalpy effects are also very important in the case of block copolymer aggregates.

VIII. CONCLUSIONS A quantitative approach to predicting micellar properties from the molecular structure of surfactant has been described in this chapter. No information derived from experiments on surfactant solutions is required for the predictive calculations. The approach combines the general thermodynamic principles of self-assembly with detailed molecular models for the various contributions to the free energy of micellization. Explicit analytical equations are developed for calculating each of these free energy contributions as a function of temperature. Methods for obtaining the few molecular constants appearing in these equations have been illustrated. A simple approach to calculating the aggregation properties via the maximum-term method, without having to perform detailed micelle size distribution calculations, is described. The quantitative approach is illustrated via predictive calculations performed on numerous surfactants, binary surfactant mixtures, and mixtures of surfactants with nonionic polymer. Surfactants having one or two tail groups, those possessing nonionic, anionic, cationic or zwitterionic headgroups, surfactants with fluorocarbon tails, and mixtures of hydrocarbon and fluorocarbon surfactants have been considered for the illustrative calculations. The formation of spherical micelles, globular micelles, rodlike micelles, and spherical bilayer vesicles have been investigated. The quantitative approach has been extended also to block copolymer aggregates taking into account the polymeric nature of both the headgroups and the tail groups of the block copolymer. The behavior of block copolymers is seen as analogous to that of surfactants, but with broader aggregation capabilities in a variety of solvent systems. The quantitative approach

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described in this chapter has a broad scope as demonstrated by its extension to treat other self-assembly phenomena such as micellization in polar nonaqueous solvents [139,140], solubilization of hydrocarbons in micelles [16,19], the formation of droplet-type and bicontinuous-type microemulsions [141], and micelle formation at solid–liquid interfaces [142,143]. An approach similar to that discussed in this chapter for calculating the aggregation behavior of surfactants and surfactant mixtures, but with some variations in the free energy expressions, has been discussed in a number of papers by Blankschtein and co-workers [39–41,144–147].

ACKNOWLEDGMENTS Professor Ruckenstein has collaborated in many parts of the work discussed here as can be inferred from the cited references. The author has benefited from numerous discussions with him.

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R. Nagarajan. ACS Polymer Preprints, 22:33, 1981. R. Nagarajan and B. Kalpakci. ACS Polymer Preprints, 23:41, 1982. R. Nagarajan. Colloids Surf., 13:1, 1985. R. Nagarajan and B. Kalpakci. In P.L. Dubin, ed. Microdomains in Polymer Solutions. New York: Plenum Press, 1985, p 368. E. Ruckenstein, G. Huber, and H. Hoffmann. Langmuir, 3:382, 1987. R. Nagarajan. J. Chem. Phys., 90:1980, 1989. R. Nagarajan. Langmuir, 9:369, 1993 Z. Tuzar and P. Kratochvil. Adv. Colloid Interface Sci., 6:201, 1976. C. Price. In I. Goodman, ed. Developments of Block Copolymers I. London: Applied Science Publishers, 1982, p 39. G. Riess, P. Bahadur, and G. Hurtrez. In Encyclopedia of Polymer Science and Engineering. New York: Wiley, 1985, p 324. P.G. de Gennes. In J. Liebert, ed. Solid State Physics. New York: Academic Press, 1978, Suppl. 14, p 1. L. Leibler, H. Orland, and J.C. Wheeler. J. Chem. Phys., 79:3550, 1983. J. Noolandi and M.H. Hong. Macromolecules, 16:1443, 1983. D. Whitmore and J. Noolandi. Macromolecules, 18:657, 1985. E. Helfand and Y. Tagami. J. Polym. Sci. B, 9:741, 1971. E. Helfand and A.M. Sapse. J. Chem. Phys., 62:1327, 1975. W.H. Stockmayer. J. Polym. Sci., 15:595, 1955. R. Nagarajan and K. Ganesh, J. Chem. Phys., 90:5843, 1989. H. Jacobson and W.H. Stockmayer. J. Chem. Phys., 18:1600, 1950. P. Bahadur, N.V. Sastry, S. Marti, and G. Riess. Colloids and Surf., 16:337, 1985. R. Nagarajan. Colloids and Surf. B Biointerfaces, 16:55, 1999. R. Nagarajan and Chien-Chung Wang. J. Colloid Interface Sci., 178:471, 1996. R. Nagarajan and Chien Chung Wang. Langmuir, 16:5242, 2000. R. Nagarajan and E. Ruckenstein. Langmuir, 16:6400, 2000. R.A. Johnson and R. Nagarajan. Colloids and Surf. A. Physicochem. Eng. Aspects, 167:31, 2000.. R.A. Johnson and R. Nagarajan. Colloids and Surf. A Physicochem. Eng. Aspects, 167, 21, 2000. A. Naor and D. Blankschtein, J. Phys. Chem., 96:7830, 1992. Y.J. Nikas and D. Blankschtein. Langmuir, 10:3512, 1994. N.J. Zoeller, D. Blankschtein. Ind. Eng. Chem. Res., 34:4150, 1995. P.K. Yuet and D. Blankschtein. Langmuir, 12:3802, 3819, 1996.

2 Modeling Association and Adsorption of Surfactants LUUK K. KOOPAL Netherlands

I.

Wageningen University, Wageningen, The

INTRODUCTION

The most characteristic property of surfactants is their strong amphipolarity caused by the primary structure of the molecules: a polar headgroup linked to an apolar hydrocarbon or fluorocarbon tail. The spacial separation between headgroup and the apolar tail and the chemical differences between both parts of the molecule give surfactants their specific properties. With the headgroups a first distinction can be made between ionic and nonionic surfactants. With the nonionic surfactants the headgroup consists of a sequence of hydrophilic or polar segments. For the ionic surfactants the headgroup is charged and coulomb interactions come into play. The specific nature of the headgroup can be emphasized by adhering specific (i.e., noncoulomb) properties to the headgroup in relation to the solvent and the surface. The tail segments of both the ionic and nonionic surfactants are apolar (hydrophobic). For the apolar tail the number of aliphatic or fluorocarbon segments and aromaticity and branching are important. Due to their primary structure, surfactants form micelles. In aqueous solution there are surfactant aggregates with an apolar core surrounded by a corona of hydrated headgroups. The micelles are formed at a specific solution concentration, the critical micelle concentration, or CMC. The detailed structure of the surfactant—and, in particular, the size and shape of the headgroup as compared to the size and shape of the tail—determine the geometry of the micelles [1,2]. Surfactants with a large headgroup as compared to their tail form spherical micelles, weakly conical surfactants form cylindrical micelles, and surfactants with a cylindrical shape form lamellar aggregates. 111

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Surfactants also have a strong tendency to adsorb on surfaces. With liquid–liquid and liquid–gas interfaces the primary structure of the surfactant determines the adsorption behavior. At solid–liquid interfaces the surface properties and the detailed structure of the surfactant determine the adsorption. In general, adsorption of surfactants at various interfaces starts at concentrations much below the CMC and it reaches a pseudosaturation value at the CMC. The fact that a pseudoplateau is reached, rather than a typical surface saturation value, is due to the fact that the concentration of free surfactant molecules hardly changes above the CMC. Provided the surfactant solution remains dilute, the ‘‘pressure’’ on the surfactants to go to the interface remains above the CMC the same. Similarly as in solution, at surfaces aggregated structures can also be present [3–14]. The structure of the surfactant will also in this case affect the size and shape of the aggregates [15]; however, for solid surfaces the surface-surfactant affinity also plays an important role. Both their behavior in solution and that at interfaces have led to a large variety of technical applications of surfactants. Solubilization of components in the micelles is one of the older applications of micelles [16]. At present, micelles are also used as, for instance, microdomains for chemical reactions or for micellar-catalyzed reactions [17–20] and in micellar chromatography [21,22] and micellar-enhanced (ultra) filtration [23,24]. In liquid–liquid systems surfactants can be used to stabilize both normal and microemulsions [25,26]. Adsorption of surfactants to solid surfaces is of technical importance because at concentrations up to the CMC surfactants can cause both stabilization and destabilization of solid–liquid dispersions [27] and they can drastically modify wetting of solids [27–29]. Apart from these applications at relatively low surfactant concentrations there is a world of applications at high surfactant concentrations [30]. In all these applications the primary structure of the surfactant determines the general behavior, whereas the detailed structure gives rise to specific behavior. To present a review of applications and practical studies of surfactant micellization and adsorption is an almost impossible task. In order to gain understanding and to be able to further explore surfactant applications, good physical insight is required. Such insight can be obtained from modeling studies on both micellization and adsorption. In primitive models of surfactant behavior the primary structure is not even made explicit. For instance, micellization can be described on the basis of the mass action law describing the situation that n monomers form a micelle [31,32], and similarly adsorption can be described as accumulation of (interacting) monomers on a surface [33]. Such models may be able to describe the behavior phenomenologically, but they are not suited to give insight in surfactant behavior in relation to their structure.

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One step better is the introduction of the head and tail group in the modeling, so that at least the primary structure is emphasized. However, in order to be able to investigate the effect of the structural characteristics of a surfactant in detail, the precise structure of both the hydrophilic and hydrophobic parts of the molecule has to be considered. It will be clear that such models are far more complex than the models that treat surfactant molecules as simple monomers or amphiphilic entities. Although the sophisticated models are complex, they have the advantage that detailed insight in the surfactant behavior in solution and near interfaces can be obtained. Besides thermodynamic models, molecular dynamic (see e.g., [34–37]) and Monte Carlo simulations [38,39] can also be useful for a better understanding of surfactant behavior. In Chapter 1 Nagarajan [40] discusses the theory of micelle formation. A slightly older overview of the theory of micellization has been presented by Nagarajan and Ruckenstein [41]. Very recently, Hines [42] has presented a current opinion on theoretical aspects of micellization. Several reviews have appeared in the field of surfactant adsorption at fluid interfaces. Lu et al. [43] review the use of neutron reflectometry to study the structure and composition of surfactant layers at the air–water interface and critically assess the results by comparing them with information obtained with other methods. Fainerman et al. [44] have discussed adsorption isotherms and equations of state for ideal and nonideal surface layers in relation to experimental results. Old and new equilibrium surface tension models have been evaluated by Prosser and Franses [45]. Aratono and Ikeda [46] discuss in Chapter 3 adsorption at the gas–liquid interface on the basis of the Gibbs equation for mixed systems. Dynamic aspects of surfactant adsorption on fluid interfaces have also attracted quite some attention. Several reviews have appeared; these are concerned partly with adsorption kinetics at static interfaces [47,48] and partly with dynamic interfaces [49–51]. Old reviews on surfactant adsorption at solid–liquid interfaces are those of Hough and Rendall [52] and Clunie and Ingram [53]. Rosen [54] has presented a practical review on synergism in mixtures of surfactants both in solution and at interfaces. Kronberg [55] has discussed mixed surfactant systems and advances in the understanding of surfactant mixtures at surfaces. Tiberg et al. [56] have discussed adsorption mechanisms and interfacial structures derived from modern measurement techniques. Manne and Gaub [57] have expressed a current opinion on imaging of surfactant micelles at surfaces and liquid films by atomic force microscopy. Cox et al. [58] discuss self-assembled surfactant patterns on solid surfaces in relation to nanolithography. The first part of the present paper presents a review of the thermodynamic models that have been used for the description of micellization and/or

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adsorption. The review on micellization is kept brief because the theory of micellization is discussed in detail in Chapter 1. Fainerman et al. [44] and Prosser and Franses [45] have recently reviewed modeling aspects of adsorption at static fluid interfaces. No recent reviews have appeared on modeling of surfactant adsorption at solid surfaces. The main attention will therefore be given to adsorption at solid surfaces. The second part of the paper is devoted to the self-consistent-field lattice model for association and adsorption of surfactants [59–70]. This model is briefly indicated by the acronym SCFA. In the SCFA model the surfactants are treated as chains of segments in solution that may adsorb at an interface and/or form micelles. Due to the number of parameters involved in such a detailed model the primary aim of this type of modeling is to study the effects of the headgroup and tail structure on the micellization and adsorption behavior and to gain further understanding of surfactant behavior. The calculated results will be compared with experimentally observed trends in micellization and adsorption. Fitting of experimental data has not been the aim of most SCFA calculations. The emphasis in the SCFA section is also on micellization and adsorption on solid–liquid interfaces. The theory is well suited to cope with fluid interfaces, but only a limited number of SCFA studies have been made on liquid–liquid [34,69] or liquid–vapor [70] interfaces.

II. MODELING MICELLIZATION A. General Classical models of micelle formation can be divided into two broad categories: the pseudophase separation models and the mass action models [71]. The range of insight that can be obtained with the pseudophase separation models as compared to the mass action models is limited. However, the relative simplicity of pseudophase models is a great advantage for their use in practice. A relatively new branch of theory is that of the self-consistent-field models. In these models the conformations of the surfactant chains in a micelle are considered in the potential field exerted by the presence of the surfactant molecules in that micelle [63].

B. Pseudophase Separation Models Phase separation models assume that the micelles are a separate pseudophase. The pseudophase commences at the CMC, which represents the saturation concentration for monomeric surfactants. The presence of the micellar phase is assumed but not explained; consequently no independent information can be obtained regarding the size, shape, or structure of the

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micelles. To be able to describe the distribution of reactants between the solution phase and the micellar pseudophase, the chemical potentials of the reactants in the micellar phase have to be equated to those of the reactants that are dispersed in the aqueous phase. By introduction of a model for the chemical potentials of the components in both the aqueous and the micellar phases (e.g., assuming that the phases may be considered as regular solutions or be treated as solutions of solvent molecules and chain molecules that can be described by the Flory–Huggings theory), several features of micellization and solubilization in micelles can be investigated. With ionic surfactants electrostatic (1) effects can be included in the activity coefficients or (2) an electrical potential is given to the micellar pseudophase together with an adjustable parameter that accounts for counterion binding and electrochemical potentials are used instead of chemical potentials. The degree of complexity of the pseudophase models is directly related to the models used for the (electro)chemical potentials. Details and a discussion of the pseudophase models can be found in [25,71–76].

C. Mass Action Models Mass action models consider surfactant aggregates that are in equilibrium with surfactant monomers in a dilute aqueous solution. In the most simple models the surfactant monomers are in equilibrium with a micelle that has a certain average size (aggregation number) and shape. The more sophisticated multiple equilibrium mass action models consider stepwise association reactions of monomers and account for polydispersity effects of micelles. The association of monomers to dimers, trimers, etc. up to n-mers now takes place under the condition that the chemical potential of the surfactant monomer in solution is equal to the chemical potential per molecule of an aggregate of any size and shape. The changes that a surfactant monomer (or a solubilizing species) in a dilute (electrolyte) solution undergoes if it is transferred to a surfactant aggregate dispersed in the same solution can be used to formulate the contributions to the standard Gibbs energy of micellization. As a result the standard free energy of the micellization reaction is related to a standard state in which the reactants (surfactants, salt ions, solubilizing species) are dispersed at infinite dilution. With this type of modeling the micelle formation is predicted from basic principles; it is reversible, and the aggregate size, aggregate polydispersity, and, depending on the model, micellar shape can be calculated. If the surfactant molecules are ions, it is often necessary to introduce counterion binding using an adjustable parameter to model the ‘‘surface’’ charge or ‘‘surface’’ potential of the micelles.

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The ‘‘flexibility’’ of the mass action models is related to the selection of the contributions to the standard Gibbs energy of micellization. Common contributions are (1) the Gibbs energy to transfer the surfactant hydrophobe to the micellar interior, (2) the interaction related to the extent of shielding of the micellar core from exposure to the aqueous solution, (3) the steric and conformational interactions related to transferring a surfactant (chain) molecule from the standard solution to the micellar phase, (4) the electrostatic interaction, and (5) the Gibbs energy of mixing. The expressions for the first four contributions are dependent on the geometry of the micelles. By introducing constraints related to the structure of the surfactant and considering different geometries it is possible to predict the preferred geometry of an aggregate. Chapter 1 [40] explains the multiple mass action models in more detail and refs. [71,72,77] review these models, with emphasis on mixed surfactant systems. References [41,78–80] can be consulted for some illustrative advanced treatments.

D. Self-Consistent-Field Models Since about 1980 several authors have used self-consistent-field (SCF) molecular thermodynamic models to gain further insight in micellization and especially the micellar structure [59–61,81–85]. Similarly as in the advanced mass action models, the detailed structure of the micelles is predicted. Special attention is given to the packing and conformations of the surfactant chains in the micelles and the relation between the molecular architecture and the micellar structure and size. Surfactant chains in a micelle are generated by a statistical step-weighted walk procedure. Each step of the walk is weighted with a Boltzmann factor that accounts for both the enthalpic and entropic (excluded volume) interactions that the segment experiences in the micelle. Similar types of interactions are taken into account as in the advanced mass action models. However, instead of applying analytical expressions for the various contributions to the standard Gibbs energy, only the interaction parameters (FH type) and the volume constraints have to be specified. The detailed equilibrium distribution of surfactant segments within the micellar structure, i.e., within a potential field exerted by the presence of the surfactant molecules themselves, is calculated by an iterative minimization of the Gibbs (Helmholtz) energy of the system. At equilibrium the segment density distribution and the potential field exerted by the segments are fully in accordance with each other. The CMC and the micelle size are found by using arguments from the small system thermodynamics as

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introduced by Hill [86] and worked out for surfactant micellization by Hall and Pethica [87]. In the early treatments the surfactant hydrophobe was assumed to be flexible and the walk was started at the plane fixing the positions of the headgroups [81–83,88–90] and hence assuming (implicitly) the geometry of the micelle. The alignment of the surfactant headgroups in one plane is a severe simplification when the aim is to describe not only the interior of the micelle but also the surface structure or ‘‘roughness’’ [91,92]. Gruen [84] and Szleifer et al. [85] relaxed this constraint in later studies but still used the ‘‘surface’’ density of the headgroups as an input parameter. The first molecular thermodynamic treatment in which these constraints were completely avoided is the SCFA theory, developed by Leermakers and Scheutjens [59–61] and applied to membrane and micelle formation. The schematic representation of a spherical micelle in the SCFA theory is shown in Fig. 1. Subsequently Van Lent and Scheutjens [93] used the SCFA model to describe micellization of block copolymers and Bo¨hmer and Koopal [62] applied it for the modeling of micellization of nonionic surfactants. After Bo¨hmer et al. [94] and Barneveld et al. [95] had shown how electrostatic interactions could be incorporated in the SCFA theory, Bo¨hmer et al. [63] have used it to model micellization of ionic surfactants. The detailed theoretical background of the SCFA model is discussed in Section IV of this chapter. Predicted results for nonionic micelles are presented in Section V, and those for micelles of ionic surfactants in Section VII.

FIG. 1 Schematic representation of a section of a spherical micelle in the SCFA treatment.

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III. MODELING ADSORPTION A. Adsorbed Layers, Surface Micelles, Hemimicelles, and Admicelles In relation to the models for micelles, most adsorption models can be classified as a kind of mass action models in which many different expressions are used for the standard Gibbs energy of adsorption. An important aspect of the adsorption models is the assumed structure of the adsorbed layer. A distinction can be made between models that treat the adsorbed layer as ‘‘smeared-out’’ and models that consider ‘‘local aggregates’’ at the surface. In principle, the best models are those that do not make any a priori assumption at all about the adsorbed layer structure, but find this structure in a self-consistent way. This kind of modeling is scarce; at present only some results obtained with the SCFA model for nonionic surfactant adsorption on a homogeneous surface are available. The smeared-out layer models can be subdivided into ‘‘monolayer’’ and ‘‘bilayer’’ models. With the local aggregate models various types of aggregates are assumed to describe the ‘‘surface micelles.’’ In the present text a distinction will be made between ‘‘surface micelles,’’ ‘‘hemimicelles,’’ and ‘‘admicelles,’’ see Fig. 2. In order to avoid confusion about the terminology, these terms will be explained first. Surface micelles are surface bound micellarlike structures in general, and more specifically the term is used when the primary structure of the surfactant plays an important role in the description of the surface aggregates. In the older literature the term hemimicelles is used for surface aggregates in general. At present, it is more common to define hemimicelles as the specific class of monolayer-type local aggregates in which the surfactants are adsorbed either with their headgroup in contact with the surface (head-

FIG. 2 Schematic representation of hemimicelles, admicelles, and surface micelles on a hydrophilic (top) and a hydrophobic surface.

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on adsorption on mineral surfaces) or with their tail in contact with the surface (tail-on adsorption on apolar surfaces). With this restricted definition the hemimicelle structure is strongly dominated by the attraction between the anchoring group and the surface. The shape of the hemimicelles is not necessarily comparable to the micellar structure that may occur in solution. The hemimicelle approximation can be compared with the monolayer approximation in the smeared-out case. Admicelles are bilayer-type local aggregates, i.e., some of the headgroups are in contact with the surface, other headgroups point toward the solution. Admicellar aggregates can be compared with bilayer adsorption in the smeared-out case. Hemimicelles on mineral surfaces (head-on adsorption) may become admicelles near the CMC, but they are not necessarily the precursor for the admicelles. At mineral surfaces smeared-out monolayer-bilayer models have often been used, in combination with the generally accepted knowledge that local aggregates may occur, to make predictions on the existence of hemimicelles or admicelles. In general, it should be realized that the attraction between the surfactant and a solid surface is an important variable that does not occur with micellization. This makes the modeling of surfactant adsorption to solid surfaces considerably more complex and more diverse than that of surfactant micellization.

