Structure and Functional Properties of Colloidal Systems, Volume 146 (Surfactant Science)

STRUCTURE AND FUNCTIONAL PROPERTIES OF COLLOIDAL SYSTEMS SURFACTANT SCIENCE SERIES FOUNDING EDITOR MARTIN J. SCHICK...

Author: Roque Hidalgo-Alvarez

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STRUCTURE AND FUNCTIONAL PROPERTIES OF COLLOIDAL SYSTEMS

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 Massachusetts Institute of Technology Cambridge, Massachusetts

ERIC W. KALER Department of Chemical Engineering 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 University of Oklahoma Norman, Oklahoma

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

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, 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 Datyner 15. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applications, edited by Ayao Kitahara and Akira 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 Naim 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 I. B. Ivanov 30. 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 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. Scamehorn and Jeffrey H. Harwell

34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69.

Cationic Surfactants: Organic Chemistry, edited by James M. Richmond Alkylene Oxides and Their Polymers, F. E. Bailey, Jr., and Joseph V. Koleske Interfacial Phenomena in Petroleum Recovery, edited by Norman R. Morrow Cationic Surfactants: Physical Chemistry, edited by Donn N. Rubingh and Paul M. Holland Kinetics and Catalysis in Microheterogeneous Systems, edited by M. Grätzel and K. Kalyanasundaram Interfacial Phenomena in Biological Systems, edited by Max Bender Analysis of Surfactants, Thomas M. Schmitt (see Volume 96) Light Scattering by Liquid Surfaces and Complementary Techniques, edited by Dominique Langevin Polymeric Surfactants, Irja Piirma Anionic Surfactants: Biochemistry, Toxicology, Dermatology, Second Edition, Revised and Expanded, edited by Christian Gloxhuber and Klaus Künstler Organized Solutions: Surfactants in Science and Technology, edited by Stig E. Friberg and Björn Lindman Defoaming: Theory and Industrial Applications, edited by P. R. Garrett Mixed Surfactant Systems, edited by Keizo Ogino and Masahiko Abe Coagulation and Flocculation: Theory and Applications, edited by Bohuslav Dobiás Biosurfactants: Production Properties Applications, edited by Naim Kosaric Wettability, edited by John C. Berg Fluorinated Surfactants: Synthesis Properties Applications, Erik Kissa Surface and Colloid Chemistry in Advanced Ceramics Processing, edited by Robert J. Pugh and Lennart Bergström Technological Applications of Dispersions, edited by Robert B. McKay Cationic Surfactants: Analytical and Biological Evaluation, edited by John Cross and Edward J. Singer Surfactants in Agrochemicals, Tharwat F. Tadros Solubilization in Surfactant Aggregates, edited by Sherril D. Christian and John F. Scamehorn Anionic Surfactants: Organic Chemistry, edited by Helmut W. Stache Foams: Theory, Measurements, and Applications, edited by Robert K. Prud’homme and Saad A. Khan The Preparation of Dispersions in Liquids, H. N. Stein Amphoteric Surfactants: Second Edition, edited by Eric G. Lomax Nonionic Surfactants: Polyoxyalkylene Block Copolymers, edited by Vaughn M. Nace Emulsions and Emulsion Stability, edited by Johan Sjöblom Vesicles, edited by Morton Rosoff Applied Surface Thermodynamics, edited by A. W. Neumann and Jan K. Spelt Surfactants in Solution, edited by Arun K. Chattopadhyay and K. L. Mittal Detergents in the Environment, edited by Milan Johann Schwuger Industrial Applications of Microemulsions, edited by Conxita Solans and Hironobu Kunieda Liquid Detergents, edited by Kuo-Yann Lai Surfactants in Cosmetics: Second Edition, Revised and Expanded, edited by Martin M. Rieger and Linda D. Rhein 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 Sørensen 80. Interfacial Phenomena in Chromatography, edited by Emile Pefferkorn 81. Solid–Liquid Dispersions, Bohuslav Dobiás, Xueping Qiu, and Wolfgang von Rybinski 82. Handbook of Detergents, editor in chief: Uri Zoller Part A: Properties, edited by Guy Broze 83. Modern 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. Silicone 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 Malgorzata Borówko 90. Adsorption on Silica Surfaces, edited by Eugène Papirer 91. Nonionic Surfactants: Alkyl Polyglucosides, edited by Dieter Balzer and Harald Lüders 92. Fine Particles: Synthesis, Characterization, and Mechanisms of Growth, edited by Tadao Sugimoto 93. Thermal Behavior of Dispersed Systems, edited 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, edited by Alexander G. 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. Friedli 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, edited 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 Carel J. van Oss 106. Interfacial Electrokinetics and Electrophoresis, edited by Ángel V. Delgado 107. Adsorption: Theory, Modeling, and Analysis, edited by József Tóth 108. Interfacial Applications in Environmental Engineering, edited by Mark A. Keane 109. Adsorption and Aggregation of Surfactants in Solution, edited by K. L. Mittal and Dinesh O. 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. Rusling 112. Structure–Performance Relationships in Surfactants: Second Edition, Revised and Expanded, edited by Kunio Esumi and Minoru Ueno 113. Liquid Interfacial Systems: Oscillations and Instability, Rudolph V. Birikh, Vladimir A. Briskman, Manuel G. Velarde, and Jean-Claude Legros 114. Novel Surfactants: Preparation, Applications, and Biodegradability: Second Edition, Revised and Expanded, edited by Krister Holmberg 115. Colloidal Polymers: Synthesis and Characterization, edited by Abdelhamid Elaissari 116. Colloidal Biomolecules, Biomaterials, and Biomedical Applications, edited by Abdelhamid Elaissari 117. Gemini Surfactants: Synthesis, Interfacial and Solution-Phase Behavior, and Applications, edited by Raoul Zana and Jiding Xia 118. Colloidal Science of Flotation, Anh V. Nguyen and Hans Joachim Schulze 119. Surface and Interfacial Tension: Measurement, Theory, and Applications, edited by Stanley Hartland 120. Microporous Media: Synthesis, Properties, and Modeling, Freddy Romm 121. Handbook of Detergents, editor in chief: Uri Zoller, Part B: Environmental Impact, edited by Uri Zoller 122. Luminous Chemical Vapor Deposition and Interface Engineering, HirotsuguYasuda 123. Handbook of Detergents, editor in chief: Uri Zoller, Part C: Analysis, edited by Heinrich Waldhoff and Rüdiger Spilker 124. Mixed Surfactant Systems: Second Edition, Revised and Expanded, edited by Masahiko Abe and John F. Scamehorn 125. Dynamics of Surfactant Self-Assemblies: Micelles, Microemulsions, Vesicles and Lyotropic Phases, edited by Raoul Zana 126. Coagulation and Flocculation: Second Edition, edited by Hansjoachim Stechemesser and Bohulav Dobiás 127. Bicontinuous Liquid Crystals, edited by Matthew L. Lynch and Patrick T. Spicer 128. Handbook of Detergents, editor in chief: Uri Zoller, Part D: Formulation, edited by Michael S. Showell 129. Liquid Detergents: Second Edition, edited by Kuo-Yann Lai 130. Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering, edited by Aleksandar M. Spasic and Jyh-Ping Hsu 131. Colloidal Silica: Fundamentals and Applications, edited by Horacio E. Bergna and William O. Roberts 132. Emulsions and Emulsion Stability, Second Edition, edited by Johan Sjöblom

133. Micellar Catalysis, Mohammad Niyaz Khan 134. Molecular and Colloidal Electro-Optics, Stoyl P. Stoylov and Maria V. Stoimenova 135. 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 Pal 137. Powders and Fibers: Interfacial Science and Applications, edited by Michel Nardin and Eugène Papirer 138. Wetting and Spreading Dynamics, edited by Victor Starov, Manuel G. Velarde, and Clayton Radke 139. Interfacial Phenomena: Equilibrium and Dynamic Effects, Second Edition, edited by Clarence A. Miller and P. Neogi 140. Giant Micelles: Properties and Applications, edited by Raoul Zana and Eric W. Kaler 141. Handbook of Detergents, editor in chief: Uri Zoller, Part E: Applications, edited by Uri Zoller 142. Handbook of Detergents, editor in chief: Uri Zoller, Part F: Production, edited by Uri Zoller and co-edited by Paul Sosis 143. Sugar-Based Surfactants: Fundamentals and Applications, edited by Cristóbal Carnero Ruiz 144. Microemulsions: Properties and Applications, edited by Monzer Fanun 145. Surface Charging and Points of Zero Charge, Marek Kosmulski 146. Structure and Functional Properties of Colloidal Systems, edited by Roque Hidalgo-Álvarez

STRUCTURE AND FUNCTIONAL PROPERTIES OF COLLOIDAL SYSTEMS

Edited by

Roque Hidalgo-Álvarez University of Granada Granada, Spain

Boca Raton London New York

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

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-8446-7 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. 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 2009 541’.345--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2009028499

Contents Preface .......................................................................................................................................... xv Editor ......................................................................................................................................... xvii Contributors ................................................................................................................................ xix

PART I: Theory Chapter 1

Colloid Dynamics and Transitions to Dynamically Arrested States ....................... 3 R. Juárez-Maldonado and M. Medina-Noyola

Chapter 2

Capillary Forces between Colloidal Particles at Fluid Interfaces .......................... 31 Alvaro Domínguez

PART II: Structure Chapter 3

Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations ........ 63 A. Martín-Molina, M. Quesada-Pérez, and R. Hidalgo-Álvarez

Chapter 4

Effective Interactions of Charged Vesicles in Aqueous Suspensions .................... 77 C. Haro-Pérez, L.F. Rojas-Ochoa, V. Trappe, R. Castañeda-Priego, J. Estelrich, M. Quesada-Pérez, José Callejas-Fernández, and R. Hidalgo-Álvarez

Chapter 5

Structure and Colloidal Properties of Extremely Bimodal Suspensions ............... 93 A.V. Delgado, M.L. Jiménez, J.L. Viota, R. Rica, M.T. López-López, and S. Ahualli

Chapter 6

Structure and Stability of Filaments Made up of Microsized Magnetic Particles ............................................................................. 117 F. Martínez-Pedrero, A. El-Harrak, María Tirado-Miranda, J. Baudry, Artur Schmitt, J. Bibette, and José Callejas-Fernández

Chapter 7

Glasses in Colloidal Systems: Attractive Interactions and Gelation .................... 135 Antonio M. Puertas and Matthias Fuchs

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Chapter 8

Contents

Phase Behavior and Structure of Colloidal Suspensions in Bulk, Confinement, and External Fields ........................................................................ 165 A.-P. Hynninen, A. Fortini, and M. Dijkstra

Chapter 9

Water–Water Interfaces ........................................................................................ 201 R. Hans Tromp

Chapter 10 Interfacial Phenomena Underlying the Behavior of Foams and Emulsions ......... 219 Julia Maldonado-Valderrama, Antonio Martín-Rodríguez, Miguel A. Cabrerizo-Vílchez, and María Jose Gálvez 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 María Tirado-Miranda, Miguel A. Rodríguez-Valverde, Artur Schmitt, José Callejas-Fernández, and Antonio Fernández-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

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Contents

Chapter 19 Droplet-Based Microfluidics: 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 Gel–Glass Dispersed Liquid Crystals Devices by Chemical Modification 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 Solè, Alejandro Fernández-Arteaga, Jordi Nolla, Núria Azemar, José Gutiérrez, Alicia Maestro, Carmen González, and Carmen M. Pey Index .......................................................................................................................................... 483

Preface This book covers important aspects of colloidal systems that have received significant 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 developments of synthesis and conditioning, while sharpening the reader’s mind for the practically unlimited possibilities of application is absolutely timely. This field 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 connection 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 field and bridges the gap between fundamentals and applications. For advanced students a book is needed to describe the connection between techniques to functionalize colloids, the characterization methods, the physical fundamentals of structure formation, diffusion dynamics, transport properties in equilibrium, the physical fundamentals of nonequilibrium 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 first two chapters (by Medina-Noyola et al. and Domínguez, respectively) deal explicitly with theoretical aspects of colloid dynamics and transitions to dynamically arrested states and capillary forces in colloidal systems at fluid interfaces. The second part covers the structural aspects of different colloidal systems. Chapters 3 and 4, by Martín-Molina et al. and Haro-Pérez et al., deal with electric double layers and effective interactions. Chapters 5 and 6, by Delgado et al. and Martínez-Pedrero et al., explore the structure of extremely bimodal suspensions and filaments 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 interactions, confinement, and external fields 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 finishes with Chapter 11, by de Vicente, which analyzes the rheological properties of structured fluids in order to establish a connection between measured material rheological 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; microfluidics by Petsev et al.; gel–glass 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.

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Preface

Finally, I gratefully acknowledge financial support from Ministerio de Educación y Ciencia (Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (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 Andalucía.

Editor Roque Hidalgo-Álvarez 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 emphasis on electrokinetic phenomena and colloidal stability. Dr. Hidalgo-Álvarez has published 212 scientific papers in international journals and has supervised 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.

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Contributors S. Ahualli Department of Applied Physics University of Granada Granada, Spain Núria Azemar Institute for Advanced Chemistry of Catalonia Consejo Superior de Investigaciones Científicas CIBER-BBN Barcelona, Spain J. Baudry Laboratoire Colloïdes et Matériaux Divisés ParisTech, ESPCI Paris, France J. Bibette Laboratoire Colloïdes et Matériaux Divisés ParisTech, ESPCI Paris, France Miguel A. Cabrerizo-Vílchez Department of Applied Physics University of Granada Granada, Spain

J. Chatterjee FAMU-FSU College of Engineering Tallahassee, Florida A.V. Delgado Department of Applied Physics University of Granada Granada, Spain M. Dijkstra Debye Institute Utrecht University Utrecht, the Netherlands Alvaro Domínguez Física Teórica Dpto. Física Atómica, Molecular y Nuclear Universidad de Sevilla Sevilla, Spain Jeremy S. Edwards Department of Molecular Genetics and Microbiology University of New Mexico Albuquerque, New Mexico

José Callejas-Fernández Department of Applied Physics University of Granada Granada, Spain

A. El-Harrak Institut de Science et d’Ingénierie Supramoléculaires Université Louis Pasteur Strasbourg, France

Nick J. Carroll Department of Chemical and Nuclear Engineering University of New Mexico Albuquerque, New Mexico

A. Elaissari Laboratoire d’Automatique et de Génie des Procédés Université de Lyon Lyon, France

R. Castañeda-Priego Department of Physics University of Guanajuato León, México

J. Estelrich Facultat de Farmàcia Universitat de Barcelona Barcelona, Spain xix

xx

Alejandro Fernández-Arteaga Institute for Advanced Chemistry of Catalonia Consejo Superior de Investigaciones Científicas CIBER-BBN Barcelona, Spain Antonio Fernández-Barbero Department of Applied Physics University of Almería Almería, Spain H. Fessi Laboratoire d’Automatique et de Génie des Procédés Université de Lyon Lyon, France Jacqueline Forcada Grupo de Ingeniería Química The University of the Basque Country San Sebastián-Donostia, Spain A. Fortini Debye Institute Utrecht University Utrecht, the Netherlands Matthias Fuchs Department of Physics University of Konstanz Konstanz, Germany

Contributors

M. Hamoudeh Laboratoire d’Automatique et de Génie des Procédés Université de Lyon Lyon, France C. Haro-Pérez Department of Applied Physics University of Granada Granada, Spain R. Hidalgo-Álvarez Department of Applied Physics University of Granada Granada, Spain A.-P. Hynninen Department of Chemical Engineering Princeton University Princeton, New Jersey Ainara Imaz Grupo de Ingeniería Química The University of the Basque Country San Sebastián-Donostia, Spain M.L. Jiménez Department of Applied Physics University of Granada Granada, Spain

