Structure Colloides

STRUCTURE AND FUNCTIONAL PROPERTIES OF COLLOIDAL SYSTEMS

DANIEL BLANKSCHTEINDepartment of Chemical

EngineeringMassachusetts Institute of

TechnologyCambridge, Massachusetts

S. KARABORNIShell International Petroleum

Company LimitedLondon, England

LISA B. QUENCERThe Dow Chemical CompanyMidland, Michigan

JOHN F. SCAMEHORNInstitute for Applied Surfactant

ResearchUniversity of OklahomaNorman, Oklahoma

P. SOMASUNDARANHenry Krumb School of MinesColumbia UniversityNew York, New York

ERIC W. KALERDepartment of Chemical

EngineeringUniversity of DelawareNewark, Delaware

CLARENCE MILLERDepartment of Chemical

EngineeringRice UniversityHouston, Texas

DON RUBINGHThe Procter & Gamble CompanyCincinnati, Ohio

BEREND SMITShell International Oil Products B.V.Amsterdam, the Netherlands

JOHN TEXTERStrider Research CorporationRochester, New York

SURFACTANT SCIENCE SERIES

FOUNDING EDITOR

MARTIN J. SCHICK19181998

SERIES EDITOR

ARTHUR T. HUBBARDSanta Barbara Science Project

Santa Barbara, California

ADVISORY BOARD

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. Lissant7. Anionic Surfactants (in two parts), edited by Warner M. Linfield (see Volume 56)8. Anionic Surfactants: Chemical Analysis, edited by John Cross9. Stabilization of Colloidal Dispersions by Polymer Adsorption, Tatsuo Sato

and Richard Ruch 10. Anionic Surfactants: Biochemistry, Toxicology, Dermatology, edited 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 Datyner15. Electrical Phenomena at Interfaces: Fundamentals, Measurements,

and Applications, edited by Ayao Kitahara and Akira Watanabe16. Surfactants in Cosmetics, edited by Martin M. Rieger (see Volume 68)17. Interfacial Phenomena: Equilibrium and Dynamic Effects, Clarence A. Miller

and P. Neogi18. Surfactant Biodegradation: Second Edition, Revised and Expanded, R. D. Swisher19. Nonionic Surfactants: Chemical Analysis, edited by John Cross20. Detergency: Theory and Technology, edited by W. Gale Cutler and Erik Kissa21. Interfacial Phenomena in Apolar Media, edited by Hans-Friedrich Eicke

and Geoffrey D. Parfitt22. Surfactant Solutions: New Methods of Investigation, edited by Raoul Zana23. Nonionic Surfactants: Physical Chemistry, edited by Martin J. Schick24. Microemulsion Systems, edited by Henri L. Rosano and Marc Clausse25. Biosurfactants and Biotechnology, edited by Naim Kosaric, W. L. Cairns,

and Neil C. C. Gray26. Surfactants in Emerging Technologies, edited by Milton J. Rosen27. Reagents in Mineral Technology, edited by P. Somasundaran and Brij M. Moudgil28. Surfactants in Chemical/Process Engineering, edited by Darsh T. Wasan,

Martin E. Ginn, and Dinesh O. Shah29. Thin Liquid Films, edited by I. B. Ivanov30. Microemulsions 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 Sato32. Interfacial Phenomena in Coal Technology, edited by Gregory D. Botsaris

and Yuli M. Glazman33. Surfactant-Based Separation Processes, edited by John F. Scamehorn

and Jeffrey H. Harwell

34. Cationic Surfactants: Organic Chemistry, edited by James M. Richmond35. Alkylene Oxides and Their Polymers, F. E. Bailey, Jr., and Joseph V. Koleske36. Interfacial Phenomena in Petroleum Recovery, edited by Norman R. Morrow37. Cationic Surfactants: Physical Chemistry, edited by Donn N. Rubingh

and Paul M. Holland38. Kinetics and Catalysis in Microheterogeneous Systems, edited by M. Grtzel

and K. Kalyanasundaram39. Interfacial Phenomena in Biological Systems, edited by Max Bender40. Analysis of Surfactants, Thomas M. Schmitt (see Volume 96)41. Light Scattering by Liquid Surfaces and Complementary Techniques, edited by

Dominique Langevin42. Polymeric Surfactants, Irja Piirma43. Anionic Surfactants: Biochemistry, Toxicology, Dermatology,

Second Edition, Revised and Expanded, edited by Christian Gloxhuber and Klaus Knstler

44. Organized Solutions: Surfactants in Science and Technology, edited by Stig E. Friberg and Bjrn Lindman

45. Defoaming: Theory and Industrial Applications, edited by P. R. Garrett46. Mixed Surfactant Systems, edited by Keizo Ogino and Masahiko Abe47. Coagulation and Flocculation: Theory and Applications, edited by

Bohuslav Dobis48. Biosurfactants: Production Properties Applications, edited by Naim Kosaric49. Wettability, edited by John C. Berg50. Fluorinated Surfactants: Synthesis Properties Applications, Erik Kissa51. Surface and Colloid Chemistry in Advanced Ceramics Processing, edited by

Robert J. Pugh and Lennart Bergstrm52. Technological Applications of Dispersions, edited by Robert B. McKay53. Cationic Surfactants: Analytical and Biological Evaluation, edited by John Cross

and Edward J. Singer54. Surfactants in Agrochemicals, Tharwat F. Tadros55. Solubilization in Surfactant Aggregates, edited by Sherril D. Christian

and John F. Scamehorn56. Anionic Surfactants: Organic Chemistry, edited by Helmut W. Stache57. Foams: Theory, Measurements, and Applications, edited by

Robert K. Prudhomme and Saad A. Khan58. The Preparation of Dispersions in Liquids, H. N. Stein59. Amphoteric Surfactants: Second Edition, edited by Eric G. Lomax60. Nonionic Surfactants: Polyoxyalkylene Block Copolymers, edited by

Vaughn M. Nace61. Emulsions and Emulsion Stability, edited by Johan Sjblom62. Vesicles, edited by Morton Rosoff63. Applied Surface Thermodynamics, edited by A. W. Neumann and Jan K. Spelt64. Surfactants in Solution, edited by Arun K. Chattopadhyay and K. L. Mittal65. Detergents in the Environment, edited by Milan Johann Schwuger66. Industrial Applications of Microemulsions, edited by Conxita Solans

and Hironobu Kunieda67. Liquid Detergents, edited by Kuo-Yann Lai68. Surfactants in Cosmetics: Second Edition, Revised and Expanded, edited by

Martin M. Rieger and Linda D. Rhein69. 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. Showell72. Nonionic Surfactants: Organic Chemistry, edited by Nico M. van Os73. Anionic Surfactants: Analytical Chemistry, Second Edition, Revised

and Expanded, edited by John Cross74. Novel Surfactants: Preparation, Applications, and Biodegradability, edited by

Krister Holmberg75. Biopolymers at Interfaces, edited by Martin Malmsten76. 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. Kwak78. Surfaces of Nanoparticles and Porous Materials, edited by James A. Schwarz

and Cristian I. Contescu79. Surface Chemistry and Electrochemistry of Membranes, edited by

Torben Smith Srensen80. Interfacial Phenomena in Chromatography, edited by Emile Pefferkorn81. SolidLiquid Dispersions, Bohuslav Dobis, Xueping Qiu,

and Wolfgang von Rybinski82. Handbook of Detergents, editor in chief: Uri Zoller Part A: Properties, edited by

Guy Broze83. Modern Characterization Methods of Surfactant Systems, edited by

Bernard P. Binks84. Dispersions: Characterization, Testing, and Measurement, Erik Kissa85. Interfacial Forces and Fields: Theory and Applications, edited by Jyh-Ping Hsu86. Silicone Surfactants, edited by Randal M. Hill87. Surface Characterization Methods: Principles, Techniques, and Applications,

edited by Andrew J. Milling88. Interfacial Dynamics, edited by Nikola Kallay89. Computational Methods in Surface and Colloid Science, edited by

Malgorzata Borwko90. Adsorption on Silica Surfaces, edited by Eugne Papirer91. Nonionic Surfactants: Alkyl Polyglucosides, edited by Dieter Balzer

and Harald Lders92. Fine Particles: Synthesis, Characterization, and Mechanisms of Growth, edited by

Tadao Sugimoto93. Thermal Behavior of Dispersed Systems, edited by Nissim Garti94. Surface Characteristics of Fibers and Textiles, edited by Christopher M. Pastore

and Paul Kiekens 95. Liquid Interfaces in Chemical, Biological, and Pharmaceutical Applications, edited

by Alexander G. Volkov96. Analysis of Surfactants: Second Edition, Revised and Expanded,

Thomas M. Schmitt97. Fluorinated Surfactants and Repellents: Second Edition, Revised

and Expanded, Erik Kissa98. Detergency of Specialty Surfactants, edited by Floyd E. Friedli99. Physical Chemistry of Polyelectrolytes, edited by Tsetska Radeva

100. Reactions and Synthesis in Surfactant Systems, edited by John Texter101. Protein-Based Surfactants: Synthesis, Physicochemical Properties,

and Applications, edited by Ifendu A. Nnanna and Jiding Xia

102. Chemical Properties of Material Surfaces, Marek Kosmulski103. Oxide Surfaces, edited by James A. Wingrave104. Polymers in Particulate Systems: Properties and Applications, edited by