B. Trends in Surfactant Adsorption on Solid Surfaces The area of surfactant adsorption to solid surfaces can be divided into subareas as indicated in Fig 3. The division is based on the type of surface, the type of surfactant, and the surfactant concentration. Broadly speaking, two types of solids and two types of surfactants can be distinguished: the surface is hydrophilic or hydrophobic and the surfactant is ionic or nonionic. The distinction between hydrophilic and hydrophobic surfaces is directly related to the distinction between the hydrophilic surfactant headgroup and the hydrophobic surfactant tail. The distinction between ionic and nonionic surfactants is important because of the special treatment required for the coulomb interactions (long-range). With adsorption on hydrophilic surfaces and low surfactant concentrations the driving force for adsorption is the attraction between the surfactant headgroup and the surface. Therefore, the surfactant adsorbs at low surfactant concentrations with its headgroup in contact with the surface. For both nonionic and ionic surfactants the attraction will have a shortrange (specific) contribution depending on the type of headgroup and the type of surface. Besides this specific attraction ionic surfactants will experi-

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FIG. 3 Subareas of surfactant adsorption on solid surfaces. From left to right the drawings give a schematic representation of the state of adsorption with increasing surfactant concentration. The important variables are shown in the middle of the figure: T (temperature), c (surfactant concentration), I (ionic strength), and pH (as indicator for the surface charge).

ence a generic electrostatic (coulomb) interaction. This interaction will be attractive if surface and surfactant are oppositely charged and repulsive when surfactant and surface carry the same charge sign. In the latter case adsorption only occurs if the specific attraction is stronger than the coulomb repulsion. In general, the coulomb interactions depend on the valency of the surfactant, the magnitude of the surface charge (pH), the ionic strength (I), and the extent of surfactant adsorption. At intermediate surfactant concentrations the hydrophobic attraction between the surfactant tails that protrude into the solution promotes further adsorption. Initially the adsorption may continue in a monolayer or be monolayerlike (hemimicelles) with the headgroup still in contact with the surface. At a certain concentration the orientation of the molecules will reverse in order to screen the hydrophobic tails from contact with the aqueous solution and a bilayer or bilayer-type aggregates (admicelles) will be formed. The concentration at which the bilayer-type adsorption starts depends on the tail and headgroup characteristics. For ionic headgroups bilayer-type adsorption is, in general, favored if the charge of the surface

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groups is compensated by the adsorbed surfactant charge. For nonionic surfactants the cross section (size) of the surfactant headgroup as compared to that of the tail is important for the concentration at which the transition occurs. Depending on the effective size of the headgroups as compared to the tail size and length, either the bilayer structures will grow upon a further increase of the surfactant concentration into complete bilayers or they will form surface micelles. The shape of the surface micelles (roughly spherical or cylindrical) will depend on the surfactant structure and the surfactant–surface interactions (distortion of the shape to pinned micelles). The top of Fig. 2 indicates the main structure of the adsorbed layer on hydrophilic surfaces as a function of the surfactant concentration. For hydrophobic surfaces the situation is simpler than on hydrophilic surfaces. With hydrophobic surfaces the adsorption at low surfactant concentration is driven by the hydrophobic attraction between the surfactant tail and the surface sites. The surfactant molecules screen the unfavorable interactions between the surface and the aqueous solution. By increasing the surfactant concentration the lateral hydrophobic attraction between the surfactant tails comes into play and a submonolayer- or monolayer-type local surfactant aggregates (hemimicelles) may form. At surfactant concentrations near the CMC the monolayer structures may gradually transform into a complete monolayer or become larger hemimicellar aggregates. Again the type of structure that will be formed depends on the surfactant characteristics and the surfactant–surface interactions. The main difference with the hydrophilic surfaces is that no bilayer-type structures are formed. The bottom part of Fig. 2 gives a schematic representation of the adsorbed structures on hydrophobic surfaces. Surfactant adsorption on aqueous LG and LL interfaces will follow a similar trend as that on hydrophobic solid surfaces, with air or the oil phase taking the position of the solid surface. The role of the ionic strength is most important for ionic surfactants; its main action is to screen the electrostatic interactions. This screening leads in the first place to lower values of the CMC. With its influence on the adsorption it should be realized that screening has a ‘‘normal’’ and a ‘‘lateral’’ component. The ‘‘normal’’ screening applies to the headgroup–surface interaction (which can be attractive or repulsive depending on the charge signs) and the ‘‘lateral’’ screening to the headgroup–headgroup repulsion. For nonionic surfactant adsorption on charged surfaces the ionic strength will affect the surface charge. Surface charge and ionic strength will together affect the specific interactions between the surfactant headgroup and the surface sites. The role of the surface charge is often synonymous with the role of the pH, because for many surfaces protons and hydroxyl ions act as chargedetermining ions.

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The effect of the temperature is important for all the interactions. However, its effect on the hydrophobic interactions is most important. For hydrophilic surfaces a first indication of the magnitude of this effect can be obtained by considering the effect of temperature on the CMC of the surfactants [96]. For hydrophobic surfaces not only the surfactant hydrophobe but also the extent of hydrophobicity of the surface will play a role. With the discussion of surfactant adsorption that follows a distinction will be made between nonionic and ionic surfactants and between smearedout models and local aggregate models. Most attention will be given to hydrophilic surfaces. In order not to confuse the discussion too much, first some general remarks will be made on surface heterogeneity effects in relation to surfactant adsorption. In the other sections surface heterogeneity is largely neglected.

C. Surface Heterogeneity and Surfactant Adsorption Surface heterogeneity plays a prominent role in the discussion about the shape of surfactant adsorption isotherms and heat of adsorption curves [97– 99]. In general, solid surfaces are heterogeneous, which is reflected by the fact that different types of surface sites have different interactions with the surfactant. With surface heterogeneity the length scale of the topography of the surface is important in relation to the length scale of the lateral interactions. Classically a distinction is made between random (molecular-length scale) and patchwise (large length scale) heterogeneity. For random surfaces the lateral interactions are averaged over the entire surface; for patchwise surfaces the lateral interactions are assumed to be ‘‘active’’ per patch only and interactions crossing the patch boundaries are mostly neglected. For nonionic surfactants the length scale of the lateral interactions is generally of the order of the size of the molecules, and the distinction between a random and a patchwise treatment can be based on the scale of the surface heterogeneity. For surfaces with a random heterogeneity the isotherm may have a somewhat lower slope than for homogeneous surfaces, but with progressive adsorption heterogeneity effects will be largely masked by the strong lateral attractions between the surfactant molecules. At low surfactant concentrations and low surface coverage, lateral attractions are absent or weak and surface heterogeneity effects may be noticeable by an initial slope of the log–log adsorption isotherm that is smaller than unity [100]. At intermediate and higher surface coverage the lateral hydrophobic attraction will most probably dominate the behavior. This is certainly the case with bilayer-type adsorption or adsorption in admicelles. Therefore, neglecting the heterogeneity may well be a reasonable approximation in

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this case. Arguments based on the initial part of the isotherm to introduce heterogeneity in adsorption models are correct, but the way in which this heterogeneity affects the rest of the adsorption depends strongly on the model for the local isotherm (i.e., the model for adsorption on a homogeneous surface). Hence, the effect of heterogeneity on the adsorption can only be fully understood when a consensus is reached about the adsorption on homogeneous surfaces. In the case of patchwise surface heterogeneity, the nonionic surfactant isotherms may be seriously affected by the heterogeneity. Each patch can be considered as a homogeneous surface and the overall adsorption is the sum of the local adsorption contributions. For instance, for well-defined large patches, with sufficiently different affinities for the surfactant, stepwise isotherms may be predicted, especially when the adsorption is plotted as a function of the logarithm of the surfactant concentration. For a wide distribution of surface heterogeneity a continuous affinity distribution should be used, and the overall adsorption equation is an integral equation based on the local isotherm and the distribution function. In this case heterogeneity will lower the slope of the semilogarithmic isotherm (adsorption versus log c), in the same way as for random heterogeneous surfaces. Plotting the isotherm double logarithmically can make a distinction between lateral attraction and heterogeneity effects. With pure surface heterogeneity the initial slope should be smaller than unity and the slope should decrease with increasing coverage. An increase in the slope of the isotherm or a slope larger than unity always points toward lateral (hydrophobic) attraction. In the case of a repulsive (coulomb) lateral interaction it is not possible to make a distinction between heterogeneity and lateral effects. Only for simple local isotherm expressions, such as the Langmuir or Frumkin– Fowler–Guggenheim (FFG) equation, and a few specific distribution functions do analytical solutions exist for the overall isotherm [100]. In general, a numerical solution is required and the outcome depends on both the chosen local isotherm and the chosen distribution function. It will be clear that many possible solutions exist in this case and that this type of treatment is poorly suited to gain insight into the adsorption process and/or the structure of the adsorbed layer. Only in the case that the surface–surfactant affinity distribution can be derived (approximately) from independent measurements is there a possibility to obtain insight into the adsorption process from the overall surfactant adsorption isotherm. For ionic surfactants the most important lateral interaction in relation to the scale of the surface heterogeneity is the coulomb repulsion between the surfactant ions. The Debye length, which is inversely proportional to the square root of the ionic strength, gives the length scale of the coulomb interactions. In practice, the Debye length may vary from a few tenths of

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an nm at high salt concentrations to several hundreds of nm in the absence of added salt. For random heterogeneous surfaces the large scale of the lateral interactions is a good reason to neglect surface heterogeneity as a first-order approximation. For patchwise surfaces the characteristic dimension of the patch has to be several times the Debye length; otherwise the surface may be considered as random heterogeneous and the heterogeneity effect is relatively small. For a surface with a patchwise heterogeneity of not too large a scale, it may be necessary to treat the heterogeneity as random (and hence neglect it) at low salt concentrations but as patchwise (and take it into account) at high salt concentrations. In many treatments involving the adsorption of ionic surfactants on patchwise heterogeneous surfaces, this aspect is poorly recognized because the coulomb interactions are accounted for in too primitive of a way [98,99,101–103]. With today’s state of the art in surfactant adsorption modeling, the understanding of surfactant adsorption will merely improve by a better understanding of what occurs on a homogeneous surface and how the detailed structure of a surfactant affects the adsorption behavior [15]. Once this issue is settled a more detailed discussion on surface heterogeneity may follow. Effects of surface heterogeneity on adsorption isotherms are, in general, known. As indicated above, for surfactant adsorption the effects will be relatively small for random-type surfaces and for patchwise surfaces they will either give steps in the isotherm (few large patches) or decrease the slope of the isotherm (wide distribution of patches). Only when independent evidence is present that the surface is composed of patches and that the isotherms predicted on the basis of surface homogeneity are too steep is it appropriate to introduce surface heterogeneity. But even then the question remains, especially in the case of ionic surfactants, of which local isotherm equation should be used. Fitting of isotherms or heats of adsorption data on the basis of surface heterogeneity and a too-simple local isotherm will not improve our understanding of the adsorption process. For the moment these arguments are a good reason to concentrate on the difficult task of improving the description of surfactant adsorption at homogeneous surfaces. The heterogeneity issue can be settled when agreement is found on the isotherm equation for the homogeneous surface.

D. Adsorption of Nonionic Surfactants 1. Adsorbed Layers A simple mean-field model that has been used to describe the adsorption of nonionic surfactants is the regular behavior model [104]. In this model the surfactant structure is completely denied and the adsorption is restricted to a smeared-out monolayer; however, nearest-neighbor contacts are taken into

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account. For adsorption from dilute solution, a situation normally found for surfactants up to the CMC, the regular behavior model simplifies to the Frumkin–Fowler–Guggenheim (FFG) [105,106] equation 1 ¼ x1 K12 exp 12 1 1
1

ð1Þ

In the case of surfactant adsorption 1 is simply the ratio between the adsorption and the adsorption at the CMC; x1 is the surfactant mole fraction (or concentration) in solution; K12 is the adsorption constant, being a measure of the preference of the surface for the surfactant with respect to the solvent; and ij is the so-called Flory–Huggins (FH) interaction parameter. The expression for ij is ij ¼

 NA Z

"ij
0:5 ð"ii þ "jj Þ RT

ð2Þ

with NA Avogrado’s number, z the number of nearest neighbors to a central molecule, and "ii , "jj , and "ij the pairwise interaction potentials (Gibbs energies) [107]. Note that according to its definition ii ¼ jj ¼ 0. Because "ii , "jj , and "ij are generally of negative sign, ij is positive if like contacts are preferred over unlike contacts and negative if unlike contacts are favored. For surfactant adsorption K12 and 12 should be considered as adjustable parameters expressing the affinity for the surface and the lateral interactions in the adsorbed layer, respectively. In order to arrive at a more realistic model for surfactant adsorption, Kronberg [108] and Koopal et al. [109,110] have used ideas expressed in polymer adsorption theories to incorporate the chain characteristics of the surfactants in the modeling. In this case volume fractions should be used rather than mole fractions and the adsorption is not restricted to a monolayer, but the surfactant segment density near the interface has a block profile. Koopal et al. [110] have shown that for dilute solutions, an equation similar to Eq. (1) results: 1 b m Wm Km exp rm 1 r ¼ 1 r ð1
1 Þ

ð3Þ

where 1 is the volume fraction of the surfactant in the adsorbed phase and b1 that in bulk solution, Km expresses the affinity, Wm is the loss in conformation entropy, r is the surfactant chain length, and m is an effective FH interaction parameter in the adsorbed layer. Km , Wm , and m are dependent on the number of segments, m, in direct contact with the surface. Although Eq. (3) is more ‘‘flexible’’ than Eq. (1), the structure of the adsorbed layer is still largely neglected. A complication of Eq. (3) is that m will depend on the adsorption itself so that Km , Wm , and m will only be constant over a limited

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range of adsorption values. In principle m can be found by minimizing the Gibbs energy of the system at a given composition. Narkiewicz-Michalek et al. [102] give an expression that can be used to find m as a function of 1 . The equation given by Kronberg is somewhat different and used in practice [108,111,112] more than Eq. (3), but for dilute solutions the differences are not essential; see [110]. Huinink et al. [113] have presented a semi-thermodynamic (ST) model for adsorption of nonionic surfactants in cylindrical hydrophilic pores based on smeared-out adsorption layers. The adsorption is treated as a transition between a dilute gaslike phase and a bilayer phase. An expression similar to the Kelvin equation has been derived to describe the influence of pore curvature on the phase transition. The effect of curvature on the phase transition increases with increasing affinity of the surfactant bilayer for the surface. The higher the affinity is, the more the formation of a bilayer in the pore is promoted as compared to that on a flat surface. Because the model is based on a sharp phase transition, it holds best if the headgroup repulsion is weak (small headgroups).

2. Adsorbed Layers: SCFA Theory Koopal et al. [62] have used the SCFA model to describe the adsorption of nonionic surfactants. For details of the SCFA theory we refer to Section IV of this chapter. For nonionic surfactants on hydrophilic surfaces the SCFA theory predicts for molecules with a small headgroup a phase transition from a dilute layer to a condensed bilayer, provided the headgroups are not too long. For small headgroups the 2D transition is from a gaslike layer to the condensed bilayer; for an intermediate headgroup size a fairly flat layer is formed before the 2D condensation to a bilayer takes place. For large headgroups the condensation step is absent. On hydrophobic surfaces surfactants with a small or intermediate headgroup undergo a phase transition from a gaslike layer to a condensed monolayer at low concentrations. For surfactants with a large headgroup a strong increase in adsorption is observed at low concentrations, but there is no phase transition. At both hydrophobic and hydrophilic surfaces the adsorption at the CMC increases with decreasing headgroup size. More details are presented in Sections VI.A and B. Huinink et al. [114,115] have extended the SCFA model to allow for adsorption of nonionic surfactants with a small headgroup on porous or rodlike hydrophilic surfaces. The phase transition that occurs on flat surfaces also appears for the curved surfaces. On porous surfaces the transition occurs at lower and on rod-shaped surfaces at higher concentrations than on flat surfaces. The prediction for the porous surface is in accordance with the results obtained with the ST theory [113]. The adsorbed amount decreases

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with decreasing pore radius, and the phase transition is shown to depend on the molecular architecture and adsorption energies. The surface tension of the bilayer can be related to the surfactant structure and the curvature constant of the bilayer.

3. Surface Aggregates Zhu and Gu [116,117] have presented a very simple model for adsorption of nonionic surfactants. They assume that the adsorbed layer is composed of surfactant aggregates. A surfactant aggregate is formed on the surface before stable aggregates are formed in solution; the aggregate is stabilized by the interaction with the surface. By applying a simple mass action model Zhu arrives at the following equation: 1 ¼ cn1 K 1
1

ð4Þ

where 1 is the adsorption divided by the adsorption at the CMC, c1 is the surfactant concentration, and n is the number of monomers in the surface micelle. Zhu uses the word ‘‘hemimicelle,’’ but in the present text this term is used in a more restricted sense. Equation (4) is also known as the Hill or Freundlich equation; it reduces to the Langmuir equation for n ¼ 1. When values of n < 1 are found with nonionic surfactants, there is good reason to believe that the surface heterogeneity is important. Equation (4) is well suited to describe cooperative adsorption and S-shaped isotherms. The main advantage of this model is that it emphasizes in a very simple way that the adsorbed layer may be composed of aggregates stabilized by the presence of the surface. In Fig. 4 adsorption data for two nonionic surfactants on silica are shown, together with the isotherms calculated with Eq. (4). Equation (4) is a simplified form of a two-stage mass action model for surface aggregate formation proposed by Zhu et al. [117–119] and used by these authors for ionic surfactants. The general model is based on the following assumptions: (1) isolated surfactant molecules adsorb at the surface according to the Langmuir model and (2) the adsorbed monomers are nuclei for the formation of surface micelles composed of in total n monomers according to the mass action law. This leads to the following equation that can be used to describe a variety of shapes of adsorption isotherms:
 1 þ k2 cn
1 k1 c1 1 n   ð5Þ 1 ¼ 1 þ k1 c1 1 þ k2 cn
1 1 In Eq. (5) 1 is the adsorption relative to that at the CMC, c1 is the monomer concentration, k1 is the affinity of the monomer for the surface, k2 is the aggregation constant, and n is the final aggregation number, or the co-

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FIG. 4 Adsorption isotherms of octylmethylsulfoxide (x) and decylmethylsulfoxide (~) on narrow pore and wide-pore (DEMS, ~) silica gels. (Redrawn from Ref. 116 with permission of Elsevier Science-NL, Amsterdam, The Netherlands.)

operativity. The value of k1 determines the initial part of the isotherm; it can give the isotherm a Langmuirlike (L) first step at low concentrations. The value of k2 determines the second step. In the presence of a two-step isotherm (LS type) n represents the ratio of adsorption levels of the two steps. S-shaped isotherms will occur when k2 is larger than k1 . The model also allows for the derivation of the fraction of the adsorption present in monomers and that present in surface micelles. Moreover, the 2D equation of state can be obtained [120]. The model gives no further information on the structure of the surface micelles; it is merely a convenient tool to describe the adsorbed state. An older, somewhat similar model is the three-stage adsorption model of Klimenko [121,122]. Another treatment that describes surface micelles is due to Israelachvili [15]. In this elegant treatment Israelachvili extends his ideas for the relations between the structural properties of the surfactant molecules and the micellar structure (three-dimensional or 3D situation) to the 2D situation of surfactant aggregates at fluid interfaces. The result is given as a nonideal 2D equation of state, and it is shown that due to the limited size of the aggregates the surface-pressure-area curves have nearly horizontal parts rather than a strictly horizontal part as expected for the 2D condensation of an infinitely large aggregate or adsorbed layer. Both the size and shape of the 2D aggregate are shown to be dependent on the surfactant structure. In principle, the treatment can be extended to adsorption at solid surfaces using the Gibbs equation to convert the 2D equation of state in the corresponding adsorption isotherm. For solid surfaces the treatment will only be

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realistic when obstruction effects caused by the solid are small. This will be the case for low-energy solids. A far more complicated model that describes surface micelles is that of Levitz [123]. In this model a mass action type of approach is used with several contributions to the standard Gibbs energy in order to relate the structure of the surface aggregates to the structure of the surfactant molecules. Also in this model the aggregates are formed at the surface before they form in solution due to the interactions of the monomers with the surface. When the interaction with the surface is weak, the aggregate structure is determined by the primary structural parameters of the surfactant, just like in solution. For strong surface–surfactant interactions the surface properties become important for the structure of the surface aggregate. Rudzinski and co-workers [98,99] have also developed models to account for surfactant aggregates on surfaces. In the first model monomers in solution are in equilibrium with a mixture of disklike or oblate aggregates of various dimensions [98]. In the second model the monomers are in equilibrium with hemimicelles and admicelles of variable size that are simultaneously present at the surface [99]. The latter model is inspired by the work of Scamehorn et al. [103] and Harwell et al. [124] on hemimicelles and admicelles, and it is suited to discuss whether at a given surfactant concentration hemimicelles or admicelles are present. In both models the scaled particle theory is used for the intermicellar (excluded-area) interactions, but the interactions with the surface and within the micelles are kept simple. The adsorption affinity and the lateral interaction have to decrease with increasing adsorption; otherwise the 2D surface aggregation occurs in an infinitely large aggregate at a concentration below the CMC. In Drach et al.’s most recent contribution [125] the intermicellar excluded-area interactions are complemented with short-range lateral interactions, and detailed attention is given to a quantitative analysis of calorimetric effects accompanying the adsorption of nonionic surfactants. A disadvantage of these models is that the structures of the aggregates and the decrease of the lateral attraction are assumed, instead of explained on the basis of the structural properties of the surfactant molecules. For the description of the isotherms of nonionic surfactants on silica (see, e.g., Fig. 3) the models of Levitz [123] and Rudzinski et al. [98,125] are not better than the simple equations of Zhu [117]. However, Rudzinski et al. also consider heats of adsorption and Levitz’s model relates the structure of the aggregate to the surfactant structure.

4. Aggregates: SCFA Theory For nonionic surfactants also isolated local aggregates on hydrophilic surfaces have been described with the SCFA theory [67]. For this purpose the

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inhomogeneity of the field has to be considered in two directions (2D SCFA theory); see Section IV.C. The resulting adsorption isotherm for surfactants with a small headgroup is very similar to that observed with the 1D SCFA calculations. For surfactants with large headgroups the adsorbed amount is considerably larger than that observed with the 1D SCFA calculations. The reason for this is that now local 2D condensation occurs and ‘‘pinned’’ micelles are formed. Section VI.C presents the detailed results. Huinink et al. [126] have considered the behavior of rodlike surfactant aggregates at surfaces by modeling polydispers rectangles with endcaps on a square lattice. The treatment goes back to the model of Taylor and Herzfeld [127]. The molecular parameters needed for the model can, in principle, be calculated from the 2D SCFA approach [67]. The aggregates grow with increasing chemical potential. If the caps become more unfavorable, the average length of the rods increases. In the adsorption isotherms this leads to an increase of the cooperativity. Above a certain rod length a second-order transition occurs from an isotropic to a nematic phase. The nematic ordering promotes the further growth of the aggregates.