Seiichi Furumi National Institute for Materials Science Tsukuba, Ibaraki, Japan

R. Juárez-Maldonado Instituto de Física “Manuel Sandoval Vallarta” Universidad Autónoma de San Luis Potosí San Luis Potoí, México

Carmen González Department of Chemical Engineering University of Barcelona Barcelona, Spain

Alexander Kamyshny Casali Institute of Applied Chemistry The Hebrew University of Jerusalem Jerusalem, Israel

Michael Grouchko Casali Institute of Applied Chemistry The Hebrew University of Jerusalem Jerusalem, Israel

David Levy Instituto de Ciencia de Materiales de Madrid, Cantoblanco Madrid, Spain

José Gutiérrez Department of Chemical Engineering University of Barcelona Barcelona, Spain

and Laboratorio de Instrumentación Espacial-LINES Torrejón de Ardoz, Madrid, Spain

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Contributors

M.T. López-López Department of Applied Physics University of Granada Granada, Spain

Rosario Pardo Instituto de Ciencia de Materiales de Madrid, Cantoblanco Madrid, Spain

Alicia Maestro Department of Chemical Engineering University of Barcelona Barcelona, Spain

and

Shlomo Magdassi Casali Institute of Applied Chemistry The Hebrew University of Jerusalem Jerusalem, Israel

Dimiter N. Petsev Department of Chemical and Nuclear Engineering University of New Mexico Albuquerque, New Mexico

Julia Maldonado-Valderrama Department of Applied Physics University of Granada Granada, Spain A. Martín-Molina Department of Applied Physics University of Granada Granada, Spain Antonio Martín-Rodríguez Department of Applied Physics University of Granada Granada, Spain F. Martínez-Pedrero Department of Applied Physics University of Granada Granada, Spain M. Medina-Noyola Instituto de Física “Manuel Sandoval Vallarta” Universidad Autónoma de San Luis Potosí San Luis Potoí, México Sergio Mendez Department of Chemical and Nuclear Engineering University of New Mexico Albuquerque, New Mexico Jordi Nolla Institute for Advanced Chemistry of Catalonia Consejo Superior de Investigaciones Científicas CIBER-BBN Barcelona, Spain

Laboratorio de Instrumentación Espacial-LINES Torrejón de Ardoz, Madrid, Spain

Carmen M. Pey Department of Chemical Engineering University of Barcelona Barcelona, Spain Antonio M. Puertas Department of Applied Physics University of Almería Almería, Andalucía, Spain M. Quesada-Pérez Department of Physics University of Jaén Linares, Jaén, Spain Jose Ramos Grupo de Ingeniería Química The University of the Basque Country San Sebastián-Donostia, Spain R. Rica Department of Applied Physics University of Granada Granada, Spain Miguel A. Rodríguez-Valverde Department of Applied Physics University of Granada Granada, Spain L.F. Rojas-Ochoa Department of Physics Cinvestav-IPN México D.F., Mexico

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Contributors

María Jose Gálvez Ruiz Department of Applied Physics University of Granada Granada, Spain

V. Trappe Department of Physics University of Fribourg Fribourg, Switzerland

Artur Schmitt Department of Applied Physics University of Granada Granada, Spain

R. Hans Tromp NIZO Food Research Kernhemseweg, the Netherlands

Conxita Solans Institute for Advanced Chemistry of Catalonia Consejo Superior de Investigaciones Científicas CIBER-BBN Barcelona, Spain Isabel Solè Institute for Advanced Chemistry of Catalonia Consejo Superior de Investigaciones Científicas CIBER-BBN Barcelona, Spain

Juan de Vicente Department of Applied Physics University of Granada Granada, Spain Serguei V. Vinogradov Department of Pharmaceutical Sciences University of Nebraska Medical Center Omaha, Nebraska J.L. Viota Department of Applied Physics University of Granada Granada, Spain

Kam C. Tam Department of Chemical Engineering University of Waterloo Waterloo, Ontario, Canada

David A. Weitz Department of Physics Harvard University Cambridge, Massachusetts

Jeremy P. K. Tan Institute of Bioengineering and Nanotechnology The Nanos, Singapore

Marcos Zayat Instituto de Ciencia de Materiales de Madrid, Cantoblanco Madrid, Spain

María Tirado-Miranda Department of Physics University of Granada Granada, Spain

and Laboratorio de Instrumentación Espacial-LINES Torrejón de Ardoz, Madrid, Spain

Part I Theory

1

Colloid Dynamics and Transitions to Dynamically Arrested States R. Juárez-Maldonado and M. Medina-Noyola

CONTENTS 1.1 1.2

1.3

1.4

Introduction ........................................................................................................................ GLE Formalism .................................................................................................................. 1.2.1 Ordinary and Generalized Langevin Equation ...................................................... 1.2.1.1 The Ordinary Langevin Equation ............................................................ 1.2.1.2 Ornstein–Uhlenbeck Processes ............................................................... 1.2.1.3 The Generalized Langevin Equation ....................................................... 1.2.2 GLE for Tracer Diffusion ....................................................................................... 1.2.2.1 Time-Dependent Friction Function: Exact Expression ............................ 1.2.2.2 Simplifying Approximations for DzT (t) ................................................... 1.2.3 Collective Diffusion ................................................................................................ 1.2.3.1 Exact Memory-Function Expressions for F(k, t) and F (S)(k, t) ................. 1.2.4 Approximate Elements of the Self-Consistent Theory ........................................... 1.2.4.1 Vineyard-Like Approximations ............................................................... 1.2.4.2 Closure Relation ....................................................................................... Self-Consistent Theory and Illustrative Applications ........................................................ 1.3.1 Model Mono-Disperse Suspensions ....................................................................... 1.3.1.1 Two-Dimensional Model System with Dipole–Dipole (r -3) Interactions ...................................................................................... 1.3.1.2 Three-Dimensional Soft Sphere Systems ................................................ 1.3.2 Extension to Colloidal Mixtures ............................................................................. 1.3.2.1 General Results ........................................................................................ 1.3.2.2 Binary Mixture of Charged Particles ...................................................... 1.3.3 Simplified SCGLE Theory ..................................................................................... 1.3.4 Diffusion of Colloidal Fluids in Random Porous Media ........................................ SCGLE Theory of Dynamic Arrest Transitions ................................................................. 1.4.1 General Results ....................................................................................................... 1.4.1.1 Nonergodicity Parameters ........................................................................ 1.4.2 Specific Systems and Comparison with Experimental Data .................................. 1.4.2.1 Hard- and Soft-Sphere Systems ............................................................... 1.4.2.2 Dispersions of Charged Particles ............................................................. 1.4.3 Multicomponent Systems ........................................................................................ 1.4.3.1 Mixtures of Hard Spheres ........................................................................ 1.4.3.2 Colloid-Polymer Mixtures .......................................................................

4 5 5 6 6 7 8 8 10 11 11 12 12 12 13 13 14 14 15 15 16 18 18 20 20 20 21 21 23 23 24 25

3

4

Structure and Functional Properties of Colloidal Systems

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

1.1 INTRODUCTION The dynamic properties of colloidal suspensions constitute an important experimental and theoretical aspect of the study of colloidal systems [1–3]. In equilibrium, and in the absence of external fields, 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 fluctuations 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)Ò, where the angular brackets indicate average over the equilibrium ensemble [2]. A closely related property is the socalled self intermediate scattering function F (S)(k, t). This is defined as F (S)(k, t) ·eik · DR(t)Ò, where DR(t) is the displacement at time t of any of the particles of the Brownian fluid. Over the years, the development of a general and practical microscopic description of colloid dynamics has proved to be a challenging task [1–3]. As a result, we have a rather diverse array of approaches, formal derivations, or physically intuitive shortcuts to the most difficult aspects of this complex manybody problem [4–15]. Taken together, these developments have provided a sound theoretical interpretation 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 constructed over the last years [16–21], leading to the first-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 function 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 selfdiffusion 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 first 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 fascinating topics of condensed matter physics, and several issues related to their microscopic description are currently a matter of discussion [24–26]. Among the various approaches to understanding the transition from an ergodic to a dynamically arrested state, the mode coupling (MC) theory [26–28] provides perhaps the most comprehensive and coherent picture. In fact, a large number of experimental observations in specific systems, particularly in the domain of colloidal systems [29–40], 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 collective 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 prediction 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 intuitive approximations referred to above. The resulting equations are simpler both, conceptually and in practice, than the corresponding MCT equations. Let us finally mention that the SCGLE theory shares with the colloid-dynamics version of MCT developed by Nägele and collaborators [10–15] 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 specific 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 [2,3,43–45]. We must also say that notable topics in both fields 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 multicomponent 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 specific 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 applications of this theory to the description of the short- and intermediate-time dynamics of specific 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 specific 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 first describe what is understood here for the GLE and then illustrate its use in the derivation of exact result for the timedependent 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 first explain what we mean here for generalized Langevin equation (GLE). For this, we summarize rather well-known concepts of the theory of Brownian motion and thermal fluctuations cast as stochastic processes, generated by linear stochastic equations with additive noise.

6

Structure and Functional Properties of Colloidal Systems

1.2.1.1 The Ordinary Langevin Equation The 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 fluctuations around this mean value. v(t) obeys a generally non_ _ linear deterministic equation of the general form M dv(t)/dt = R[v(t)], similar to the equation that describes the hydrodynamic resistance on a small macroscopic ball settling in a liquid. Langevin’s assumption was that the instantaneous velocity v(t) obeys just the same equation, but with an added stochastic term f(t), which represents the random thermal fluctuations 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(t) = 0), this equation may be linearized to read M ddv(t)/dt = [∂R[v]/∂v] v=0 ∙ dv(t) + f(t). For a 0 -[∂R[v]/∂v] spherical particle in the of external fi elds the friction tensor z v=0 is diagonal absence and isotropic, that is, z0 = z0I. Thus, since v(t) = dv(t), the previous equation can be recognized as the celebrated Langevin equation, M

dv(t ) = -z 0 v(t ) + f (t ), dt

(1.1)

in which z0 is the friction coefficient 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 fluctuations, and delta-correlated with itself, f(t)f(t¢) = g 2d(t - t¢) (i.e., it is a “purely random,” or “white,” noise). The stationarity condition is in reality equivalent to the fluctuation–dissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fixes 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 coincide with the equilibrium average ·vvÒ = (k BT/M)I given by the equipartition theorem (with I being the 3 × 3 Cartesian unit tensor), and this fixes the value of g to g = k BTz0I . 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 specific results that follow from this simple mathematical model regarding properties such as the velocity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48]. 1.2.1.2 Ornstein–Uhlenbeck Processes The 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 defines a mathematical model for the description of thermal fluctuations, 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 deterministic equation for its mean value a¯(t) and a linear stochastic equation with additive noise for the fluctuations 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 a¯ (t ) = R[a¯ (t )], dt

(1.2)

7

Colloid Dynamics and Transitions to Dynamically Arrested States

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 relaxation equation above, denoted by a¯ss, which solve the equation d a¯ ss = R[a¯ ss ] = 0. dt

(1.3)

One then postulates that the fluctuations da(t) = a(t) – a¯ss will satisfy a linearized stochastic Langevin-type version of Equation 1.2, namely, dda (t ) = H[a¯ ss ] ∞ da(t ) + f (t ), dt

(1.4)

where the C × C matrix H[a¯ss] is defined as È ∂R [ a ] ˘ Hij [a¯ ss ] ∫ Í i ˙ Î ∂a j ˚ a = a¯

(i, j = 1, 2, … , C ),

(1.5)

ss

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) ________ †(0) = 0, the dagger meaning transpose], and purely random, that is, of the fl uctuations [ f(t)da _______ f(t)f †(t¢) = g2d(t - t¢) with g being a C × C matrix. The stationarity condition then fixes the value of g by means of a fluctuation–dissipation relation, that reads g = -{H[a¯ ss ]s + sH† [a¯ ss ]},

(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) defined by the stochastic differential equation in Equation 1.4 with f(t) having the properties just described, is referred to as an Ornstein–Uhlenbeck process [49], and just like the solution of the ordinary Langevin equation, it is also a Gaussian stationary Markovian stochastic process. Many time-dependent fluctuation phenomena can be cast in terms of an Ornstein– Uhlenbeck process, including the fluctuating 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 Equation The 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” Ornstein–Uhlenbeck process, defined as the solution of the most general linear stochastic equation with additive noise, which we write as t

dda (t ) = dt ¢H(t - t ¢ ) ∞ da (t ¢ )dt ¢ + f (t ), dt

Ú

(1.7)

0

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

8

Structure and Functional Properties of Colloidal Systems

_______

f(t)f †(t¢) = L(t - t¢), and expect that the time-dependent matrix L(t) will be related with the timedependent relaxation kernel H(t) by some form of fluctuation–dissipation 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 fluctuation– 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, t

dda (t ) = - w ∞ s -1 ∞ da (t ) - dt ¢L (t - t ¢ ) ∞ s -1 ∞ da (t ¢ ) + f (t ), dt

Ú

(1.8)

0

where w is an antisymmetric matrix, w† = -w, and the matrix L(t) satisfies the fluctuation– dissipation relation L (t ) = L† (-t ) = f (t )f † (0).

(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 projection operator techniques to describe the time-dependent thermal fluctuations of systems in thermodynamic 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 fluctuations 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 useful approximate expression. 1.2.2.1 Time-Dependent Friction Function: Exact Expression Let 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]

M

dvT (t ) = -zT0 vT (t ) + f (t ) + dt

N

Â — u(| r - r |), i

i =1

i

T

(1.10)

9

Colloid Dynamics and Transitions to Dynamically Arrested States

which may be re-written exactly as

M

dvT (t ) = -zT0 vT (t ) + fT (t ) + d 3r[—u(r )]dn* (r, t ), dt

Ú

(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 particle) from its equilibrium average neq(r). Thus, the direct interactions, represented by the pair potential 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] t

∂dn*(r, t ) = [—neq (r )]◊ vT (t ) - dt ¢ d 3r ¢ d 3r ¢¢L (r, r ¢; t - t ¢ )s -1 (r ¢, r ¢¢)dn*(r ¢¢, t ¢ ) + f (r, t ), dt

Ú Ú Ú 0

(1.12) where the first term on the right__________ hand side is a linearized streaming term and f (r, t) is a fluctuating 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)Ò, that is, it is the solution of Ú d3r¢s -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 particle coupled to the fluctuations 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 tracer’s 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

M

t dvT (t ) = -zT0 vT (t ) + fT (t ) - Ú dt ¢ DzT (t - t ¢ ) ◊ vT (t ¢ ) + FT (t ), dt 0

(1.13)

where the new fluctuating force FT (t) is related with the time-dependent friction tensor DzT (t) _________ through FT (t)FT (0) = Mk BT DzT (t), with DzT (t) given by the following exact result: DzT (t ) = - d 3r d 3r ¢[—u(r )]X *(r, r ¢; t )[— ¢neq (r ¢ )],

Ú Ú

(1.14)

where X*(r, r¢; t) is the propagator, or Green’s function, of the diffusion equation in Equation 1.12, that is, it solves the equation t

∂X *(r, r ¢; t ) = - dt ¢ d 3r ¢ d 3r ¢¢L (r, r ¢¢; t - t ¢ )s -1 (r ¢¢, r ≤¢ )X *(r ≤¢, r ¢; t ¢ ), ∂t

Ú Ú Ú

(1.15)

0

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)

10

Structure and Functional Properties of Colloidal Systems

·dn*(r, t)dn*(r¢, 0)Ò 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” field u(r) of the tracer particle fixed at the origin, and as described from the tracer particle’s reference frame, which executes Brownian motion. 1.2.2.2 Simplifying Approximations for DzT (t) To simplify the notation, let us rewrite Equation 1.14 as DzT (t ) = - [—u† ] ∞ X *(t ) ∞[—neq ],

(1.16)

where the inner product A B, between two arbitrary functions A and B, is defined by the convolution Ú d3r≤A(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 Wertheim–Lovett relation [23,56] of the equilibrium theory of inhomogeneous fluids, [—neq ] = - bs ∞[—u],

(1.17)

with b (k BT)-1 This relation, along with the definition 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: DzT (t ) = kBT [—neq† ] ∞ s -1 ∞ G*(t ) ∞ s -1 ∞[—neq ],

(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 DzT (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 twoparticle distribution function of the colloidal particles surrounding the tracer particle which are, hence, subjected to the “external” field 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” field, can one write s(r, r¢) = s(|r - r¢|) nd(r - r¢) + n2[g(|r - r¢|) - 1], where g(r) is the bulk radial distribution 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 fluid approximation” [17], which allows us N to write G*(r, r¢; t) = [n/(2p)3] Ú d3k exp[-ik · r]F*(k, t), with F*(k, t) N-1/·Âi,j exp[ik · [ri(t) - rj(0)]]Ò. Notice also that in particular G*(r, r¢; t = 0) = s(|r - r¢|) = [n/(2p)3] Ú d 3k exp[-ik · r]S(k), where S(k) 1 + n Ú d3r exp[-ik · r][g(r) - 1] is the static structure factor. The function F*(k, t) just defined 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-fixed reference frame, we may re-write F*(k, t) as È F *(k , t ) ∫ [exp(ik ◊ [ xT (t ) - xT (0)])] ◊ Í N -1 ÍÎ

N

Â i, j

˘ exp(ik ◊ [ x i (t ) - xj (0)])]˙ , ˙˚

(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].