Vincent A. Hackley, P. Somasundaran, and Jennifer A. Lewis105. Colloid and Surface Properties of Clays and Related Minerals, Rossman F. Giese

and Carel J. van Oss106. Interfacial Electrokinetics and Electrophoresis, edited by ngel V. Delgado107. Adsorption: Theory, Modeling, and Analysis, edited by Jzsef Tth108. Interfacial Applications in Environmental Engineering, edited by Mark A. Keane109. Adsorption and Aggregation of Surfactants in Solution, edited by K. L. Mittal

and Dinesh O. Shah110. Biopolymers at Interfaces: Second Edition, Revised and Expanded, edited by

Martin Malmsten111. Biomolecular Films: Design, Function, and Applications, edited by

James F. Rusling112. StructurePerformance Relationships in Surfactants: Second Edition, Revised

and Expanded, edited by Kunio Esumi and Minoru Ueno113. Liquid Interfacial Systems: Oscillations and Instability, Rudolph V. Birikh,

Vladimir A. Briskman, Manuel G. Velarde, and Jean-Claude Legros114. Novel Surfactants: Preparation, Applications, and Biodegradability:

Second Edition, Revised and Expanded, edited by Krister Holmberg115. Colloidal Polymers: Synthesis and Characterization, edited by

Abdelhamid Elaissari116. Colloidal Biomolecules, Biomaterials, and Biomedical Applications, edited by

Abdelhamid Elaissari117. Gemini Surfactants: Synthesis, Interfacial and Solution-Phase Behavior,

and Applications, edited by Raoul Zana and Jiding Xia118. Colloidal Science of Flotation, Anh V. Nguyen and Hans Joachim Schulze119. Surface and Interfacial Tension: Measurement, Theory, and Applications, edited by

Stanley Hartland120. Microporous Media: Synthesis, Properties, and Modeling, Freddy Romm121. Handbook of Detergents, editor in chief: Uri Zoller, Part B: Environmental Impact,

edited by Uri Zoller122. Luminous Chemical Vapor Deposition and Interface Engineering, HirotsuguYasuda123. Handbook of Detergents, editor in chief: Uri Zoller, Part C: Analysis, edited by

Heinrich Waldhoff and Rdiger Spilker124. Mixed Surfactant Systems: Second Edition, Revised and Expanded, edited by

Masahiko Abe and John F. Scamehorn125. Dynamics of Surfactant Self-Assemblies: Micelles, Microemulsions, Vesicles

and Lyotropic Phases, edited by Raoul Zana126. Coagulation and Flocculation: Second Edition, edited by

Hansjoachim Stechemesser and Bohulav Dobis127. Bicontinuous Liquid Crystals, edited by Matthew L. Lynch and Patrick T. Spicer128. Handbook of Detergents, editor in chief: Uri Zoller, Part D: Formulation, edited by

Michael S. Showell129. Liquid Detergents: Second Edition, edited by Kuo-Yann Lai130. Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering, edited by

Aleksandar M. Spasic and Jyh-Ping Hsu131. Colloidal Silica: Fundamentals and Applications, edited by Horacio E. Bergna

and William O. Roberts132. Emulsions and Emulsion Stability, Second Edition, edited by Johan Sjblom

133. Micellar Catalysis, Mohammad Niyaz Khan134. Molecular and Colloidal Electro-Optics, Stoyl P. Stoylov and Maria V. Stoimenova135. Surfactants in Personal Care Products and Decorative Cosmetics, Third Edition,

edited by Linda D. Rhein, Mitchell Schlossman, Anthony O'Lenick, and P. Somasundaran

136. Rheology of Particulate Dispersions and Composites, Rajinder Pal137. Powders and Fibers: Interfacial Science and Applications, edited by Michel Nardin

and Eugne Papirer138. Wetting and Spreading Dynamics, edited by Victor Starov, Manuel G. Velarde,

and Clayton Radke139. Interfacial Phenomena: Equilibrium and Dynamic Effects, Second Edition,

edited by Clarence A. Miller and P. Neogi140. Giant Micelles: Properties and Applications, edited by Raoul Zana

and Eric W. Kaler141. Handbook of Detergents, editor in chief: Uri Zoller, Part E: Applications, edited by

Uri Zoller142. Handbook of Detergents, editor in chief: Uri Zoller, Part F: Production, edited by

Uri Zoller and co-edited by Paul Sosis143. Sugar-Based Surfactants: Fundamentals and Applications, edited by

Cristbal Carnero Ruiz144. Microemulsions: Properties and Applications, edited by Monzer Fanun145. Surface Charging and Points of Zero Charge, Marek Kosmulski146. Structure and Functional Properties of Colloidal Systems, edited by

Roque Hidalgo-lvarez

STRUCTURE ANDFUNCTIONAL PROPERTIESOF COLLOIDAL SYSTEMS

Edited by

Roque Hidalgo-lvarezUniversity of Granada

Granada, Spain

CRC Press is an imprint of theTaylor & Francis Group, an informa business

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Library of Congress Cataloging-in-Publication Data

Structure and functional properties of colloidal systems / editor, Roque Hidalgo-Alvarez.p. cm. -- (Surfactant science series ; v. 146)

Includes bibliographical references and index.ISBN 978-1-4200-8446-7 (hardcover : alk. paper)1. Colloids. 2. Surface tension. I. Hidalgo-Alvarez, Roque.

QD549.S793 2009541.345--dc22 2009028499

Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.com

xi

Contents

Preface .......................................................................................................................................... xvEditor ......................................................................................................................................... xviiContributors ................................................................................................................................ xix

PART I: Theory

Chapter 1 Colloid Dynamics and Transitions to Dynamically Arrested States ....................... 3

R. Jurez-Maldonado and M. Medina-Noyola

Chapter 2 Capillary Forces between Colloidal Particles at Fluid Interfaces .......................... 31

Alvaro Domnguez

PART II: Structure

Chapter 3 Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations ........ 63

A. Martn-Molina, M. Quesada-Prez, and R. Hidalgo-lvarez

Chapter 4 Effective Interactions of Charged Vesicles in Aqueous Suspensions .................... 77

C. Haro-Prez, L.F. Rojas-Ochoa, V. Trappe, R. Castaeda-Priego, J. Estelrich, M. Quesada-Prez, Jos Callejas-Fernndez, and R. Hidalgo-lvarez

Chapter 5 Structure and Colloidal Properties of Extremely Bimodal Suspensions ............... 93

A.V. Delgado, M.L. Jimnez, J.L. Viota, R. Rica, M.T. Lpez-Lpez, and S. Ahualli

Chapter 6 Structure and Stability of Filaments Made up of Microsized Magnetic Particles ............................................................................. 117

F. Martnez-Pedrero, A. El-Harrak, Mara Tirado-Miranda, J. Baudry, Artur Schmitt, J. Bibette, and Jos Callejas-Fernndez

Chapter 7 Glasses in Colloidal Systems: Attractive Interactions and Gelation .................... 135

Antonio M. Puertas and Matthias Fuchs

xii Contents

Chapter 8 Phase Behavior and Structure of Colloidal Suspensions in Bulk, Confi nement, and External Fields ........................................................................ 165

A.-P. Hynninen, A. Fortini, and M. Dijkstra

Chapter 9 WaterWater Interfaces ........................................................................................ 201

R. Hans Tromp

Chapter 10 Interfacial Phenomena Underlying the Behavior of Foams and Emulsions ......... 219

Julia Maldonado-Valderrama, Antonio Martn-Rodrguez, Miguel A. Cabrerizo-Vlchez, and Mara Jose Glvez Ruiz

Chapter 11 Rheological Models for Structured Fluids ........................................................... 235

Juan de Vicente

PART III: Functional Materials

Chapter 12 Surface Functionalization of Latex Particles ....................................................... 263

Ainara Imaz, Jose Ramos, and Jacqueline Forcada

Chapter 13 Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles .......................................................... 289

Mara Tirado-Miranda, Miguel A. Rodrguez-Valverde, Artur Schmitt, Jos Callejas-Fernndez, and Antonio Fernndez-Barbero

Chapter 14 Advances in the Preparation and Biomedical Applications of Magnetic Colloids ............................................................................................ 315

A. Elaissari, J. Chatterjee, M. Hamoudeh, and H. Fessi

Chapter 15 Colloidal Dispersion of Metallic Nanoparticles: Formation and Functional Properties ................................................................... 339

Shlomo Magdassi, Michael Grouchko, and Alexander Kamyshny

Chapter 16 Hydrophilic Colloidal Networks (Micro- and Nanogels) in Drug Delivery and Diagnostics ........................................................................ 367

Serguei V. Vinogradov

Chapter 17 Responsive Microgels for Drug Delivery Applications ....................................... 387

Jeremy P.K. Tan and Kam C. Tam

Chapter 18 Colloidal Photonic Crystals and Laser Applications ........................................... 415

Seiichi Furumi

Contents xiii

Chapter 19 Droplet-Based Microfl uidics: Picoliter-Sized Reactors for Mesoporous Microparticle Synthesis ................................................................... 429

Nick J. Carroll, Sergio Mendez, Jeremy S. Edwards, David A. Weitz, and Dimiter N. Petsev

Chapter 20 Electro-Optical Properties of GelGlass Dispersed Liquid Crystals Devices by Chemical Modifi cation of the LC/Matrix Interface .......................... 447

Marcos Zayat, Rosario Pardo, and David Levy

Chapter 21 Nano-Emulsion Formation by Low-Energy Methods and Functional Properties ........................................................................................... 457

Conxita Solans, Isabel Sol, Alejandro Fernndez-Arteaga, Jordi Nolla, Nria Azemar, Jos Gutirrez, Alicia Maestro, Carmen Gonzlez, and Carmen M. Pey

Index .......................................................................................................................................... 483

xv

Preface

This book covers important aspects of colloidal systems that have received signifi cant inputs and deserve a collective presentation. The unique purpose of this book is to present as clearly as possible the connection between structure and functional properties in colloid and interface science.