D. Adsorption of Ionic Surfactants 1. Adsorbed Layers For ionic surfactants the situation is more complicated than for the nonionics; besides specific interactions, the coulomb interactions are important. Due to the physical discreteness of the charges, an adsorbing ionic species experiences an electrostatic potential that can be thought of as composed of a mean ‘‘smeared-out’’ potential, a , and a self-atmosphere potential, a . In principle, both a and a depend on the magnitude of the surface charge and the adsorbed layer charge. However, in practice it is often assumed that the self-atmosphere potential is constant. In this case self-atmosphere potential effects can be incorporated into the specific affinity constant and only the smeared-out potential is considered explicitly. Now a Boltzmann factor including the smeared-out potential can be added to the expression for the standard Gibbs energy of adsorption or aggregate formation to account for the coulomb interactions. Equations (1) and (3) can thus be adapted for the description of the adsorption of ionic surfactants by adding such a Boltzmann factor, and Eq. (1) becomes

1 1 F a ¼ x1 K12 exp 12 1
ð6Þ 1
1 RT where a is the mean electrostatic potential in the adsorbed monolayer and 1 is the valency (sign included) of the surfactant ion. As a first-order approximation, and in the absence of specific adsorption of salt ions to

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either the surface or the surfactant, a can be equated to the electrokinetic or potential [128]. The potential of the adsorbent particles can be measured as a function of the surfactant adsorption or surfactant concentration. In the older literature on surfactant adsorption [129,130] Eq. (6), or reduced forms of Eq. (6), have been used frequently in combination with potential measurements to gain qualitative understanding of ionic surfactant adsorption on charged surfaces. Equation (6) can also be used in combination with a double-layer model such as the Gouy–Chapman or the Debye–Hu¨ckel model [131], to show that in the absence of specific adsorption of salt ions a is related to the surface charge, s , the charge due to the adsorbed surfactant, 1 , and the ionic strength [33]. The advantages of the simple Debye–Hu¨ckel (DH) model are that charge and potential are directly proportional to each other and that the equation that results from a combination of Eq. (6) and the DH model is simple and illustrative:
 1 ð þ 1 Þ 1 F ¼ x1 K12 exp 12 1
s ð7Þ "0 "r  RT 1
1 where "0 is the permittivity of vacuum, "r the permittivity of the solution, and  the inverse Debye length that is proportional to the square root of the ionic strength. Although Eq. (6) is based on the Debye–Hu¨ckel approximation ( a < 50 mV), it has the advantage over Eq. (5) that it clearly shows how specific and coulomb interactions affect the adsorption. All specific interactions (including the self-atmosphere potential) between surface and surfactant are included in K12 . Specific and hydrophobic lateral interactions between the surfactant molecules are incorporated through 12 . The coulomb interactions are governed by the surface charge, s , the adsorbed surfactant charge, 1 , the valency of the surfactant, 1 , and the square root of the ionic strength, . In general, the surface charge depends on (1) the concentration of surface charge-determining ions (mostly pH) [97], (2) the ionic strength, and (3) the surfactant adsorption. Only for low potentials, constant-equilibrium pH, constant ionic strength, and when it is assumed that the surface charge is fixed or that it adapts to the adsorbed surfactant charge in a linear manner is it possible to simplify Eq. (7) to an FFG equation (in which K and  have a compounded character) [33]. On the one hand, this illustrates that surfactant isotherms measured at constant-equilibrium pH and ionic strength will be easier to understand than when these conditions are not controlled. On the other hand, it follows that a whole range of conditions has to be fulfilled in order to arrive at an FFG-type adsorption isotherm for ionic surfactants. The advantage of Eq. (7) is that it already shows that the presence of the coulomb interactions has important consequences for surfactant adsorption.

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When surface and surfactant carry the same charge sign, the adsorption is strongly and progressively inhibited. Only when the specific affinity is larger than the coulomb repulsion does the adsorption start. With hydrophobic surfaces the tails form the anchoring groups and the headgroups will point to the solution; with hydrophilic surfaces either very limited or no adsorption will occur. When the surfactant and the surface are oppositely charged, the coulomb interaction promotes the adsorption as long as the surface charge is not yet compensated by the surfactant charge; see Eq. (7). The charged headgroups of the surfactant will prefer contact with the surface (head-on adsorption), also because there is mostly a specific contribution to the headgroup surface attraction. In the iso-electric point (or iep) the surfactant charge just compensates the surface charge (provided absence of specific adsorption of salt ions) and the coulomb attraction vanishes. In the case of superequivalent adsorption the coulomb interaction is repulsive and the charged headgroups tend to stay away from the surface (head-out adsorption) because it is easier to compensate the headgroup charge at the solution side of the adsorbed layer than at the surface side. The main driving force for further adsorption is now the hydrophobic attraction between the surfactant tails. For surfactant adsorption on an oppositely charged surface the surfactant orientation tends to go with increasing adsorption from ‘‘head-on’’ to ‘‘head-out,’’ with the iep as the turning point. In general, this applies to both hydrophilic and hydrophobic surfaces. The above trend can be rephrased as follows: below the iep (local) ‘‘monolayer’’ adsorption will prevail, whereas above the iep (local) ‘‘bilayer’’ adsorption starts. Moreover, below the iep the coulomb interactions promote adsorption, whereas above the iep these interactions inhibit adsorption. Therefore, screening of the coulomb interactions by salt addition will lead below the iep to a decrease and above the iep to an increase in adsorption. At the iep the coulomb interactions are absent and salt addition has no effect on the adsorption. The conclusion has to be that at high salt concentrations the adsorption starts at higher surfactant concentrations, but the slope of the isotherm is steeper than at low salt concentrations. As a consequence, surfactant isotherms measured at different salt concentrations should intersect when super equivalent adsorption occurs, and the common intersection point (or cip) should correspond with the point where the coulomb interactions are absent, i.e., the iep. The first verification of these trends has been made by De Keizer et al. [132], who also provide the thermodynamic interpretation. De Keizer et al. show that the adsorption isotherms of sodium nonyl-benzenesulfonate on positively charged rutile and that dodecyl-pyridinium chloride and dodecyltrimethylammonium bromide on kaolinite, measured at constant-equili-

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brium pH and three different salt concentrations, all show a cip corresponding to the iep that was obtained by microelectrophoresis. Figure 5 gives an illustration. The phenomenon of the cip has been a focus point for further investigations by Koopal et al. [64,65,68]. A similar cip has been observed by Bitting and Harwell [133] for the adsorption of dodecylsulfate on Al2O3 in the presence of 0.15 M of different monovalent electrolytes. In this case the behavior is due to specific effects, but the explanation of the phenomenon is the same. The stronger the screening ability of an ion is, the weaker are the coulomb attraction (low adsorption) and repulsion (high adsorption). In the cip the screening ability is unimportant and the isotherms for the different ions cross each other. Koopal et al. [109,110] have also discussed Eq. (3) extended with the Boltzmann factor for the coulomb interactions in relation to the adsorption

FIG. 5 Adsorption isotherms of dodecyl trimethylammonium bromide on kaolinite (a) and electrophoretic mobility of kaolinite (b) at pH ¼ 5 and different NaCl concentrations. (Redrawn from Ref. 132 with permission of Elsevier Science-NL, Amsterdam, The Netherlands.)

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of ionic surfactants. A main problem of this more complicated treatment as compared to Eq. (6) or (7) is that decisions have to be made (1) about the location of the headgroups in the adsorbed layer and (2) about the distribution of the salt ions in the surfactant layer. A more general relation between a and the surfactant charge, the surface charge, and the ionic strength in the case that not all headgroups are located at one plane is discussed in Refs. [97,109]. For a qualitative understanding of surfactant adsorption behavior, Eq. (7) is, due to its elegant simplicity, however, equally well suited. Narkiewicsz-Michalek et al. [102] have discussed Koopal’s approach and generalized it for the case of adsorption on a heterogeneous surface. As part of the discussion an expression has been derived that allows for the calculation of m as a function of 1 . Also, a simple extension is presented for the description of bilayers with this model. According to Narkiewicsz-Michalek both Koopal’s equation and the generalized form for a heterogeneous surface predict a strong 2D condensation to occur. It should be noted, however, that the actual shape of the isotherms is determined by the values of the parameters, and by making a different choice the 2D condensation is not so strong. Koopal (unpublished results) has calculated isotherms for different values of m in combination with a Gibbs energy minimization to find the correct value of m at a given value of 1 numerically. These calculations, with FH parameter values commonly used for the SCFA, show a more detailed isotherm composed of a Henry region and a 2D condensation step followed by a gradual increase of the adsorption. In the ideal case the results of calculations based on Eq. (3), extended with coulomb interactions, should give results compatible to those observed with the SCFA calculations (see Section VIII.B, Fig. 22) because the underlying model is the same. Cases and co-workers [101] use the FFG Eq. (1) by assuming that the affinity (K12 ) and lateral interaction parameter ð12 Þ also account for the coulomb interactions. As has been indicated above with the discussion of Eq. (7) this is only approximately correct for low potentials and when both salt concentration and pH are kept constant. Cases et al. [134] also assume that the net lateral attraction parameter for surfactant chains with 8 or more CH2 groups is so large that strong 2D surface condensation occurs, so that the isotherm can be replaced by a step function. The fact that, in general, experimental surfactant isotherms increase stepwise or gradually is now entirely explained by patchwise surface heterogeneity. Although the stepwise nature of some of the isotherms as measured by Cases et al. [101] indeed may point toward surface heterogeneity, the arguments used cannot be applied in general. For a firm conclusion that heterogeneity indeed determines the shape of the isotherm, a more advanced form of local isotherm is required.

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In order to find out whether the adsorbed layer is hemimicelle-like or admicelle-like, Scamehorn et al. [103] extend the approach of Cases et al. [101] by considering bilayer formation on the basis of a BET-type extension of the FFG equation, without treating the coulomb interactions explicitly. In this model the affinities of the surfactant for the first and second layer differ, but the lateral interactions are assumed to be the same. For a homogeneous surface (patch) and their choice of parameters the model predicts a strong 2D phase transition at low surfactant concentrations. Based on these results the authors conclude that a bilayer rather than a monolayer will form at the 2D phase transition. Following Cases et al. the shape of experimental isotherms is entirely explained by assuming that surfaces are patchwise heterogeneous. The size of the patches determines the size of the local aggregates. In a follow-up of the work of Scamehorn, Harwell et al. [124] introduce the ‘‘admicelle’’ concept and use a pseudophase model to describe the admicelles. In this model both hydrophobic and coulomb interactions are included, but the electrostatic interactions are treated with an overdose of detail as compared to the specific interactions. Again the limited size of the admicelles in this treatment is due to the surface heterogeneity. Yeskie and Harwell [135] have addressed the issue of whether hemimicelles or admicelles will be present at the surface in more detail, using the same model. According to Yeskie et al. there is a range of conditions under which hemimicelles are preferred over admicelles due to a strong repulsion between the headgroups present at the solution side of the adsorbed layer. Admicelles will form only at fairly high salt concentrations. In the section on the SCFA model, we will return to this issue. Mehrian et al. [136] have reconsidered the bilayer FFG model of Scamehorn et al. [103] by arguing that in the case of coulomb interactions not only the affinities for the first and second layer should be different, but also the lateral interactions. In view of the discussion of Eq. (7) given above it will be clear that the bilayer FFG model is only suited for the description of experiments done at constant pH and constant salt concentration. Mehrian et al. [136] have shown that under these conditions surfactant adsorption on kaolinite could be fitted with the bilayer model without the introduction of surface heterogeneity. Wilson et al. [137–139] have presented several models for surfactant adsorption in which the lateral interactions have been introduced on the basis of the ‘‘quasichemical’’ approximation [137] instead of with the mean-field or Bragg–Williams approximation used for the FFG-type equations. Coulomb interactions are accounted for on the basis of the Poisson– Boltzmann equation assuming a flat surface and taking into account the finite volume of the ions [139]. Wilson and Kennedy [138] have also derived an equation for bilayer adsorption. The disadvantage of these models is that

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some aspects are treated in great detail and others very crudely. Combination of the regular solution model for charged surfactants, Eq. (6), with either the Debye–Hu¨ckel or the Gouy–Chapman double-layer model is much simpler and leads to an equally good semiquantitative understanding. Wa¨ngnerud and Jo¨nsson [140,141] have proposed two theoretical models for adsorption of ionic surfactants on oppositely charged surfaces, one for very low surfactant concentrations, the other for concentrations close to the CMC [140]. In the model for low concentrations dimerization of the surfactant molecules in bulk solution is assumed. Due to this association the charge number of the surfactant ions becomes higher than that of the ions of the background electrolyte. This leads to a larger coulomb attraction and to preferential adsorption of the surfactant dimers over the other counterions. In the model for concentrations close to the CMC [141] it is assumed that a close-packed (bi)layer is formed at a certain surfactant concentration. The model takes into account the effects of hydrophobic, electrostatic, solvation, and steric interactions. The thickness of the layer increases with increasing surfactant concentration in solution.

2. Adsorbed Layers: SCFA Theory Koopal et al. [64–66,68,142–144] have extensively used the SCFA model to describe the adsorption of ionic surfactants. For details of the SCFA theory consult Section IV of this chapter. With ionic surfactants it has been possible only to study adsorbed layers; modeling of the coulomb interactions of (interacting) axially symmetric aggregates of an unidentified shape is rather complicated. The calculations apply to ionic surfactant adsorption on hydrophilic surfaces with a charge opposite that of the surfactant molecules. The main emphasis has been given to effects of the ionic strength and surface charge on (the shape of) the isotherm. The FH parameters given to the segment–solvent interactions in bulk solution are obtained from modeling micellization of the surfactants [63]. In most calculations the specific interaction between a headgroup segment and the surface has been given an attractive value, whereas for all other species the FH parameters are put equal to zero. This implies that tail segments have an affinity for the surface because they dislike the solution and that the ions of the background electrolyte are indifferent. The first calculations have been made for surfaces with a constant charge [64]. This approximation may hold for plate surfaces of clays when the plates do not contain impurities. As purely constant surfaces are scarce, further calculations have been made for constant potential surfaces [65,66,142–144]. These surfaces adjust the surface charge upon surfactant adsorption in order to keep the surface potential constant. The constant surface potential approximation is a reasonable approximation for mineral

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oxide surfaces [97]. Both for constant-charge and constant-potential surfaces surfactant isotherms have a complex shape composed of a Henry region, a 2D condensation step, a gradual increase of the adsorption beyond the condensation step, and an adsorption pseudosaturation value at the CMC. Isotherms calculated at different values of the ionic strength show a common intersection point in agreement with experimental results as presented in Fig. 5. The segment distribution profiles indicate that beyond this point bilayer-type adsorption occurs. The adsorption at constant-charge surfaces differs from that at a constant-potential surface by the fact that after compensation of the surface charge a considerable hesitation occurs before the adsorption continues. Next to the calculations for constantcharge and constant-potential surfaces some calculations were also done on charge- and potential-regulating surfaces, in order to mimic the adsorption of surfactants on silica. Qualitatively the same trends are found as for the constant-charge surfaces, but the 2D condensation step in the isotherm is much smaller or absent. The overall conclusion that can be derived from the SCFA studies is that the isotherms of ionic surfactants on oppositely charged homogeneous surfaces are far from being a simple 2D condensation phenomenon. A more detailed comparison of the calculated and experimental results can be found in sections VIII.A, B, and C.

3. Surface Aggregates Although the hemimicelle–admicelle issue has been discussed on the basis of the models mentioned in the previous section, real aggregates could not be described; only the monolayer or bilayer character of the adsorption could be discussed. Models specifically developed to describe surface aggregates are those of Chander et al. [145], Zhu et al. [117–119], Rudzinski et al. [98,99], and Li and Ruckenstein [146]. Chander et al. [145] have derived a model comparable to the simple model of Zhu as described in the nonionic section. In this model the electrostatic interactions are considered by realizing that the standard Gibbs energy has two contributions—one specific, the other electrostatic (coulomb)—but the treatment is rather primitive. Zhu et al. [117–119] have applied the family of models presented in Section III.C (surface aggregates of nonionic surfactants) also to describe the adsorption of ionic surfactants. The disadvantage of this practice is that the electrostatic interactions are not taken into account explicitly. This means that for every pH and or ionic strength a new parameter set is required. Although a good fit of separate experimental results can be achieved by parameter fitting, the model is not suited to make predictions. The physical interpretation of the parameter values is quite complicated, if not impossible. In principle, the monomer affinity will be affected by the

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degree of surface charge neutralization, the adsorbed micelles will repel each other more strongly when the surface density of micelles is increasing, and all electrostatic interactions are a function of the ionic strength. These aspects are hidden in the fitted values of all three parameters. The oblate and hemimicelle and admicelle models derived by Rudzinski et al. [98,99] and discussed in Section III.C (nonionic surface aggregates) have also been used by Rudzinski et al. to describe the adsorption of ionic surfactants. Figure 6 gives a schematic picture of the surface phase and the description of the adsorption of dodecyl trimethylammonium bromide on silica. In the model the interactions between the headgroup and the surface and lateral interaction between the monomers in the aggregates are both linearly decreasing with adsorption density. To some extent this could be due to the coulomb interactions, but it will be a realistic approximation only when the potentials are sufficiently low and the experiments are carried out at constant-equilibrium pH and salt concentration. The interaction between the aggregated structures is neglected in the model except for the excluded volume effect. Because of the electrostatic repulsion between the aggregates, this excluded volume is likely to be given by the Debye length at low salt concentrations and at high salt concentrations by the size of the aggregates (excluded area). Hence, we may not expect the model to be very accurate for ionic surfactants. By introduction of surface heterogeneity, as done by Rudzinski et al., all effects not properly accounted for will show up as heterogeneity. In relation to this aspect it should be realized that an underestimation of the coulomb repulsion leads to too steep an isotherm; increasing this repulsion may lead to a similar improvement as by invoking

FIG. 6 Schematic diagram of the surface phase composed of monomers, hemimicelles, and admicelles (panel a) and the contributions of hemimicelles (- - - -) and admicelles (— — —) to the total (——) adsorption of trimethylammonium bromide on silica (*) (panel b). (Redrawn with permission from Ref. 99. Copyright 1996 American Chemical Society.)

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heterogeneity. In any case, models that take into account the coulomb interactions implicitly or very poorly are not very well suited to make firm conclusions about the role of surface heterogeneity. It may be clear that the above models for the adsorption of surface micelles composed of ionic surfactants are still very primitive and not likely to provide detailed information on the surfactant aggregation behavior. They merely served as first steps on the way to model surface aggregation of ionic surfactants. A much more advanced surface aggregation model for ionic surfactants on oppositely charged hydrophilic surfaces that takes the coulomb interactions explicitly into account is the model by Li and Ruckenstein [146]. This model has much in common with the description of micellization of ionic surfactants in bulk solution as done by Nagarajan and Ruckenstein [41], but it also reflects the scaled particle theory of Rudzinski et al. [98,99]. Similarly as in the model by Rudzinski et al. the surface is covered by solvent molecules, surfactant monomers, and monolayer- and bilayer-type surfactant aggregates of various sizes. The competition between the enthalpic and entropic contributions to the Gibbs energy in the adsorbed phase is responsible for the composition of the adsorbed phase. Similarly as done by Nagarajan and Ruckenstein [41] the standard Gibbs energy change in going from the solution to the surface aggregates is calculated by considering five contributions: hydrophobic, conformational, electrostatic, steric, and interfacial. The electrostatic contributions are treated within the framework of the Poisson–Boltzmann equation for flat plates. Calculated results of the model, based on reasonable estimates of the parameter values, are compared with experimental results obtained on mineral surfaces. Figure 7 shows a typical example of the good quality of the prediction. From Fig. 7 a similar conclusion can be drawn as from the results of the SCFA theory: the isotherm for ionic surfactants adsorbing in surface aggregates on an oppositely charged homogeneous surface is not a simple step function. Also, the present model predicts that the isotherm is composed of four regions. So far, Li et al. have not taken into account the surface charge adjustment, nor has attention been paid to the effect of the ionic strength and the common intersection point.

E. Conclusion The general conclusion of the section on adsorption of ionic surfactants is that both the SCFA result for adsorbed layers and the surface micellization model of Li and Ruckenstein [146] show very similar trends. The isotherm starts at low concentrations with a Henry region. The 2D condensation step that occurs in the SCFA results above the Henry region shows up in the

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FIG. 7 Comparison between the isotherms predicted with the model of Li and Ruckenstein and measured adsorption isotherms for sodium alkylbenzenesulfonates on aluminum oxide at pH 7.7 and 0.171 mol/L NaCl. The results are plotted in a double logarithmic plot. (Redrawn from Ref. 146. Copyright 1996 American Chemical Society.)

results of Li and Ruckenstein as a steep but not vertical increase. This corresponds very well with the fact that with the SCFA theory the aggregate is infinitely large, whereas the surface aggregates in the model of Li and Ruckenstein have a limited size. In both models the third region of the isotherm increases less steeply than the second region and beyond the CMC a pseudoplateau is reached. In both models also a monolayer-tobilayer transition takes place. This gives faith to the results of both models, and although the SCFA theory is not capable of describing local surfactant aggregates, it will still give a very good representation of the trends in surfactant adsorption, especially if it is realized that the steepness of the second region is overestimated. Therefore, in the second part of this review more detailed attention is paid to the SCFA theory and a more detailed comparison is made between calculated SCFA results and experimental findings.