11

Colloid Dynamics and Transitions to Dynamically Arrested States

The two approximations just described may now be introduced in Equation 1.18, and this leads to an approximate expression for Dz (t). Since for spherical particles, the tensor Dz T T (t) must be diagonal, DzT (t) = DzT (t)I, such an approximate result can be written as 2

Dz*T (t ) ∫ DzT (t ) / zT0 =

DT0 È k[ S (k ) - 1] ˘ (S ) dk Í ˙ F (k, t )F (k, t ), 3(2 p)3 n Î S (k ) ˚

Ú

(1.20)

where DT0 k BT/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 function 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 intermediate scattering function F(k, t) of a colloidal dispersion in the absence of external fields. For this, consider again Equation 1.8, but now with the vector da(t) defined as da(t) [dn(k, t), djl(k, t)] with dn(k, t) being the FT of the fluctuations dn(r, t) and djl(k, t) = jl(k, t) k · j(k, t) being the longitudinal 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)Ò = S(k), where S(k) is the static structure factor of the bulk suspension. The continuity equation, ∂dn(k, t ) = ikdjl (k, t ), ∂t

(1.21)

couples the time derivative of dn(k, t) with the current fluctuations. 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 ∂djl (k, t ) Êk Tˆ Ê M ˆ = ik Á B ˜ S -1 (k )dn(k, t ) - Á ∂t M Ë ¯ Ë kBT ˜¯

t

Ú L(k, t - t ¢)dj (k, t ¢) dt ¢ + f (k, t ), l

l

(1.22)

0

_____________

where f l(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 (k BTz0/M2)2d(t). Thus, we write L(k, t) = (k BTz0/M 2)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

F (k, z ) =

S (k ) , z + {[ k D0 S (k )] / [1 + C (k, z )]} 2

-1

(1.23)

12

Structure and Functional Properties of Colloidal Systems

with D 0 KBT/z0 and C(k, z) D 0 M2b2DL(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 F ( S ) (k, z ) =

1 . z + {[ k 2 D0 ] / [1 + C ( S ) (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 additional independent relations, which we now discuss. 1.2.4.1 Vineyard-Like Approximations The first 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 first 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, defined as È C SEXP (k, z ) ˘ ( S ) C (k, z ) = Í ( S )SEXP ˙ C (k, z ), (k, z ) ˚ ÎC

(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 advantage, however, is only meaningful in the short-time regime. At longer times the additive approximation, defined as C (k, t ) = C ( S ) (k, t ) + [C SEXP (k, t ) - C ( S )SEXP (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 Relation Either 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 function 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

Colloid Dynamics and Transitions to Dynamically Arrested States

13

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 wavelength limit of C(S)(k, t), that is, lim kÆ0 C(S)(k, t) = Dz*(t)/z0. Thus, the proposal was to interpolate C(S)(k, t) between its two exact limits, namely, C ( S ) (k, t ) = C ( S )SEXP (k, t ) + [ Dz*(t ) - C ( S )SEXP (k, t )]l (k ),

(1.27)

where l(k) is a phenomenological interpolating function such that l(k Æ 0) = 1 and l(k Æ •) = 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 first minimum that follows the main peak of the static structure factor S(k). Thus, the interpolating function employed in the closure relation above reads l(k ) ∫ [1 + (k/kmin )2 ]-1.

(1.28)

1.3 SELF-CONSISTENT THEORY AND ILLUSTRATIVE APPLICATIONS We now have all the elements needed to define 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 approximations 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 determined by the methods of equilibrium statistical thermodynamics. However, as we will see below, a simplified 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 first 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 dipole–dipole interaction). The third one was the three-dimensional weakly screened repulsive Yukawa potential (whose twodimensional version had been studied in reference [18]). The last system considered involved shortranged, 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 simulations (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

14

Structure and Functional Properties of Colloidal Systems

isolate one of the most important effects in the relaxation of the concentration fluctuations in colloidal 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 Dipole–Dipole (r -3) Interactions The first example involves a two-dimensional model system with dipole–dipole (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], defined 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 k BT. The specific 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 figure 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 figure, 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 differences 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 Systems The second illustrative example refers to a three-dimensional system of Brownian particles interacting through a strongly repulsive and short-ranged, pair potential u(r) of the form bu(r ) =

1 2 +1 (r /s s )2 m (r /s s )m

(1.29)

2.5

2.0

F(k,t)

1.5

1.0

0.5

0.0 0

1

2 ks

3

4

FIGURE 1.1 Intermediate scattering function F(k, t) as of the two-dimensional fluid 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.)

15

Colloid Dynamics and Transitions to Dynamically Arrested States 2.5

F(k,t)

2.0

1.5

1.0

0.5

0.0

0

10

20 ks

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.)

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 specific case m = 18 at a soft-sphere volume fraction f s 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/D 0).

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, defined as Fab (k, t) ·dn a (k, t)dn b (-k, 0)Ò, where ___ N dn a (k, t) (1/√Na ) Âi=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 define the self component of Fab (k, t), referred to as the self-intermediate scattering (S) function, as Fab(k, t) d ab ·exp[ik · DR(a)(t)]Ò, where DR(a)(t) is the displacement of any of the Na particles of species a over a time t, and d ab is Kronecker’s 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 Results The 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

Dz*a (t ) ∫ Dza (t ) / za0 =

Da0 d 3 kk 2 [ F ( S ) (t )]aa ÈÎc nF (t )S -1 nh ˘˚ , aa 3(2 p)3

Ú

(1.30)

where the k-dependent elements of the n × n matrices F(t), F(S)(t), S, h and c are, respectively, the (S) collective and self-intermediate scattering functions Fab (k, t), F ab (k, t), the partial static structure factors Sab (k) = Fab (k, t = 0), and the FTs hab (k) and cab (k) of the Ornstein–Zernike total and direct __ __ __ __ correlation functions, respectively. Thus, h and c are related to S by S = I + √ n h√ n = [I - √ nc√ n ]-1,

16

Structure and Functional Properties of Colloidal Systems __

__

__

with the matrix √ n defined as [√ n ] ab d ab√ n a . In these equations, n a is the number concentration of species a and D a0 = k BT/za0 is the free-diffusion coefficient 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 F (k, z ) = {z + ( I + C (k, z ))-1 k 2 DS -1 (k )}-1 S (k )

(1.31)

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

(1.32)

and

where D is the diagonal matrix D ab d ab D a0. Notice also that all the matrices involved in the equation 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), defined 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 C (k, z ) = C ( S ) (k, z ) + [C SEXP(k, z ) - C ( S )SEXP (k, z)],

(1.33)

where CSEXP(k, z) and C(S)SEXP(k, z) are the single exponential approximation of these memory functions, also defined 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 Ca( S ) (k, z ) = Ca( S )SEXP (k, z ) + [ Dz*a (k ) - Ca( S )SEXP (k, z )]l a (k ),

(1.34)

where l a (k) is a phenomenological interpolating function defined as -1

( ) 2˘ l a (k ) = È1 + (k/kmin ) , Î ˚ a

(1.35)

(a)

with kmin being the position of the first minimum (beyond the main peak) of the partial static structure factor S aa (k). The time-dependent friction Dz*(t) is a diagonal matrix whose diagonal element Dz*a (t) is given by Equation 1.30. The set of Equations 1.30 through 1.35 thus constitute the multicomponent extension of the SCGLE theory, whose applications are now reviewed. 1.3.2.2 Binary Mixture of Charged Particles The 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

buab (r ) = K a Kb

e - z[(r / s ) -1] . r /s

(1.36)

17

Colloid Dynamics and Transitions to Dynamically Arrested States

The dimensionless parameters that define 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 relative concentrations x a n a /n, and the potential parameters K1, K2, and z. The free-diffusion coefficients D a0 are also assumed identical for both species, that is, D10 = D 20 = D 0. Explicit values of the parameters s and D 0 are not needed, since the dimensionless dynamic properties, such as Fab (k, t), only depend on the dimensionless parameters specified above, when expressed in terms of the scaled variables ks and t/t0, where t0 s2/D 0. Besides solving the SCGLE scheme, in reference [21] Brownian dynamics simulations were generated for the static and dynamic properties of the system above. (S) In this manner all the dynamic properties that derive from Fab (k, z) and Fab(k, z) can be calcu(D) (S) lated. These include the distinct intermediate scattering functions F ab (k, z) Fab (k, z) - F ab(k, z), (a) (a) 2 the msd W (t) ·(Dr (t)) Ò of particles of species a, and the time-dependent diffusion coefficient D a (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

0.5

1

0.8 D(t)/D0

F d2(k,t)

0

0.6 –0.5

0.4

0

2

t/t0

4

6

kσ

1

1.5

1

1.5

1.5 BD SEXP SCGLE-Vadd SCGLE-Vmult

1

F s2(k,t)

F s2(k,t)

1

0.5

0.5

0.5

0

0

0.5

1 kσ

1.5

2

0

0.5

kσ

__

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 coefficients D1(t) and D 2(t), are shown normalized with their initial value D a0 (the upper curves corresponding to species 1). The self, the distinct, and the total intermediate scattering 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 figure compare results of the SCGLE theory within the additive (solid lines) and the multiplicative (dot-dashed lines) Vineyard-like approximations. For reference, also the results of the SEXP approximation (dashed lines) are presented. (From Chávez-Rojo, M. A. and Medina-Noyola, M. 2005. Phys. Rev. E 72: 031107; Phys. Rev. E 76: 039902, 2007. With permission.)

18

Structure and Functional Properties of Colloidal Systems

that the two versions of the SCGLE bracket the simulation data. More detailed calculations, however, 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 difficulties of the SCGLE theory is the involvement 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 simplified version of this theory, in which this short-time information is eliminated, leads to essentially the same results at intermediate and long times. The simplified 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)

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

(1.38)

and

It is not difficult to see that the original self-consistent set of equations (involving Equations 1.33 and 1.34) shares the same long-time asymptotic solutions as its simplified version. It is then natural to ask what the consequences would be of replacing Equations 1.33 and 1.34 of the full SCGLE set of equations by the simpler approximations in Equations 1.37 and 1.38, that no longer contain the functions CSEXP(k, t) and C(S)SEXP(k, t). The proposal of a simplified version of the SCGLE theory consists precisely of this replacement. In reference [59], a systematic comparison of the various dynamic properties involved in the SCGLE theory was performed. Surprisingly enough, it was found that the degree of accuracy of the simplified theory in the short- and intermediate-time regimes was remarkably good as well. This led to the general conclusions that the simplified SCGLE theory provides a description of the relaxation of concentration fluctuations in colloidal suspensions qualitatively and quantitatively virtually identical to the full SCGLE theory. Its practical implementation, however, is much simpler than either the full SCGLE or the MCT schemes. This simplified enormously the application of the SCGLE theory to additional topics, such as the discussion of the dynamics of colloidal fluids adsorbed in model porous media and dynamic arrest in colloidal systems, as described in what follows.

1.3.4

DIFFUSION OF COLLOIDAL FLUIDS IN RANDOM POROUS MEDIA

A porous medium is sometimes modeled as a random array of locally regular pores to incorporate the intrinsic randomness of most natural or synthetic porous materials [60]. One may adopt, instead, a simplified model of a random porous medium, namely, a matrix of spherical particles with random but fixed positions. This matrix is permeated by a colloidal liquid, whose dynamics we wish to understand. Such model systems have been employed to describe mostly equilibrium structural properties [61,62], although simple experimental realizations of this system have been prepared

19

Colloid Dynamics and Transitions to Dynamically Arrested States

[63], in which the dynamic properties of the mobile species are also measured. One possible approach to the interpretation of such measurements is to use available theories of the dynamic properties of bulk colloidal mixtures, such as the SCGLE theory [21,59], in which the mobility of one of the species is artificially set equal to zero. Another possibility is to first reformulate these theories to explicitly consider the porous matrix as a random external field [64,65]. In recent work [66], it was demonstrated that the first of these approaches suffices to provide a simple but correct first-order prediction of the main features of the dynamics of the permeating fluid. As a concrete example, illustrated in Figure 1.4, the same binary colloidal mixture of particles interacting through the screened Coulomb potentials in Equation 1.36 was considered in reference [66], in which one of the two species plays the role of the porous matrix. Two kinds of computer experiments were carried out, which differ in the manner in which the structure of the porous matrix was generated. In the first kind, the N particles of both species are allowed to thermalize

1 0.5

0.5

0

0 BD SCGLE

0.5

1

F(k,t)

D*(t)

0.5 0

0 1

0.5

0.5

0

0 1

0.5

0.5

0 0

5 t/t0

0

0

1

2

kσ

FIGURE 1.4 Time-dependent self diffusion coefficient D*(t) ·(Dr(t))2 Ò/(6D 0 t) (left column) and partial intermediate scattering function F(k, t) for t = 0, t0, and 10t0 (right column) of the diffusive species permeating the porous matrix, interacting with the repulsive Yukawa potential with fixed screening parameter z = 0.15 and volume fractions f1 = f2 = 2.2 × 10-4, but with parameters K1 and K2 given by K1 = K2 = 100 (first row), K1 = K2 = 500 (second row), and K1 = 100 and K2 = 500 (third and fourth rows). The symbols represent Brownian dynamics results and the solid lines are the SCGLE theoretical predictions. The first three rows describe experiments of the first kind and the fourth row describes one experiment of the second kind. (From Chávez-Rojo, M. A. et al. 2008. Phys. Rev. E 77: 040401(R). With permission.)