The idea of having a book relating the physical fundamentals of colloid science and new develop-ments of synthesis and conditioning, while sharpening the readers mind for the practically unlim-ited possibilities of application is absolutely timely. This fi eld is evolving so rapidly and successfully that good guidance is utterly needed.

The intended audience is scientists who are interested in understanding more about the connec-tion between the structure, in two and three dimensions of colloidal systems and the functional properties of those systems, combining fundamental research with technical applications. It addresses an important and explosively expanding fi eld and bridges the gap between fundamentals and applications.

For advanced students a book is needed to describe the connection between techniques to func-tionalize colloids, the characterization methods, the physical fundamentals of structure formation, diffusion dynamics, transport properties in equilibrium, the physical fundamentals of nonequilib-rium systems, the measuring principles to exploit these properties in applications, the differences in designing lab experiments and devices, and a few selected application examples.

In order to try to achieve these objectives, several issues have been addressed. This book is organized into three parts: theory, structure, and functional materials.

The fi rst two chapters (by Medina-Noyola et al. and Domnguez, respectively) deal explicitly with theoretical aspects of colloid dynamics and transitions to dynamically arrested states and capil-lary forces in colloidal systems at fl uid interfaces.

The second part covers the structural aspects of different colloidal systems. Chapters 3 and 4, by Martn-Molina et al. and Haro-Prez et al., deal with electric double layers and effective inter-actions. Chapters 5 and 6, by Delgado et al. and Martnez-Pedrero et al., explore the structure of extremely bimodal suspensions and fi laments made up of microsized magnetic particles. Chapters 7 and 8, by Puertas and Fuchs, and Hynninen et al., analyze the role played by the attractive interac-tions, confi nement, and external fi elds on the structure of colloidal systems. Chapters 9 and 10, by Tromp and Maldonado-Valderrama et al., cover some structural aspects in food emulsions. This second part of the book fi nishes with Chapter 11, by de Vicente, which analyzes the rheological properties of structured fl uids in order to establish a connection between measured material rheo-logical functions and structural properties.

The last part of this book is devoted to functional colloids. Examples treated in this part of the book are as follows: polymer colloids by Imaz et al.; protein-functionalized colloidal particles by Tirado-Miranda et al.; magnetic particles by Elaissari et al.; metallic nanoparticles by Magdassi et al.; micro- and nanogels and responsive microgels by Vinogradov and Tan and Tam, respectively; colloidal photonic crystals by Furumi; microfl uidics by Petsev et al.; gelglass dispersed liquid crystals (GDLCs) devices by Zayat et al.; and nano-emulsions by Solans et al.

My sincere gratitude to all referees and participating authors for their support and involvement, which has made my job as editor an easy and satisfying one.

xvi Preface

Finally, I gratefully acknowledge fi nancial support from Ministerio de Educacin y Ciencia (Plan Nacional de Investigacin Cientfi ca, Desarrollo e Innovacin Tecnolgica (I + D + i), Projects MAT 2006-13646-C03-03 and MAT2006-12918-C05-01), by the European Regional Development Fund (ERDF), and by the Project P07-FQM-2496 from Junta de Andaluca.

xvii

Editor

Roque Hidalgo-lvarez received a master in science and a PhD in physics from the University of Granada. He joined the physics department at the University of Granada in September 1975. He was a postdoctoral fellow at Wageningen University, the Netherlands from 1984 to 1985. His research and teaching interests lie in the general area of colloid and interface sciences with a special empha-sis on electrokinetic phenomena and colloidal stability.

Dr. Hidalgo-lvarez has published 212 scientifi c papers in international journals and has super-vised 21 PhD theses. Currently, he is the president of the Group of Colloid and Interface Science associated with the Royal Societies of Chemistry and Physics in Spain.

xix

Contributors

S. AhualliDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

Nria AzemarInstitute for Advanced Chemistry

of CataloniaConsejo Superior de Investigaciones

Cientfi casCIBER-BBNBarcelona, Spain

J. Baudry Laboratoire Collodes et Matriaux DivissParisTech, ESPCIParis, France

J. Bibette Laboratoire Collodes et Matriaux DivissParisTech, ESPCIParis, France

Miguel A. Cabrerizo-Vlchez Department of Applied Physics University of GranadaGranada, Spain

Jos Callejas-FernndezDepartment of Applied Physics University of GranadaGranada, Spain

Nick J. Carroll Department of Chemical and

Nuclear Engineering University of New Mexico Albuquerque, New Mexico

R. Castaeda-PriegoDepartment of Physics University of GuanajuatoLen, Mxico

J. ChatterjeeFAMU-FSU College of EngineeringTallahassee, Florida

A.V. DelgadoDepartment of Applied Physics University of GranadaGranada, Spain

M. DijkstraDebye InstituteUtrecht UniversityUtrecht, the Netherlands

Alvaro DomnguezFsica TericaDpto. Fsica Atmica,

Molecular y NuclearUniversidad de Sevilla Sevilla, Spain

Jeremy S. EdwardsDepartment of Molecular Genetics

and MicrobiologyUniversity of New MexicoAlbuquerque, New Mexico

A. El-Harrak Institut de Science et dIngnierie

Supramolculaires Universit Louis PasteurStrasbourg, France

A. ElaissariLaboratoire dAutomatique

et de Gnie des ProcdsUniversit de LyonLyon, France

J. EstelrichFacultat de Farmcia Universitat de BarcelonaBarcelona, Spain

xx Contributors

Alejandro Fernndez-ArteagaInstitute for Advanced Chemistry


Cientfi casCIBER-BBN Barcelona, Spain

Antonio Fernndez-BarberoDepartment of Applied Physics University of AlmeraAlmera, Spain

H. FessiLaboratoire dAutomatique


Jacqueline ForcadaGrupo de Ingeniera Qumica The University of the Basque CountrySan Sebastin-Donostia, Spain

A. Fortini Debye InstituteUtrecht UniversityUtrecht, the Netherlands

Matthias FuchsDepartment of PhysicsUniversity of KonstanzKonstanz, Germany

Seiichi FurumiNational Institute for Materials ScienceTsukuba, Ibaraki, Japan

Carmen GonzlezDepartment of Chemical EngineeringUniversity of BarcelonaBarcelona, Spain

Michael GrouchkoCasali Institute of Applied ChemistryThe Hebrew University of Jerusalem Jerusalem, Israel

Jos GutirrezDepartment of Chemical EngineeringUniversity of BarcelonaBarcelona, Spain

M. HamoudehLaboratoire dAutomatique


C. Haro-PrezDepartment of Applied Physics University of GranadaGranada, Spain

R. Hidalgo-lvarezDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

A.-P. HynninenDepartment of Chemical EngineeringPrinceton UniversityPrinceton, New Jersey

Ainara ImazGrupo de Ingeniera Qumica The University of the Basque CountrySan Sebastin-Donostia, Spain

M.L. JimnezDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

R. Jurez-MaldonadoInstituto de Fsica Manuel Sandoval

VallartaUniversidad Autnoma de San Luis PotosSan Luis Poto, Mxico

Alexander KamyshnyCasali Institute of Applied ChemistryThe Hebrew University of JerusalemJerusalem, Israel

David LevyInstituto de Ciencia de Materiales

de Madrid, CantoblancoMadrid, Spain

and

Laboratorio de Instrumentacin Espacial-LINES

Torrejn de Ardoz, Madrid, Spain

Contributors xxi

M.T. Lpez-LpezDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

Alicia MaestroDepartment of Chemical EngineeringUniversity of Barcelona Barcelona, Spain

Shlomo MagdassiCasali Institute of Applied ChemistryThe Hebrew University of JerusalemJerusalem, Israel

Julia Maldonado-ValderramaDepartment of Applied Physics University of GranadaGranada, Spain

A. Martn-Molina Department of Applied PhysicsUniversity of GranadaGranada, Spain

Antonio Martn-Rodrguez Department of Applied Physics University of GranadaGranada, Spain

F. Martnez-PedreroDepartment of Applied Physics University of GranadaGranada, Spain

M. Medina-NoyolaInstituto de Fsica Manuel Sandoval VallartaUniversidad Autnoma de San Luis PotosSan Luis Poto, Mxico

Sergio MendezDepartment of Chemical and Nuclear

EngineeringUniversity of New MexicoAlbuquerque, New Mexico

Jordi NollaInstitute for Advanced Chemistry of

CataloniaConsejo Superior de Investigaciones Cientfi casCIBER-BBNBarcelona, Spain

Rosario Pardo Instituto de Ciencia de Materiales


and



Dimiter N. Petsev Department of Chemical and Nuclear

EngineeringUniversity of New MexicoAlbuquerque, New Mexico

Carmen M. PeyDepartment of Chemical

EngineeringUniversity of BarcelonaBarcelona, Spain

Antonio M. PuertasDepartment of Applied Physics University of AlmeraAlmera, Andaluca, Spain

M. Quesada-PrezDepartment of PhysicsUniversity of JanLinares, Jan, Spain

Jose RamosGrupo de Ingeniera QumicaThe University of the Basque

CountrySan Sebastin-Donostia, Spain

R. RicaDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

Miguel A. Rodrguez-ValverdeDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

L.F. Rojas-OchoaDepartment of Physics Cinvestav-IPNMxico D.F., Mexico

xxii Contributors

Mara Jose Glvez RuizDepartment of Applied Physics University of GranadaGranada, Spain

Artur SchmittDepartment of Applied PhysicsUniversity of Granada Granada, Spain

Conxita SolansInstitute for Advanced Chemistry



Isabel SolInstitute for Advanced Chemistry



Kam C. TamDepartment of Chemical EngineeringUniversity of WaterlooWaterloo, Ontario, Canada