IV. SCFA THEORY A. General Outline The SCF theory was originally developed by Scheutjens and Fleer [147,148] for polymer adsorption and is reviewed and discussed in relation to other

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polymer adsorption treatments in a recent monograph [149]. In the last decades gradually other applications [34,59–61,69,70,93–95,150–152] appeared, and Koopal et al. [62–68,142–144] have used the model for a systematic investigation of the adsorption of both nonionic and ionic surfactants. The acronym SCFA for the theory is not widespread but is used in the surfactant applications. The SCFA theory involves a kind of ab initio treatment. Starting from system characteristics such as the number of different types of molecule and the amount of each of them, together with molecular properties such as numbers of segments per molecule ð 1Þ, structure and charge of the molecules, and the interactions between the various segments, the equilibrium distribution of molecules is calculated. The theory involves a lattice that is completely filled with the molecules (and possibly vacancies) present in the system. The theory is an extension of the theory of polymer solutions by Flory [107] to systems that have inhomogeneities in one direction (1D SCFA), i.e., perpendicular to the surface or the center of the lattice, or in two directions (2D SCFA), i.e., perpendicular to the surface and in concentric rings parallel to the surface. The layers are numbered z ¼ 1 for the layer in the center of the lattice (spherical and cylindrical lattice) or that closest to the surface (flat lattice) to z ¼ M. At the outer boundary of M a reflecting mirror is placed. The rings in the 2D system are numbered starting from R ¼ 1 in the central ring. At the outer boundary of the last ring a second reflecting mirror is placed. In the 1D SCFA option a mean-field approximation is used in every lattice layer around to the center of the lattice or parallel to the surface. In the 2D case the mean-field approximation is used in every ring. The contact interactions are incorporated in the SCFA theory similarly as in the Flory theory. The electrostatic interactions are calculated using a multiplate condenser model. The conformational statistics of the chains are evaluated using Boltzmann statistics. Basically the theory calculates, by minimizing the free energy of the system, iteratively, the equilibrium distribution of segments in a potential field excerpted by the presence of the segments themselves. Once this is done the structure of the aggregates or that of the adsorbed layer and the thermodynamic properties of the system can be obtained. The advantage of the SCFA theory is that both surfactant micellization and adsorption can be considered. When micellization is predicted with a certain set of parameters, the same set of parameters should be used to predict the adsorption. The only new parameters for the adsorption calculation are those that describe the interactions of the segments with the solid surface.

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B. One-Dimensional SCFA 1. Volume Fractions of Free Segments To study the shapes of the aggregates in bulk solution the 1D SCFA theory is used with a flat, a cylindrical, or a spherical lattice. For adsorption studies a flat lattice is considered. Let us first consider the 1D case. In a flat lattice the fraction of contacts of a lattice site in layer z with sites in layer z
1, denoted as 
1 , equals the fraction of contacts with sites in layer z þ 1, denoted as 1 . These fractions are independent of z. In spherical and cylindrical lattices the number of lattice sites is not the same in every lattice layer; therefore, 
1 , 0 , and 1 will be functions of z. These functions have been given elsewhere [93,153]. For spherical and cylindrical symmetries the layers will be numbered starting from the center of the lattice. To each segment of type x, in each layer z a volume fraction x ðzÞ is assigned. This implies that only inhomogeneities in the direction perpendicular to the lattice layers are considered. The calculated aggregate is in equilibrium with the bulk solution, where the volume fractions of molecules i are denoted by bi . As the volume fractions in the layers z are not the same as those in the bulk solution, a Helmholtz (or Gibbs) energy difference per segment (or potential of mean force), ux ðzÞ, exists for every type of segment with respect to the bulk solution. The expression for ux ðzÞ follows from statistical thermodynamics [94,153]: ux ðzÞ ¼ u 0 ðzÞ þ kT

X

ð< y ðzÞ >
by Þxy þ vx e ðzÞ

ð8Þ

y

Three terms contribute to ux ðzÞ, u 0 ðzÞ quantifies the local hard-core interaction with respect to the bulk solution and must be the same for every segment type in layer z. u 0 ðzÞ is related to the conformational entropy of the aggregate or adsorbed layer. As no explicit expression for u0 (z) is available, it has to be numerically adjusted to ensure that the sum of the volume fractions equals unity [153]. Physically, u 0 ðzÞ equals þ1 or
1 if the sum of the volume fractions in a layer is not unity. The second term on the right-hand side contains the contact interactions: xy is the FH parameter between segments x and y; < y ðzÞ > is the contact volume fraction of y in layer z, which has contributions from layers z
1, z, and z þ 1. The last term in Eq. (8) covers the electrostatic interactions: vx is the valency of segment x, ðzÞ is the electrostatic potential in layer z, and e is the elementary charge. From ux ðzÞ, segment-weighting factors, Gx ðzÞ, are calculated: Gx ðzÞ ¼ exp ½
ux ðzÞ=kT 

ð9Þ

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The Boltzmann factor Gx ðzÞ indicates the probability to find an unbound segment x in layer z relative to finding it in a layer in homogeneous equilibrium solution. In the case of a molecule containing just one segment in equilibrium with the bulk solution the volume fraction xz of a segment of type x in layer z is calculated from its bulk volume fraction bi and the segment-weighting factor: x ðzÞ ¼ bx Gx ðzÞ

ð10Þ

2. Volume Fractions of Segments Belonging to a Chain To obtain the volume fractions of segments belonging to chain molecules, one must take into account that the segments of the chain are connected to each other. Several types of chain statistics can be applied to arrive at the segment distribution of chain segments. The simplest form used in the SCFA theory is first-order Markov statistics, where the chain segments follow a step-weighted walk and backfolding is not forbidden. For shortchain molecules, such as surfactants, a rotational isomeric state (RIS) scheme [60,61] is also used. This RIS method precludes backfolding in a series of five consecutive segments of a chain and allows for a distinction between trans and gauche conformations. The application of RIS results in a decrease of chain flexibility compared to first-order Markov statistics. This is important if a CH2 unit is considered as one segment, because these groups are too small to be regarded as statistical chain elements. To calculate the volume fractions of segments s of chains i containing ri segments, the so-called chain segment-weighting factor, Gi ðz; s j1 : rÞ, is required [94]. The chain segment-weighting factor combines two chain end-segment distribution functions. One describes a walk starting at segment 1 located at an arbitrary position in the system. The start value is given by G1 ðzÞ (the subscript 1 refers to segment 1, which is of type x), and it ends at segment s in layer z after a walk of s
1 steps along the chain—the index 1 in Gi ðz; s j1 : rÞ refers to this walk. The other chain end-segment distribution function starts at the other chain end, i.e., segment ri , where Gr ðzÞ is the start value. This walk ends at the same segment s in layer z after a walk of r
s
1 steps along the chain. The index r in Gi ðz; s j1 : rÞ refers to this starting point. In each step of the walk along the chain a Boltzmann factor Gr ðzÞ appears. If the RIS method is applied, the chain segment-weighting factor, Gi ðz; s j1 : rÞ, of segments s of chains i containing ri segments depends not only on ux ðzÞ and the positions of the neighboring segments of a chain, but also on the orientation of the bonds between these segments. End-segment distribution functions have to be calculated for every orientation [60,61]. The RIS procedure has been generalized to branched chains [60].

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From the chain segment-weighting factors the volume fractions of segments s, belonging to chain i, can be calculated, using the analog of Eq. (10) for free chain segments [94,153]: i ðz; sÞ ¼ Ci Gi ðz; sj1 : rÞ

ð11Þ bi =ri

or, more generally, i =ri Gi ðrj1Þ, The normalization constant Ci equals where i is the total amount of chain i and Gi ðrj1Þ is the sum of the chain end-segment-weighting factors for segment ri , over the total number of layers, defined as M. Equating Ci to bi =ri is equivalent to taking a special case of i =ri Gi ðrj1Þ, because in a homogeneous bulk solution i ¼ M  bi and Gi ðrj1Þ = M  1 . The volume fraction, i ðzÞ, that each molecule type i has in layer z can be obtained by a summation of i ðz; sÞ over all segments that this molecule type has in layer z: i ðzÞ ¼

r X

i ðz; sÞ

ð12Þ

s¼1

The volume fractions of segments of type x, belonging to molecules i, xi ðzÞ can be obtained if the sum is restricted to segment s of type x. Because of the use of volume fractions instead of number of segments (or molecules), the definition of the total amount, i ; in the system is not entirely straightforward. For spherical geometry the number of lattice sites per layer, LðzÞ, can be calculated from the difference between volume V(z) and volume V(z
1), where V(z) equals 4 z3 =3 and the total amount is most conveniently expressed as X i ¼ LðzÞi ðzÞ ð13aÞ z

For lattices with cylindrical geometry only the relative change in the number of lattice sites with layer number is known, and not the total number of lattice sites, since the cylinder has an infinite length. The volume of a disk with the thickness equal to the length of one lattice site is given by VðzÞ ¼  z2 . In Eq. (13a) we now use for L(z) the difference in volume of disks with radius z and z
1. For flat geometry the amount i is defined per unit cross section of a lattice site and is thus simply given by X i ¼ i ðzÞ ð13bÞ z

3. Electrostatic Interactions The uneven distribution of charged molecules leads to the development of electrostatic potentials. These potentials are calculated using a multiplate

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condenser model and depend on the net charge and the electrostatic capacitance in each layer [94,154]. The charge is assumed to be exclusively located on the midplanes of the lattice layers, as shown schematically in Fig. 8. The charge density in each layer, ðzÞ, follows directly from X vx ex ðzÞ=as ð14Þ ðzÞ ¼ x

where as is the cross section of a lattice site. For the permittivity of layer z, "ðzÞ, the density weighted average can be used as an approximation: "ðzÞ ¼

X

"x x ðzÞ

ð15Þ

x

FIG. 8 Schematic representation of an electrostatic potential profile in a flat lattice. The lattice consists of z layers, the electrostatic charges, s, are located on the midplanes in each lattice layer. For the calculation of the electrostatic potentials, , the charges, s, the distance, l, between the midplanes, and the average permittivity, "ðzÞ, in each layer have to be known. For l a value of 0.31 nm is chosen and the distance from the surface is now measured by (nm). A discontinuous change in the field strength occurs at the midplanes where the charge is present and at the boundary planes between two lattice layers where "ðzÞ change. (Reprinted with permission from Ref. 64. Copyright 1996 American Chemical Society.)

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where "x is the permittivity of species x ("x ¼ "0 "r;x ). The field strength E as a function of the distance may change discontinuously at the midplanes of the lattice layers, which are separated by a distance l, located at z ¼ 1; 2; . . . ( ¼ 1=2l; 3=2l; . . .) due to the presence of charge and at z ¼ 1=2; 3=2; . . . ð ¼ 0; l; . . .) due to a change in permittivity; see Fig. 8. In addition, in curved lattices a continuous decrease of Eð Þ exists due to the divergence of the electric field. The field strength at distance from the central point (spherical), line (cylindrical), or plane (flat) geometry of the lattice follows Þ from Gauss’s law, EdAs ¼ q=", where q is the charge and As the area. Application to the different geometries yields: flat:

intð =lþ X 2Þ 1 ðz 0 Þ Eð Þ ¼ "ð Þ z 0 ¼1

ð16aÞ

cylinder:

intð =lþ2Þ X 1 Eð Þ ¼ Lðz 0 Þðz 0 Þ 2 "ð Þ z 0 ¼1

ð16bÞ

sphere:

intð =lþ2Þ X 1 Lðz 0 Þðz 0 Þ Eð Þ ¼ 2 4 "ð Þ z 0 ¼1

ð16cÞ

1

1

1

The upper boundary of the summation is the Entier function of =l þ 1=2 indicated as intð =l þ 1=2Þ. Because the charges are located on the midplanes of the lattice layers, only the potentials on these midplanes are relevant for the calculation. The potential difference between two neighboring midplanes at z þ 1 and z is given by

¼ðzþ1=2Þl ð

ðz þ 1Þ
ðzÞ ¼


Eð Þd

ð17Þ

¼ðz
1=2Þl

Using Eq. (17) we obtain for the three geometries:  X z l 1 1 þ ðz 0 Þ ð18aÞ flat: ðz þ 1Þ ¼ ðzÞ
2 "ðzÞ "ðz þ 1Þ z 0 ¼1 "
 1 1 z 1 ln cylinder: ðz þ 1Þ ¼ ðzÞ
þ 2 "ðzÞ z
1=2 "ðz þ 1Þ
# X z z þ 1=2 ln Lðz 0 Þðz 0 Þ z z 0 ¼1 ð18bÞ

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 1 1 ðz þ 1Þ ¼ ðzÞ
8l "ðzÞðz
1=2Þz X z 1 þ Lðz 0 Þðz 0 Þ "ðz þ 1Þzðz þ 1=2Þ z 0 ¼1

sphere:

ð18cÞ

In all geometries there is a contribution of layer z where the permittivity is "ðzÞ and a contribution of layer z þ 1 where the permittivity is "ðz þ 1Þ. In the middle of layer z the distance to the center of the lattice is ðz
1=2Þ ; this causes the ðz
1=2Þ-term and the absence of a ðz þ 1)-term in Eqs. (18b) and (18c). Except for cylindrical geometry the distance between two midplanes, l, has to be quantified to obtain the potential difference. For uncharged systems this distance is arbitrary. To calculate the potentials with respect to the bulk solution a starting point at layer M is needed; for this purpose the electroneutrality condition has to be used [94].

4. Self-Consistent Solution The circular definition of  in terms of u, which in itself is defined in terms of , makes it necessary to iterate to find a solution to the set of equations that describe the system. When a set of fux g is obtained that generates a set of fx g, which is used to find the same set of fux g (within the desired numerical precision), a self-consistent solution is obtained. Because only equilibrium potentials can be calculated, only equilibrium systems can be described.

5. Excess Free Energy for the Creation of a Micelle and Excess Amounts In the SCFA theory thermodynamic data are readily available. The derivation of the equations has been presented by Bo¨hmer et al. [94] and by Evers et al. [153]. In the case of micellization an important quantity is the excess free energy for the creation of a micelle with a given aggregation number and a fixed position: Am

¼ kT

X z

( LðzÞ

X x

x ðzÞ ln Gx ðzÞ


X i ðzÞ
bi i

ri

h    i 1XX xy x ðzÞ < y ðzÞ >
by
bx y ðzÞ
by 2 x y ) ðzÞ ðzÞ þ 2 þ

ð19Þ

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The segment density profiles are used to calculate the excess amount of molecules in either the aggregate or the adsorbed layer. For the adsorbed layer the excess amount of molecules per surface site equals i Xh nexc ¼ i ðzÞ
bi =ri ð20Þ i z

The excess number of molecules i in the system with respect to the equilibrium solution, due to the presence of a micelle, is h i X b ¼ LðzÞ  ðzÞ
 nexc ð21Þ i i i z

may differ from the The number of aggregated molecules in a micelle, nagg i excess number of molecules in the system because in layers adjacent to the micelle a depletion of surfactant molecules can occur. To calculate the aggregation number of a micelle, Eq. (21) can be used for layers where ½i ðzÞ
bi  is positive: h i h i X nagg ¼ LðzÞ i ðzÞ
bi for i ðzÞ
bi > 0 ð22Þ i z

C. Two-Dimensional SCFA The lattice used for the 2D SCFA calculations is shown schematically in Fig. 9. The number of lattice sites in a ring, L(R), varies as a function of R [155]. Within every ring the mean-field approximation is applied and the volume fraction of segment x is now a function of both z and R : x ðz; RÞ. The sum of the volume fractions of the segments in each ring (z, R) is again unity. Just as in the case of the 1D SCFA theory the Gibbs energy per segment, ux ðz; RÞ, which is now also a function of R, has to be calculated: X ux ðz; RÞ ¼ u 0 ðz; RÞ þ kT xy ð< y ðz; RÞ >
by Þ ð23Þ y

In this equation u 0 ðz; RÞ is independent of the segment type and it ensures the complete filling of ring ðz; RÞ. The angular brackets () indicate a weighted average of the volume fraction, which has contributions from the volume fractions in neighboring rings: ðz
1; R
1Þ, ðz
1; RÞ, and ðz
1; R þ 1Þ from the previous layer; ðz; R
1Þ, ðz; RÞ, and ðz; R þ 1Þ from layer z; and ðz þ 1; R
1Þ, ðz þ 1; RÞ, and ðz þ 1; R þ 1Þ from the next layer. As the number of lattice sites varies with R, so does the weight of the contributions from the neighboring rings. The ux ðz; RÞ function is calculated with respect to the bulk solution, where the equilibrium volume fraction of x is bx .

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FIG. 9 Representation of the 2D, cylindrical, symmetrical lattice. The layers contain concentric rings. The hatched plane represents the surface. (Reprinted with permission from Ref. 67. Copyright 1996 American Chemical Society.)

The weight of each conformation is calculated using a step-weighted walk procedure as discussed above, i.e., each step depends on the position of the previous segment and is weighted with the segment-weighting factor, Gx ðz; RÞ: Gx ðz; RÞ ¼ exp½
ux ðz; RÞ=kT 

ð24Þ

For monomers and chain molecules the volume fractions can be calculated from the segment-weighting factors by using the equivalents of Eq. (10) and Eq. (11), respectively. Also in this case the solution of the set of equations can only be found numerically.

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From the equilibrium volume fraction profile the excess adsorbed amount of molecule i with chain length ri can be calculated using   XX ¼ LðRÞ i ðz; RÞ
bi =ri ð25Þ nexc i z

R

In bulk, surfactant molecules associate in micelles. For concentrations higher than the CMC, bx in the above equations represents the bulk volume fraction of free molecules. It is therefore necessary to know the CMC, which can be calculated with the 1D SCFA theory.

D. Thermodynamics of Small Systems Using the SCFA theory one can calculate the equilibrium structure of a single micelle. In combination with the thermodynamics of small systems [59,93,86,87] the composition of a micellar solution, consisting of a number of identical micelles, can be found. To this end, the solution is divided into a number of subsystems with volume Vs , where every subsystem contains one micelle, with volume Vm . To distinguish between the equilibrium volume fraction of i in homogeneous bulk solution, bi , and the average volume fraction of i in both the system and each subsystem, the latter is denoted exc by i . The excess free energy of the subsystem,  As consists of two parts: the translational entropy of the micelle, k lnðVm Vs Þ (micelles may move freely in the solution) and the excess free energy for creation of a micelle with a given aggregation number and a fixed position, Am ; hence:   Aexc ð26Þ s ¼ Am þ kT lnðVm Vs Þ For micelles in equilibrium with a homogeneous solution Aexc is zero. s for a given value of V and Am . As Equation (26) allows calculation of V s m   k lnðVm Vs Þ is always negative, micelles can only exist if Am is sufficiently positive. To obtain Vm for a charged micelle a cutoff distance for the electrostatic interactions between the micelles has to be chosen. A quite common choice for the effective range of the electrostatic interactions is the distance at which the electrostatic potential has dropped to 1/e of its maximum value. Consequently, the potential decay depends on the ionic strength and so does the chosen cutoff distance. Once Vm is determined, Vs can be calculated with Eq. (26), and i follows from i ¼

nexc i r þ bi Vs

ð27Þ

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Contrary to small spheres, structures like infinitely long membranes and cylinders have a negligible kinetic energy. Therefore, for these geometries the translational entropy term can be neglected so that for equilibrium membranes and cylinders Am ¼ 0.

E. General Parameter Values In principle, there is no limit to the number of different molecules and segment types that can be handled in an SCFA calculation. For practical reasons the molecular structures and interaction parameters to model the surfactants in aqueous solutions and near interfaces are chosen as simple as possible. Only the main interactions involved in micellization and adsorption have been incorporated. The surfactants are modeled as An Bm chains, where A represents an aliphatic segment and B a hydrophilic segment. Water is modeled as a monomer W. The contact interactions, including the hydrophobic interaction, are modeled through the values of the  parameters. With these parameters the properties of surfactants in water, such as their capability of forming micelles, should be reproduced. Temperature effects are not considered; all results refer to room temperature. Ion specificity, hydration of ions, and size differences between the salt ions have been neglected, too; they are regarded as second-order effects. The parameter AW should reflect the poor solubility of segments A in water W. Unless stated otherwise each CH2 group is considered as an A segment and AW = 2. The parameter BW depends on the type of headgroup and should reflect that water is a good or moderately good solvent for B. In the case of ionic surfactants BW ¼ 0, for ethylene oxide- (EO) type surfactants the measured value [156] BW ¼ 0:4 is used. The interaction between the hydrophilic segments B and the hydrophobic segments A should be repulsive. In general, a value AB ¼ 2 is chosen to ensure spatial separation of headgroup segments and tail segments, which is necessary for the formation of micelles. This set of parameters, in combination with our choice to treat one CH2 group as a segment, enables a correct prediction of the change in CMC with aliphatic chain length for nonionic surfactants. An alternative way to obtain the AW value is to calculate the partition equilibria for homologous series of alkanes between a water and an oil phase for which experimental data were reported by Tanford [157] with the multicomponent Flory equations [93]. This yields an only slightly higher value, AW ¼ 2:3. The segment-surface xS parameters will be specified with the results. It should be noted that the adsorption is a displacement process in which a segment x replaces a fraction 1 of its contacts with the solvent for contacts with the surface so that xS is not the ‘‘adsorption energy.’’ The ‘‘adsorption

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energy’’ or, more correctly, the free energy of displacement, dðxÞ , of a water molecule by a segment of type x is defined as dðxÞ ¼ 1 ðxS
xW
WS þ WW Þ

ð28Þ

Equation (28) shows that xS
WS is important rather than each of these parameters. By using WS ¼ 0, dðxÞ is determined by xS
xW because WW = 0, by definition.