20

Structure and Functional Properties of Colloidal Systems

according to conventional Brownian dynamics with the same free-diffusion coefficient D10 = D20 = D 0 until equilibrium is reached. At this point, the motion of the particles of species 2 is artificially arrested by setting D20 = 0 at an arbitrary configuration; a more limited version of this exercise, referring only to tracer diffusion phenomena, had been carried out by Viramontes-Gamboa et al. [67,68] early in 1995. In the second kind of experiments, a preexisting matrix is formed in the absence of the mobile species, by choosing the arrested configurations according to a prescribed distribution, afterward “pouring” the mobile particles into this matrix of obstacles. The prescribed average structure of the matrix considered in the example in Figure 1.4 corresponds to the structure of an equilibrium mono-component fluid of species 2. In both cases, after choosing a particular configuration of the matrix, the mobile species is allowed to equilibrate in the external field of the fixed particles at that particular frozen configuration, and then the dynamic properties of interest are calculated. In both cases the radial distribution functions between the two species are recorded, to be employed as the static input required by the SCGLE theory, thus avoiding the use of liquid state approximations [3]. From information such as that in the figure, one concludes that the SCGLE theory, devised to describe the dynamics of equilibrium colloidal mixtures, provides a reasonable description of the dynamics of a mono-disperse suspension permeating a porous medium formed by a random array of fixed particles.

1.4 SCGLE THEORY OF DYNAMIC ARREST TRANSITIONS For “SCGLE theory of dynamic arrest,” we mean the straightforward application of the SCGLE theory just reviewed to the specific study of colloidal systems near their dynamic arrest transitions [69–74]. Such transitions are characterized by dynamic “order parameters,” such as the long-time self-diffusion coefficient DL (limtÆ• D(t)), reaching a critical value, in this case DL = 0. This indicates that, on average, the constituent particles have been immobilized, and any local concentration fluctuation will no longer be able to relax to equilibrium, remaining frozen due to the inability of the particles to efficiently sample the configurational space of the system. Thus, if we have a theory that predicts the value of DL for a given system (i.e., given interparticle interactions) and given state (i.e., given concentration and temperature), then it will be enough to scan the state space monitoring this order parameter to detect the location of a dynamic arrest transition.

1.4.1

GENERAL RESULTS

To illustrate these ideas let us summarize the general system of equations that constitute the SCGLE theory. In principle, these are the exact results for Dz(t), F(k, t), and F (S)(k, t) in Equations 1.20, 1.23, and 1.24, complemented with the simplified Vineyard approximation in Equation 1.37 and the simplified interpolating closure in Equation 1.38. This set of equations define the SCGLE theory of colloid dynamics. Its full solution also yields the value of the long-time self-diffusion coefficient DL , which is the order parameter appropriate to detect the glass transition from the fluid side. This is, however, not the only method to detect dynamic arrest transition, as we now explain. 1.4.1.1 Nonergodicity Parameters A complementary criterion emerges if we approach the glass transition from the region of dynamically arrested, or “nonergodic,” states. The dynamic properties F(k, t), F (S)(k, t), C(k, t), C(S)(k, t), and Dz*(t) decay to zero in an ergodic (i.e., equilibrium) state, and decay to finite asymptotic values in a nonergodic state. These asymptotic values are referred to as the nonergodicity parameters, which we denote, respectively, by f(k)S(k), f (S)(k), c(k), c (S)(k), and Dz*(•). One can then rewrite Equations 1.20, 1.23, 1.24, 1.37, and 1.38 in terms of these asymptotic values plus a regular contribution that does decay to zero. Taking the long-time limit of the resulting equations leads to a system of five equations for these five unknown nonergodicity parameters [59,70]. Such a system of equations,

Colloid Dynamics and Transitions to Dynamically Arrested States

21

which in MCT literatures known as bifurcation equations [26], can easily be reduced in our case to a single equation for the scalar parameter Dz*(•), written as 1 1 = g 6p 2 n

Ú

•

0

dkk 4

[ S (k ) - 1]2 l 2 (k ) , [l(k )S (k ) + k 2 g ][l(k ) + k 2 g]

(1.39)

*(•) being the msd of a particle localized by the arrested cage formed by its neighwith g D 0 /Dz __ bors, that is, √ g as the localization length of a tracer particle in the glass [70]. The form of this criterion exhibits its simplicity: Given the effective inter-particle forces, statistical thermodynamic methods allow one to determine S(k), and the absence or existence of finite real solutions of this equation will indicate if the system remains in the ergodic phase or not. The other four equations for the nonergodicity parameters can then be used to express those quantities in terms of g. The equation for the nonergodicity parameters f(k), for example, reads f (k ) =

1.4.2

l ( k )S ( k ) . l ( k )S ( k ) + k 2 g

(1.40)

SPECIFIC SYSTEMS AND COMPARISON WITH EXPERIMENTAL DATA

The theoretical developments just reviewed lead to interesting predictions when applied to specific model systems, which must be validated by their comparison with experimental data. Let us review some of those comparisons involving mono-disperse suspensions. 1.4.2.1 Hard- and Soft-Sphere Systems A virtually exact representation of the static structure factor of a hard-sphere system is provided by the Percus–Yevick (PY) approximation [75,76] with the Verlet–Weis (VW) correction [77]. The use of this approximation allows the solution of the full SCGLE theory (Equations 1.20, 1.23, 1.24, 1.37, and 1.38), which can then be compared [73] with the experimental results, as illustrated in Figure 1.5 with the data of van Megen and Underwood [30]. The results of the SCGLE theory are calculated at a volume fraction f corresponding to the same separation parameter e (f- fg)/fg as the corresponding experimental volume fraction. The glass transition of the hard-sphere system occurs, according to the experimental report, at fg = 0.575 [30], whereas the SCGLE theory predicts fg = 0.563. In addition,_______ from Equation 1.39 one calculates that at fg = 0.563, g = 1.060 × 10-2 s, leading to a ratio d √·[Dx(t)]2 Ò/d of the localization length g1/2 to the mean inter-particle distance d n-1/3 of 0.105, strongly reminiscent of the Lindemann criterion of melting [78]. Finally, the collective nonergodicity parameter f(k) was also calculated in reference [70], and compared with the corresponding experimental data of reference [29]. Such comparison, not shown here, is very good concerning the height and position of the first maximum of f(k), although one observes that at smaller and larger wave-vectors the agreement between theory and experiment for f(k) deteriorates appreciably [70]. In reference [73], a comparison similar to that in Figure 1.5 was also performed for a soft-sphere system, namely, a dispersion of microgel particles studied by Bartsch et al. [32], who indicate a value of the soft-sphere diameter s(m) of 1.0 mm, and report the glass transition to occur at a volume fraction f(m) g 0.644. In reference [73], this system was modeled with the soft potential in Equation 1.29. Unfortunately, the experimental report does not define or quantify the degree of softness of the particles, in a manner that serves to determine the parameter m of the model. In reference [73], however, the glass transition volume fraction f(m) = pns(m)3/6 was calculated for each n using Equation 1.39. This determines the glass transition line in the softness-concentration state space [i.e., in the (m, f(m) plane)]. By observing this “glass transition phase diagram,” the value of m whose glass transition volume fraction fg(m) coincides with the experimentally reported value of 0.644, it was concluded that

22

Structure and Functional Properties of Colloidal Systems 1

0.8

f (kexp,t)

0.6

0.4

0.2

0

10–4

10–2

100

102

t/t0

FIGURE 1.5 Comparison of the SCGLE collective correlator, f(k, t) F(k, t)/S(k), of the hard-sphere system at the position kmax of the main peak of S(k) (solid lines) with the experimental results of van Megen and Underwood [24] (symbols) corresponding to the experimental volume fractions f = 0.494, 0.528, 0.535, 0.574, 0.581, and 0.587 (from bottom to top). The SCGLE theory predicts fg = 0.563, and hence, the comparison is made at the same values of the separation parameter e = (f - fg)/fg. (From Ramírez-González, P. and MedinaNoyola, M. 2009. J. Phys.: Condens. Matter. 21: 075101. With permission.)

m = 14 is the softness parameter that best represents the experimental system. The dynamic properties of the soft-sphere system corresponding to m = 14 were then theoretically calculated solving the full SCGLE theory and then compared with the experimental data of Bartsch et al. [32]. The resulting comparison is contained in Figure 1.6. Just like in the hard-sphere case, here too the parameter t0 was treated as the only adjustable parameter. The comparison in this case and in that of the hard-sphere

1.2 1

f(kexp,t)

0.8 0,6 0.4 0.2 0

10–4

10–2

100 t/t0

102

104

106

FIGURE 1.6 Comparison of the SCGLE collective correlator f(k, t) F(k, t)/S(k) of the soft-sphere system in Equation 1.29 with m = 14 at the position kmax of the main peak of S(k) in the vicinity of the glass transition, for the volume fractions f (m) pns(m)3/6 = 0.642, 0.667, and 0.7 (from bottom to top, solid lines). The symbols are the corresponding experimental results of Bartsch et al. [26] at the same volume fractions. (From RamírezGonzález, P. and Medina-Noyola, M. 2009. J. Phys.: Condens. Matter. 21: 075101. With permission.)

23

Colloid Dynamics and Transitions to Dynamically Arrested States (a) 7

(b) 1 1

6

0.8

f (kσ)

S(kσ)

5 4 3

0.4 0

2

4

6

8

0.2

2 1 0 2

0.5

0.6

00 3

4

5 kσ

6

7

5

10

kσ

15

20

25

8

FIGURE 1.7 (a) Theoretical fit (solid line) of the static structure factor, and (b) SCGLE theoretical predictions (solid line) for the nonergodicity parameter f(k) of the mono-disperse charged sphere system of reference [33], modeled by the pair potential of Equation 1.36 at f = 0.27, z = 3.1587, and K21 = 11.66. The symbols correspond to the experimental data. The inset in (b) enlarges the region where experimental data for f(k) are available. (From Yeomans-Reyna, L. et al. 2007. Phys. Rev. E 76: 041504. With permission.)

system in Figure 1.5 is very good considering that no adjustable parameters were introducer other than t0 and, in the hard-sphere case, the use of the separation parameter. 1.4.2.2 Dispersions of Charged Particles Let us finally illustrate the application of the SCGLE theory to a mono-disperse suspension of charged colloidal particles. Experimental data for such systems were reported in reference [33]; for one of the samples, the hard-sphere diameter and the volume fraction f are experimentally determined to be s = 272 nm and f = 0.27, and data were provided for the static structure factor. In order to use these data as the static input in the dynamic theory, one needs to have a smooth representation of the experimentally measured S(k). In reference [69], the measured static structure factor was fitted with a standard liquid theory approximation, and the solid line in Figure 1.7a corresponds to such fit. This static structure factor was then employed as the input of Equation 1.39 to calculate g, with the result g = 2.85 × 10-3 s2. The nonergodicity parameter f(k) was then calculated according to the general result in Equation 1.40, and the results are compared with the experimental data in Figure 1.7b. As one can see from this comparison, the agreement of the SCGLE theoretical predictions with the experimental data for the nonergodicity parameter turns out to be very good for all the wave-vectors reported in the experiment.

1.4.3

MULTICOMPONENT SYSTEMS

The multicomponent extension of the SCGLE theory [21] may be summarized, in its simplified form [59], by Equations 1.30 through 1.32, 1.37, and 1.38. It was applied rather recently [72] to the description of dynamic arrest in two simple model colloidal mixtures, namely, the hard-sphere and the repulsive Yukawa binary mixtures. The main contribution of reference [72], however, is the extension to mixtures of Equation 1.39. Thus, the resulting equation for g a , the localization length squared of particles of species a, was shown to be 1 1 d 3 kk 2 {[ I + k 2 gl -1 (k )]-1}aa {c n [ I + k 2 gl -1 (k )S -1 (k )]-1 nh}aa , = g a 3(2 p)3

Ú

(1.41)

24

Structure and Functional Properties of Colloidal Systems

written in the notation introduced in Equations 1.31 through 1.35. The corresponding extension of Equation 1.40 for the nonergodicity parameter y(k) limtÆ• F(k, t)S-1(k) reads y(k) = [I + k2 gl -1(k)S-1(k)]-1. Thus, for a given system one first determines S ab (k), as well as the matrices c, h, and l needed in these equations, and then numerically solve the n equations for the n parameters g a to classify the resulting state. For a binary mixture, for example, the solution g1 = g2 = • corresponds to a fully ergodic state, whereas a finite solution for both of these parameters corresponds to a fully arrested state. Under some conditions we also expect mixed states in which the particles of one species are arrested (e.g., finite g2), while the other particles remain mobile (g1 = •). In this manner one may scan the state space to determine the regions where these different dynamic states occur, and the boundaries between them. Of course, mixtures with more than two components will present richer dynamic arrest phase diagrams. 1.4.3.1 Mixtures of Hard Spheres This exercise was reported in reference [72] for the simplest example of a colloidal mixture, namely, a binary mixture of hard-spheres with diameters s1 and s2, and number concentrations n1 and n2, within the PY [79] approximation for S ab (k). The asymmetry parameter d ( s1/s2 £ 1), the total volume fraction f f1 + f2 (with f a pn a s a3/6), and the molar fraction x1 = n1/(n1 + n2) of the smaller spheres span the state space (d, f, x1). Figure 1.8a illustrates the regime of negligible or moderate size-disparities, d 1, by plotting the glass transition lines (solid curves) corresponding to the values of the asymmetry parameter d = 1.0, 0.8, 0.6 and 0.4. The glass transition line moves to higher total volume fractions fg as the size disparity increases, as experimentally observed and described by Williams and van Megen [31] who melted an originally mono-disperse glass by means of the replacement of a fraction of its particles by particles of a smaller size keeping the same total volume fraction. The regime illustrated in Figure 1.8a was also studied by Götze and Voigtmann with MCT [80–83], with predictions qualitatively similar to ours for intermediate size-disparities (d 0.65), but with conflicting predictions for milder size disparities (d 0.8). Figure 1.8b illustrates the much more interesting regime of large size disparities. The SCGLE theory predicts that

(b)

(a) 1.1

0.2

1.25

1.08

1.2

φ/φg(m)

1.04

φp

0.15

1.06

φ/φg(m)

1.3

E

0.1

0.05

1.15

0 0.5

0.55

1.1

0.6 φc

0.65

0.7

1.02 1.05 1 0

0.2

0.4

x1

0.6

0.8

1

1 0

0.2

0.4

x1

0.6

0.8

1

FIGURE 1.8 Dynamic arrest phase diagram of the binary hard-sphere mixture in the plane (f, x1) for fixed size-disparity d. Each solid curve is the boundary between fully ergodic states (below the curve) and fully arrested states (above the curve) for different values of d. The dashed lines indicate the border between fully ergodic and mixed states (shaded area). The dotted lines are the border of mixed states with fully arrested states. The total volume fraction f has been scaled with the volume fraction fg(m) of the ideal glass transition of the mono-disperse hard-sphere system. The glass transition curves in (a) correspond to d = 1.0 (horizontal line f = fg(m), 0.8, 0.6, and 0.4 (d c). The dynamic arrest phase diagram in (b) corresponds to d = 0.2, and illustrates the presence of the region of mixed states (shaded area); the inset of (b) redraws the same diagram in the plane (f1, f2), but for d = 0.09 and emphasizing the region near the bifurcation point E.