Jeremy P. K. TanInstitute of Bioengineering and

NanotechnologyThe Nanos, Singapore

Mara Tirado-Miranda Department of Physics University of GranadaGranada, Spain

V. TrappeDepartment of Physics University of FribourgFribourg, Switzerland

R. Hans TrompNIZO Food ResearchKernhemseweg, the Netherlands

Juan de VicenteDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

Serguei V. Vinogradov Department of Pharmaceutical

SciencesUniversity of Nebraska Medical

CenterOmaha, Nebraska

J.L. ViotaDepartment of Applied PhysicsUniversity of GranadaGranada, Spain

David A. WeitzDepartment of PhysicsHarvard UniversityCambridge, Massachusetts

Marcos ZayatInstituto de Ciencia de Materiales


and



Part I

Theory

31 Colloid Dynamics and Transitions to Dynamically Arrested States

R. Jurez-Maldonado and M. Medina-Noyola

CONTENTS

1.1 Introduction ........................................................................................................................ 41.2 GLE Formalism .................................................................................................................. 5

1.2.1 Ordinary and Generalized Langevin Equation ...................................................... 51.2.1.1 The Ordinary Langevin Equation ............................................................ 61.2.1.2 OrnsteinUhlenbeck Processes ............................................................... 61.2.1.3 The Generalized Langevin Equation ....................................................... 7

1.2.2 GLE for Tracer Diffusion ....................................................................................... 81.2.2.1 Time-Dependent Friction Function: Exact Expression ............................ 81.2.2.2 Simplifying Approximations for DzT (t) ................................................... 10

1.2.3 Collective Diffusion ................................................................................................ 111.2.3.1 Exact Memory-Function Expressions for F(k, t) and F(S)(k, t) ................. 11

1.2.4 Approximate Elements of the Self-Consistent Theory ........................................... 121.2.4.1 Vineyard-Like Approximations ............................................................... 121.2.4.2 Closure Relation ....................................................................................... 12

1.3 Self-Consistent Theory and Illustrative Applications ........................................................ 131.3.1 Model Mono-Disperse Suspensions ....................................................................... 13

1.3.1.1 Two-Dimensional Model System with DipoleDipole (r -3) Interactions ...................................................................................... 14

1.3.1.2 Three-Dimensional Soft Sphere Systems ................................................ 141.3.2 Extension to Colloidal Mixtures ............................................................................. 15

1.3.2.1 General Results ........................................................................................ 151.3.2.2 Binary Mixture of Charged Particles ...................................................... 16

1.3.3 Simplifi ed SCGLE Theory ..................................................................................... 181.3.4 Diffusion of Colloidal Fluids in Random Porous Media ........................................ 18

1.4 SCGLE Theory of Dynamic Arrest Transitions ................................................................. 201.4.1 General Results ....................................................................................................... 20

1.4.1.1 Nonergodicity Parameters ........................................................................ 201.4.2 Specifi c Systems and Comparison with Experimental Data .................................. 21

1.4.2.1 Hard- and Soft-Sphere Systems ............................................................... 211.4.2.2 Dispersions of Charged Particles ............................................................. 23

1.4.3 Multicomponent Systems ........................................................................................ 231.4.3.1 Mixtures of Hard Spheres ........................................................................ 241.4.3.2 Colloid-Polymer Mixtures ....................................................................... 25

4 Structure and Functional Properties of Colloidal Systems

1.5 Summary and Perspectives ................................................................................................. 26Acknowledgment ......................................................................................................................... 27References .................................................................................................................................... 27

1.1 INTRODUCTION

The dynamic properties of colloidal suspensions constitute an important experimental and theoreti-cal aspect of the study of colloidal systems [13]. In equilibrium, and in the absence of external fi elds, the most relevant dynamic information of such systems is contained in the intermediate scattering function F(k, t) [2]. This function is the spatial Fourier transform (FT) of the van Hove function G(r, t), which measures the spatial and temporal correlations of the thermal fl uctuations dn(r, t) n(r, t) n of the local concentration n(r, t) of colloidal particles at position r and time t around its equilibrium bulk average n, that is, nG(|r - r|; t) dn(r, t)dn(r, 0), where the angular brackets indicate average over the equilibrium ensemble [2]. A closely related property is the so-called self intermediate scattering function F (S)(k, t). This is defi ned as F (S)(k, t) eik DR(t), where DR(t) is the displacement at time t of any of the particles of the Brownian fl uid.

Over the years, the development of a general and practical microscopic description of colloid dynam-ics has proved to be a challenging task [13]. As a result, we have a rather diverse array of approaches, formal derivations, or physically intuitive shortcuts to the most diffi cult aspects of this complex many-body problem [415]. Taken together, these developments have provided a sound theoretical interpreta-tion of a large number of experimental facts. These involve important effects present in everyday colloidal suspensions, such as charge effects in electrostatically stabilized suspensions and the effects of direct and hydrodynamic interactions in hard-sphere-like suspensions. These quantitative tests have involved the description of both, self- or tracer-diffusion and collective-diffusion phenomena.

The present work reviews the development and applications of a theory of colloid dynamics con-structed over the last years [1621], leading to the fi rst-principles calculation of the dynamic properties above. This theory, referred to as the self-consistent generalized Langevin equation (SCGLE) theory, is based on general and exact expressions for F(k, t) and F (S)(k, t), and for the time-dependent friction func-tion Dz(t), the added friction on a tracer particle due to its direct interactions with the other colloidal particles. These three exact results, derived within the generalized Langevin equation (GLE) formalism [22,23], are complemented by two physically intuitive notions, namely, that collective diffusion should be related in a simple manner to self-diffusion, and that space-dependent self-diffusion, in turn, should be related in a simple manner to the mean-squared displacement (msd) [or other k-independent self-diffusion property, such as Dz(t)]. The intrinsic accuracy and limitations of the resulting approximate scheme under the simplest possible conditions (model monodisperse suspensions of spherical particles with no hydrodynamic interactions) have been systematically assessed by the comparison of its predictions with the corresponding Brownian dynamics computer simulation data in the short- and intermediate-time regimes [19]. The same theoretical scheme has also been extended to describe the dynamics of colloidal mixtures [20,21]. The purpose of the present review is two-fold. In the fi rst place, we provide a summary of the conceptual basis of the SCGLE theory as well as an illustrative selection of the applications just referred to. Second, we review the application of this theory to one particularly interesting area, namely, the description of dynamic arrest phenomena in colloidal systems.

The fundamental understanding of dynamically arrested states of matter is one of the most fasci-nating topics of condensed matter physics, and several issues related to their microscopic description are currently a matter of discussion [2426]. Among the various approaches to understanding the transition from an ergodic to a dynamically arrested state, the mode coupling (MC) theory [2628] provides perhaps the most comprehensive and coherent picture. In fact, a large number of experimen-tal observations in specifi c systems, particularly in the domain of colloidal systems [2940], relate to the predictions of this theory. The mode coupling theory (MCT) of the ideal glass transition emerged originally in the framework of the dynamics of molecular (not colloidal) liquids. Although one can

Colloid Dynamics and Transitions to Dynamically Arrested States 5

expect that the phenomenology of dynamic arrest does not depend on the short-time motion (which distinguishes between molecular and colloidal dynamics), it is convenient to base a theory for the glass transition of colloidal systems on the diffusive microscopic dynamics characteristic of these systems.

As indicated above, the SCGLE theory was originally devised to describe the tracer and collec-tive diffusion properties of colloidal dispersions in the short- and intermediate-time regimes. Its self-consistent character, however, introduces a nonlinear dynamic feedback, leading to the predic-tion of dynamic arrest, similar to that exhibited by the MCT of the ideal glass transition [26]. The SCGLE theory, however, is not another version of MCT, and it differs also from recent variants [41,42] of MCT mainly aimed at improving the performance of the original theory concerning the description of the ideal glass transition.

As indicated above, the SCGLE theory is based on three exact results. Hence, it should not be a surprise that the same results also appear in the formulation of MCT, although their derivation [22,23] is completely different from the standard projection operator derivation followed by MCT [26,27]. The most fundamental difference between these two theories is, however, the manner in which the SCGLE complements those three exact results with the two physically intui-tive approximations referred to above. The resulting equations are simpler both, conceptually and in practice, than the corresponding MCT equations. Let us fi nally mention that the SCGLE theory shares with the colloid-dynamics version of MCT developed by Ngele and collaborators [1015] the original intention of describing accurately the short- and intermediate-time dynamics of colloidal systems.

The present chapter is aimed at reviewing the development and the specifi c applications of the SCGLE theory of colloid dynamics and of dynamic arrest. Thus, it is not aimed at reviewing the state-of-the-art in either of these research areas, for which excellent reviews are available [2,3,4345]. We must also say that notable topics in both fi elds are barely or never mentioned here. This includes, for example, the structural, mechanical, and rheological properties, and the effects of hydrodynamic interactions. Instead, we focus on the treatment of the effects of direct conservative interactions in simple colloidal systems. Thus, here we shall primarily deal with monodisperse suspensions of spherical particles in the absence of hydrodynamic interactions, although the extension to multi-component systems will also be an important aspect of this review.

This work is divided into three parts, mapped onto the following three sections. Section 1.2 is devoted to reviewing the fundamental basis of the SCGLE theory of colloid dynamics. Readers more interested in the specifi c applications of this theory than in the theoretical arguments or derivations leading to it may proceed directly to Section 1.3, where we review the actual appli-cations of this theory to the description of the short- and intermediate-time dynamics of specifi c model colloidal systems. Readers more interested in dynamic arrest phenomena might even prefer proceeding directly to Section 1.3, which reviews the SCGLE theory of dynamic arrest and its applications to the interpretation of specifi c experimental results in colloidal systems. The chapter concludes with a brief section on the perspectives and possible extensions of the SCGLE theory.