V. SCFA AND MICELLIZATION OF NONIONIC SURFACTANTS A. Aggregate Shape To study the association behavior of nonionics, calculations have been performed for a series of A10 Bn molecules in lattices with spherical and flat geometry [62]. B segments are used as a model for EO units, whereas the A segments mimic CH2 groups. It is obvious that an EO unit is much bigger than a CH2 group; nevertheless, one segment of type A for a CH2 group and one segment of type B for an EO unit have been used in the first calculations. Three segments for an EO unit would overestimate the chain flexibility, and 3 CH2 groups in one A segment require a very large positive value of AW which leads to large lattice artifacts. First-order Markov statistics are used for the chain statistics. Results for the series A10 Bn as a function of n are shown in Fig. 10. The volume fraction divided by the total chain length of the surfactant is a measure for the surfactant concentration. The concentration where phase separation occurs can be calculated using the extended Flory–Huggins theory [93]. Figure 10 shows that phase separation takes place when the number of B segments is smaller than 3. If the number of B segments is more than 3, the concentration at which micelles are formed is lower than the concentration required for membrane formation, so that micelles are preferred over membranes. Phase separation occurs at much higher concentrations. Due to the conical structure of the surfactant molecules the spherical geometry is the preferred shape of the aggregate. With increasing length of the headgroup the steric hindrance between the headgroups in an aggregate increases. This effect is especially evident from the growing difference between the critical membrane concentration and the critical micelle concentration with increasing length of the B block. The CMC increases approximately linearly with the number of segments of type B. This agrees with experimental results for octylphenol polyoxyethylenes [158,159] and decyl polyoxyethylenes [160]. With the interaction parameters chosen the increase in the CMC with increasing length of the headgroup Bn is well predicted for octylphenol polyoxyethylenes: the CMC

Modeling Association/Adsorption of Surfactants

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FIG. 10 Equilibrium volume fractions of A10Bn molecules, for phase separation, membrane formation, and micelle formation, as a function of the B block length. (Reprinted with permission from Ref. 62. Copyright 1996 American Chemical Society.)

doubles if the number of headgroup segments increases from 12 to 30. For small B blocks the calculated slope is somewhat smaller than that found experimentally for monodisperse decyl polyoxyethylenes.

B. Structure of the Micelles In Fig. 11 the segment density profiles of the A and B segments in A10 B6 and A10 B40 micelles are shown at the same overall concentration (=r is 0.510
3 ) of surfactant. The segments of type A are in the center of the micelle while the B segments are on the outside. Some solvent is still present in the interior of the micelle (not shown in the figure), which is due to the poor model used for water: a monomer without a preferential orientation. The hydrophobic core of the micelles formed by the A10 B6 molecules is much larger than that formed by the A10 B40 amphiphiles. Steric hindrance between the hydrophilic B blocks prevents the formation of large micelles of A10 B40 . The number of molecules per micelle at a constant overall concentration (=r is 0:5  10
3 ) decreases sharply for n < 10 and gradually for large n; see Fig. 12. This trend compares well with the experimental trend in the aggregation numbers measured for octylphenol polyoxyethylenes [158,159] using pyrene solubilization.

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FIG. 11 Volume fraction profiles for A10 B6 (a) and A10 B40 (b) micelles. The volume fractions for segments of type A and type B are indicated. The overall concentration =r is 5  10
4 . (Reprinted with permission from Ref. 62. Copyright 1996 American Chemical Society.)

FIG. 12 The aggregation number of A10 Bn micelles at a constant overall concentration of 5  10
4 as a function of the number of B segments in the molecule. (Reprinted with permission from Ref. 62. Copyright 1996 American Chemical Society.)

Modeling Association/Adsorption of Surfactants

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VI. SCFA AND ADSORPTION OF NONIONIC SURFACTANTS A. Adsorption Layers on Hydrophilic Surfaces 1. Adsorption Isotherms Adsorption isotherms have been calculated for the same homologous series of surfactants as used with the micellization study [62]. In general, the hydrophilic segments will have a relatively strong interaction with the surface and the hydrophobic a weak interaction. It is assumed that BS ¼
6 and that the other FH parameters with the surface are zero. In this case the ‘‘adsorption energy’’ for a B segment is
1.6 kT and that of an A segment
0.5 kT, see Eq. (28). Results are presented in Fig. 13 for six different values of n. The adsorption is presented as nexc , the excess number of molecules per lattice site. On the horizontal axis the overall concentration is given as =r. The CMCs are indicated with an asterisk. For a B block considerably longer than the A block (i.e., A10 B30 and A10 B40 ) the adsorption reaches a near-saturation value before the CMC is reached, and the shape of the isotherms is similar to that of a homopolymer of type B. The small irregularities in the adsorption isotherms are caused by layering transitions, a kind of lattice artifacts. At very low-volume fractions the affinity increases with increasing length of the B block since chains with more B segments have a higher adsorption energy. With increasing volume fraction the cooperativity of the adsorption

FIG. 13 Calculated adsorption isotherms for a series of amphiphilic chain molecules on a hydrophilic surface. The CMC values are indicated with an asterisk. (Reprinted with permission from Ref. 62. Copyright 1996 American Chemical Society.)

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becomes important and the adsorption increases with decreasing B block size. The increase in adsorption is due to unfavorable interactions between A segments and solvent: the A segments associate to reduce the number of contacts with the solvent. The association is more pronounced if the A/B ratio increases, a trend comparable with the influence of the A/B ratio on the CMC. The A/B ratio also affects the maximum adsorption; see Fig. 13. The higher the A/B ratio is, the higher is the adsorption at the CMC. For short B blocks the isotherm shows a 2D phase separation in the adsorbed layer, due to association of the A segments, and a surfactant bilayer is formed. If the B block is long, steric hindrance prevents such bilayer formation. This compares well with the fact that the difference between the critical micelle concentration and the critical membrane concentration increases if the B block becomes longer (Fig. 10).

2. Structure of the Adsorbed Layer The volume fraction profiles, x ðzÞ, in the plateau of the isotherm are given in Fig. 14 for segments of type A and type B for A10 B6 and A10 B40 . For A10 B6 a thick bilayer is formed: B segments are present both at the surface and at the solution side, spatially separated from each other by a layer of segments of type A. For A10 B40 no maximum in the volume fraction of B segments at the solution side is found, no bilayer is formed. Adsorbed and

FIG. 14 Volume fraction profiles of A10 B6 (a) and A10 B40 (b) in the adsorbed layer at the plateau of the isotherm on a hydrophilic surface. The volume fractions of segments of type A and of type B are indicated. (Reprinted with permission from Ref. 62. Copyright 1996 American Chemical Society.)

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free surfactant bilayers become more and more unfavorable as the number of B segments rises. The trends observed from the calculated isotherms compare well with experimental results by Levitz and Van Damme [158,159], Partyka et al. [161], and Tiberg et al. [162,163] for adsorption of a series of alkylphenol polyoxyethylenes on hydrophilic silica. Also for other nonionic surfactants, like decyl methylsulfoxide, adsorbed onto silica similar isotherms were found [164]. Levitz [158,159] also studied the structure of the adsorbed layer of octylphenol polyoxyethylenes, using pyrene as a fluorescent probe. Pyrene solubilization was found to occur for all chain lengths studied, indicating that nonpolar domains exist in the adsorbed layer. For surfactants with small headgroups a homogeneous adsorbed layer may be formed near the CMC. For surfactants with longer headgroups aggregates were formed on the surface with sizes comparable to the sizes of micelles. These results, together those obtained with neutron reflection [67] and the fact that Fig. 10 shows that bilayers are not likely when the A=B ratio is small, all strongly suggest that the 1D SCFA calculations for the adsorbed layer are not adequate. We will return to this issue when the 2D SCFA calculations are discussed.

B. Adsorption Layers on Hydrophobic Surfaces In the study the adsorption of A10 Bn molecules on hydrophobic surfaces, SA is set to
4 and SB = 0 [64]. In this case the free energy of displacement of a water molecule on the surface by an A segment is
1.5 and that of a B segment
0.1 kT. The adsorption of A10 Bn versus a linear concentration axis approximates the L-type for all values of n studied ð6  n  40Þ. However, plotting the results semilogarithmically reveals that this is not the case: as soon as some material is present at the surface a strong increase in the adsorption occurs due to the association of the hydrophobic chains. The increase in adsorption with concentration levels off before the CMC is reached and saturation adsorption is reached around the CMC. The adsorption plateau decreases with increasing B block size. These trends agree with experimental isotherms of a series of nonylphenol polyoxyethylenes on polystyrene and poly(methyl methacrylate) [112,165].

C. Aggregates on Hydrophilic Surfaces 1. Parameter Values In order to investigate the problem of local aggregate formation in more detail, the adsorption of two surfactants, C12 ðEOÞ6 and C12 ðEOÞ25 , and an EO oligomer, EO22, have been studied experimentally and the results were compared with calculations using both the 1D and 2D SCFA theory [67]. In

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these calculations a somewhat different choice of segments has been made. The C 12 ðEOÞ6 and C12 ðEOÞ25 are modeled as A4 B6 and A4 B25 , i.e., one A segment is composed of 3 CH2 groups and each EO unit equals a B segment. The EO oligomer is modeled as a chain of 25 B segments. The present choice leads to a more realistic size ratio between a B and a A segment and makes it possible to perform the 2D calculations (fewer segments). With these large aliphatic segments a value AW ¼ 4:3 is required to reproduce approximately the correct change in CMC as a function of aliphatic chain length. For the 2D calculations a lattice with 16 layers and 10 rings has been used. This is rather small with respect to the chain length of the surfactant, but to deal with the problem a large number of equations have to be solved. At the time of the calculations this was the maximum size of the system that could be handled. For the B segments that are assumed to adsorb on the surface a value BS ¼
3 is used rather than
6 as done above. The latter choice gave rise to high-affinity isotherms for surfactants with long headgroups, which is not in agreement with the experimental findings. For the interaction between A and S a repulsive value AS ¼ 4:3 is chosen. This value equals AW , and it ensures that no preferential adsorption of A occurs on the surface, because the interaction of A with the surface is just as unfavorable as the interaction of A with W.

2. Comparison of 1D and 2D SCFA Calculations The adsorption isotherms of A4 B6 , A4 B25 , and B25 calculated with the 1D SCFA theory are shown in Fig. 15. An arrow indicates the CMC, beyond which no further increase in adsorption occurs. For A4 B6 (Fig. 15a) the initial adsorption is very low; just before the CMC a phase separation takes place in the adsorbed layer, leading to a large condensation step. The dashed curve is the calculated isotherm; the full curve indicates the condensation step. Layering transitions are especially evident because high x-values are used. For A4 B25 , Fig. 15b, a more gradual increase in adsorption is found and a much lower plateau value is reached than for A4 B6 (note the scale difference with Fig. 15a). The differences between A4 B25 and B25 are much smaller than observed experimentally for C12 ðEOÞ25 and the ðEOÞ22 oligomer. Due to the different choice for the A segments and the different parameter values, the shapes of the isotherms have changed slightly; compare Figs. 13 and 15. The A4 B6 isotherm starts to increase sharply at a concentration close to the CMC, whereas the A10 B6 isotherm in Fig. 13 increased well below the CMC due to the higher value of BS . For the surfactant with long headgroup the high-affinity isotherm has disappeared, in agreement with experimental work [158,159]. The difference in plateau values of the two isotherms

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FIG. 15 Calculated 1D SCFA adsorption isotherm of A4 B6 (panel a). The dashed curve represents the calculated points; the full curve, which coincides with the dashed curve at high coverage, shows the phase separation. The results for A4 B25 and B25 are given in panel b. An arrow indicates the CMC. (Reprinted with permission from Ref. 67. Copyright 1996 American Chemical Society.)

and the volume fraction profiles of the A and B segments at the CMC are, however, very similar for the two sets of parameters. As indicated above, with the very long headgroup of A4 B25 as compared to the tail, it is unlikely that a homogeneous A4 B25 bilayer will form. Most likely the A segments will form hydrophobic clusters in the adsorbed layer. This effect can be studied using the 2D SCFA theory. For A4 B6 the adsorption isotherm calculated with the 2D SCFA theory is given in Fig. 16a. Due to the aforementioned layering transitions the shape of the isotherm is not perfect and small irregularities should be neglected, but the first condensation loop is realistic. It is clear that the instability region has become much smaller than with the 1D calculations. This is due to the fact that a critical concentration in a certain layer is no longer needed to get local phase separation. This critical concentration is needed in just one ring. The adsorption isotherm increases almost vertically, close to the step in the 1D adsorption isotherm. A drawback of the 2D calculations is that at present only a fairly small aggregate and its growth with bulk solution concentration could be studied. In the case of A4 B6 and high amounts of surfactant in the system, the surfactant molecules also start to accumulate near the system boundaries (near ring 10) and then the calculated volume fraction profile depends strongly on the number of rings used. Because of this behavior the calculated isotherm is truncated at nexc ¼ 0:3. As the 2D isotherm follows about

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FIG. 16 Calculated 1D and 2D SCFA adsorption isotherms of A4 B6 (panel a), A4 B25 , and B25 (panel b). (Reprinted with permission from Ref. 67. Copyright 1996 American Chemical Society.)

the condensation step in the 1D isotherm it may be concluded that the local aggregates have to be very large bilayer structures (double ‘‘pancakes’’). For A4 B25 the difference between the 1D and 2D isotherms is quite significant; see Fig. 16b. Again only the first (and largest) instability in the isotherm is realistic; irregularities for nexc > 0:025 are layering transitions and should be neglected. The adsorbed amount calculated with the 2D theory is much higher than that with the 1D theory. Moreover, the adsorption is much higher than that of B25 . These differences and the high adsorption values indicate that strong association must have taken place.

3. Surface Micelles The calculated structure of the surfactant aggregate is illustrated in Fig. 17, where the volume fraction profiles of the adsorbed aggregate of A4 B25 at nexc ¼ 0:1 are shown. The B segments are located at the surface and on the outside of the core formed by the A segments. The aggregate consists of about 30 molecules. Note that in layer 10 and ring 1 a significant volume fraction of B segments is still present. The shape of the isotherm C12 ðEOÞ25 is in agreement with experimental work [67,158,159,163]. From theory bilayer formation on a hydrophilic surface is evident. Experimentally a complete bilayer of C12 ðEOÞ25 on the surface is not found [8,9,67,163,166], but the large condensation step in the isotherm indicates that fairly large bilayer fragments are present. For A4 B25 the 2D SCFA theory predicts the formation of aggregates on the surface. The presence of these aggregates is responsible for the large adsorbed

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FIG. 17 Volume fraction profiles of B and A segments of A4 B25 for an adsorbed aggregate near the CMC. (Reprinted with permission from Ref. 67. Copyright 1996 American Chemical Society.)

amounts in comparison with those for the polymer B25 . These differences were also found experimentally between C12 ðEOÞ25 and ðEOÞ22 [67]. Assuming that the difference between the 1D and 2D isotherm is due to aggregation and that all aggregates are equal in size, an estimate can be made of the minimum aggregation number. The maximum aggregation number follows from the 2D isotherm. The obtained numbers range from 10–15 at low surfactant concentration to 26–34 near the CMC. The aggregation number of the adsorbed aggregate of A4 B25 at the CMC is about the same as that for A4 B25 micelles, calculated at high surfactant concentrations. This was also found experimentally by Levitz et al. [123,160] for surfactants with long EO chains adsorbed on silica, using pyrene excimer

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formation. For a surfactant with an ðEOÞ13 headgroup Levitz et al. found a growth in aggregate size with the concentration. Calorimetric data of Denoyel et al. [167] with the same systems show two domains: at low adsorbed amounts the heat effect is exothermal, and the contacts between surface and surfactant segments are enthalpically favorable. At higher adsorption values an endothermal effect is observed, with a molar enthalpy of the same order of magnitude as the molar enthalpy of micellization, indicating the role of hydrophobic lateral interaction. With an increase in EO length the heat effect at low coverage is enlarged, whereas at higher coverages the endothermic effect is diminished. These trends are in agreement with the present calculations, where also two domains in the adsorption isotherm are found. The first part is dominated by the adsorption of B segments on the surface, whereas the second part stems from the accumulation of A segments. In general, it may be concluded that a satisfactory agreement between 2D SCFA calculations and experimental data is observed.

VII. SCFA AND MICELLIZATION OF IONIC SURFACTANTS A. Parameter Values In the study of the ionic surfactants, the surfactants have been treated as chains consisting of series aliphatic segments, A, and three sequential headgroup segments B: An B3 [63–66,142,143]. The headgroup is considered to be about three times as large as a CH2 or A segment. A slightly different choice of the number of headgroups segments (2 or 4) did not change the results significantly. RIS statistics are used for the chain conformations. The energy difference between a trans and a gauche conformation in the chain is taken to be 1 kT. The nonelectrostatic interaction parameters between the A segments and the other segments in solution are set to AW ¼ AB ¼ AC ¼ AD ¼ 2. The remaining -values are 0. The valency of the headgroup, B3 , equals
1, or
1/3 charge per B segment. Cations and anions ðC; DÞ are modeled as monomers with a valency of +1 and
1, respectively. Because segments B, C, and D are defined as charged segments, their charge is not indicated. The relative permittivity of the segments of type A is 2 and that of water and all other segments is 80. Salt ions only differ from the solvent by their charge. Apart from the fact that salt ions and solvent have a volume, this situation is comparable to that in diffuse double-layer theories such as the Gouy Chapman theory. The distance l between two midplanes of lattice layers, which must be quantified in the computation in the case of electrostatic interactions, is 0.31

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nm and the cross section of a lattice site is assumed to be l2. Using l3 as the volume of a lattice site, this choice leads to 55.5 moles of lattice sites per dm3, i.e., every monomer is assumed to have the size of a water molecule. This size of a lattice site corresponds approximately with the volume of a CH2 group as calculated from the bulk density of alkanes. However, the length of a C–C bond is overestimated.

B. Aggregate Shape and Structure of the Micelles To obtain information on the preferred aggregate shape, calculations for spherical, cylindrical, and flat lattices have been compared for various model surfactants [63]. The effect of chain length of a homologous series of An B3 on the equilibrium volume fractions of aggregate formation for spherical aggregates (near the theoretical and practical CMC, i ¼ 0:01), infinitely long cylinders, and infinite membranes, all at a salt concentration of 0.1 M CD, is presented in Fig. 18. Because the spherical structures are in equilibrium with lower bulk volume fractions than the cylinders and the membranes, the first-formed aggregates are spherical for all chain lengths studied. The equilibrium volume fractions for aggregate formation decrease with increasing chain length with a slope that is hardly different for the different aggregate shapes.

FIG. 18 Equilibrium bulk volume fractions as a function of the number, n, of aliphatic segments, A, of spherical, cylindrical, and flat structures formed by AnB3 molecules, with a valency of
1 on the B3 headgroup (
1/3 per B) in a univalent symmetrical electrolyte (C+ and D
) solution with cs ¼ 0:1 M, solvent W. (Redrawn with permission from Ref. 63. Copyright 1996 American Chemical Society.)

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This slope is mainly dependent on the choice of the Ay parameter and the segment size. The calculated slopes correspond to that measured for a homologous series of ionic surfactants at high ionic strength [157], indicating an appropriate combination of parameter values. As n increases the equilibrium volume fractions at which cylinders and spheres (i = 0.01) are formed approach each other. In this situation it is to be expected that deviations from the spherical shape will occur and intermediate forms, such as ellipsoids, may be present and the aggregates will be polydisperse. The salt concentration also affects the CMC and aggregation number. In Fig. 19 the CMC (panel a) and nagg (panel b) of spherical micelles are shown as a function of the aliphatic tail length, in the absence and presence of 0.1 M indifferent salt. For both situations the CMC decreases and the aggre-

FIG. 19 Effect of the chain length on the cm (panel a) and the aggregation number (panel b) at i = 0.01 at cs = 0 and cs = 0.1 M for AnB3 molecules. The calculated values are indicated with markers; the drawn lines are fits through the calculated points. (Redrawn with permission from Ref. 63. Copyright 1996 American Chemical Society.)

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gation number increases with increasing chain length, but the effects are larger in the presence of salt. In the absence of added electrolyte, the ionic strength is determined by the surfactant. A surfactant with a longer aliphatic tail has a stronger tendency to associate than one with a short tail due to the stronger hydrophobic interactions. This leads to a lower CMC. At this low CMC, however, the repulsion between the headgroups is strong because the ionic strength, determined by the CMC, is low. This repulsion counteracts the hydrophobic attraction and an increase of the chain length with one CH2 segment has a relatively small effect on the CMC and the aggregation number. In the presence of 0.1M electrolyte the ionic strength is not significantly affected by the surfactant; the headgroup repulsion is relatively small and independent of the surfactant chain length. An increase in chain length manifests itself exclusively in the increase of the hydrophobic attraction and much larger micelles are formed and their size more strongly increases with n. These trends are in reasonable agreement with experimental data [157,168–171]. Volume fraction profiles, i.e., x as a function of z, the distance to the center of the micelles, for a spherical A12 B3 micelle in the absence of added electrolyte, are plotted in Fig. 20. In panel a the results are plotted on a linear scale and in panel b on a logarithmic scale. The profiles show that the aliphatic segments form the core of the micelle and that the headgroup segments are located on the outside of the aggregate. The interface between the hydrophobic segments A and the aqueous

FIG. 20 Volume fraction profiles of W, A, B, and C for an A12 B3 spherical micelle in the absence of added electrolyte. In panel a a cross section through a micelle is plotted on a linear scale; in panel b the same on a logarithmic scale (starting from the center of the micelle). (Redrawn with permission from Ref. 63. Copyright 1996 American Chemical Society.)

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phase W is rather sharp: it drops from about 0.9 to less than 0.1 over only 3 lattice layers (1 nm). This distance is in agreement with MD simulations [92,172–174]. The charged headgroups are located on the outside of the aggregate. The headgroup distribution is less sharp than the ‘‘A=W interface’’; it extends over about five lattice layers. Calculations performed for other headgroup sizes ðB; B2 ; B4 Þ gave almost identical results. The width of both the A=W interface and the headgroup distribution is hardly affected by the salt concentration. The volume fraction profile of the counterions is barely visible on the linear scale, but a clear picture results from the logarithmic plot. The counterions, C, accumulate mainly in the same layers as the charged headgroups. This implies an effective partial neutralization of the micelle. To a good approximation micelles behave as spheres with a constant net charge density. Upon a tenfold increase of the salt concentration, the electric potential in the headgroup region is decreased in absolute value by about 59 mV to maintain this charge density. This result corresponds with experimentally obtained surface potentials as a function of the salt concentration [175]. Outside the micelle, at about z ¼ 12 a diffuse ionic layer has developed. In the diffuse layer the counterions are positively adsorbed and the surfactant anions are depleted. The difference between B and C at large distance is due to the fact that each B segment has a charge of 1/3 and each ion C a unit charge.