Colloid Dynamics and Transitions to Dynamically Arrested States

25

below the threshold asymmetry d c ~ 0.4, the region of mixed states appear, represented by the shaded area in the results for d = 0.2. 1.4.3.2 Colloid-Polymer Mixtures One of the most spectacular successes of MCT is the prediction of the reentrant glass transition in mono-disperse colloidal suspensions of particles with generic short-ranged attractive interactions. MCT predicts that, independently of the origin of these forces, and starting with a hard-sphere glass, increasing the strength of the attractions (or lowering the effective temperature) at fixed volume fraction may lead to the restoration of ergodicity, followed by the re-entrance to new (“attractive”) glass states [84–85]. The SCGLE theory of dynamic arrest also predicts a similar scenario, as reported in references [69,71]. The natural physical realization of this phenomenon should be sought in strictly mono-component systems in which the attractions between colloidal particles are caused, for example, by some form of solvophobic effect, as in copolymer micelles [34–37] or in (grafted) polymerstabilized colloids [38] in a marginal solvent. Effective attractive interactions between colloidal particles can also be produced by the addition of a second colloidal component [86]. The best-known experimental examples of systems with these so-called depletion interactions involve hard-sphere-like colloids with added nonadsorbing polymer [39]. At fixed colloid concentration, and upon the addition of polymer, reentrant behavior similar to that described by MCT for mono-component systems has been observed and documented with sufficient detail in these systems [40]. Thus, it is a widespread and virtually unquestioned belief that this bicomponent system provides just another physical realization of the phenomenon predicted for strictly mono-component systems. A natural question is, however, whether the multicomponent extension of the available (MC or SCGLE) theories of dynamic arrest, applied to a bicomponent model of colloid-polymer mixtures, will predict this experimentally-observed re-entrant scenario. In recent work [87], the multicomponent MCT was applied to a binary hard sphere mixture with nonadditive diameters, in which the interactions between small spheres (representing the polymer) are neglected, but the interactions between large particles (representing the colloid) and between large and small particles remain the same as in the additive hard sphere mixture of Figure 1.8; this new model mixture is referred to as the Asakura–Oosawa (AO) mixture. The conclusion of reference [87] is that the bicomponent MCT fails to predict the experimentally observed re-entrance for this model. Concerning the multicomponent extension of the SCGLE theory, already from the inset of Figure 1.8b we see that the dashed line corresponds to the dynamic arrest of only the large particles and the solid curve to the simultaneous dynamic arrest of both species. The union of these two lines then represents the glass transition line of the large particles, independently of their mode of arrest, and clearly exhibits reentrance. In fact, the asymmetry d = 0.09 is the asymmetry of the experimental dynamic arrest phase diagram of reference [40], with which the quantitative agreement turns out to be remarkably good [74]. The hard sphere mixture, however, may not be the best representation of the colloid polymer mixture, since the polymer–polymer interactions are clearly overemphasized. In Figure 1.9a, however, the same calculations are presented for the AO mixture, in which these interactions are totally neglected. The first relevant conclusion is that for both models the SCGLE theory predicts essentially the same basic topology of the dynamic arrest phase diagram, including the same reentrant glass transition scenario for the colloidal species. This similarity is enhanced if the volume fractions are scaled with the corresponding volume fractions of the respective end point E, as done in Figure 1.9b. The symbols in this figure are the experimental data of reference [40], also scaled in the same manner. Thus, the second relevant conclusion is that, at least in the region of the reentrance, the theoretical predictions of the colloid dynamic arrest transition are rather insensitive to the treatment of the polymer–polymer interactions, and the overall prediction is in remarkable agreement with the experimental data. More important, however, is the fact that the SCGLE theoretical results suggests a new and unexpected scenario of the dynamic arrest in colloid-polymer mixtures [74]. According to this scenario the mono-component representation of this system, which

26

Structure and Functional Properties of Colloidal Systems (b)

(a) 0.12

0.08

3

(E) φp/φp

φp

2

Ε 0.04

E

1

B

0 0.5

0.55

0.6 φc

0.65

0.7

0 0.5

0.6

0.7

0.8 0.9 ) φc/φ(E c

1

1.1

FIGURE 1.9 (a) Glass transition phase diagram of the AO binary mixture of size-asymmetry d = 0.09. The solid, dashed, and dotted lines have the same meaning as in Figure 1.8. The dark area is the spinodal region of the AO model. The theoretical glass transition line of the colloidal particles are shown in (b) for the hardsphere (solid line) and for the AO (dashed line) mixtures using reduced units (f c/f c(E), f p/f p(E)); they are compared with the experimental data of reference [40], with the solid symbols representing repulsive (diamonds) and attractive (circles) glass states, and with the empty symbols representing ergodic states.

assigns the polymer the role of a background component that only renormalizes colloid–colloid interactions, only makes sense at low polymer concentrations, but not in the regime of attractive glasses, where its slow dynamics becomes an essential aspect of the dynamic arrest of the colloidal species. These predictions await experimental confirmation, and possibly will generate interest and direction for further experimental investigations.

1.5

SUMMARY AND PERSPECTIVES

In this chapter we have reviewed the fundamental basis and applications of the SCGLE theoretical approach to the description of the dynamic properties of colloidal systems, with emphasis on their relation with transitions to dynamically arrested states in these systems. The well-known MCT is conventionally regarded as some form of canonical theory of the ideal glass transition. Due to the emerging interest of experimental groups in the detailed observation of these phenomena, the time seems opportune to explore alternative perspectives to go beyond the limitations of this canonical theory. The SCGLE theory is one of such attempts. The applications reviewed in this chapter, particularly those involving specific experimental results, indicate that it provides useful tools to describe important aspects of these phenomena. We refer, for example, to the remarkably simple bifurcation equations illustrated by Equation 1.41 for the localization length of the particles of a species that undergoes dynamic arrest. Just like any approximate theory, the SCGLE theory relies on approximations that must be tested in order to improve them. For example, further analysis is required to identify the nature and physical meaning of important elements of the theory, such as the interpolating function l a (k) and to explain the appearance of equilibrium structural properties in a theory of supposedly nonequilibrium states. Reviewing the theoretical basis of this approach allows the possibility of relaxing or extending some of these approximations, thus opening the possibility of describing more complex phenomena, such as the effects of external fields on the dynamic arrest transitions and the processes of ageing of colloidal glasses and gels. Preliminary steps in these directions seem encouraging, and the conceptual simplicity of the approach reviewed here is expected to constitute an important asset in these efforts.

Colloid Dynamics and Transitions to Dynamically Arrested States

27

ACKNOWLEDGMENT This work was supported by CONACYT, México, through grants 47611 and 84076.

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29. van Megen, W., Underwood, S. M., and Pusey, P. N. 1991. Nonergodicity parameters of colloidal glasses. Phys. Rev. Lett. 67: 1586. 30. van Megen, W. and Underwood, S. M. 1994. Glass transition in colloidal hard spheres: Measurement and mode-coupling-theory analysis of the coherent intermediate scattering function. Phys. Rev. E 49: 4206. 31. Williams, S. R. and van Megen, W. 2001. Motions in binary mixtures of hard colloidal spheres: Melting of the glasses. Phys. Rev. E 64: 041502. 32. Bartsch, E. et al. 1997. The glass transition dynamics of polymer micronetwork colloids: A mode coupling analysis. J. Chem. Phys. 106: 3743. 33. Beck, C., Härtl, W., and Hempelmann, R. 1999. The glass transition of charged and hard sphere silica colloids. J. Chem. Phys. 111: 8209. 34. Mallamace, F. et al. 2000. Kinetic glass transition in a miscellar system with short range attractive interaction. Phys. Rev. Lett. 84: 5431. 35. Chen, S. H. et al. 2003. The glass-to-glass transition and its end point in a copolymer micellar system. Science 300: 619. 36. Chen, W. R. et al. 2003. Neutron- and light-scattering studies of the liquid-to-glass and glass-to-glass transitions in dense copolymer miscellar solutions. Phys. Rev. E 68: 041402. 37. Grandjean, J. and Mourchid, A. 2004. Re-entrant glass transition and logarithmic decay in a jammed miscellar system: Rheology and dynamics investigation. Europhys. Lett. 65: 712. 38. Pontoni, D., Finet, S., Narayanan, T., and Rennie, A. R. 2003. Interactions and kinetic arrest in an adhesive hard-sphere colloidal system. J. Chem. Phys. 119: 6157. 39. Pham, K. N. et al. 2002. Multiple glassy states in a simple model system. Science 296: 104. 40. Pham, K. N. et al. 2004. Glasses in hard spheres with short-range attraction. Phys. Rev. E 69: 011503. 41. Szamel, G. 2003. Colloidal glass transition: Beyond the mode-coupling theory. Phys. Rev. Lett. 90: 228301. 42. Wu, J. and Cao, J. 2005. High-order mode-coupling theory for the colloidal glass transition. Phys. Rev. Lett. 95: 078301. 43. Cipelletti, L. and Ramos, L. 2005. Slow dynamics in glassy soft matter. J. Phys.: Condens. Matter 17: R253. 44. Sciortino, F. and Tartaglia, P. 2005. Glassy colloidal systems. Adv. Phys. 54: 471. 45. Zaccarelli, E. 2005. Colloidal gels: Equilibrium and non-equilibrium routes. J. Phys.: Condens. Matter 19: 323101. 46. Langevin, P. 1908. Sur la théorie du mouvement brownien. Comptes Rendus Acad. Sci. (Paris) 146: 530. 47. Chandrasekhar, S. 1943. Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15: 1. 48. McQuarrie, D. A. 1975. Statistical Mechanics. New York: Harper and Row. 49. Keizer, J. 1987. Statistical Thermodynamics of Nonequilibrium Processes. New York: Springer-Verlag. 50. Fox, R. F. and Uhlenbeck, G. E. 1970. Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical Fluctuations. Phys. Fluids 13: 1893. 51. Fox, R. F. and Uhlenbeck, G. E. 1970. Contributions to non-equilibrium thermodynamics. II. Fluctuation theory for the Boltzmann equation. Phys. Fluids 13: 2881. 52. Landau, L. D. and Lifshitz, E. M. 1959. Fluid Mechanics. Oxford: Pergamon Press. 53. Medina-Noyola, M. and Keizer, J. 1981. Spatial correlations in non-equilibrium systems: The effect of diffusion. Physica A 107A: 437. 54. Berne, B. J. 1977. In B. J. Berne (Ed.), Projection operator techniques in the theory of fluctuations, Statistical Mechanics, Part B: Time-Dependent Processes. New York: Plenum. 55. Vizcarra-Rendón, A. et al. 1989. Brownian motion in complex fluids: Venerable field and frontier of modern physics. Rev. Mex. Fis. 35: 517. 56. Evans, R. 1979. The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids. Adv. Phys. 28: 143. 57. Zahn, K., Méndez-Alcaraz, J. M., and Maret, G. 1997. Hydrodynamic interactions may enhance the selfdiffusion of colloidal particles. Phys. Rev. Lett. 79: 175. 58. Hernandez-Contreras, M., Medina-Noyola, M., and Vizcarra-Rendon, A. 1996. General theory of tracerdiffusion in colloidal suspensions. Physica A 234: 271. 59. Juárez-Maldonado, R., Chávez-Rojo, M. A., Ramírez-González, P. E., Yeomans-Reyna, L., and MedinaNoyola, M. 2007. Simplified self-consistent theory of colloid dynamics. Phys. Rev. E 76: 062502. 60. Sahimi, M. 1993. Flow phenomena in rocks: From continuum models to fractals, percolation, cellular automata, and simulating annealing. Rev. Mod. Phys. 65: 1393. 61. Madden, W. G. and Glandt, E. D. 1988. Distribution functions for fluids in random media. J. Stat. Phys. 51: 537.

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62. Given, J. and Stell, G. 1992. Comment on fluids distributions in two phase random media: Arbitrary matrices. J. Chem. Phys. 97: 4573. 63. Cruz de Leon, G. et al. 1998. Colloidal interactions in partially quenched suspensions of charged particles. Phys. Rev. Lett. 81: 1122. 64. Krakoviack, V. 2007. Mode-coupling theory for the slow collective dynamics of fluids absorbed in disordered porous media. Phys. Rev. E 75: 031503. 65. Krakoviack, V. 2005. Liquid-glass transition of a fluid confined in a disordered porous matrix: A modecoupling theory. Phys. Rev. Lett. 94: 065703. 66. Chávez-Rojo, M. A., Juárez-Maldonado, R., and Medina-Noyola, M. 2008. Diffusion of colloid fluids in random porous media. Phys. Rev. E 77: 040401(R). 67. Viramontes-Gamboa, G., Arauz-Lara, J. L., and Medina-Noyola, M. 1995. Tracer diffusion in a Brownian fluid permeating a porous medium. Phys. Rev. Lett. 75: 759. 68. Viramontes-Gamboa, G., Medina-Noyola, M., and Arauz-Lara, J. L. 1995. Phys. Rev. E 52: 4035. 69. Ramírez-González, P., Juárez-Maldonado, R., Yeomans-Reyna, L., Chávez-Rojo, M. A., Chávez-Páez, M., Vizcarra-Rendón, A., and Medina-Noyola, M. 2007. First principles predictor of the location of ergodic/non-ergodic transitions. Rev. Mex. Fís. 53: 327. 70. Yeomans-Reyna, L., Chávez-Rojo, M. A., Ramírez-González, P. E., Juárez-Maldonado, R., Chávez-Páez, M., and Medina-Noyola, M. 2007. Dynamic arrest within the self-consistent generalized Langevin equation of colloid dynamics. Phys. Rev. E 76: 041504. 71. Ramírez-González, P., Vizcarra-Rendón, A., Guevara-Rodríguez, F., de J., and Medina-Noyola, M. 2008. Glass-liquid-glass reentrance in mono-component colloidal dispersions. J. Phys.: Condens. Matter 20: 205104. 72. Juárez-Maldonado, R. and Medina-Noyola, M. 2008. Theory of dynamic arrest in colloidal mixtures. Phys. Rev. E 77: 051503. 73. Ramírez-González, P. and Medina-Noyola, M. 2009. Glass transition in soft-sphere dispertions. J. Phys.: Condens. Matter 21: 075101. 74. Juárez-Maldonado, R. and Medina-Noyola, M. 2008. Alternative view of dynamic arrest in colloid-polymer mixtures. Phys. Rev. Lett. 101: 267801. 75. Percus, J. K. and Yevick, G. J. 1957. Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. 110: 1. 76. Wertheim, M. S. 1963. Exact solution of the Percus–Yevick integral equations for hard spheres. Phys. Rev. Lett. 10: 321. 77. Verlet, L. and Weis, J. J. 1972. Equilibrium theory of simple liquids. Phys. Rev. A 5: 939. 78. Lindemann, F. A. 1911. The calculation of molecular natural frequencies. Phys. Z. 11: 609. 79. Lebowitz, J. L. 1964. Exact solution of generalized Percus–Yevick equation for a mixture of hard spheres. Phys. Rev. A 133: 895. 80. Götze, W. and Voigtmann, Th. 2003. Effect of composition changes on the structural relaxation of a binary mixture. Phys. Rev. E 67: 021502. 81. Voigtmann, Th. 2003. Dynamics of colloidal glass-forming mixtures. Phys. Rev. E 68: 051401. 82. Foffi, G., Götze, W., Sciortino, F., Tartaglia, P., and Voigtmann, Th. 2004. a-relaxation processes in binary hard-sphere mixtures. Phys. Rev. E 69: 011505. 83 Foffi, G., Götze, W., Sciortino, F., Tartaglia, P., and Voigtmann, Th. 2003. Mixing effects for the structural relaxation in binary hare-sphere liquids. Phys. Rev. Lett. 91: 085701. 84. Bergenholtz, J. and Fuchs, M. 1999. Nonergodicity transitions in colloidal suspensions with attractive interactions. Phys. Rev. E 59: 5706. 85. Fabbian, L., Götze, W., Sciortino, F., Tartaglia, P., and Thiery F. 1999. Ideal glass-glass transitions and logarithmic decay of correlations in a simple system. Phys. Rev. E 59: R1347. Erratum. 1999. Phys. Rev. E 60: 2430. 86. Asakura, S. and Oosawa, F. 1958. Interaction between particles suspended in solutions of macromolecules. J. Polym. Sci. 33: 183. 87. Zaccarelli, E. et al. 2004. Is there a reentrant glass in binary mixtures? Phys. Rev. Lett. 92: 225703.