1.2 GLE FORMALISM

This section deals with the fundamental basis of the SCGLE theory. We fi rst describe what is understood here for the GLE and then illustrate its use in the derivation of exact result for the time-dependent friction function Dz(t), and for the collective and self intermediate scattering functions. In addition, we discuss two additional approximations that convert these exact results into a closed self-consistent system of equations.

1.2.1 ORDINARY AND GENERALIZED LANGEVIN EQUATION

Let us fi rst explain what we mean here for generalized Langevin equation (GLE). For this, we sum-marize rather well-known concepts of the theory of Brownian motion and thermal fl uctuations cast as stochastic processes, generated by linear stochastic equations with additive noise.


1.2.1.1 The Ordinary Langevin EquationThe Brownian motion of an isolated colloidal particle is described by the ordinary Langevin equation [46,47]. Let M be the mass and v(t) the instantaneous velocity of such a particle, which we write as the sum of two terms, v(t) =

_ v (t) + dv(t), where

_ v (t) is the macroscopically observed mean velocity and

dv(t) are the instantaneous thermal fl uctuations around this mean value. _ v (t) obeys a generally non-

linear deterministic equation of the general form M d _ v (t)/dt = R[

_ v (t)], similar to the equation that

describes the hydrodynamic resistance on a small macroscopic ball settling in a liquid. Langevins assumption was that the instantaneous velocity v(t) obeys just the same equation, but with an added sto-chastic term f(t), which represents the random thermal fl uctuations of the total force that the solvent exerts on the particle. Thus, v(t) is the solution of the stochastic equation M dv(t)/dt = R[v(t)] + f(t). If, however, the colloidal particle and the supporting solvent are in thermodynamic equilibrium (_ v v (t) = 0), this equation may be linearized to read M ddv(t)/dt = [R[v]/v]v=0 dv(t) + f(t). For a

spherical particle in the absence of external fi elds the friction tensor z

0 -[R[v]/v]v=0 is diagonal and isotropic, that is, z

0 = z0I

. Thus, since v(t) = dv(t), the previous equation can be recognized as the

celebrated Langevin equation,

0d ( ) ( ) ( ),

dt

M t tt

= -z +v v f

(1.1)

in which z0 is the friction coeffi cient of the particle.The statistical properties of the random force f(t) are modeled with an extreme economy of

assumptions: f(t) is assumed to be a stationary and Gaussian stochastic process, with zero mean ( ___

f(t) = 0), uncorrelated with the initial value v(t = 0) of the velocity fl uctuations, and delta-correlated with itself,

______ f(t)f(t) = g2d(t - t) (i.e., it is a purely random, or white, noise). The stationarity

condition is in reality equivalent to the fl uctuationdissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fi xes the value of g. In fact, from Equation 1.1 and the assumed properties of f(t), we can derive the expression

______ v(t)v(t) = exp-2t/tB[

_______

_ v (0)v(0) -

g/Mz0] + g /Mz0, where tB M/z0. In equilibrium, the long-time asymptotic value g/Mz0 must coin-

cide with the equilibrium average vv = (kBT/M)I given by the equipartition theorem (with I

being

the 3 3 Cartesian unit tensor), and this fi xes the value of g to g = kBTz0I.

This set of assumptions on the statistical properties of f(t) determines the statistical properties of the solution v(t) of the stochastic differential equation in Equation 1.1, which are summarized saying that v(t) is a Gaussian stationary Markov stochastic process, that is, it is generally not delta-correlated. The specifi c results that follow from this simple mathematical model regarding properties such as the veloc-ity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48].

1.2.1.2 OrnsteinUhlenbeck ProcessesThe physical value of the Langevin equation is twofold. Firstly, it is a simple model for the Brownian motion of an isolated colloidal particle. Secondly, and far more fundamental, it defi nes a mathematical model for the description of thermal fl uctuations, which can be generalized in several directions. Thus, consider a system whose macroscopic state is described by a set of C macroscopic variables ai(t), i = 1, 2, . . . , C, which we group as the components of a C-component (column) vector a(t). The fundamental postulate of the statistical thermodynamic theory of nonequilibrium processes [49] is that the dynamics of the state vector a(t) = a(t) + da(t) constitutes a multivariate stochastic process, composed of a deter-ministic equation for its mean value a(t) and a linear stochastic equation with additive noise for the fl uctuations da(t). It is assumed that a(t) coincides with the macroscopically measured value, and that its time evolution is described by a phenomenological equation of the general form

d ( )[ ( )],

dt

tt

=

a

aR

(1.2)


where the (generally nonlinear and temporally-nonlocal) functional dependence of the C-component vector R[a(t)] on a(t) includes both, dissipative and mechanical (i.e., conservative) terms [49].

We will restrict our discussion to stationary states, that is, to the stationary solutions of the relax-ation equation above, denoted by ass, which solve the equation

d[ ] 0.

d

ssss

t= =

aaR

(1.3)

One then postulates that the fl uctuations da(t) = a(t) ass will satisfy a linearized stochastic Langevin-type version of Equation 1.2, namely,

d ( )[ ] ( ) ( ),

dsst t t

td = d +

aa a fH

(1.4)

where the C C matrix H[ass] is defi ned as

[ ][ ] ( , 1, 2, , ),

ss

ss iij

j

i j Ca

=

= a a

a

aR

H

(1.5)

and where indicates matrix product. In Equation 1.4, f(t) is a C-component stochastic vector assumed stationary, Gaussian, with zero mean (

___ f(t) = 0), uncorrelated with the initial value da(0)

of the fl uctuations [ ________

f(t)da(0) = 0, the dagger meaning transpose], and purely random, that is, _______ f(t)f (t) = g2d(t - t) with g being a C C matrix. The stationarity condition then fi xes the value of

g by means of a fl uctuationdissipation relation, that reads

{ [ ] [ ]},ss ssg = - s + s a aH H

(1.6)

in which the C C matrix s is the equal-time stationary correlation function s _________

da(t)da(t) ss. The stochastic process da(t) defi ned by the stochastic differential equation in Equation 1.4 with f(t) having the properties just described, is referred to as an OrnsteinUhlenbeck process [49], and just like the solution of the ordinary Langevin equation, it is also a Gaussian stationary Markovian stochastic process. Many time-dependent fl uctuation phenomena can be cast in terms of an OrnsteinUhlenbeck process, including the fl uctuating version of the hydrodynamic equations [50,52], the Boltzmann equation [49], the diffusion equation [53], and so on.

1.2.1.3 The Generalized Langevin EquationThe assumption that f(t) is purely random may actually be a highly restrictive and unnecessary limitation. In fact, many physical processes cannot be described by Equation 1.4 simply because they exhibit memory effects. Thus, we must introduce an extended OrnsteinUhlenbeck process, defi ned as the solution of the most general linear stochastic equation with additive noise, which we write as

0

d ( )d ( ) ( )d ( ),

d

tt

t t t t t tt

d = - d + a a fH

(1.7)

with f(t) being, as before, a stationary Gaussian stochastic process with zero mean, but no longer purely random; instead, we assume in general that its time correlation function is given by


_______ f(t)f (t) = L(t - t), and expect that the time-dependent matrix L(t) will be related with the time-

dependent relaxation kernel H(t) by some form of fl uctuationdissipation relation. Such a relation can indeed be demonstrated [22], and the resulting stochastic process da(t) then turns out to be a stationary, Gaussian, and non-Markov process.

In fact, the demonstration that the stationary condition leads to such a general fl uctuation dissipation relation also leads to very stringent and rigid conditions, of purely mathematical nature, on the structure of the relaxation matrix H(t). Thus, the so-called theorem of stationarity, states [22] that the stationarity condition alone is in fact equivalent to the condition that Equation 1.7 must be such that it can be formatted as the following general equation,

1 1

0

d ( )( ) d ( ) ( ) ( ),

d

tt

t t L t t t tt

- -d = -w s d - - s d + a a a f

(1.8)

where w is an antisymmetric matrix, w = -w, and the matrix L(t) satisfi es the fl uctuation dissipation relation

( ) ( ) ( ) (0).L t L t t= - = f f (1.9)

The linear stochastic equation with additive noise and with the structure of Equation 1.8 is referred to as the GLE. Most frequently this term is associated with the stochastic equation formally derived from a N-particle microscopic (Newtonian or Brownian) dynamic description by means of projec-tion operator techniques to describe the time-dependent thermal fl uctuations of systems in thermo-dynamic equilibrium [54]. Indeed, such an equation has exactly the same format as Equation 1.8. It is important to insist, however, that this format has a purely mathematical origin, imposed by the stationarity condition, and is certainly NOT a consequence of formally deriving this equation from an underlying microscopic level of description. In any case, the mathematical structure of the GLE, and the selection rules imposed by the symmetry properties of the matrices w and L(t) (along with other selection rules imposed by additional symmetries [22]) allow a fruitful use of the rigid format of this equation to describe complex dynamic phenomena in a rather simple manner, with complete independence of the detailed N-particle microscopic dynamics underlying the time evolution of the fl uctuations da(t). We illustrate the use of this general approach by deriving the three exact results upon which the SCGLE theory has been built.

1.2.2 GLE FOR TRACER DIFFUSION

Let us now apply the general concepts above to the description of the Brownian motion of a tracer particle that interacts with the other particles of a colloidal dispersion [23,55]. In this manner we will derive an exact result for the time-dependent friction function Dz(t), which is later given a use-ful approximate expression.