C. Chain Branching and Micellization Besides the chain length, branching of surfactants is also known to affect the shape of the micelles, the CMC, and the aggregation number. Some results have been obtained for an isomeric series of A12 B3 surfactants for which the segment of attachment of aliphatic chain to the headgroup is varied [63]. The results are presented in Figs. 21 and 22. For all surfactants the tail is connected to the third of the three headgroup segments. A linear chain is obtained when the first segment of the aliphatic chain, with segment ranking number SB =1, is connected to the headgroup. The longest branch length is attained when the segment with SB = 6 is attached to the headgroup. The results in Fig. 21 show that the CMC increases and that nagg decreases with increasing branch length. For the most strongly branched molecule, with SB ¼ 6, the CMC is about twice as high as for linear A12 B3 and nagg decreases from 37 to 26. The reason for this behavior is that for molecules with a thick but short hydrophobic part the packing into a spherical micelle is more difficult than for long and thin molecules. The calculated shifts are in agreement with experimental findings for isomers of sodium alkylbenzene sulfonates [176], but for sodium alkyl sulfates the effect of branching is more pronounced [177]. Calculations where the surfactant

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FIG. 21 Effect of branching on the cm (panel a) and aggregation numbers at i = 0.01 (panel b) of molecules with 12 A segments, no salt added. The segment number to which the headgroup is attached, SB, is given on the horizontal axis. The calculated values are indicated with markers; the drawn lines are fits through the calculated points. (Redrawn with permission from Ref. 35. Copyright 1996 American Chemical Society.)

was modeled as A12 B with a charge of
1 on the B headgroup showed a more pronounced effect of branching. The effect of branching and salt concentration on the shape of the micelles is shown in Fig. 22, where the equilibrium volume fractions for spherical micelles (near the CMC, and at i ¼ 0:01), cylindrical micelles, and flat membranes are compared for A12 B3 with SB ¼ 1 and SB ¼ 6. For the linear chains and low salt concentrations, spherical micelles are favored over cylindrical aggregates and membranes. At higher salt concentrations, the equilibrium bulk volume fractions for spheres (i = 0.01), cylinders, and membranes are rather close to each other. Spherical micelles are still preferred at low concentrations but at higher overall concentrations the system may be very polydisperse because the equilibrium volume fractions for spheres and cylinders are hardly different.

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FIG. 22 Comparison between aggregates of different shapes. Effect of the salt concentration on the equilibrium volume fraction of linear A12B3 (panel a) and branched A12B3 with SB = 6 (panel b). Equilibrium volume fractions at the CMC and i = 0.01 for spheres are shown together with those for cylinders and membranes. (Redrawn with permission from Ref. 63. Copyright 1996 American Chemical Society.)

For the branched isomer (A12 B3 , SB = 6) spherical aggregates are preferred at low salt and surfactant concentration, i.e., the micelles first formed are spherical. If more surfactant is added, globular micelles remain preferred at low salt concentration. However, at higher salt concentration, about 0.2 M in this case, membranes and cylinders are preferred over spherical micelles: a transition from a spherical to a cylindrical shape occurs as function of salt or surfactant concentration. The calculated results are in agreement with data for alkylbenzene sulfonate micelles [178] that reveal that in the absence of salt the branched isomers have lower aggregation numbers than linear molecules. At high salt concentrations the aggregation numbers of the branched surfactants increase to very high values, while the aggregation number of the linear alkylbenzene sulfonate shows a limited increase.

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169

SCFA AND ADSORPTION OF IONIC SURFACTANTS

Adsorption calculations have been made for constant-charge [64], constantpotential [65,66,142,143], and charge-regulating surfaces [68]. All bulk parameters are the same as in the micellization study. For the constant-charge and constant-potential case, the noncoulomb interactions with the surface are expressed by BS ¼
10 and xS ¼ 0 for all other segments. For the A and B segments this leads to a displacement free energy per segment of
0.5 kT and
2.5 kT, respectively. For the other segments the ‘‘adsorption energy’’ is zero. For the regulating surface (representing silica) the surface interaction parameters will be specified later.

A. Adsorption on Constant-Charge Surfaces 1. Adsorption Isotherms and Structure of the Layer Calculated adsorption isotherms of A12 B3 on a surface with a fixed charge of 0.1 charges per lattice site (0.17 C/m2) and at three salt concentrations are shown in Fig. 23 [64]. To show the low coverage part more clearly, the isotherms are also plotted on log–log scales. The excess number of surfactant molecules per surface site is plotted as a function of the overall volume fraction, , of surfactant. The point where the CMC is reached is indicated with an asterisk. At low and intermediate salt concentrations the adsorption isotherms show two distinct steps, but at a salt concentration of 0.1 M the stepwise nature of the isotherm has almost disappeared. At low surfactant and low salt concentrations the initial adsorption is higher than at a high salt concentration. At high surfactant concentrations the situation is reversed, i.e., the adsorption is larger at high than at a low salt concentration. Due to this behavior the isotherms at different salt concentrations intersect. The intersection point of the three isotherms marks the point where the surface charge is compensated by the adsorbed surfactant (iep). This behavior was already predicted on qualitative grounds by Eq. (7). The strong increase in adsorption, just before the first plateau is reached, is due to hydrophobic attraction. This interaction is already substantial at rather low adsorption values. The second step in the isotherm occurs at about 0.1 of the CMC; this rise in the adsorption is also due to hydrophobic attraction, but this time a ‘‘bilayer’’ is formed. Experimentally this type of two-step isotherms has been found for surfaces such as mica [179], polystyrene [52], biotite [180], and spheron [52]. In cases where the adsorption was studied as a function of the salt concentration, the two-step shape disappeared at high salt concentrations [52]. Also for surfactant adsorption on silica a two-step isotherm has been found [181],

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FIG. 23 Theoretical adsorption isotherms of A12 B3 on a surface with a fixed charge density of 0.1 charges per lattice site at three salt concentrations. The CMCs are indicated with an asterisk. Isotherms are plotted in two ways: log–log (a) and lin–log (b). (Redrawn with permission from Refs. 64 and 144. Copyright 1996 American Chemical Society.)

but silica is not a constant-charge surface and the conclusions drawn in [64] with respect to silica should be regarded as premature. In the section on adsorption on charge-regulating surfaces, the silica behavior will be discussed in more detail. The structure of the adsorbed A12 B3 layer on the constant-charge surface followed from the calculated volume fraction profiles [64]. At submonolayer coverage the surface charge is mainly compensated by the salt ions. After the first step, in the plateau region, the surface charge is almost exclusively compensated by surfactant ions. The A segments are accumulated at the solution side of the adsorbed layer, so that the particles have become hydrophobic. At coverages above nexc = 0.1, a (partial) surfactant bilayer is present on the surface. Due to the presence of headgroups at both sides of the adsorbed layer the system becomes hydrophilic again. In general, the bilayer is asymmetric: the headgroups near the surface are distributed over three layers, whereas at the solution side the headgroup distribution is much wider

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and it closely resembles the headgroup distribution in a micelle. This similarity is also reflected in the electrostatic properties of the bilayer: a low and approximately constant net charge density occurs in the outer headgroup region. The potential in this region strongly depends on the salt concentration. For a detailed discussion on the charge and potential distribution, consult Ref. [64]. Besides the electrostatic, the specific or contact interactions between B segments and A segments with the surface determine the shape of the adsorption isotherm. Some typical examples to illustrate this for a charge density of 0.1 charges per site and a salt concentration of 10
3 M can be found in Ref. [64].

2. Remarks on the SCFA Model Finally, a remark should be made with respect to the failure of the 1D SCFA theory to model surface aggregates. In the case of constant-charge surfaces where the charges are most probably distributed fairly regular over the surface (due to the electrostatic repulsion), the errors made with the 1D SCFA theory might be small, especially at high-surface-charge densities and strong noncoulomb interactions between the headgroup and the surface. In this case the charges at the surface act as initial nuclei for the surfactant adsorption, and this would result in a smeared-out (sub) monolayer of surfactants. Once the nuclei start to grow substantially, the aggregates formed may interact and form a bilayer. For low-charge densities and weak noncoulomb interactions with the surface, independent aggregates may be formed and the calculations should be considered with some reservation.

B. Adsorption on Constant Potential (Variable-Charge) Surfaces 1. Adsorption Isotherms Many metal oxide surfaces can be regarded, to a first-order approximation, as constant-potential surfaces, provided the pH in solution is fixed [97,183]. Theoretical investigations of these systems are therefore quite relevant. Adsorption isotherms of A12 B3 on a surface with a potential of 100 mV have been calculated [65,66,142,143]. Except for the surface charge and potential, all parameters are the same as for the constant-charge case. Results are shown in Fig. 24, where the excess number of surfactant molecules per surface site is plotted versus the overall volume fraction  of surfactant. Three common ways of presentation are used. The log–log isotherms, presented in panel a, strongly emphasize the lower part of the isotherms. They show four regions, in qualitative agreement with

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FIG. 24 Theoretical adsorption isotherms of A12 B3 on a surface with a fixed potential of 100 mV at three CD concentrations. The isotherms are plotted in two ways: log–log (a) and lin–log (b). (Redrawn with permission from Refs. 65 and 144. Copyright 1996 American Chemical Society.)

experimental results [28,65,66,142,143]. In region I the slope is unity (Henry region) and the adsorption decreases with increasing salt concentration. Region II shows a phase transition indicating that two coexisting phases are present at the surface at a given volume fraction. The condensation step occurs at all salt concentrations. Region II starts at a coverage of a few percent of the adsorbed amount at the CMC and ends at about nexc = 0.2. Region III has a slope much smaller than region II and the adsorption increases with increasing salt concentration. Above the CMC, in region IV, the adsorption reaches a plateau value. The lin–log plot of the isotherms (panel b) give equal weight to all adsorption values and show the effects in the upper part of the isotherm more clearly than the log–log plots. In the lin–log plot the four regions of

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the isotherm can also be distinguished. Region II ends when the surface charge can no longer adapt to the surfactant adsorption. The lin–lin plots are not shown, because they are rather featureless and the information at low-surfactant concentrations is lost. Comparison of the adsorption isotherms on a constant-potential surface (panel b) with those on a constant-charge surface (Fig. 23) shows that the behavior is rather different: in the constant-potential case at low-surfactant concentration a strong phase transition occurs, the first plateau is absent, and the adsorption increases more gradually to its value at the CMC in the second part of the isotherm. Moreover, the surface charge increases upon surfactant adsorption from below 0.05 to about 0.2 charge units per site, depending on the salt concentration. The fact that the surface responds to the surfactant adsorption explains why the first plateau is missing: the coulomb interaction with the surface remains attractive much longer due to the fact that new charges are formed at the surface. The common intersection point of the three isotherms is located at nexc ¼ 0:1 (see Fig. 24), and it corresponds with the iep. After the iep is reached, the surface still increases its charge with increasing surfactant adsorption in order to screen the charges of the head-on adsorbed surfactant molecules. However, a part of the surfactant molecules adsorbs in the bilayer, and this adsorption gives the particles a net (negative) charge.

2. Structure of the Adsorbed Layer The volume fraction profiles in each of the four regions of the isotherm are presented in Fig. 25 for a salt concentration of 0.01 M [65]. In region I, panel a, the molecules adsorb fairly flat on the surface, the tails are squeezed out of the water, and the headgroups adsorb strongly due to specific and coulomb attraction. The surface charge is screened by both surfactant and salt ions. In region II, at the end of condensation step (nexc ¼ 0:2), panel b, the aliphatic segments on the surface are displaced by B segments; head-on adsorbed surfactant is present on the surface and the surface is hydrophobized by the A segments at the solution side of the adsorbed layer. The volume fraction of B segments at the solution side is still very low. In region III (nexc ¼ 0:3, panel c) the second layer grows significantly and the solution side of the layer becomes gradually more hydrophilic. This process stops when the CMC is reached, region IV. For adsorption values above the cip (iep) the hydrocarbon core of the bilayer separates two net uncharged parts. In regions III and IV the headgroup charges at the solution side are partly compensated by accumulation of counterions in between the headgroups, partly by the diffuse double layer. Again the outside of the bilayer is very similar to the outside of micelles. Some accumulation of counterions occurs at the surface side of the bilayer.

174

Koopal

FIG. 25 Adsorbed layer structure for A12 B3 on a constant-potential surface at three different coverages corresponding with regions I, II, and III of the isotherm. Volume fraction profiles of A, B, C, and D are shown (lin–lin and log–lin plots). The volume fractions of C and D are best shown in the log–lin plots. (Redrawn with permission from Refs. 65 and 144. Copyright 1996 American Chemical Society.)

In general, the calculations show that head-out adsorption starts after the cip has been reached [142,143]. Before the cip head-on adsorption is exclusive, after the cip both head-on and head-out adsorption occur, but the head-out adsorption increases more strongly. Increasing the salt

Modeling Association/Adsorption of Surfactants

175

concentration favors head-out adsorption due to screening of the lateral repulsion. Above the cip head-on adsorption is discouraged by salt addition because of the enhanced screening of the coulomb headgroup surface attraction. Below the cip the head-on adsorption is hardly affected in the case that BS ¼
10.

3. Chain Branching and Adsorption The effects of branching on the adsorption isotherms of three A12 B3 isomers are presented in Fig. 26 [66]. In the log–log plot, panel a, differences between the adsorption isotherms show up mainly in region I. Through the choice of AW the tail segments have some affinity for the surface. The isomers with longer side chains will have less initial adsorption, because the entropy loss for a branched molecule to lie flat on the surface will be larger than for a linear one. As expected, region II of the calculated adsorption isotherms shows a phase transition. The trends in the upper part of the isotherms are more clearly shown in the lin–log plot; see panel b. The differences in the adsorbed amounts show that packing of the molecules becomes more difficult when the chains are strongly branched.

FIG. 26 Effect of branching on the adsorption of A12B3 molecules on a surface with potential of 50 mV. The results are given on a log–log (a) and a log–lin (b) scale. The CMC values are indicated with an asterisk. (Reprinted with permission from Ref. 66. Copyright 1996 American Chemical Society.)

176

Koopal

4. Anionic Surfactant Adsorption on Metal Oxides Adsorption isotherms of anionic surfactants on positively charged metal oxide surfaces have been investigated in numerous studies [3,28,65,66, 103,124,129,133,135,142,143,145]. The general shape of the isotherm is well established. A typical result is shown in Fig. 27, where the adsorption of sodium nonylbenzene sulfonate on rutile is plotted in three different ways [65]. The experimental isotherms are in qualitative agreement with the theoretical results. In both cases a four-region isotherm is found; along the isotherm an inversion of the salt effect is observed and a common intersection point is present. The predicted trend that the adsorption increases slightly with increasing salt concentration in region IV also corresponds with experimental findings [65,132,133]. A qualitative agreement between 1D SCFA

FIG. 27 Four-region isotherm. Adsorption of sodium nonylbenzene sulfonate (SNBS) on rutile at pH ¼ 4.1 at three NaCl concentrations. The isotherms are plotted in two ways: log–log (a) and lin–log (b). The arrows indicate the CMC values. (Redrawn with permission from Refs. 65 and 144. Copyright 1996 American Chemical Society.)

Modeling Association/Adsorption of Surfactants

177

calculations and experimental results also exists with respect to the effects of chain length and branching of the surfactant on adsorption [66]. Moreover, the calculated shifts of the isotherms for different surface potentials compare well with experimentally measured shifts of isotherms obtained at different pH values [65,142]. Qualitatively the calculated volume fraction profiles relate closely to experimentally observed maximums in the hydrophobicity [28] and flotation recovery [183] of particles as a function of surfactant concentration: at low adsorbed amounts the particles are hydrophobized, whereas at high adsorbed amounts the particles are hydrophilic again. Around the iep the colloidal stability will be at its minimum, not only because the particles carry no net charge, but also because they are hydrophobized.

5. Surface Charge and Surfactant Adsorption An important feature of constant-potential surfaces is that the surface charge is adapted when specifically adsorbing ions, such as surfactant ions, are present. Due to the noncoulomb attraction, surfactant ions screen the surface charge better than the indifferent ions present in the diffuse layer. The surface charge adjustment upon surfactant adsorption has been investigated in detail [65,142,143]. Results of the theoretical calculations are shown in Fig. 28. The surface charge, expressed in number of charges per lattice site, is plotted versus the surfactant concentration for two salt concentrations and one surface potential. The initial surface charge for all conditions is very low ( _ c

Viscoelastic Surfactant Solutions

473

FIG. 32 The shear viscosity against the shear rate _ for mixtures of C14DMAO:SDS ¼ 6:4 with various total concentrations in a capillary viscometer at 258C. (From Ref. 15.)

the solutions now show some shear thickening behavior. Obviously something dramatic has happened to the micelles in the solutions. Some conclusions about what has happened can be drawn from flow birefringence measurements. Some typical results of flow measurements from a Couette system are shown in Fig. 33 [15]. We note a sudden increase of the flow birefringence at a critical shear rate. For _ > _ c , no flow birefringence could be detected. In Fig. 34 viscosity and flow birefringence measurements for other surfactant systems with shear-induced structures are shown [37]. In these experiments a constant shear rate above the threshold value was abruptly applied in the solution at rest; this shear rate was kept constant for a certain time and then suddenly dropped to zero. Under these conditions the viscosity and the flow birefringence increase with characteristic time constants to their corresponding plateau values at which the viscosity passes over a maximum; both quantities relax with characteristic relaxation times to their original value when the shear rate dropped to zero. In flow experiments, besides the birefringence it is also easy to measure the angle of extinction, which is the angle between the direction of flow and the mean orientation of the rods. In normal flow orientation this angle

474

Hoffmann

FIG. 33 The flow birefringence
n against the shear rate _ for the same solutions as in Fig. 32. (From Ref. 15.)

decreases smoothly from 458 to zero with increasing flow rate because the rods become more and more aligned. In the drag-reducing solution the situation is very different. In the Newtonian region for _ < _ c , the solution remains isotropic and no preferential alignment can be detected. When _ > _ c , however, the angle of extinction is close to zero. This experimental

FIG. 34 The shear viscosity at a shear rate _ of 50 s
1 for a solution of 0.9-mM CPySal at 208C (a) and the flow birefringence
n at a shear rate of 10 s
1 for a solution of 20-mM C10F21CO2N(CH3)4 at 308C (b) as a function of the time t (_ jumped from 0 s
1 to the corresponding value and dropped after a certain time again to zero). (From Ref. 37.)

Viscoelastic Surfactant Solutions

475

fact tells us that the new structures that are produced by the shear are completely aligned. Obviously the new-found structures must be much larger than the original small rods that were not aligned. Figure 35 shows this situation for a selected surfactant system together with the viscosity, the first normal stress difference, and the flow birefringence as functions of the shear rate [16]. It is now believed that the small, rodlike micelles undergo collisions in the shear flow, and their interfacial properties are such that they stick together for some time. In this way long necklace-type structures are formed under shear which at the same time become aligned in the shear flow. This situation is schematically sketched in Fig. 36 [16]. These neck-

FIG. 35 The zero shear viscosity and the complex viscosity j *| (a), the first normal stress difference p11–p22 (b), the flow birefringence
n (c), and the extinction angle  (d), as functions of the shear rate _ or the oscillation frequency !, respectively, for a solution of 5-mM C14TMASal at 208C. (From Ref. 16.)

476

Hoffmann

FIG. 36 Model to explain shear-induced micellar structures: The small rodlike micelles can form long necklace-type structures under shear. (From Ref. 16.)

lace-type structures act like high-molecular-weight polymers and give rise to drag reduction. One may ask why the small rods stick together under shear while they do not in the solution at rest. It is conceivable that the reason lies in a sheardependent equilibrium constant K for the aggregation equilibrium A1 þ A1 Ð A2 ½A2  K¼ ½A1   ½A1  where A1 are the small rods. The situation could be comparable to the micelle formation from monomers. If the situation is such that K  A1 < 1, where the A1 are now the monomers, there are practically no micelles in the solution. When, however, K is increased somewhat so that K  A1 > 1, then micelles will form because the micellar concentration is proportional to A1 ðK  A1 Þn , where n is an aggregation number. If the necklace-type aggregates become large enough, they will then be completely aligned because the product of rotation time and shear rate became larger than 1 ( _ > 1). The

Viscoelastic Surfactant Solutions

477

stability of the necklace-type aggregates could lie in the chemical composition of the micellar interfaces. All the known systems that give rise to shearinduced structures (SIS) formation form small rods on which the micellar surfaces are covered with methyl, ethyl, or other hydrophobic groups. The surfaces are thus hydrophobic even when they are charged. It is likely that if such two surfaces collide and make contact, they should form a sticky contact even though the total interaction energy between the particles is repulsive. Support for this model has recently been found. It was shown that if the surface of the small rods was modified by hydrophilic groups that protrude from the micellar interphase and a contact of the two surfaces was no longer possible, the shear-induced structures were no longer observed [113]. These results explain nicely why micellar systems where the micelles have hydrophilic surfaces do not form SIS. In a shear rate range just above the critical shear rate _c , the flow birefringence increases with _ before it becomes constant. This is an indication that not all the surfactants above the CMC are involved in the SIS and that there are still small rods in equilibrium with the SIS. The scattering anisotropy of SANS measurements under shear points out the same. Phenomenologically the situation is similar as for first-order phase transitions when a thermodynamic equilibrium parameter is varied. The systems changes then from a single-phase to a two-phase and finally again to a onephase situation. In this respect, it was shown recently that the systems not only behaves in this way but the systems in the two-phase region separates actually macroscopically in flow in a Couette system. The steady state in the two-phase region is characterized by two coexisting states separated by a cylindrical interface. Near the inner cylinder viscous SIS were observed, while near the outer cylinder, a much less viscous fluid similar to the original micellar solution was observed [114]. It furthermore was shown in this work that for much higher shear stresses the SIS can break again and the system switches back to the original state. Furthermore, it was shown that the system does not only form two macroscopic separate phases but several rings and periodic structures that change with time. This can lead to periodic oscillations in the birefringence [115]. In many systems the lifetime of the SIS is so short that it disappears as soon as the shear is stopped. For longer chain length surfactants, the SIS can still be present after the shear has been turned off. For such a situation the SIS have recently been made visible by FF TEM [116–118].