2

Capillary Forces between Colloidal Particles at Fluid Interfaces Alvaro Domínguez

CONTENTS 2.1 2.2 2.3

Introduction ........................................................................................................................ 2D Colloids at Fluid Interfaces ........................................................................................... Deformation of the Fluid Interface and Capillary Forces .................................................. 2.3.1 Small Deformations of a Flat Interface .................................................................. 2.3.1.1 Pressure-Free Interface ............................................................................ 2.3.1.2 Interface under the Effect of a Pressure Field ......................................... 2.3.2 Small Deformations of a Curved Interface ............................................................. 2.3.2.1 Monopole ................................................................................................. 2.3.2.2 Quadrupole .............................................................................................. 2.3.2.3 Spherical Charged Particle ...................................................................... 2.4 Conclusions ......................................................................................................................... References .................................................................................................................................... Appendix ......................................................................................................................................

31 33 34 36 38 46 50 51 51 53 54 55 57

2.1 INTRODUCTION In recent years, the research of two-dimensional (2D) colloids at fluid interfaces has received increased attention (see, e.g., reference [1]). The reasons for this can be classified roughly into two groups: (i) New technologies offer a higher degree of control of the system’s physical parameters, thus expanding the spectrum of systematic experimental studies. One can mention the controlled engineering of colloidal particles with the desired physicochemical properties, the techniques of particle positioning and real-time tracking with submicrometer resolution, and the measurement of forces with piconewton resolution. (ii) 2D colloids have an intrinsic interest from the point of view of fundamental questions and applications: one can address, for example, the influence of dimensionality on physical properties, the controlled assembly of microstructures out of 2D colloids, or the particle flow at fluid interfaces in microfluidic devices. Finally, 2D colloids at fluid interfaces may serve as simple models for naturally occurring complex systems like the membrane of living cells. The presence of a fluid interface affects the interaction forces between the colloidal particles compared to the bulk (see, e.g., the recent reviews in references [2,3]). There is the specific capillary force due to the deformable nature of the fluid interface. Furthermore, other forces can decay with

31

32

Structure and Functional Properties of Colloidal Systems

B

Air/oil ++ ++ ++ ++ ++ ++ ++ + +++ + + ++ +++

Water

++ ++ ++ ++ ++ ++ ++ + +++ + + ++ +++

FIGURE 2.1 Left: Electrically charged colloids at the fluid interface between a dielectric (usually air or oil) and an electrolyte (usually water, possibly with added salt). The electric interaction between the particles is described asymptotically at large separations by electric dipoles perpendicular to the interface, as indicated in the figure. Right: Superparamagnetic colloidal particles at a fluid interface. The externally tunable magnetic field B induces magnetic moments and consequently a dipole–dipole force between the particles. The tilt of B with respect to the interface determines the anisotropy of the interaction.

separation at a different rate than in bulk: Electric forces arise naturally as the colloidal particles usually have a net surface charge (due to the process of preparation) or acquire it when chemical radicals at the surface dissociate into water. Magnetic forces appear when the particles are fabricated with a superparamagnetic core and an external magnetic field is switched on. In both cases, the interaction between two particles is asymptotically described by a repulsive potential decaying like 1/d3 (dipole–dipole repulsion) (Figure 2.1). Elastic stresses can arise if, for example, one of the fluids is in a nematic phase and the particles distort the nematic director field. This leads to an effective interaction which, according to reference [4], is described by a repulsive, effective potential decaying asymptotically like 1/d5 (quadrupole–quadrupole repulsion). The capillary force between particles when the interface is deformed by their weight is known long ago and was studied theoretically in the pioneering work by Nicolson [5]. In the 1990s, major theoretical research was conducted by Kralchevsky and coworkers motivated by the new experiments involving submillimeter particles at fluid interfaces; a review of their work can be found in reference [6], part of which will be discussed here. These earlier works address the deformation of the interface when it is caused by either the effect of gravity (particle buoyancy) or the effect of the wetting properties of the particles (reflected in the boundary conditions at the particle-interface contact line).* However, recent experiments conducted by different groups [7–14] provided evidence that was interpreted as a long-range attraction between electrically charged micrometer-sized particles at a fluid interface in spite of their expected electrostatic repulsion; the attraction was proposed in reference [11] to be of capillary origin due to interfacial deformations by electric stresses. This motivated the recent theoretical research of the capillary force in the presence of the electrostatic pressure, which is also reviewed below. The conclusion of this investigation is that the capillary attraction can unlikely explain the observations. Actually, it has been proposed [15] that the apparent attraction could be an artifact of interfacial contamination with oil. An apparent less controversial experimental evidence of capillary attraction at the micrometer scale has been reported recently in systems where the interfacial deformation is induced by nonspherical particles [16] or by electric stresses caused by an externally controlled electric field [17]. The main goal of this contribution is a review of the latest achievements in the theoretical investigation of the capillary force and the relevance of this work in the interpretation of some recent experiments. When possible we will dispense with the heavy mathematical apparatus involved in the calculations and provide instead a justification of the results based on the analogy of capillary deformation with 2D electrostatics and on rough estimates; the detailed computations can be found in the bibliography. Additionally, we will be concerned only with static properties, that is, the interface and the particles are always assumed to be in equilibrium; dynamical phenomena are out of the scope. *

In systems of particles at a thin liquid film on a solid substrate, the relevant disjoining pressure can be formally approximated by an effective gravitational field [36] (see Section 2.3.1.1.).

33

Capillary Forces between Colloidal Particles at Fluid Interfaces

2.2 2D COLLOIDS AT FLUID INTERFACES Many experiments exhibit the formation of monolayers of colloidal particles at the interface between two fluids. A common way of achieving this is by means of colloidal particles that are only partially wetted by the fluids. Let us consider a particle at a flat interface neglecting any kind of other forces (e.g., gravity or electric fields): g denotes the surface tension of the interface between the fluids and g1(2) denotes the surface tension of the interface between the particle and the upper (lower) fluid phase. The energy associated with the wetting of the particle is Ewet = g1 A1 + g2 A2 - gAint,

(2.1)

where A1(2) is the area of the particle in contact with the upper (lower) fluid phase and Aint is the area of the fluid interface cut out by the particle. The contact angle qY is defined by Young’s equation, cos qY = (g1 - g2)/g. In the simplest model [18] of a perfectly spherical particle of radius R, which is only partially wetted by the fluid phases (0 < qY < p), Ewet exhibits an absolute minimum when q = qY (Figure 2.2). The depth of this energy well (detachment energy) is DE = pgR2(1 - |cos qY|)2 (see, e.g., reference [1]). With a typical value of the surface tension g 0.05 N/m ~ 107 kT/mm2 at room temperature T = 300 K (and k is Boltzmann’s constant) and for a particle with qY not very close to 0 or p, this gives a typical detachment energy DE ~ 107 kT (R/mm)2. Therefore, particles with a size above the nanometer can be considered irreversibly trapped at the interface and for all practical purposes one is dealing with a 2D colloid at the fluid interface. The conclusions of this simple model could be altered by a series of issues. The cases of nonspherical particles or particles with a chemically heterogeneous surface (e.g., “Janus” particles [19]) introduce in general the complication of an additional term in Ewet accounting for the interfacial deformation away from flatness. This is precisely the main topic of this contribution, which will be studied at length in the following sections. Another feature worth mentioning is the relevance of the line tension associated to the three-phase contact line where the particle and the fluid phases meet: if R lies in the range of a few nanometers, the line tension seems to play a relevant role in determining the equilibrium configuration [20–22]. Engineering of particle’s wettability is not the only manner of producing 2D colloids: even if the particle tends to be completely immersed in one of the fluid phases, the interplay with a bulk force pushing it toward the interface may yield a thermally robust equilibrium position very close to the interface and thus an effective 2D colloid. An example of this scenario is the experimental realization of 2D colloids of superparamagnetic particles [23], where gravity lets the micrometer-sized hydrophilic particles stay not farther than about 0.1 mm from the interface according to reference [24]. Similarly, the interplay of electric forces and wetting properties seems to explain certain recent observations of nonwetting colloidal particles bound to a water–oil interface [24,25].

Ewet θY

γ θ R

ΔE 0

π/4

π/2 θ

3π/4

π

FIGURE 2.2 Left: The vertical position of a spherical particle intersecting a flat interface is parametrized by the angle q. Right: Plot of the wetting energy, Equation 2.1, as a function of q.

34

Structure and Functional Properties of Colloidal Systems

R R

R+h

2h Substrate

FIGURE 2.3 A colloidal particle in a thin liquid film, either sustained by a solid substrate (left sketch, film thickness R + h) or forming a liquid bridge (right sketch, film thickness 2h).

Finally, another experimentally relevant situation is that of a thin fluid film. The particles lie on a solid substrate that also sustains a thin fluid film or, alternatively, are trapped inside a liquid bridge (Figure 2.3). If the average thickness of the film is less than the particle diameter, the particles can get in touch with the fluid interface and capillary forces can play a role. In this case, the wettability properties of the particle’s surface are not so critical and the interfacial deformation can be brought about also by geometrical constraints of the setup, for example, conservation of volume of the liquid film.

2.3 DEFORMATION OF THE FLUID INTERFACE AND CAPILLARY FORCES In the macroscopic approach we will adopt, the fluid interface is characterized by the surface tension g. This quantity can be defined in two alternative but equivalent ways. In the “energy approach,” the (free) energy Einter associated with an interface S is proportional to its area:

Ú

Einter = g dA,

(2.2)

S

where dA is the element of area. In the “force approach,” one considers an arbitrary closed curve C in the interface: The net force Finter exerted at C by the part of the interface exterior to this curve is given by the following line integral (see, e.g., reference [26]): Finter = g

Ú d e

t

¥ en ,

(2.3)

C

where d is the element of arc, and en and et are defined in Figure 2.4. Usual values of the surface tension have an order of magnitude of 0.01–0.1 N/m (the addition of surfactants can reduce this value by a factor of about 10) and this sets the characteristic energy scale of interfacial deformation. At room temperature T = 300 K, this range corresponds to (106 –107) kT/mm2. Therefore, thermal fluctuations of the interface (capillary waves) can be neglected as long as the relevant length scales involved are well above the nanometer and they will not be considered here. We refer the reader to some recent works [27–30], which investigate the significance of the thermally excited capillary waves in the computation of capillary forces as they can give rise to an effective attraction akin to the Casimir force. Usually, there is a pressure field acting on the interface (due to, e.g., an external gravitational field, electric fields by charged particles, an osmotic imbalance between the fluid phases, etc.). We define P(r) as the force per unit area acting at point r of the interface in the direction of its normal en. The equilibrium states of the interface are determined, in the “force approach,” by the condition that the pressure is balanced by the interfacial line force in every patch S of the interface S: g

Ú d e ∂S

t

Ú

¥ e n + dA e n P(r ) = 0, "S Ã S , S

(2.4)

35

Capillary Forces between Colloidal Particles at Fluid Interfaces

where ∂S is the contour of the patch S. (One could write a similar equation for the balance of torques. However, since Equation 2.4 is valid for arbitrary patches S, a local version of force balance will follow, see Equation 2.6, so that torque balance is automatically satisfied and does not provide an independent constraint.) Alternatively, in the “energy approach,” the equilibrium states are determined by an extremal condition: Upon an arbitrary, infinitesimal variation r Æ r + dr of an arbitrary point r of the interface, it must hold that Ê ˆ d Á g dA˜ Ë S ¯

Ú

Ú dAP(r)e · dr = 0

(2.5)

n

S

for given boundary conditions. Note that the second term does not have the form of the total variation of a functional: It is actually a topic of current research whether the effect of a general force on the interface can be cast in a functional of quantities pertaining solely the interface [31]; this is so in some particular cases such as, for example, a constant pressure P(r) = P [31,32] or for small interfacial deformations, see Equation 2.11. The variation of the surface area in Equation 2.5 can be calculated with the tools of differential geometry (see, e.g., references [32,33]) and a relationship is obtained which links the geometrical properties of the interface with the pressure field (the Young–Laplace equation): 2gH(r) = P(r),

(2.6)

where H(r) is the mean curvature of the interface (with the convention H > 0 when the normal en points from the concave to the convex side of the surface, Figure 2.4). In the special case of a spherical interface of radius R (and thus H = 1/R) under a constant pressure P, this expression reduces to 2g/R = P, that is, the Laplace equation of capillarity. In the case of a vanishing pressure field, the interface is determined by the condition H(r) = 0, that is, a minimal surface. Likewise, when Equation 2.4 is applied to an infinitesimal patch S, the same relationship Equation 2.6 is obtained

en Σ

Z

et Z

C et ¥ en Y

Π t u n X

FIGURE 2.4 Left: In a fluid interface, en is defined as the unit vector normal to the interface. For an arbitrary region S enclosed by the contour C, et is defined as the unit vector tangent to it in the sense such that et × en points to the exterior of C. In the small-deformation regime, C and S are projected onto the reference XY-plane as C|| and S||, respectively. In this plane, the unit vector n normal to C|| lies along the projection of the unit vector et × en, and the unit vector t tangent to C|| points along the projection of et. Right: The small deformation (exaggerated in the schematic drawing) of an interface away from the reference XY-plane is parametrized by the 2D field u(r||) (Monge representation), where r|| = xex + yey is the 2D position vector in the XY-plane. The field P(r||) is the force per unit area in the normal direction, almost coincident with the Z-direction.

36

Structure and Functional Properties of Colloidal Systems

(the proof of this statement is given in Section 2.3.1 for small deformations about a flat interface; since any smooth interface looks locally like a small deformation of its local tangent plane, that proof is actually general). Although both the energy and the force approaches are equivalent at a fundamental level, use of one or the other may be more advantageous at a practical level depending on the circumstances. The capillary force as usually measured in experiments and as relevant in theoretical calculations calling for an effective interparticle force is actually a so-called mean force, that is, the result after all the degrees of freedom other than the particle position and orientation have been integrated out. Consider, for example, the simplest case of two identical, spherical particles a distance d apart. Let Ecap(d ) denote the parametric d-dependence of the free energy stemming from the terms affected by interfacial deformation (see, e.g., Equation 2.11). Then, Ecap(d ) is a potential of mean force and the capillary force is defined as the derivative Fcap = -E¢cap(d ). In this respect, two observations are in order concerning particularly the case P(r) π 0: 1. The capillary force Fcap acting on a particle is in general different from the interfacial force Finter defined in Equation 2.3: Additional to the pull by the interface at the contact line, there can be other forces whose effect depends on the state of the interface, for example, an inhomogeneous pressure field in the fluid phases gives a net force affected by the position of the contact line. This contribution, however, is in many cases negligible or subdominant at the small length scales we are interested in (see, e.g., the discussion concerning Equation 2.18), so that for simplicity we will identify here the capillary force with the interfacial force in Equation 2.3. With this approximation, one also benefits from less cumbersome calculations. 2. As a mean force, the capillary force also includes the force that the pressure field exerts on the interface. Therefore, the effective force that is assigned to a particle is not just the net force felt by the particle itself but there is also this contribution from the interface surrounding it, so that a more appropriate description would be that of an “effective particle” comprising the true particle plus the neighboring interface. We discuss an example of this phenomenon in Section 2.3.1.2.