1.2.2.1 Time-Dependent Friction Function: Exact ExpressionLet us go back to Equation 1.1, but now imagine that the Brownian particle diffuses in a colloidal dispersion formed by other N particles in the volume V with which it interacts by means of pairwise direct (i.e., conservative) interactions but in the absence of hydrodynamic interactions. The pairwise force that this tracer particle (T ) exerts on particle i is given by FTi = -iu(|ri - rT|), so that Equation 1.1 now reads [1,2]

0

1

d ( )( ) ( ) ( ),

d

NT

T T i i T

i

tM t t u

t=

= -z + + | - |v v f r r

(1.10)


which may be re-written exactly as

0 3d ( ) ( ) ( ) d [ ( )] * ( , ),

dT

T T T

tM t t u r n t

t= -z + + dv v f r r (1.11)

where dn*(r, t) n*(r, t) - neq(r) is the departure of the instantaneous local concentration n*(r, t) of the other colloidal particles at time t and position r (referred to the position xT(t) of the tracer par-ticle) from its equilibrium average neq(r). Thus, the direct interactions, represented by the pair poten-tial u(r), couple exactly the motion of the tracer particle with the motion of the other particles only through the collective variable dn*(r, t), without explicitly involving the detailed position of each of the N colloidal particles.

We now need a relaxation equation that couples the time derivative of the variable dn*(r, t) with this variable and with vT (t). This must be a linear version of a generalized diffusion equation, whose structure is dictated by the rigid format of the GLE of Equation 1.8 applied to the vector da(t) [vT (t), dn*(t)], which leads to [24]

3 3 1

0

*( , )[ ( )] ( ) d d d ( , ; ) ( , ) *( , ) ( , ),

d

t

eqT

n tn r t t r r L t t n t f t

t-d = - - s d + r v r r r r r r

(1.12)

where the fi rst term on the right hand side is a linearized streaming term and f (r, t) is a fl uctuating term, related to L(r, r; t) by

__________ f(r, t)f(r, t) = L(r, r; t - t), with s-1(r, r) being the inverse of

s(r, r) dn*(r, 0)dn*(r, 0), that is, it is the solution of d3rs-1(r, r)s(r, r) = d(r - r).Equations 1.11 and 1.12 provide an exact description of the Brownian motion of the tracer par-

ticle coupled to the fl uctuations of the local concentration n*(r, t) in terms of the components of the vector da(t) [vT(t), dn*(t)]. If one is interested only in the tracers velocity, one would have to eliminate dn*(t) from this description, and the corresponding process is referred to as a contraction of the description. In the present case, this is achieved by formally solving Equation 1.12 for dn*(r, t) and substituting the solution in Equation 1.11, which then becomes

= -z + - Dz - +

vv f v F

0

0

d ( ) ( ) ( ) d ( ) ( ) ( ),d

tT

T T T T T TtM t t t t t t t

t

(1.13)

where the new fl uctuating force FT(t) is related with the time-dependent friction tensor Dz

T(t) through

_________ FT(t)FT(0) = MkBT Dz

T(t), with Dz

T(t) given by the following exact result:

3 3( ) d d [ ( )] *( , ; )[ ( )],eqT t r r u r t n rDz = - r r

X

(1.14)

where X*(r, r; t) is the propagator, or Greens function, of the diffusion equation in Equation 1.12, that is, it solves the equation

3 3 1

0

*( , ; )d d d ( , ; ) ( , ) *( , ; ),

tt

t r r L t t tt

- = - - s r r

r r r r r rX

X

(1.15)

with initial condition X*(r, r; t = 0) = d(r r). Notice that, since the initial value dn*(r, t = 0) is statistically independent of vT(t) and f(r, t), the time-correlation function G*(r, r; t)


dn*(r, t)dn*(r, 0) is also a solution of the same equation with initial value G*(r, r; t = 0) = s(r, r). The function G*(r, r; t) is just the van Hove function of the colloidal particles in the presence of the external fi eld u(r) of the tracer particle fi xed at the origin, and as described from the tracer parti-cles reference frame, which executes Brownian motion.

1.2.2.2 Simplifying Approximations for DzDzT(t)To simplify the notation, let us rewrite Equation 1.14 as

( ) [ ] *( ) [ ],eqT t u t nDz = -

X (1.16)

where the inner product A B, between two arbitrary functions A and B, is defi ned by the convolu-tion d3rA(r)B(r). With this notation, let us recall an additional exact relation between the vectors u, neq and the matrix s. This is the so-called WertheimLovett relation [23,56] of the equilibrium theory of inhomogeneous fl uids,

[ ] [ ],eqn u = - bs (1.17)

with b (kBT)-1 This relation, along with the defi nition of the inverse matrix s-1 (s -1 s = I, with I being the identity matrix), allows us to write Equation 1.14 in a variety of different but equivalent and exact manners. In particular, let us consider the following:

1 1

B( ) [ ] *( ) [ ],eq eq

T t k T n G t n- -Dz = s s

(1.18)

where we have used the fact that the van Hove function G*(t) can be written as G*(t) = X*(t) s.This exact result for Dz

T(t) may be given a more practical form by introducing some simplifying

assumptions on the general properties of the functions G*(r, r; t) and s(r, r). The latter is the two-particle distribution function of the colloidal particles surrounding the tracer particle which are, hence, subjected to the external fi eld u(r) exerted by this tracer particle. Thus, it is effectively a three-particle correlation function. Only if one ignores the effects of such an external fi eld, can one write s(r, r) = s(|r - r|) nd(r - r) + n2[g(|r - r|) - 1], where g(r) is the bulk radial distri-bution function of the colloidal particles. Similarly, we may also approximate G*(r, r; t) by nG*(|r - r|; t). This is referred to as the homogeneous fl uid approximation [17], which allows us to write G*(r, r; t) = [n/(2p)3] d3k exp[-ik r]F*(k, t), with F*(k, t) N-1/ N i,j exp[ik [ri(t) - rj(0)]]. Notice also that in particular G*(r, r; t = 0) = s(|r - r|) = [n/(2p)3] d3k exp[-ik r]S(k), where S(k) 1 + n d3r exp[-ik r][g(r) - 1] is the static structure factor.

The function F*(k, t) just defi ned is the intermediate scattering function, except for the asterisk, which indicates that the position vectors ri(t) and rj(0) have their origin in the center of the tracer particle. Denoting by xT(t), the position of the tracer particle referred to a laboratory-fi xed reference frame, we may re-write F*(k, t) as

1

,

*( , ) [exp( [ ( ) (0)])] exp( [ ( ) (0)])] ,N

T T i j

i j

F k t i t N i t-

- -

k x x k x x

(1.19)

where xi(t) is the position of the ith particle in the laboratory reference frame. Approximating the average of the product in this expression by the product of the averages, leads to F*(k, t) = F(k, t)F(S)(k, t), which we refer to as the decoupling approximation [23].


The two approximations just described may now be introduced in Equation 1.18, and this leads to an approximate expression for Dz

T(t). Since for spherical particles, the tensor Dz

T(t) must be

diagonal, Dz

T(t) = DzT(t)I, such an approximate result can be written as

20* 0 ( )

3

[ ( ) 1]( ) ( ) / d ( , ) ( , ),

( )3(2 )ST

T T T

D k S kt t F k t F k t

S kn

- Dz Dz z = p k (1.20)

where DT0 kBT/zT0. This result will be employed below as one of the three main ingredients of the

SCGLE theory.

1.2.3 COLLECTIVE DIFFUSION

Let us now describe the application of the GLE formalism to the description of collective diffusion. As a result, we shall derive exact memory-function expressions for the intermediate scattering func-tion F(k, t) and for its self-diffusion counterpart F (S)(k, t). We then explain the approximations that transform these exact results in an approximate self-consistent system of equations for these properties.

1.2.3.1 Exact Memory-Function Expressions for F(k, t) and F (S )(k, t)One can also use the GLE method to derive the most general expression for the collective intermedi-ate scattering function F(k, t) of a colloidal dispersion in the absence of external fi elds. For this, consider again Equation 1.8, but now with the vector da(t) defi ned as da(t) [dn(k, t), djl(k, t)] with dn(k, t) being the FT of the fl uctuations dn(r, t) and djl(k, t) = jl(k, t) k j(k, t) being the longitu-dinal component of the current j( k, t). The variable dn(k, t) is normalized such that its equal time correlation is dn(k, 0)dn(-k, 0) = S(k), where S(k) is the static structure factor of the bulk suspen-sion. The continuity equation,

d = d( , )

( , ),ln t

ik j ttk

k

(1.21)

couples the time derivative of dn(k, t) with the current fl uctuations. This suggests using the format of Equation 1.10 to determine the most general time evolution equation of the current (as carried out in detail in reference [17]), with the following result

1B

B0

( , )( ) ( , ) ( , ) ( , ) d ( , ),

t

ll l

j t k T Mik S k n t L k t t j t t f t

t M k T-d = d - - d +

kk k k

(1.22)

where fl(k, t) is a stochastic term whose time-correlation function is given by _____________

fl(k, t)fl(-k, t) = L(k, t - t). It can be shown [11], that in the absence of interactions, the exact value of the memory function L(k, t) is (kBTz0/M2)2d(t). Thus, we write L(k, t) = (kBTz0/M2)2d(t) + DL(k, z), where the last term represents the contribution of the direct interactions to the particle current relaxation.

The next step is to contract the description, that is, solve Equation 1.22 for djl(k, t) and insert the solution in the continuity equation, to obtain an equation for dn(k, t) alone. In the resulting equation, we must take the limit of overdamping, t tB M/z0, since we are interested in describing only the diffusive regime. The resulting equation is also an equation for F(k, t), which in Laplace space reads

2 1

0

( )( , ) ,

{[ ( )] / [1 ( , )]}

S kF k z

z k D S k C k z-=

+ + (1.23)


with D0 KBT/z0 and C(k, z) D0M2b2DL(k, z). This equation describes the diffusive collective dynamics of the suspension.