C. SANS Measurements on Shear-Induced Structures The growth process of the large micellar structures (which are strongly aligned) has been studied in more detail by transient SANS experiments

478

Hoffmann

where the shear rate for the samples was raised stepwise from zero to a certain finite value. These experiments showed that the large micelles grow according to the Avrami law cðtÞ ¼ cinf  ½1
expð
kt Þ

ð20Þ

that originally was used to describe nucleation and growth in metals and alloys. For the system C14TMA-Sal the exponent was found to be 2–2.5 [119]. The exponent should be i þ 1 for i-dimensional growth, which means that the growth process observed here is close to unidimensional as should be expected since the micelles just grow in length without otherwise changing their dimensions. Such SIS are an interesting phenomena in particular for self-aggregating systems like micelles, where the equilibrium structure often depends very subtly on small energetic changes. These structural changes have a profound influence on the properties of these systems, especially on their flow behavior. The scattering patterns of the SANS experiments exhibit a strong anisotropy. This shear-induced effect will already occur at _  rot  1 ( rot ¼ rotational time constant of the small type of micelles), i.e., in a range where the shear field should not be able to orient the small rodlike aggregates significantly. This means that the observed anisotropy is not due to the orientation of these originally present micelles but that instead larger oriented micellar aggregates have to be present in the solutions. So far the mechanism for formation of the shear-induced state is not fully understood, and several different mechanisms have been postulated [120–122]. One particular system—a mixture of tetradecyldimethylaminoxide (C14DMAO) and sodiumdodecylsulfate (SDS)—has been studied in much detail. This system shows a pronounced SIS formation around a molar mixing ratio of 7:3 for C14DMAO/SDS [15]—and it might also be noted here that for this composition the nematic phase that is found for that systems extends to the lowest surfactant concentration [123]. In order to find relations between the macroscopic behavior of the system and the structure of the micellar aggregates present, a SANS study was performed [124]. Here by changing the contrast condition in the micellar aggregate by using both hydrogenated and deuterated SDS, detailed information regarding the structure of the micelles could be obtained. The SANS experiments show that at the mixing ratios where SIS is observed, elongated micelles are present that are best described by a three-axes prolate ellipsoid, and where the length of these aggregates reduces with increasing SDS content. From a contrast variation experiment using both deuterated and hydrogenated SDS, it could be concluded that the buildup of these micelles is homogeneous and no internal segregation of the surfactant molecules within the aggregate could be deduced [124]. Such an internal segregation by having an

Viscoelastic Surfactant Solutions

479

enrichment of SDS at the endcaps (here the relative area per molecule is necessarily larger because of the larger curvature and one could imagine that the SDS with its larger hydrophilic headgroup would preferentially be located at this position) would have been conceivable and, of course, such a buildup that would contain two more highly charged ends would have been an important factor to consider for the explanation of the SIS. However, this evidently is not the case, and SIS formation has to be explained starting from originally short, homogeneously charged, elongated aggregates.

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483

H. Yuntao and E.F. Matthys. Rheol. Acta 35(5):470, 1996, L. Chu-heng and D.J. Pine. Phys. Rev. Lett. 77(10):2121, 1996. I. Wunderlich, H. Hoffmann, and H. Rehage. Rheol. Acta 26:532, 1987. S. Hofmann and H. Hoffmann. J. Phys. Chem. B 102(29):5614, 1998. Y.T. Hu, P. Boltenhagen, and D.J. Pine. J. Rheol. 42(5):1185, 1998. E.K. Wheeler, P. Fischer, and G.G. Fuller. J. Non-Newtonian Fluid Mech. 75(2,3):193, 1998. S.L. Keller, P. Boltenhagen, D.J. Pine, and J.A. Zasadzinski. Phys. Rev. Lett. 80(12):2725, 1998. R. Oda, P. Panizza, M. Schmutz, and F. Lequeux. Langmuir 13(24):6407, 1997. R. Oda, P. Panizza, and F. Lequeux. Prog. Colloid Polym. Sci. 105 (Trends in Colloid and Interface Science XI), 276, 1997. C. Mu¨nch, H. Hoffmann, K. Ibel, J. Kalus, G. Neubauer, U. Schmelzer, and J. Selbach. J. Phys. Chem. 97:4514, 1993. M.E. Cates and M.S. Turner, Europhys. Lett. 11:681, 1990. S.Q. Wang. J. Phys. Chem. 94:8381, 1990. R. Bruinsma, W.M. Gelbart, and A. Ben-Shaul. J. Chem. Phys. 96:7710, 1992. H. Hoffmann, S. Hofmann, and J.C. Illner. Prog. Colloid Polym. Sci. 97:103, 1994. H. Pilsl, H. Hoffmann, S. Hofmann, J. Kalus, A.W. Kencono, P. Lindner, and W. Ulbricht. J. Phys. Chem. 97:2745, 1993.

11 Microstructures of Nonionic Surfactant–Water Systems From Dilute Micellar Solution to Liquid Crystal Phase TADASHI KATO

I.

Tokyo Metropolitan University, Tokyo, Japan

INTRODUCTION

Surfactant systems exhibit a variety of phase behaviors. When one of these phases is transformed into another phase, not only the arrangement but also the shape of building block itself is changed. This makes it difficult but fascinating to clarify the mechanism of the phase transition in surfactant systems. Variation in the shape of building block occurs even far from the transition point. So the equilibrium structure near the phase boundary should be elucidated to clarify the mechanism of the phase transition. It is, however, very difficult to determine the structure uniquely at one point in the temperature–concentration diagram due to the complex structure. Thus variation in structures with concentration and temperature should be investigated systematically. Moreover, a variety of techniques is necessary to observe the structures from different length scales. From these points of view, the author studies polyoxyethylene surfactant–water systems in a wide concentration and temperature range by using scattering techniques and pulsed-gradient spin echo (PGSE). Polyoxyethylene surfactants are used because (1) the hydrophilic-lipophilic balance of molecules can be systematically changed, (2) theoretical treatment is much simpler than in ionic systems in the sense that it is unnecessary to take into account electrostatic effects, (3) various types of phase transitions can be observed merely by changing temperature, and (4) the transverse relaxation time of 1H is much longer than that of ionic surfactants due to the fast motion of the oxyethylene groups; this is very important for the PGSE measurements. 485

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Kato

In Section II, we discuss the problems encountered in the determination of micelle size by using conventional techniques. A novel method is then described that gives information on intermicellar interactions and micelle size distribution separately. Finally, thermodynamic models for micellar growth are discussed based on the concentration and temperature dependence of micelle aggregation number obtained from light scattering measurements in extremely dilute solutions where effects of intermicellar interactions can be neglected. Section III covers the structure and dynamics of semidilute and concentrated solutions. After the analogy with polymer solutions is described, dynamical aspects of entangled wormlike micelles are discussed based on the surfactant self-diffusion coefficient and the structural relaxation time obtained by using PGSE and dynamic light scattering, respectively. In the last part of this section, the gradual transformation from entangled wormlike micelles to a three-dimensional network with a stable connection with increasing concentration and temperature is demonstrated. Variation in the structure occurs also in the lamellar phase, as is shown in Section IV. We discuss correlations among the structures of the concentrated micellar phase, the lamellar phase, and the bicontinuous cubic phase as well as relations with phase behaviors. Finally, our recent studies on effects of shear flow and adding fatty acid on the structure of the lamellar phase are briefly described.

II. DILUTE SOLUTIONS A. Micellar Growth with Increasing Temperature The most fundamental properties of micellar solutions may be the critical micelle concentration (CMC) and the mean aggregation number of micelles. For determining the latter, static light scattering is one of the most useful methods because it does not need any probe molecules. However, the light scattering intensity depends on not only the aggregation number but also the intermicellar interactions. For globular micelles, the effects of intermicellar interactions can be eliminated by extrapolating data to the CMC because the aggregation number depends on the concentration only slightly. In the case of elongated micelles, however, the aggregation number usually depends on the concentration and so it is difficult to separate these two contributions (see below). In nonionic surfactant systems, light scattering intensities increase with increasing temperature above a certain temperature. From the reason described above, this does not always indicate the increase in the aggregation number with temperature, because they usually have the lower critical

Microstructures of Nonionic Surfactant–Water Systems

487

solution temperature, that is, attractive interactions between micelles (or the critical effects) may affect the light scattering intensity. Corti and Degiorgio [5] pointed out this problem for the first time. After their work, various studies have been performed in order to elucidate the influence of critical effects [1,3]. Among them are important the studies using techniques that are less affected by the critical effects such as NMR self-diffusion [6–8], fluorescence decay [9], dynamic neutron scattering [3,10], light scattering from semidilute solutions [11,12], and so on. From the results of these studies (summarized by Lindman and Wennerstro¨m [13]), it can be concluded that micellar growth actually occurs, but the extent of micelle size variation depends strongly on the specific surfactants in question. This conclusion has been confirmed by cryo-TEM (for C16E6) [14] and small-angle neutron scattering (SANS) using direct analysis by Fourier transformation (for C12E5) [15] and generalized indirect Fourier transformation (for C8E3, C8E4, C8E5, C10E4, C12E5, and C12E6) [16].

B. Intermicellar Interactions 1. Virial Coefficients Obtained from Conventional Methods Usually, thermodynamic arguments for interparticle interactions are developed on using the virial coefficients obtained from the osmotic pressure, , or the light scattering intensities. For micellar solutions, the absolute intensity of the scattered light (Rayleigh ratio) extrapolated to the zero scattering vector, R(0), can be expressed as follows [3,17]: Rð0Þ ¼ R0 ð0Þ þ HMðc
c0 ÞSð0Þ
 42 2 @ 2 H 4  NA @c T;P

ð1Þ

where M is the weight-averaged micellar weight, c is the weight concentration (g cm
3 ) of surfactant, c0 is the CMC in g cm
3 , R(0) and R0(0) are the Rayleigh ratios at zero scattering vector at c and c0, respectively,  is the wavelength in vacuum, is the refractive index of solution, NA is Avogadro’s number, and S(0) is the structure factor at zero scattering vector. In dilute solutions, S(0) can be related to the second virial coefficient, A2, through the equation Sð0Þ
1 ¼ 1 þ kðc
c0 Þ þ   

ð2Þ

where k 2A2 M. Substitution of Eq. (2) into Eq. (1) gives Hðc
c0 Þ 1 1 1 ¼ ½1 þ kðc
c0 Þ þ    ¼ þ 2A2 ðc
c0 Þ þ    ¼ Rð0Þ
R0 MSð0Þ M M ð3Þ

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In the case where the micellar weight is independent of concentration, the second virial coefficient can be obtained from the slope of the Debye plot, that is, H(c
c0)/[R(0)
R0(0)] versus c
c0. However, situations become complicated when the micellar weight depends on concentration. As a typical example, we consider the case where intermicellar interactions can be neglected and the concentration dependence of M is expressed as M ¼ M0 ½1 þ kM ðc
c0 Þ þ   

ð4Þ

From Eqs. (3) and (4), we obtain Hðc
c0 Þ 1 1 ¼ ¼ ½1 þ ðk
kM Þðc
c0 Þ þ    Rð0Þ
R0 MSð0Þ M0

ð5Þ

Therefore, the apparent second virial coefficient, k
kM, becomes smaller than the ‘‘true’’ one, k, and so it is no longer possible to discuss intermicellar interactions on the basis of light scattering data. Concentration dependence of the micellar weight also affects the collective (or mutual) diffusion coefficient (Dc) obtained from dynamic light scattering, which can be expressed as [18] kT Sð0Þ C 1 1 ¼ ½1 þ kC ðc
c0 Þ þ   

C 6 0 r0

DC ¼

ð6Þ ð7Þ

where 0 is the solvent viscosity, r0 is the hydrodynamic radius of micelles, and kc represent hydrodynamic interactions between micelles. Substituting Eq. (2) into Eq. (7), we obtain DC ¼

kT ½1 þ ðk þ kc Þðc
c0 Þ þ     6 0 r0

ð8Þ

Therefore, the concentration dependence of Dc is dominated by both the micelle size term, r0, and the interaction term, k + kc, when the micellar weight depends on concentration.

2. Information Obtained from the Ratio of Mutual Diffusion to Self-Diffusion Coefficient As expressed in the following equation, the surfactant self-diffusion coefficient (DS) also depends on intermicellar interactions although the effect of it is much smaller than that of Dc.

Microstructures of Nonionic Surfactant–Water Systems

kT

S 1 1 ¼ ½1 þ kS ðc
c0 Þ þ    &s 6 0 r0

DS ¼

489

ð9aÞ ð9bÞ

On the other hand, the ratio Dc/Ds can be expressed as DC 1 &S ¼ ¼ 1 þ k0 ðc
c0 Þ þ    DS Sð0Þ &C k0 k þ kC
kS

ð10aÞ ð10bÞ

As the relation k  ðkc
ks Þ usually holds [8], the coefficient k 0 is dominated by k. Then we can discuss intermicellar interactions on the basis of k 0 at least qualitatively; k 0 is positive/negative when repulsive/attractive interactions are dominant. It should be noted, however, that k 0 depends on concentration. When the aggregation number increases with concentration, the size distribution becomes broad, which also affects the ratio Dc /Ds . In this case, the following equation can be used instead of Eq. (10) (see below): DC =DS ¼ 1 þ k0 ðc
c0 Þ þ    lim DC =DS

ð11Þ

c!c0

Figure 1b shows temperature dependence of k 0 for C12E5, C12E6, and C12E8 systems obtained by using dynamic light scattering and PGSE [19]. One sees that below Tc
30K, k 0 depends on temperature only slightly, which is consistent with the fact that micelles remain small in this temperature range [see the apparent hydrodynamic radii RH in Fig. 1a obtained from DS through the relation Ds ¼ kT=ð6t RH Þ. Above Tc
30K, k 0 increases with temperature and takes a maximum at about Tc
15K. The increase in k 0 indicates that repulsive interactions becomes strong as the temperature increases. This appears to be an unexpected result because attention has been paid to only attractive interactions in nonionic systems. If micellar growth actually occurs, however, the increase of k is a natural consequence. It is well known that S(0)
1 or the second virial coefficient increases as the particle shape deviates more and more from a sphere under a constant volume fraction of particles [19–22]. If the particle is hemispherocylinder, for example, k can be expressed as " # ðL=Þ2 k ¼ 8 g 1 þ ð12Þ 8=3 þ 4ðL=Þ

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Kato

FIG. 1 Temperature dependence of the apparent hydrodynamic radii (a), intermicellar interaction parameter k 0 in Eq. (10) (b), size distribution parameter in Eq. (13), and line width of the main proton NMR signals for the methylene groups (d) for D2O solutions of C12E5 (*), C12E6 (~), and C12E8 (!). (From Ref. 8, 19, and 20.)

where g is the specific volume of micelles (cm3 g
1) and L and s are the length and diameter of the cylinder part, respectively. As particles are elongated, L/s increases, which results in an increase of k. The decrease in k originates from the rapid increase in attractive interactions (i.e., increase in critical concentration fluctuations) and/or the decrease in the excluded volume effects. We discuss these two factors in turn.

Microstructures of Nonionic Surfactant–Water Systems

491

(a) Decrease in the Excluded Volume Effects. The decrease in the excluded volume effects can be attributed to (1) change in micellar shape from rod to disk and/or (2) increase in flexibility of elongated micelles. Interpretation (1) is consistent with the prediction of Tiddy and co-workers based on systematic studies on phase behaviors of polyoxyethylene surfactants–water systems [23]. On the other hand, interpretation (2) is supported by NMR data of Nilsson and co-workers [6]. They have observed the line width
v1=2 of the main proton NMR signal for the methylene groups of C12E5 and found that the line width takes a maximum at 10– 158C. They have noted that the increase of
v1/2 arises from micellar growth, whereas the decrease of
v1/2 is due to fluctuations in aggregate size and shape. As these measurements have been performed at relatively high concentrations (about 50 g dm
3 ), the author and co-workers have measured the line widths of the same signals at 6 g dm
3 . The results are shown in Fig. 1d, where the line width of C12E6 at 15 g dm
3 is also presented. As can be seen from this figure, k and
v1/2 reach maxima at almost the same temperature, namely about Tc
15K. (b) Critical Effects. Figure 1c demonstrates also that k 0 becomes negative above Tc
ð3  5ÞK. This cannot be explained by the decrease in the excluded-volume effects alone. As these systems have the LCST, attractive interactions should increase with increasing temperature. Even if the excluded-volume effects continue to increase, rapid increase in attractive interactions may result in decreasing k 0 . Claesson and co-workers [24] have directly measured the force between two surfaces coated with C12E5 in water as a function of surface separation in the temperature range 15– 378C. They have shown that at 158C the interaction is repulsive at all separations. Above 208C, however, an attractive minimum appears and the attraction increases rapidly with temperature. Note that the temperature range 15–208C, where interactions are changed from repulsive to attractive, is close to the temperature that gives a maximum for k 0 , Tc
15K. This may support a rapid increase in attractive interactions above this temperature. However, a definite conclusion cannot be drawn until such direct measurements of surface force are performed for other homologues. Thus, at this stage, we cannot distinguish between these two effects. In either case, however, the most important result in Fig. 1c is that k 0 is positive below Tc
ð3  5ÞK. In other words, net interactions are not attractive until the temperature increases up to a few degrees below Tc, contrary to the ordinary recognition. This specific temperature where k changes from positive to negative corresponds to the y temperature in polymer solutions. Note that if the conventional method is applied to these systems, the y temperature is obtained about Tc
30K, as previously described.

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The above result that the  temperature is close to Tc is confirmed by the scaling analysis in semidilute region (see Section IV) and the following studies. Wilcoxon and Kaler [25,26] have made precise measurements of the angular distribution of scattered light from the C12E6 system and obtained the temperature dependences of the osmotic compressibility and the static and dynamic correlation lengths. They show that in the range Tc
4K to Tc , these quantities lie reasonably close to the predicted results for critical scattering. Similar conclusions have been obtained for the same system by Brown and co-workers [7,27] and Strey and Pakusch [28].

C. Micelle Size Distribution Figure 1b shows the limiting value for Dc /Ds as c ! 0. As the CMC is much lower than the concentration range studied, the limiting value corresponded to Dc /Ds , not for the infinitely dilute solution (only monomers exist) but for the solution where intermicellar interactions can be neglected. Thus, we obtain [8] P P 2 DC mn D < m >D m m nm D i = Pm m m ¼ lim ¼ P ð13Þ 2 c!0 DS < m >W m m nm = m mnm where nm is the number of micelles having aggregation number m and diffusion coefficient Dm. As Dm decreases with increasing m, the limiting value for Dc /Ds is always equal to or smaller than unity. If size distribution becomes broad, therefore, the limiting value decreases. See also the discussion by Brown and co-workers [7]. One sees from Fig. 1c that the limiting value is close to unity below about Tc
30K, indicating a narrow distribution of micellar size. This is consistent with the existence of globular micelles in this temperature range. Figure 1c demonstrates also that the limiting value decreases with increasing temperature above Tc
30K and then becomes almost constant (about 0.5 for C12E5 and 0.55 for C12E6). This suggests that micelle size distribution becomes broad. As shown in Ref. [8], the limiting value becomes 0.5 if stepwise selfassociation and rodlike micelles are assumed. The results for the C12E5 and C12E6 systems in the higher temperature range are consistent with this model.

D. Mean Aggregation Number in Extremely Dilute Solutions and Thermodynamic Models for Micellar Growth 1. Concentration and Temperature Dependence of Mean Aggregation Number in Extremely Dilute Solutions As emphasized before, light scattering intensity depends on not only the aggregation number, but also the intermicellar interactions, and so it is

Microstructures of Nonionic Surfactant–Water Systems

493

difficult to obtain the aggregation number of elongated micelles from light scattering measurements. However, there is a way to obtain the aggregation number as a function of concentration: perform measurements in the extremely dilute region, where the effects of intermicellar interactions can be neglected. Such a measurement is difficult because of weak scattering intensities. In fact, most of the measurements have been made at concentrations above about 10
3g cm
3 , which are much higher than the CMC for most elongated micelles. Table 1 summarizes available light scattering data on C12E6, a typical nonionic surfactant. The author reported light scattering intensities from this system (C12E6) at concentrations as low as 10
5 g cm
3 , which is two orders of magnitude lower than those of published data [33]. Figure 2 shows double logarithmic plots of light scattering intensity against concentration. The breakpoint in the plot at 258C is close to the CMC determined by the surface tension measurement. The figure demonstrates that the CMC decreases with the elevation of temperature as is expected from the published data [34]. To the author’s knowledge, however, light scattering data around the CMC have not yet been reported for the C12E6 system or for any surfactant solution with the CMC as low as that of C12E6. In Fig. 2, the data of Balmbra and co-workers [29] are also included. They report that the plot of the excess turbidity against concentration is linear below 2  10
2g cm
3 but intersects the concentration axis at about 5  10
4 cm
3, which is about 10 times higher than the CMC. From these results, they propose that there is secondary aggregation of spherical micelles. Figure 2 demonstrates that such a discussion based on the data in a range much higher than the CMC is meaningless. However, it can be

TABLE 1 Light Scattering Data on Dilute Solutions of C12E6 Concentration (10
3 g cm
3 Þ 2–20 1–15 3–20 2–10 3–10 4–10 12.5 0.01–10 0.01–2.5

Temperature (8C)

Ref.