2.3.1

SMALL DEFORMATIONS OF A FLAT INTERFACE

The first approach to the understanding of the capillary forces in equilibrium states involves the approximation of small interfacial deformations about a reference flat state. It will be understood under this approximation that both the departures from the flat interface and the spatial derivatives of the departure are as small as necessary to retain only the lowest-order terms in an expansion in those small quantities.* This approximation simplifies considerably the analytical calculations and it also seems to describe well many of the experiments performed so far: Because of the typically large values of g, considerable large forces already come into play upon tiny interfacial deformations; additionally, the geometrical configurations usually considered are simple enough that no large derivatives appear. The height of the interface over the reference flat interface is described by a 2D scalar field u(r||) (see Figure 2.4 for the definitions). The force Finter defined in Equation 2.3 is easily evaluated up to second order in the small-deformation approximation [34]: Finter = - ge z

Ú d n ◊ — u(r ) - Ú d n ◊ T , ||

C ||

*

||

||

||

C ||

This also implies in particular that the deformed interface does not have overhangs.

||

(2.7)

Capillary Forces between Colloidal Particles at Fluid Interfaces

37

where —|| = ex ∂x + ey ∂y, and T|| is a second-rank tensor (1|| is the unit tensor in the XY-plane), 2 1 È ˘ T|| = g Í(—||u )(—||u ) - —||u 1|| ˙ , 2 Î ˚

so that the second term in Equation 2.7 is actually normal to ez (lateral capillary force). Using this expression in the force balance, Equation 2.4, one gets a condition for the component along ez, which yields the linearized version of the Young–Laplace relationship (Equation 2.6) after application of Gauss’ theorem: g — ||2u = -P,

(2.8)

where H = -—2|| u/2 for small deformations. On the other hand, the projection of the force balance onto the XY-plane yields in the small-deformation limit

Ú d n ◊ T ||

||

=-

∂S ||

Ú dA P— u. ||

||

(2.9)

S ||

We have just described the linearized theory of capillarity. In the electrostatic analogy the field u(r||) is identified with a 2D electrostatic potential (“capillary potential”) and P(r||) with a charge density (“capillary charge”): Equation 2.8 reduces to the Poisson equation of electrostatics and Equation 2.9 relates the tensor T||, which has the form of Maxwell’s stress tensor, with the “electric force” exerted on the “capillary charge” P(r||) (also the usual boundary conditions imposed on the interface have a close electrostatic analogy [34,35]). This analogy is almost perfect, except for a single but important difference: The capillary force in the XY-plane, given by the second term in Equation 2.7, is minus the flux of the stress tensor T||, so that capillary charges of equal (different) sign will attract (repel). Unlike in electrostatic, where Equation 2.9 serves as definition of the stress tensor, here it is a balance condition between two forces of different origin (interfacial force and pressure): The true definition of T|| and its physical relationship with a force is actually Equation 2.7. Apart from this, the electrostatic analogy provides a transparent visualization of small interfacial deformations and ensuing forces in terms of an equivalent 2D electrostatic problem and many results can be carried over without change. This idea has been widely used and the notion of “capillary charge” is not uncommon in the literature (see, e.g., reference [6]).* The electrostatic analogy can be employed also in situations when the interfacial deformation is not necessarily small everywhere: The “nonlinear patches” of the interface can be isolated by contours outside of which the deformations are small and the analogy holds. The effect of the nonlinear patches on the rest of the interface is replaced by boundary conditions at these contours, that is, by an appropriate distribution of “virtual capillary charges” in the nonlinear regions. As we will see, in this manner one can generalize conclusions obtained only when the deformation is small everywhere. An important result in this respect is the value of the capillary monopole and dipole of a bounded region of the interface. By Gauss’ theorem in electrostatics, the capillary monopole Q of a patch S is given by Q = -g

Ú d n ◊ — u, ||

||

∂S ||

which, according to Equation. 2.7, is precisely the component of the capillary force at the contour ∂S|| in the direction of ez. On the other hand, the fully nonlinear condition of force balance establishes that *

We also note the alternative magnetostatic analogy, in which ezu(r||) is identified with a magnetostatic vector potential and ezP(r||) with a current density. This establishes an interesting and unexplored connection with the physics of 2D vortices appearing in other areas of statistical physics, for example, fluid turbulence [74].

38

Structure and Functional Properties of Colloidal Systems

the capillary force (Equation 2.3) at the contour just balances whatever other forces Fext are acting in the interior of the patch of interface (including the pressure field P(r||) on the interface, but also any other volumic forces acting on colloidal particles possibly contained in the patch): Finter + Fext = 0. Therefore one has Q = ez · Fext,

(2.10)

and the only condition for validity of this expression is that the interface deforms slightly from a flat one at the contour of the patch, independently of how complicated the deformation is inside it. A similar relationship can be obtained between the capillary dipole P (with respect to a point) of the patch S and the torque Mext (with respect to the same point) by forces other than the interfacial one [34]: P = ez × Mext is valid under the same conditions as Equation 2.10. For completeness, we mention briefly the equivalent energy approach. In the small-deformation limit, the interfacial deformation is derived from variations at fixed pressure field P(r) of the energy functional Ecap =

1 2 g dA|| (—u ) 2

Ú S||

Ú dA Pu + E ||

wet

+ Eext

(2.11)

S||

(see, e.g., references [35–37]). Here, the first term is the small-deformation approximation of Einter defined in Equation 2.2, the second term is the work done by P(r) during a small vertical displacement u(r) (see Equation 2.5), the third term is the wetting energy of the particle (Equation 2.1), and the last term accounts for the possible external forces and torques on the particle.* The variation of the first two terms gives Equation 2.8, while the last two terms yield boundary conditions at the particle-interface contact line and constraints on the particle’s position and orientation. 2.3.1.1 Pressure-Free Interface As a first application of these results, consider the simplest case of an interface in the absence of a pressure field (P = 0) but deformed by the presence of particles. The information about the particle relevant for the interfacial deformation is encoded in the complex-valued capillary multipoles Qs, s = 0, 1, 2, . . . (see the Appendix). These quantities are determined, via the boundary conditions at the particle-interface contact line, by the shape and wettability properties of the particle, and by the forces and torques acting on it. We have already seen, in particular, how the monopole Q0 and the dipole Q1 are actually independent of geometrical and chemical details and given solely in terms of forces and torques. In two dimensions the multipoles Qs are characterized by a real-valued strength qs and an orientation 0 < qs < 2p: Qs = qseisqs. The physical interpretation of the quantities qs and qs follows from the interface deformation described by Equation 2.12 below. One can now apply the electrostatic analogy advantageously to compute the interfacial deformation and the capillary forces. Given an isolated particle and introducing polar coordinates (r, j) in the XY-plane centered at some point of the particle, one can easily write the interfacial deformation as a multipole expansion (see the Appendix):

uisol (r , j; {Qs }) =

q0 L 1 ln + 2 pg r 2 pg

•

qs cos s(j - q s ) , sr s s =1

Â

(2.12)

where L is a constant determined by the distant boundary conditions, which set the zero point of the “capillary potential” u (see the discussion of Equation 2.17 below). *

For example, a net vertical force Fextez leads to a term E ext = -Fext Dh, where Dh is the vertical displacement of the center of mass of the particle.

39

Capillary Forces between Colloidal Particles at Fluid Interfaces

Consider now a collection of N particles. The electrostatic analogy establishes that the interfacial deformation can be written as a superposition of expansions of the form of Equation 2.12, one centered (i) at each particle, but with certain multipole charges Q(i) s + dQs (i = 1, 2, . . . , N), which do not have to be identical with the charges as if each particle were isolated: N

u(r ) =

Âu

isol

(r - ri ; {Q (si ) + dQ (si ) }).

(2.13)

i =1

The charges Q(i) s which the ith particle has when isolated at the interface can be termed permanent capillary multipoles, as opposed to the capillary multipoles dQ(i) s induced by the presence of the other particles. The induced multipoles arise in general because it is obvious that a superposition of profiles as if the particles were isolated cannot satisfy in general the boundary conditions at all the contact lines (see the sketch in Figure 2.5). In general, the induced multipoles dQ(i) s will depend on the three-dimensional (3D) positions and orientations of all the particles through the boundary conditions at each particle. Thus, there is not a simple “constitutive relation” between dQ(i) s and the deformation field, and the electrostatic analogy seems of limited use in this respect; instead, dQ(i) s would have to be determined by a detailed solution of the problem near the particles. Note that, because of the implicit dependence of dQ(i) s on the interfacial deformation brought about by the other particles, Equation 2.13 is not a linear superposition. However, by definition, the corrections dQ(i) s vanish in the limit of well-separated particles, |ri – rj| Æ •. As a consequence, asymptotically for large separations the deformation is dominated by the linear superposition of the deformation due to the lowest-order nonvanishing permanent capillary charge of each particle. This is the so-called superposition approximation introduced in reference [5] and frequently used in the literature (either in the force or in the energy approaches). As we see, it is a straightforward consequence of the electrostatic analogy in the absence of induced capillary multipoles; violations of this approximation are possible in the context of the linearized theory of capillary due to “capillary polarization” effects. The discussion concerning the capillary force, that is, the effective force between particles due to the interfacial deformation that they induce, proceeds in an analogous manner. We consider two particles in a fixed configuration determined on the XY-plane by their separation d and the orientation j of the joining line with respect to the coordinates’ axes. In this configuration, each particle is ˜ n, respectively, which characterized by the (permanent plus induced) capillary multipoles Qs and Q depend on their full 3D position and orientation. One introduces an effective interaction potential of the form (see Appendix)

U eff = -

q0 q0 L 1 ln 2pg d 2pg

•

Â

n, s = 0 ( n , s ) π (0,0)

qs q n

(-1)n (s + n - 1)! cos ÎÈ(n + s )j - sq s - nq n ˚˘. (2.14) s ! n! d n + s

The capillary force and the capillary torque can be computed by virtual variations of Ueff, that is, ˜ n were taking into account only the dependence on d and j explicit in Equation 2.14 as if Qs and Q rc zc ψc

Δu R

u(contact)

θY

u+h

h

R FIGURE 2.5 Left: Sketch of a contact line tilted by the presence of a second particle. Right: Definition of some geometrical quantities at the contact line.

40

Structure and Functional Properties of Colloidal Systems

fi xed. When there are more than two particles, Ueff has the form of a sum of terms like Equation 2.14, one for each pair of particles, and the capillary multipoles will depend on the 3D positions and orientations of all the particles. As discussed with Equation 2.13, this does not imply pairwise additivity of forces and torques because of this implicit dependence in the multipoles. However, if it happens that there are no induced capillary charges or, more generally, when the separation between all the particles grows and the induced charges vanish, the potential Ueff is dominated by the lowest-order nonvanishing permanent capillary charge of each particle and the capillary forces and torques between particles are asymptotically pairwise additive. Finally, we remark on two closely related, but conceptually different issues: On the one hand, there are the capillary deformations themselves, described by the form of the expansions in Equations 2.12 and 2.14, which follow solely from the general properties of a fluid interface, that is, from the energy functional (Equation 2.2) in the small-deformation regime. On the other hand, there are the sources of the deformation, encoded in the values of the capillary multipoles Qs and which are determined by the physical properties of the particles, for example, by the wetting energy (Equation 2.1). These two aspects of the problem get usually merged in most calculations. Aside from the monopole and the dipole, there is generically no way to compute the value of the multipoles without solving the detailed problem of capillary deformation near the particles with the appropriate boundary conditions. However, in many instances it suffices with an estimate of the order of magnitude of the multipoles. Additionally, if one employs the superposition approximation, as is almost invariably the case in analytical calculations, one requires by consistency just the knowledge of the lowest-order permanent capillary charge, determined in turn by the much simpler problem of an isolated particle. 2.3.1.1.1 Monopole–Monopole Interaction The simplest situation is the interaction between particles with a permanent capillary monopole. By Equation 2.10, it means that the particles are under the action of an external vertical force: the bestknown example is the buoyancy force due to gravity. The potential of mean force between two particles under the action of vertical buoyancy forces Fbuoy and F˜buoy, respectively, is asymptotically (i.e., in the superposition approximation)

U eff = -

Fbuoy Fbuoy L ln . 2 pg d

(2.15)

_____

The capillary length l := √g/gDr is defined in terms of the acceleration of gravity g and the difference Dr in mass density of the fluid phases; l 3 mm for the air–water interface. For spherical colloidal particles of radius R and with a mass density of the order of Dr, the buoyancy force is given approximately by Fbuoy (4p/3)R3gDr (4p/3)gR3/l2, and the effective potential is Ueff -(8p/9) gR2(R/l)4 ln(L/d) -10-6 kT (R/mm)6 ln(L/d), for typical values of the parameters at room temperature. Hence, this interaction is relevant compared to the effect of the thermal agitation only for particle sizes above 10 mm roughly. Equation 2.15 holds when P = 0, but it has to be modified to account for the effect of gravity on the fluid phases, showing up in the form of a pressure field:

P grav = - g

u . l2

(2.16)

The corrected interaction energy reads as

U eff = -

Fbuoy Fbuoy Ê dˆ K0 Á ˜ 2 pg Ë l¯

(2.17)

Capillary Forces between Colloidal Particles at Fluid Interfaces

41

in terms of Bessel’s function K0. The capillary length acts as a natural cutoff for the logarithmic divergence, with a crossover to an exponential decay beyond d l. However, on much smaller length scales (d l), the simpler expression 2.15 is valid with L = 2le-ge 1.12l (ge is the Euler– Mascheroni constant). This potential of mean force was first derived using the energy approach and the superposition approximation [5,38] with the additional hypothesis that the interfacial deformation is small everywhere. As discussed after Equation 2.9, however, Equation 2.15 is valid provided the particles are far enough from each other, even if the interfacial deformation is large in their neighborhoods: This was actually an experimental finding reported in reference [39] for the force measured between two < d/l ~ < 4 (this phenomenon was termed vertical large cylinders of radius 0.4 mm in the range 1 ~ the “nonlinear superposition approximation” in reference [39]). The behavior predicted by Equation 2.17 was observed only at the largest separations; departures at short distances appeared because then nonlinear effects beyond the small-deformation approximation were important in the region between the cylinders. In this experiment, the capillary monopole q0 is not a consequence of gravity (the vertical height of the cylinders is irrelevant), but rather of their wettability properties. More recently, Vassileva et al. [40] reported the measurement of the force between freely floating submil< d/l ~ <2 limeter spherical particles. For a particle radius R 0.3 mm and separations 0.3 ~ < d/R ~ < 24), the force agrees within experimental errors with the prediction of Equation 2.17. (4 ~ Concerning these works, we note that the force measured in these experiments is not only due to the interfacial tension since there is also a contribution by the vertical gradient of hydrostatic pressure caused by gravity: If the particle-interface contact line is tilted with respect to the vertical by a small amount Du R (see Figure 2.5), the lateral pressure imbalance across the particle is of the order of g(Dr) u(contact) and acts on a surface of size ~R(Du). By the superposition approximation (i.e., as R/d Æ 0), one can estimate u(contact) ~ uisol(R), Du ~ uisol(d + R) - uisol(d) ~ Ru¢isol(d), with the monopole deformation field uisol(r) = (Fbuoy/2pg)K0(r/l) for an isolated particle. Thus, there is a lateral pressure force 2

Ê Rˆ Fp ª g(Dρ) R(Du)u(contact) ª γ Á ˜ uisol( R)uisol ¢ (d ). Ë l¯

(2.18)

This is a factor ~(R/l)2 smaller than the interfacial force Finter = Fbuoyu¢isol(d) following from Equation 2.15 and thus negligible for submillimeter particles compared to the interfacial force. This rough estimate is confirmed by exact calculations in references [41,42]. In a different setup, the presence of an external electric field E perpendicular to the undistorted flat interface polarizes the particles, which repel each other laterally due to the electrostatic interaction. However, the vertical pull on the particles by both gravity and the external electric field generates a capillary monopole and therefore a lateral capillary attraction. The equilibrium separation between the particles can be varied with the controllable field E, since sizable capillary forces already appear for not too large values of the electric field [37]; this has been demonstrated in recent experiments [17,43]. In a homogeneous field, a net electric force on the dielectrics arises only at the discontinuities in the permittivity, where there is a polarization charge. Since the polarization charge of the particle is proportional to e0R2E (e0 is the permittivity of the vacuum), the vertical electric force on the particle is of the form Felec = e0R2E2felec, where felec is a (positive or negative) dimensionless factor accounting for the detailed dependence on the dielectric constants and the geometry of the problem (relative position of the particle and the interface). The buoyancy force on the particle can be likewise written as Fbuoy = gR3(Dr)fbuoy, and the lateral capillary interaction force in the monopole approximation is therefore Fcap = (Felec + Fbuoy)2/2pgd. Similarly, the total electric dipole induced in the particle scales like e0R3E, and the lateral dipole–dipole repulsion between two

42

Structure and Functional Properties of Colloidal Systems

particles a distance d apart is Frep = e0R2E2(R/d)4frep, where frep > 0, collects the detailed dependence on the geometry. The equilibrium separation deq between two particles is determined by the balance of the capillary attraction Fcap and the electrostatic repulsion Frep: ˆ deq Ê 2 pg e 0 E 2 frep =Á 2 2˜ R Ë R(e 0 E felec + ( Dr)gRfbuoy ) ¯

1/3

.