Let us mention that by proceeding in an entirely analogous manner one can also derive a similar result for the self intermediate scattering function F(S)(k, t). Such an equation reads

( )2 ( )

0

1( , ) .

{[ ] / [1 ( , )]}S

SF k z z k D C k z=

+ + (1.24)

In the diffusive regime, Equations 1.23 and 1.24 are exact expressions for F(k, t) and F (S)(k, t) in terms of the so-called irreducible memory functions C(k, t) and C(S)(k, t).

1.2.4 APPROXIMATE ELEMENTS OF THE SELF-CONSISTENT THEORY

The exact expressions for F(k, t) and F(S)(k, t) (Equations 1.23 and 1.24) involve the two unknown memory functions C(k, z) and C(S)(k, z). The determination of these properties requires two addi-tional independent relations, which we now discuss.

1.2.4.1 Vineyard-Like ApproximationsThe fi rst such relation involving the irreducible memory functions is based on a physically intuitive notion: Brownian motion and diffusion are two intimately related concepts; we might say that collective diffusion is the macroscopic superposition of the Brownian motion of many individual colloidal particles. It is then natural to expect that collective diffusion should be related in a simple manner to self-diffusion. In the original proposal of the SCGLE theory [18], such connections were made at the level of the memory functions. Two main possibilities were then considered, referred to as the additive and the multiplicative Vineyard-like approximations. The fi rst approximates the difference [C(k, z) - C(S)(k, z)], and the second the ratio [C(k, z)/C(S)(k, z)], of the memory functions, by their exact short-time limits, using the fact that the exact short-time values, CSEXP(k, t) and C(S)SEXP(k, t), of these memory functions are known in terms of equilibrium structural properties [18]. The label SEXP refers to the single exponential time dependence of these memory functions.

The multiplicative approximation, defi ned as

SEXP( )

( )SEXP

( , )( , ) ( , ),

( , )S

S

C k zC k z C k z

C k z

=

(1.25)

was devised to describe more accurately the very early relaxation of F(k, t) [19] by incorporating the exact short-time behavior up to order t3 in the resulting intermediate scattering function. This advan-tage, however, is only meaningful in the short-time regime. At longer times the additive approxima-tion, defi ned as

( ) SEXP ( )SEXP( , ) ( , ) [ ( , ) ( , )],S SC k t C k t C k t C k t= + - (1.26)

was found to provide similar results. The main advantage of the additive approximation appears, however, when these approximations are applied to the description of dynamic arrest phenomena.

1.2.4.2 Closure RelationEither of these Vineyard-like approximations, along with an additional closure relation, will allow the exact results for Dz(t), F(k, t), and F (S)(k, t) to constitute a closed set of equations. The closure relation consists of an independent approximate determination of the self irreducible memory func-tion C(S)(k, t). One intuitive notion behind the proposed closure relation is the expectation that the k-dependent self-diffusion properties, such as F(S)(k, t) itself or its memory function C(S)(k, t), should


be simply related to the properties that describe the Brownian motion of individual particles, just like the Gaussian approximation [1,2] expresses F(S)(k, t) in terms of the msd

______ (Dx(t))2 as

F(S)(k, t) = exp[-k2( ______

Dx(t))2 /2]. An analogous approximate connection is considered here, but at the level of their respective memory functions. A memory function of

______ (Dx(t))2 is the time-dependent

friction function Dz(t). This function, normalized by the solvent friction z0, is the exact long wave-length limit of C(S)(k, t), that is, limk0 C(S)(k, t) = Dz*(t)/z0. Thus, the proposal was to interpolate C(S)(k, t) between its two exact limits, namely,

= + Dz - l( ) ( )SEXP ( )SEXP( , ) ( , ) [ *( ) ( , )] ( ),S S SC k t C k t t C k t k (1.27)

where l(k) is a phenomenological interpolating function such that l(k 0) = 1 and l(k ) = 0. In the absence of rigorous fundamental guidelines to construct this interpolating function, l(k) was chosen to represent the optimum mixing of these two limits of C(S)(k, t) in the simplest possible analytical manner. Guided by these practical considerations, in reference [18] the proposal was made to model l(k) by the functional form l(k) [1 + (k/kc)n]-1, and the parameters n and kc were empirically calibrated to the values n = 2 and kc = kmin, with kmin being the position of the fi rst mini-mum that follows the main peak of the static structure factor S(k). Thus, the interpolating function employed in the closure relation above reads

2 1

min( ) [1 ( / ) ] .k k k-l + (1.28)

1.3 SELF-CONSISTENT THEORY AND ILLUSTRATIVE APPLICATIONS

We now have all the elements needed to defi ne a self-consistent system of equations to describe the full dynamic properties of a colloidal dispersion in the absence of hydrodynamic interactions. In this section we summarize the relevant equations for both, mono-disperse and multicomponent suspensions, and review some illustrative applications. The general results for Dz(t), F(k, t), and F(S)(k, t) in Equations 1.20, 1.23, and 1.24, complemented by either one of the Vineyard-like approxi-mations in Equations 1.25 and 1.26, and with the closure relation in Equation 1.27, constitute the full self-consistent GLE theory of colloid dynamics for monodisperse systems. Besides the unknown dynamic properties, it involves the properties S(k), CSEXP(k, t), and C(S)SEXP(k, t), assumed to be deter-mined by the methods of equilibrium statistical thermodynamics. However, as we will see below, a simplifi ed version of the SCGLE theory, in which the short-time memory functions CSEXP(k, t) and C(S)SEXP(k, t) are neglected, happens to be essentially as accurate, but much simpler in practice, since it requires only S(k) as input.

1.3.1 MODEL MONO-DISPERSE SUSPENSIONS

In reference [19], a systematic comparison between the predictions of the SCGLE theory and the corresponding computer simulation data for four idealized model systems was reported. The fi rst two were two-dimensional systems with power law pair interaction, bu(r) = A/rn, with n = 50 (i.e., strongly repulsive, almost hard-disk like) and with n = 3 (long-range dipoledipole interaction). The third one was the three-dimensional weakly screened repulsive Yukawa potential (whose two- dimensional version had been studied in reference [18]). The last system considered involved short-ranged, soft-core repulsive interactions, whose dynamic equivalence with the strictly hard-sphere system allowed discussion of the properties of the latter reference system. For all these systems G(r, t) and/or F(k, t) were calculated from the self-consistent theory, and Brownian dynamics simu-lations (without hydrodynamic interactions) were performed to carry out extensive quantitative comparisons. In all those cases, the static structural information [i.e., g(r) and S(k)] needed as an input in the dynamic theories was provided by the simulations. The aim of that exercise was to


isolate one of the most important effects in the relaxation of the concentration fl uctuations in col-loidal liquids, namely, the conservative direct interaction forces between the colloidal particles. In what follows we review two illustrative examples of the comparisons just described.

1.3.1.1 Two-Dimensional Model System with DipoleDipole (r -3) InteractionsThe fi rst example involves a two-dimensional model system with dipoledipole (r-3) interactions. This model corresponds, as far as the interactions are concerned, to the quasi-two-dimensional system of paramagnetic colloidal particles studied by Zahn et al. [57], defi ned by the pair potential bu(r) = G(l/r)3, with l being the mean interparticle distance l n-1/2 and G being the ratio of the potential at mean distance in units of kBT. The specifi c conditions considered below refer to a highly interacting (G = 4.4) and very dilute (n* ns2 = 0.041) suspension.

For this system G(r, t) and F(k, t) were calculated theoretically for short and intermediate times, and compared with the corresponding simulation data illustrates a comparison for F(k, t) typical of the intermediate-time regime (t/t0 = 0.0833 and 1.666, with t0 = s2/D0). This fi gure also includes the results of the single exponential (SEXP) approximation, which corresponds to setting l(k) = 0 in the SCGLE theory. For the conditions of the fi gure, the limitations of this simpler theory are already quite evident. This comparison indicates that, although there are small systematic differences with respect to the exact (simulation) data, these are not appreciable within the resolution of Figure 1.1. Analogous differ-ences were also virtually negligible in the case of the two-dimensional repulsive Yukawa system that was employed in reference [18] to calibrate the only element of the theory that could not be determined from more basic principles, namely, the interpolating function l(k) of Equation 1.28.

1.3.1.2 Three-Dimensional Soft Sphere SystemsThe second illustrative example refers to a three-dimensional system of Brownian particles interact-ing through a strongly repulsive and short-ranged, pair potential u(r) of the form

2

1 2( ) 1

( / ) ( / )s su r

r rm mb = - +

s s (1.29)

FIGURE 1.1 Intermediate scattering function F(k, t) as of the two-dimensional fl uid of magnetic particles with pairwise interactions given by bu(r) = G(l/r)3 with G = 4.4 and reduced number concentration n* = 0.041 function of k for t = 0 (static structure factor S(k), dotted line), t/t0 = 0.0833, and t/t0 = 1.666, according to the SCGLE theory (solid line) and the SEXP approximation (dashed line), compared with the corresponding Brownian dynamics results (symbols). (From Yeomans-Reyna, L. et al. 2003. Phys. Rev. E 67: 021108. With permission.)