15,25 35,45 5 10–45 25 5–45 24.8–49.9 15,25 35,45

29 29 30 7 31 32 5 33 33

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FIG. 2 Double logarithmic plots of the Rayleigh ratio against concentration for aqueous solutions of C12E6. Open and closed symbols are the data of the author (from Ref. 33) and Balmbra et al. (from Ref. 29), respectively. Solid lines indicate least-square fittings based on Eq. (16) (from Ref. 35). (From Ref. 33.)

seen from the figure that the slope of the plot begins to increase at a certain concentration cSP, which decreases with increasing temperature in the range 15–358C (cSP seems very close to the CMC at 458C). As the concentration increases further, the Rayleigh ratio becomes proportional to c3/2. Such results cannot be seen from the data of Balmbra and co-workers. It should be noted that their measurements are made down to a concentration that is the lowest for any published data (see Table 1). As S(0) in Eq. (1) is unity in the concentration and temperature range studied (see the discussion in Refs. [33] and [35], the light scattering intensity at the zero scattering vector can be expressed as Rð0Þ ¼ R0 ð0Þ þ HM1 ½c1 þ hmiw ðc
c1 Þ  R0 ð0Þ þ HM1 hmiw ðc
c0 Þ

ð14Þ

where R 0 (0) is the Rayleigh ratio for the fluctuations other than the concentration fluctuation, M1 is the molecular weight of surfactant monomers, c1 is the weight concentration of monomers, and hmiw is the weight-averaged aggregation number of micelles. In the second relation of Eq. (14), the monomer concentration is assumed to be equal to the CMC and so R0 ð0Þ ¼ R 0 ð0Þ þ HM1 c0 . Thus, we can determine hmiw at each concentration from R(0) by using Eq. (14). In Fig. 3, observed hmiw is plotted against the square root of the mole fraction of the surfactant molecules forming micelles. It can be seen from the

Microstructures of Nonionic Surfactant–Water Systems

495

FIG. 3 Concentration dependence of weight-averaged aggregation number of micelles obtained from the Rayleigh ratio for aqueous solutions of C12E6. X is the mole fraction of surfactant and Xcmc is the CMC expressed by the mole fraction. The solid lines represent theoretical values obtained from the least-square fitting for the light scattering intensity based on Eq. (16) shown in Fig. 2. (From Ref. 33.)

figure that hmiw at the CMC is small ( 110) and depends on temperature only slightly. According to the results around the critical composition (cC  10
2 g cm
3 ) described in the previous section, micelles are globular below about TC
30K (or TC
35K) and grow with increasing temperature above this specific temperature. Figure 3 demonstrates that this does not hold true in the extremely dilute region (note that the specific temperature just mentioned is 15–208C in the case of C12E6). As the concentration exceeds a certain concentration (cSP), hmiw becomes proportional to (X
XCMC)1/2  X1/2 / c1/2, as can be expected from the concentration dependence of R(0) in Fig. 2 and Eq. (14).

2. Thermodynamic Models for Micellar Growth The light scattering data for the C12E6 system at extremely low concentrations enable us to choose an appropriate model for micellar growth. The author calculated concentration dependence of light scattering intensity using various thermodynamic models for micellar growth and compared these results with the observed one. The most simple and frequently used model for rodlike micelles may be the ‘‘stepwise aggregation model’’ (or ‘‘ladder model’’). In this model (referred to as model A), it is assumed that a rodlike micelle of aggregation number m is constructed from two hemispherical endcaps (each cap is formed by s/2 monomers) and a cylindrical region. The thermodynamic

496

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states for the surfactant molecules in two endcaps and in the cylindrical part are the same as that in the spherical micelle of aggregation number s and the rodlike micelle of infinite aggregation number, respectively. Then the standard chemical potential of the surfactant molecules in rodlike micelles of aggregation number m, g0m , is expressed as [36] g0m ¼

s 0 m
s 0 s gs þ g1 ¼ g01 þ ðg0s
g01 Þ m m m

ð15Þ

where g0s and g01 are the standard chemical potentials for spherical micelles of aggregation number s and rodlike micelles of infinite aggregation number, respectively. According to this model, the mean aggregation number of micelles is proportional to the square root of concentration of surfactant molecules forming micelles except for the concentrations near the CMC [36,37]. As can be seen from Fig. 3, the data at 458C in the C12E6 system can be explained by this model. At 258C and 358C, however, the slope of the observed plot abruptly increases at a certain concentration, Csp, which decreases as the temperature increases from 258C to 358C (at 458C, Csp may be further decreased and become close to the CMC). This model cannot explain these features. Such a tendency can be seen more clearly in the double logarithmic plot of hmiw versus c shown in Fig. 4. The limitation of this model is that ‘‘immediate environment of a monomer in the more or less hemispherical ends of a big micelle is just the same as the one encountered by the monomer in the small globular micelle of minimum size’’ [38]. Note that the radius of the cylindrical part is not always equal to that of the hemispherical end, and so there may exist a junction zone between them [39]. Moreover, the micelle of minimum size is not always spherical [39–42]. Thus, the above assumption may be doubtful, as pointed out by Porte and co-workers [38]. The apparent success of this model reported for various systems comes from the fact that the concentration range of the observed data is much higher than the CMC. Second, calculations were made for the model where a rodlike micelle is assumed to be formed by the secondary aggregation of p small (probably spherical) micelles of aggregation number s [43–46], that is, only rodlike micelles of aggregation number ps coexist with the small micelle of aggregation number s. This model (referred to as model B) can explain the observed results at 258C and 358C in the sense that hmiw increases abruptly at a certain concentration. However, the calculated hmiw is not proportional to the square root of the concentration for any parameter values and converges to q in the higher concentration limit. These features are different from the observed concentration dependence at 458C. A model that can explain the observed data in the entire concentration and temperature range with the least fitting parameters was found to be that

Microstructures of Nonionic Surfactant–Water Systems

497

FIG. 4 Double logarithmic plots of the weight-averaged aggregation number of micelles against concentration for aqueous solutions of C12E6. The solid lines represent theoretical values obtained from the least-square fitting for the light scattering intensity based on Eq. (16) shown in Fig. 2. The arrows indicate the CMC at 15, 25, 35, and 458C from left to right.

where the chemical potential difference g0m
g01 in Eq. (15) changes above a certain aggregation number, q, that is, s g0m ¼ g01 þ ðg0s
g0q Þ for s  m  q ð16aÞ m i sh 0 q
s 0 ðgs
g0q Þ þ ðgq
g01 Þ for q  m ð16bÞ g0m ¼ g01 þ m m Least-square fitting based on this model was performed not for hmiw obtained from Eq. (14) but for the light scattering intensity (see the solid lines in Fig. 2) because we need not replace the monomer concentration by the CMC. The calculated hmiw and distribution of aggregation number for the best-fit parameters are shown in Figs. 3–5. It has been found from the fitting procedure that the q-value is relatively close to s, the aggregation number of the small (globular) micelle (ffi 110) formed at the CMC, as can be seen from the distribution of the aggregation number shown in Fig. 5. This suggests that micellar shape changes significantly when globular micelles begin to grow into rodlike micelles. Such early steps of the elongation process have been previously suggested by Porte and co-workers [47,48] using magnetic birefringence and viscosity data for concentrated solutions

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FIG. 5 Distribution of aggregation number for aqueous solutions of C12E6 calculated from the least-square fitting for the light scattering intensity based on Eq. (16) shown in Fig. 2. (From Ref. 35.)

of a cationic surfactant. However, the author’s study is the first one that analyzes the experimental data quantitatively without any problems concerning treatment of intermicellar interactions. The results are considered to become the basis of more detailed thermodynamic theory taking into account intermicellar interactions [49–53]. Eriksson and Lunggren [54,55] have developed a more detailed thermodynamic theory for rodlike micelles formed by ionic surfactants. They consider that the radius of the spherical micelle is significantly larger than the corresponding radius of the cylindrical micelle and so there should be a junction zone. It is also possible that the change of g0s
g01 described above comes from the existence of such a junction zone. Even if the system being studied is limited to the nonionic micelles, the mechanism of micellar growth has not yet been established. Most investigators use model A or model B for analyzing their experimental data obtained in a limited range of concentration or temperature. As shown in the previous section, the results in the higher temperature range (458C in the C12E6 system) can be explained by model A, whereas model B can explain the results in the lower temperature range (258C). Balmbra et al. [29] as described in Section III.E propose the secondary aggregation of spherical micelles on the basis of light scattering data on the C12E6 system in the concentration range much higher than CMC.

Microstructures of Nonionic Surfactant–Water Systems

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Funasaki et al. [46] have investigated the aggregation properties of C12E6 at 258C in the concentration range 1.7–50 mM (7:6  10
3
2:3  10
2 g cm
3 ) by gel filtration chromatography (GFC). Computer simulations of the GFC data reveal strong evidence of the coexistence of small micelles with large micelles. From this, they have considered that model B is more favorable. As described before, our data at 258C can be explained by model B. However, this model cannot explain the data at higher temperatures. Strey and Pakusch [28] have measured the relaxation time of temperature jump experiments on the same system (C12E6). They found that a plot of the reciprocal relaxation time against the square root of (c
CMC)/CMC becomes a straight line intersecting the abscissa at finite concentrations; the higher the temperature, the lower the concentration. The intersection has been regarded as the crossover concentration between globular and large micelles. Also, they consider the possibility of both models A and B. The author’s results are not inconsistent with theirs. In fact, the Csp-value at each temperature is not much different from the value of the intersection at the corresponding temperature. However, note that these measurements were performed above about 10
3 g cm
3 , which is still 20–50 times higher than the CMC.

III. SEMIDILUTE AND CONCENTRATED SOLUTIONS A. Analogy with Polymer Solutions Exhibiting Critical Demixing During the 10 years after 1985, applications of the scaling theory for entangled polymers to semidilute solutions of surfactants had been reported [11,12,56–63]. According to this theory, the correlation length of concentration fluctuations, , and the osmotic compressibility, @=@c, follow the power laws above the overlap concentration c at which polymer chains begin to entangle with each other [64,65]:  / c =ð1
3 Þ @ / c1=ð3
1Þ @c

ð17aÞ ð17bÞ

where c is the concentration of the polymer and is 0.5 and 0.588 for  solvent and good solvent, respectively. Candau and co-workers [56–59] have applied this theory to micellar solutions for the first time. They show that light scattering data on aqueous KBr solutions of cetyltrimethylammonium bromide (CTAB) can be explained by Eqs. (17a) and (17b).

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The osmotic compressibility is related to the scattering intensity at the zero scattering vector as @ RTHðc
c0 Þ RT ¼ ¼ @c Rð0Þ
R0 ð0Þ mapp MSð0Þ mapp

M1

ð18aÞ ð18bÞ

where mapp is the apparent aggregation number, which becomes equal to the weight-averaged aggregation number if intermicellar interactions can be neglected. Combining Eqs. (17b) and (18a), we obtain mapp / c1=ð1
3vÞ

ð19Þ

Figure 6 shows double logarithmic plots of mapp against c for D2O solutions of C12E5 [12]. This figure demonstrates that the data at 17.9, 25.0, and 27.08C above c can be fitted by a straight line that breaks at a certain concentration, c . From the slope of the log–log plots, -values can be obtained using Eq. (17). The -values obtained from the data below and above c are close to those for good solvent and  solvent, respectively. When the concentration exceeds about 150 g dm
3 , the log–log plots deviate

FIG. 6 Double logarithmic plots of the apparent aggregation number obtained from light scattering intensities against surfactant concentration for D2O solutions of C12E5. (From Ref. 12.)

Microstructures of Nonionic Surfactant–Water Systems

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from these straight lines. Similar results have been obtained for C16E7 [66], C14E6 [67], and C14E7 [67] systems. Daoud and Jannink [68] have deduced a temperature–concentration (T– C) diagram of a flexible polymer solution that exhibits phase separations. They have divided the one-phase region into four different subregions: dilute region (region I); dilute  region (region I 0 ); semidilute region (region II); and semidilute  region (region III). Experimental determination of these regions has been performed by Cotton and co-workers [69]. The above regions are characterized by the -value; 0.588 for regions I and II and 0.5 for regions I 0 and III. Therefore, c and c for the C12E5 system may correspond to the crossover concentration between regions I and II and that between regions II and III, respectively. In polymer systems, the  temperature exists between Tc and the temperature at which region II disappears. In the C12E5 system, TC is equal to 30.5  0.58C and region II disappears above 278C [12]. So it can be inferred that there exists the  temperature at 27–308C, which is in good agreement with that obtained from the ratio Dc =Ds in dilute solutions (see Section II.B), i.e., TC
ð3  5Þ8C. These results confirm the analogy between semidilute solutions of nonionic surfactants and entangled solutions of flexible polymers that exhibit the critical demixing. It should be noted that the results below about TC
15K cannot be explained by the scaling theory. Also, the C12E8 system does not show analogy with polymer systems in all temperature ranges. These results are also reasonable because micelles are not so elongated to entangle each other under these conditions. The existence of the crossover concentration c in surfactant systems has been reported by Imae [61,62] for nonionic surfactant systems with salt. She has also calculated c - and c - values by using the theory of Ying and Chu [70]. Agreement with experimental results is good for c , whereas calculated values of c are much smaller than the observed ones. The scaling theory assumes monodisperse polymers, whereas wormlike micelles have broad distribution in aggregation number and, equivalently, molecular weight, or length. Ohta and co-workers [71,72] have developed conformation space renormalization theory of entangled polymers in good solvent taking into account the polydispersity. Schurtenberger and co-workers [73–75] have applied the results of this theory to wormlike micelles taking into account the increase in the micelle aggregation number with concentration by assuming a power law Nw / c . From the least-square fitting of the apparent micellar weight [proportional to mapp in Eq. (18)] in dilute and semidilute regions (10
4–10
2 g cm
3 ), they have obtained  ¼ 1:2 for the C16E6 system, which is much larger than that found in extremely dilute solutions, 0.5, described previously.

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Other interpretations of light scattering data than the scaling analysis have been reported. Brown and co-workers [27,76] have investigated C12Em (m ¼ 5; 7; 8) in the concentration range 0.05–15% by using static and dynamic light scattering and PGSE. They have noted that mixture of micelles and loose clusters of micelles is formed for C12E7 and C12E8 systems. For the C12E5 system, however, they have noted that such clusters are not formed even at temperatures close to the cloud point and that micellar growth occurs. Richtering et al. [77] have studied a C14E8 system by measuring viscosity, static and dynamic light scattering, and PGNMR. From the analysis of the friction coefficient, the second virial coefficient, and the osmotic compressibility, they have concluded that small micelles aggregate to random percolation clusters.

B. Self-Diffusion Processes in Entangled Wormlike Micelles The results described in the previous section suggest that static properties of semidilute solutions of surfactants show similar behavior to that observed in semidilute solutions of flexible polymers in a certain concentration and temperature range. On the other hand, dynamical properties of wormlike micelles are expected to be different from those of polymers. For ionic surfactant systems, in fact, such a discrepancy has been reported mainly for viscoelastic behaviors. At present, however, a theoretical treatment for the dynamical aspects of entanglement has not yet been established. The surfactant self-diffusion coefficient is a useful property for discussing dynamics of entanglement from the ‘‘microscopic’’ point of view, because it gives direct information on the translational motion of surfactant molecules forming micelles themselves; this will be shown later. Measurement of the self-diffusion coefficient of the ionic surfactant in the semidilute region by PGSE is rather difficult because of a very short transverse relaxation time (T2) due to the restricted motion of the hydrophobic chain. For surfactants of polyoxyethylene type, on the other hand, the polyoxyethylene chain can move more freely than the hydrophobic chain does and so gives longer T2. In this section, self-diffusion behaviors in entangled nonionic micelles are described after a brief review of dynamical properties for other systems.

1. Self-Diffusion Behaviors in Entangled Polymers and Ionic Micelles In the case of entangled polymers, the self-diffusion coefficient follows the power law [64,65,78] D / N
2 cð
2Þð3
1Þ

ð20Þ

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503

where N is the number of monomers per chain. In micellar solutions, N corresponds to the aggregation number and so increases with concentration. If we assume the stepwise aggregation model (model A in Section II.D), the concentration dependence of N can be written as N / c1=2

ð21Þ

for the concentration range much larger than CMC. Combining Eqs. (20) and (21), we obtain D / cx

ð22Þ

where x ¼
2:85 and
4 for ¼ 0:588 and 0.5, respectively. Cates [79,80] has proposed a theoretical model (‘‘living polymers’’) taking into account chain breakage and recombination. In his theory, the survival time of a chain before it breaks into two pieces,  break, is introduced in addition to the classical reptation time,  rep. In the case where  break   rep, he obtains the exponent x in Eq. (19) to be
1.6 for ¼ 0:6 and  ¼ 0:5. Messager and co-workers [81] have measured the self-diffusion coefficient of the photobleaching dyes dissolved in micelles for KBr solutions of cetyltrimethylammonium bromide (CTAB) by using the technique of fluorescence recovery after fringe-pattern photo bleaching (FRAP). They have shown that x varies from
1.57 to
4.6 depending on the salinity. Safran and co-workers [82,83] have attributed this variation of x to an increase in the ionic strength with surfactant concentration. Such effects can be neglected for ionic surfactants in water at high salinity, ionic surfactants in organic solvent, and nonionic surfactants. Ott and co-workers [84] have performed a self-diffusion study on reverse micelles of lecithin in isooctane with a very small amount of water by using FRAP. They obtained x ¼
1:32 and
1.35 for w0 ¼ 1 and 2, respectively, where w0 is the water-to-lecithin molar ratio. These results indicate that the exponent depends on conditions such as salinity, which cannot be explained by the living polymer theory.

2. Self-Diffusion Behaviors in Nonionic Surfactant Systems Figure 7 shows the self-diffusion coefficients of C16E7 obtained by using PGSE at the concentrations and temperatures where entanglement of micelles is suggested from the light scattering results [66,85,86]. In the lower concentration range, the self-diffusion coefficient decreases with concentration, and the slope was found to be gentler than for polymers, as was observed in the CTAB system. As the concentration increases further, however, the diffusion coefficient increases after taking a minimum. Such selfdiffusion behavior is completely different from the prediction of the theories for polymers and even for the ‘‘living polymers.’’ Similar results have been obtained for C14E6 and C14E7 systems [85]. Also, it has been found that the

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FIG. 7 Self-diffusion coefficient of C16E7 in D2O obtained by using pulsed-gradient spin echo. (From Ref. 86.)

slope in the higher concentration range tends to approach almost the same value, that is, about 2/3, regardless of the temperature and surfactants. The existence of the minimum in the concentration dependence of the self-diffusion coefficient was found first for the C12E5 and C12E8 systems by Nilsson and co-workers [6] before the scaling analyses of light scattering results were performed by the author in these systems. They deduced that the increase in the self-diffusion coefficient is due to the enhanced exchange of surfactant monomers between different micelles. The author has confirmed the existence of intermicellar migration by measuring the dependence of the self-diffusion coefficient on diffusion time; it has been found that surfactant molecules migrate to another micelles in the diffusion time (0.1–0.3 s). Subsequently, a diffusion model for entangled wormlike micelles has been proposed [66,85]. In this model, it is assumed that (1) a surfactant molecule diffuses in a micelle along its contour during the time  m and then migrates to adjacent micelles at one of the entanglement points (see Fig. 8a) and (2)  m satisfies the condition R2g =DL  m 
, where Rg is the radius of gyration of wormlike micelles, DL is the intramicellar (lateral) diffusion coefficient, and
is the diffusion time. Then the mean-square displacement

Microstructures of Nonionic Surfactant–Water Systems

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of surfactant molecules for the time much longer than  m can be estimated by random walks where a jump of length d is performed at each time  m (see Fig. 8b), which gives the surfactant self-diffusion coefficient as D ¼ DM þ hdi2 =ð6m Þ

ð23Þ

where DM is the self-diffusion coefficient of micelles, hdi is the mean distance between the centers of mass of adjacent micelles. As hdi3 is inversely proportional to the number density of micelles, hdi can be expressed in terms of the surfactant concentration (c) and the mean aggregation number of micelles (hmi) as hdi / ðc=hmiÞ
1=3 / cð
1Þ=3

ð24Þ

In the second relation of Eq. (24), the power law for the mean aggregation number, hmi / c , is assumed. Equation (24) indicates that hdi depends on the concentration only slightly; hdi / c1=6 for  ¼ 1=2. On the other

FIG. 8 Diffusion model taking into account intermicellar migration of surfactant molecules (from Ref. 66). (a) A surfactant molecule diffuses in a wormlike micelle along its contour for the time  m (the white line indicates the path) and migrates to adjacent micelles at one of the entanglement points (see the solid circle). The dotted circles indicate the region where the surfactant molecule exists for the time  m . (b) If  m satisfies the condition R2g =DL  m 
(Rg is the radius of gyration of wormlike micelles, DL is the intramicellar (lateral) diffusion coefficient, and
is the diffusion time), the mean-square displacement of surfactant molecules for the time much longer than  m can be estimated by random walks where a jump of length d is performed at each time  m .

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hand,  m is expected to decrease as the concentration increases (see later). Thus, the second term of Eq. (23) increases with increasing concentration. In dilute solutions (but still much higher than the CMC where contributions from the monomer diffusion can be neglected), the self-diffusion coefficient is dominated by the first term of Eq. (23), DM . As the concentration increases, DM decreases rapidly due to the entanglement of wormlike micelles, whereas the second term of Eq. (23) increases. Thus, we can explain the existence of the minimum in the plot of D versus c by using this model. By assuming power laws for the concentration dependence of hmi and  m, we succeeded in reproducing the observed concentration dependence of D [85]. At almost the same time as the author’s study was published, Cates and co-workers modified the living polymer theory where ‘‘bond interchange’’ and ‘‘end interchange’’ reactions are taken into account as well as the reversible scission [87]. The bond interchange reactions occur when the two chains come into contact and react at some point along the arc length, chosen at random. When the end of one chain ‘‘bites into’’ a second chain at a random position along its length, the end interchange reactions occur. They have calculated the scaling of the stress relaxation time, the zero shear viscosity, and the monomer diffusion coefficient. However, their theory predicts only a monotonous decrease of the diffusion coefficient with increasing concentration because neither lateral diffusion nor intermicellar migration of surfactant molecules is taken into account. After the author’s study, several models have been reported taking into account these processes including more general cases [88–90].

C. Structural Relaxation and Self-Diffusion Processes As described in the previous section, the observed concentration dependence of the surfactant self-diffusion coefficient can be explained by the author’s model if appropriate power laws are assumed for the concentration dependence of hmi and  m. However, this does not always indicate the validity of these power laws (note that the thermodynamic models for micellar growth described in the previous section do not always hold when micelles are entangled [73], as described in Section III.A). This motivated the author to perform dynamic light scattering measurements paying attention to the slow mode that may give information on  m [86,91,92]. Figure 9 shows time correlation functions of the scattered field at different scattering angles for the C16E7 system [91]. One sees that the relaxation is bimodal and that the correlation time  s of the slow relaxation is independent of the scattering angle. Such behaviors were not observed for dilute solutions (