0.00 1 2

ks3 4

0.5

1.0

F(k,t

)

1.5

2.0

2.5


for 0 < r < ss, and assumed to vanishes for r > ss. In this equation, m is a positive integer. This potential and its derivative strictly vanish at, and beyond, ss. In reference [19], the specifi c case m = 18 at a soft-sphere volume fraction fs pnss3/6 = 0.5146 was considered. In Figure 1.2, the comparison between the results of the SCGLE theory and the Brownian dynamics simulations for F(k, t) is presented at three different values of the correlation time (in units of ts ss2/D0).

1.3.2 EXTENSION TO COLLOIDAL MIXTURES

For a colloidal mixture with n species, the dynamic properties can be described in terms of the partial intermediate scattering functions, defi ned as Fab(k, t) dna(k, t)dnb(-k, 0), where dna(k, t) (1/

___ Na )

N i=l exp[ik ri

(a)(t)], with ri(a)(t) being the position of particle i of species a at time t. One can also defi ne the self component of Fab(k, t), referred to as the self-intermediate scattering function, as F (S) ab (k, t) dabexp[ik DR

(a)(t)], where DR(a)(t) is the displacement of any of the Na particles of species a over a time t, and dab is Kroneckers delta function. Here we summarize the extension of the SCGLE theory to multicomponent systems, which was developed in reference [21].

1.3.2.1 General ResultsThe general result for the time-dependent friction function DzT(t) in Equation 1.20 was extended to colloidal mixtures in reference [58]. Thus, if Dza(t) is the time-dependent friction function on a tracer particle of species a due to its direct interactions with the other particles in the mixture, such extension reads

-aa a a aa aa

Dz Dz z = p 0

* 0 3 2 ( ) 13( ) ( ) / d [ ( )] ( ) ,3(2 )

SDt t kk F t c nF t S nh

(1.30)

where the k-dependent elements of the n n matrices F(t), F(S)(t), S, h and c are, respectively, the collective and self-intermediate scattering functions Fab(k, t), F

(S) ab (k, t), the partial static structure

factors Sab(k) = Fab(k, t = 0), and the FTs hab(k) and cab(k) of the OrnsteinZernike total and direct correlation functions, respectively. Thus, h and c are related to S by S = I +

__ n h

__ n = [I -

__ nc

__ n ]-1,

FIGURE 1.2 Intermediate scattering function F(k, t) for the soft sphere system in Equation 1.29 with m = 18 and f = 0.5146 at t/t0 = 0.006559, 0.02623, and 0.05247. SCGLE theory: solid, dashed, and dotted lines; BD results: open circles, squares, and triangles. (From Yeomans-Reyna, L. et al. 2003. Phys. Rev. E 67: 021108. With permission.)

0.00 10 20

ks

0.5

1.0

F(k,t

) 1.5

2.0

2.5


with the matrix __

n defi ned as [ __

n ]ab dab __

n a. In these equations, na is the number concentration of species a and Da0 = kBT/za0 is the free-diffusion coeffi cient of particles of that species.

The exact memory function expressions for F(k, t) and F(S)(k, t) in Equations 1.23 and 1.24 were extended to colloidal mixtures in reference [20]. Written in matrix form and in Laplace space, these exact expressions for the matrices F(k, t) and F (S)(k, t) (when convenient, their k-dependence will be explicitly exhibited) in terms of the corresponding irreducible memory function matrices C(k, t) and C(S)(k, t) read

1 2 1 1( , ) { ( ( , )) ( )} ( )F k z z I C k z k DS k S k- - -= + + (1.31)

and

- -= + +( ) ( ) 1 2 1( , ) { ( ( , )) } ,S SF k z z I C k z k D (1.32)

where D is the diagonal matrix Dab dabDa0. Notice also that all the matrices involved in the equa-tion for F(S)(k, z) are diagonal. Thus, we shall denote by Fa(S)(k, z) and Ca(S)(k, z) the ath diagonal element of F(S)(k, z) and C(S)(k, z), respectively.

These exact results are complemented with either the multiplicative Vineyard-like connection between C(k, z) and C(S)(k, z), defi ned as C(k, z)CSEXP(k, z)-1 = C(S)(k, z)C(S)SEXP(k, z)-1, or with the additive Vineyard approximation, which reads

( ) SEXP ( )SEXP( , ) ( , ) [ ( , ) ( , )],S SC k z C k z C k z C k z= + - (1.33)

where CSEXP(k, z) and C(S)SEXP(k, z) are the single exponential approximation of these memory func-tions, also defi ned in reference [21]. Following the mono-component version of the self-consistent theory [18], the proposal was made in reference [21] to interpolate Ca(S)(k, z) between its two exact limits by means of the following interpolating formula

( ) ( )SEXP ( )SEXP*( , ) ( , ) [ ( ) ( , )] ( ),S S SC k z C k z k C k z ka a a a a= + Dz - l (1.34)

where la(k) is a phenomenological interpolating function defi ned as

( ) 12min( ) 1 ( / ) ,k k k

-aa

l = + (1.35)

with k (a) min being the position of the fi rst minimum (beyond the main peak) of the partial static struc-ture factor Saa(k). The time-dependent friction Dz*(t) is a diagonal matrix whose diagonal element Dza*(t) is given by Equation 1.30. The set of Equations 1.30 through 1.35 thus constitute the multi-component extension of the SCGLE theory, whose applications are now reviewed.

1.3.2.2 Binary Mixture of Charged ParticlesThe dynamics of colloidal mixtures provided by the previous extension of the SCGLE theory was illustrated in reference [21] with its application to a binary mixture of particles interacting through a hard-core pair potential of diameter a (assumed to be the same for both species), and a repulsive Yukawa tail of the form

- s -

ab a bb = s

[( / ) 1]

( ) ./

z reu r K K

r (1.36)


The dimensionless parameters that defi ne the thermodynamic state of this system are the total volume fraction f (p/6)ns3 (with n being the total number concentration, n = n1 + n2), the rela-tive concentrations xa na/n, and the potential parameters K1, K2, and z. The free-diffusion coef-fi cients Da0 are also assumed identical for both species, that is, D10 = D20 = D0. Explicit values of the parameters s and D0 are not needed, since the dimensionless dynamic properties, such as Fab(k, t), only depend on the dimensionless parameters specifi ed above, when expressed in terms of the scaled variables ks and t/t0, where t0 s2/D0. Besides solving the SCGLE scheme, in reference [21] Brownian dynamics simulations were generated for the static and dynamic properties of the system above.

In this manner all the dynamic properties that derive from Fab(k, z) and F (S)

ab (k, z) can be calcu-lated. These include the distinct intermediate scattering functions F (D) ab (k, z) Fab(k, z) - F

(S) ab (k, z),

the msd W (a)(t) (Dr(a)(t))2 of particles of species a, and the time-dependent diffusion coeffi cient Da(t) Wa(t)/6t. The solution of the multiplicative and the additive versions of the SCGLE theory were also calculated in order to see the actual quantitative superiority of either of them. Figure 1.3 presents the results for the dynamics of the more interacting species of a mixture with K1 = 10 and K2 = 10

__ 5 . Concerning the decay of the intermediate scattering functions, at times of the order of

t = 10t0, we see that the SEXP approximation exhibits large departures from the simulated data and

FIGURE 1.3 Application of the SCGLE to a bidisperse system with K1 = 10, K2 = 10 __

5 , and f1 = f2 = 2.2 10-4. The time-dependent diffusion coeffi cients D1(t) and D2(t), are shown normalized with their initial value Da0 (the upper curves corresponding to species 1). The self, the distinct, and the total intermediate scat-tering functions of the more interacting species (species 2) are shown, for the times t = t0 (upper set of curves for FS(k, t) and F(k, t), and more structured curves for Fd(k, t)) and t = 10t0. This fi gure compare results of the SCGLE theory within the additive (solid lines) and the multiplicative (dot-dashed lines) Vineyard-like approx-imations. For reference, also the results of the SEXP approximation (dashed lines) are presented. (From Chvez-Rojo, M. A. and Medina-Noyola, M. 2005. Phys. Rev. E 72: 031107; Phys. Rev. E 76: 039902, 2007. With permission.)

420 6t/t00.4

0.6

0.8

1

D(t)/

D0

0.5 1 1.5

0.5

0

0.5

Fd2(

k,t)

0 0.5 1 1.5 2k k

k

0

0.5

1

Fs2(

k,t)

Fs2(

k,t)

0.5 1 1.50

0.5

1

1.5BDSEXPSCGLE-VaddSCGLE-Vmult


that the two versions of the SCGLE bracket the simulation data. More detailed calculations, how-ever, reveal a slight superiority of the simpler additive SCGLE theory. This type of comparisons was made for other systems, with similar conclusion.

1.3.3 SIMPLIFIED SCGLE THEORY

The main conclusion of the previous comparisons is that, except for very short times, in reality there is no practical reason for preferring the multiplicative version of the SCGLE theory over its additive counterpart, particularly if we are interested in intermediate and long times. Since the additive approximation is numerically simpler to implement, we shall no longer refer to the multiplicative approximation. Still, one of the remaining practical diffi culties of the SCGLE theory is the involve-ment of the SEXP irreducible memory functions CSEXP(k, z) and C(S)SEXP(k, z); the need to previously calculate these properties constitutes a considerable practical barrier for the application of the SCGLE theory. It was recently discovered [59], however, that a simplifi ed version of this theory, in which this short-time information is eliminated, leads to essentially the same results at intermediate and long times. The simplifi ed version is suggested by the form that the distinctive equations of the theory, that is, the Vineyard approximation, Equation 1.33, and the closure relation, Equation 1.34, attain for times longer than the relaxation time of the functions CSEXP(k, t) and C(S)SEXP(k, t). For such long times, these equations become, respectively,

C(k, t) = C(S)(k, t) (1.37)

and

C(S)(k, t) = [Dz*(t)]l(

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