(2.19)

Two limiting regimes can be distinguished: (i) large particles or small electric fields: the electric contribution Felec to the capillary force is negligible and deq/R ~ E2/3/R; (ii) small particles or large electric fields: Fbuoy is negligible and deq/R ~ E -2/3R-1/3. The parameter region (Rcross, E cross) where the crossover occurs can be estimated very roughly by the condition Felec = F buoy: for particles with Dr ~ 1 g/cm3, one has [E cross/(V/m)]2 109 (Rcross/mm). Thus, with micrometer-sized particles one can switch from one regime to the other in a range of experimentally accessible values of the electric fields. For submicrometer particles, the limiting case (ii) is particularly relevant, since the effect of buoyancy is swept out by thermal fluctuations anyhow. By dropping the buoyancy term in Equation 2.19, one has deq/R 4 · 105 [E/(V/m)]-2/3 (R/mm)-1/3 for a value g = 0.05 N/m. With R = 50 nm and E = 106 V/m, this gives a fairly large equilibrium separation deq 100R. The analysis we have just discussed was presented in reference [17], where the electrostatic problem of two spherical particles at a flat interface in an external homogeneous field E was solved numerically and a detailed study of the dimensionless coefficients felec and frep was carried out for different values of the dielectric constants and the particle-interface contact angle. The results for deq were compared with experimental measurements: an interface between two dielectric fluids (air and oil) contains a 2D closely packed assembly of uncharged glass particles (radius R ~ 50 mm) and is subjected to an external homogeneous field E ~ 105 V/m; with these parameters, the system is in the crossover regime and the lattice constant of the 2D assembly can be varied with the field E. The measurements of deq quoted in Figure 12 of reference [17] do not refer, however, to the lattice constant but to the equilibrium separation between two isolated particles (N. Aubry, private communication). In the range 2 < deq/R < 4.5 a satisfactory agreement with the prediction given by Equation 2.19 was obtained. The theoretical derivation neglected the pressure field P(r) due to the electric stresses by the polarized particle on the interface. This additional effect is shown in Section 2.3.1.2 to be subdominant for large separations d (see Equation 2.32). Another example of a relevant monopole capillary force, which will also serve to illustrate the effect of induced capillary charges, is the case of spherical particles on a solid substrate sustaining a thin liquid film (Figure 2.3). First we mention briefly the role played by the disjoining pressure Pdisj exerted by the substrate on the film. As an illustration, consider the simplest case of van der Waals forces, which give Pdisj = -2H/h3 inside the film as a function of the height h over the flat substrate, where H is the Hamaker constant. For small deformations of the interface, this pressure enters Equation 2.8 as a term [35,36] Pdisj(r||) = -gu(r||)/l2disj, formally identical to the gravitational pressure, _______ Equation 2.16, but with an effective capillary length l disj: = √ gh 4/(6H) . With typical values g = 0.05 N/m, H = 10-20 J, this gives ldisj/1 mm (h/1 mm)2. Therefore, for liquid films with a thickness in the micrometer range, the effect is comparable to that of gravity, and thus negligible for submicrometer particles. In the substrate–liquid film configuration of Figure 2.3, the interfacial deformation at a distance r from an isolated spherical particle in the small-deformation limit is uisol(r) = (q0/2pg)K0(r/l), with qs = 0 for s > 0 by spherical symmetry. Instead of the value of q0 (the force exerted by the substrate), one knows the fixed thickness R + h of the film far from the particle. The relationship between these two quantities is given by the boundary condition that the particle-interface contact angle is qY.

43

Capillary Forces between Colloidal Particles at Fluid Interfaces

Together with some geometric relationships (see Figure 2.5 for the definitions of the geometrical quantities rc, zc, and yc), one has at the contact line duisol (r ) = - tan(y c - q Y ), dr c rc = R sin y c ,

uisol (rc ) = R - zc - h ,

(2.20)

zc = R (1 - cos y c ) .

(2.21)

Simplifying these expressions in the small-deformation limit, |yc – qY| 1, one finds h ª R cos q Y - uisol (rc ) + (rc )

duisol q R sin q Y (r ) ª R cos q Y + 0 ln , dr c 2 pg 2le1- g

(2.22)

e

when R l, assuming that the particle is partially wetted by the fluids, that is, qY π 0, p (we address the case of complete wetting later). When there is a second particle a distance d apart, the capillary monopole gets modified to q0 + dq0(d), meaning by Equation 2.10 that the vertical external force on the particle does depend on the second particle: This is so because the external force exerted by the substrate is not fixed but is actually a reaction to the pulling force exerted by the variable meniscus of the interface. The induced capillary monopole dq0(d) is determined by the same boundary conditions as before but replacing uisol(r) in Equations 2.20 and 2.21 by the full deformation field u(r). In the limit d rc, one neglects the higher-order induced multipoles and makes the simplest approximation u(r) uisol(r) + uisol(d) near each particle; in the small-deformation limit, Equation 2.22 is then replaced by h ª R cos q Y +

q0 + dq0 (d ) È R sin q Y Ê dˆ˘ ln - K0 Á ˜ ˙ . Í 1- ge 2 pg Ë l¯˚ Î 2le

(2.23)

Therefore, the induced capillary monopole reads as È Í K 0 (d /l ) q0 + dq0 (d ) = q0 Í1 R sin q Y Í ln 2le1- ge ÎÍ

-1

˘ ˙ ˙ , ˙ ˚˙

(2.24)

and the capillary force between the two monopoles is Fcap = [q0 + dq0(d)]2/2pgd. The pure monopole– monopole force is observed only asymptotically for d l. In the opposite limit, q0 + dq0(d) exhibits a mild logarithmic dependence that modulates the 1/d-decay; at contact, d = 2R<< l, Equation 2.24 yields q0 + dq0(d) 0.5q0. For values of the radius R and the thickness h of the same order and in the range from the nanometer to the micrometer, Equation 2.22 gives |q0| / 2pg ~ 0.1R roughly, and the order of magnitude of the force is consequently Fcap (105 kT/d) (R/mm)2 at room temperature and thus relevant in a broad range of particle sizes, even when the monopole contribution by buoyancy is completely negligible. In reference [36], an implicit expression for the force between two spheres in a liquid film is computed, which is formally exact within the small-deformation approximation but so involved that it is useful mainly for numerical evaluation. The discussion presented here rationalizes the result of that work in simple terms. The modulation of the 1/d-decay has been observed in a recent experiment [44], where the particles are trapped in a thin liquid film hanging at a frame. Now there are two fluid interfaces (the upper and the lower interfaces of the film, see Figure 2.3), for each of which the previous discussion applies in the limit d, R, h l (so that gravity is negligible and both interfaces behave identically).

44

Structure and Functional Properties of Colloidal Systems

The only difference is that the particles used in the experiment are completely wetted by the liquid, qY = 0, and the simplification of the boundary conditions (Equations 2.20 and 2.21) in the small-deformation limit, |yc | 1, gives a slightly different equation for the capillary monopole: In the limit l Æ • one arrives at QlnQ = -e, in terms of the dimensionless quantities Q := C(q0 + dq0)d2R/4pgl4 and e := Cd2R(R - h)/l4 with the numerical constant C := e4ge-1/8 0.462 . . . . The lengths d, R, and h lie in the micrometer range, while l is about a millimeter, so that the quantities Q and e are very small (~10-12). The solution Q(e) can be searched as a logarithmic expansion around the contact point d = 2R, resulting in a power-law dependence a

Ê d ˆ q0 + dq0 (d ) ª [q0 + dq0 (d = 2 R)] Á , Ë 2 R ˜¯

aª-

2 , ln Q(d = 2 R )

which is nevertheless rather mild because the exponent is small: For realistic values of e in the range [10-13, 10-10], it gives 0.060 < a < 0.076. Therefore, the capillary force between two identical particles in the liquid film is predicted to decay with separation like 2

Èq0 + dq0 (d )˘˚ Fcap = Î μ d 2a -1 , pgd that is, with an exponent in the approximate range [-0.88, -0.85] for micrometer-sized particles (the force is multiplied by a factor 2 to account for the contributions of the upper and lower fluid interfaces of the film). This result was derived in reference [44], where the capillary force between two particles (radius R = 2.8 mm) in a liquid film (h ~ 1 mm) at separations 2R < d < 20R has been measured and a power-law decay with d was obtained. According to this work, the best fit to a power law with a and h as free parameters gives 1 - 2a = 0.86 and h = 2.2 mm. This good agreement with the theoretical calculations even at separations close to contact indicates that, as far as the capillary force is concerned in this setup, the approximation that the relevant effect of the wetting energy (Equation 2.1) can be accounted for by an induced capillary monopole and that higher-order induced multipoles can be neglected, holds beyond the condition R << d. 2.3.1.1.2 Quadrupole–Quadrupole Interaction If there is no external force or torque acting on the particles, Fext = 0, Mext = 0, then the capillary monopole and the capillary dipole vanish, see Equation 2.10. Therefore, the lowest-order nonvanishing capillary charge will be generically the quadrupole, which arises naturally when the particle is not spherical, because a flat interface will unlikely satisfy the corresponding boundary condition at the contact line. The quadrupole also arises generically if the interface is pinned in an irregular manner at the particle’s surface or if the contact angle qY depends on the position at a heterogeneous particle’s surface. The interfacial deformation by an isolated quadrupole is u2(r, j) = (q2/4pgr 2) cos 2(j - q2), which yields a maximum deformation at the contact line of the order of H = q2/4pgR2 for a particle characterized by a typical size R; it is useful to parametrize q2 by the length scale H. The capillary interaction between these particles will be described asymptotically by the quadrupole–quadrupole term of Equation 2.14, which can be rewritten as [9] 4

Ê Rˆ U eff = - 12 pgHH Á ˜ cos 2(q2 + q 2 ), Ë d¯

(2.25)

after taking j = 0, that is, the orientations q2, q˜2 are measured with respect to the radius vector joining the quadrupoles. Therefore, the quadrupoles will experience a torque that aligns them parallel to

Capillary Forces between Colloidal Particles at Fluid Interfaces

45

each other (q˜2 = -q2, p - q2) but at an arbitrary tilt with respect to the radius vector. When aligned like this, they will experience an attractive force decaying like 1/d5. With a typical value g ~ 107 kT/ mm2, the depth of the effective potential at contact (d = 2R) is of the order of 107 kT (H/mm)2 at room temperature. Hence, even tiny deformations at contact in the nanometer range induce forces overcoming the effects of thermal agitation by far. One can thus expect the effectively irreversible formation of clusters, maybe with a glass-like structure due to the huge strength of the attraction compared to thermal motion. Furthermore, the anisotropic nature of the force will lead to highly branched clusters at low particle densities, and to anisotropic solid-like phases at high particle densities [45]. The possible relevance of this quadrupole–quadrupole force at colloidal length scales was advanced in references [9,46,47]. As emphasized in reference [9], the natural surface roughness in the nanometer scale of a micrometer-sized spherical particle could conceivably cause a force of this kind. It was also proposed as an explanation of the structures observed experimentally [47] in 2D colloids of nonspherical micrometer particles at a fluid interface. This motivated the theoretical investigation of the anisotropic capillary forces: The main difficulty lies in the determination of the capillary charges Qs in terms of given properties of the particles (e.g., wettability and shape). Different simplifications were applied: the contact line is approximated by a circle [48–50] or by an expansion in small eccentricity [45]; highly elongated shapes are dealt with numerically [51]. Loudet and coworkers [16,52] report recent experimental investigations on 2D colloids of particles with the shape of prolate ellipsoids (long axis in the range of 10 mm, short axis about 1 mm). The measured interfacial deformation around an isolated particle seems compatible with a quadrupole of amplitude H ~ 0.5 mm. The interpretation of the results is, however, somewhat problematic because the value of the contact angle qY derived from the interfacial deformation by numerically solving Equation 2.8 turns out to depend on the aspect ratio of the particle’s shape; this unexpected result is ascribed in reference [52] to the fabrication procedure of the particles. In reference [16], the irreversible formation of highly branched clusters was observed. The effective pair interaction potential for separations 20 mm < d < 55 mm was inferred from the particle motion: The potential well was deeper than 104 kT and a dependence 1/d4 was obtained when the ellipsoidal particles approached tip-to-tip in very good agreement with Equation 2.25. However, the side-to-side approach gave a dependence 1/d3.1; a possible explanation has been advanced in reference [51] and is discussed below. 2.3.1.1.3 Interaction between Higher-Order Multipoles As we have seen, the monopole or the quadrupole can dominate asymptotically in easily realizable setups. The dipole would dominate if the particles were under the effect of an external torque with vanishing net external force; so far there does not seem to be any clear experimental realization of this possibility. Therefore, higher-order multipoles will appear in the expansions (Equations 2.12 and 2.14) as subdominant in general and their contribution has to be addressed beyond the superposition approximation. In reference [40] the correction to the monopole capillary force (Equation 2.17) of freely floating particles was studied numerically for a pair of spherical particles and a pair of cylinders. As expected, the error of the approximation increases with decreasing separation. However, it is very good for spheres, with a maximum error of ~10% at contact for particles with a contact angle of qY = p/2. In the case of cylinders, however, the approximation is very bad unless the separation is a few times the radius of the cylinders. Since in both cases the deformation profile of isolated particles is monopolelike by rotational symmetry, the corrections are due exclusively to induced capillary charges. Their sensitivity to the boundary conditions could explain the difference observed between spherical and cylindrical particles: Since the vertical position of spherical particles is an unconstrained and relevant degree of freedom (as opposed to vertical cylinders), Vassileva et al. [40] suggest that this may reduce the effect of a contact line changing with separation and, therefore, of violations of the superposition approximation. In references [48,50], the force between two particles, approximated as vertical

46

Structure and Functional Properties of Colloidal Systems

cylinders, was addressed analytically using bipolar coordinates. The chosen boundary conditions describe a single permanent capillary charge, that is, the interfacial deformation of an isolated particle is given by just one of the terms in Equation 2.12, and departures are to be assigned entirely to induced multipoles. The deformation profile is written as a “bipolar” multipole expansion, viewed as a resummation of the spherical multipole expansions in the general solution (Equation 2.13). The most interesting conclusion is that, for certain orientations of the multipoles, the capillary force is not monotonous with separation but crosses over from attraction to repulsion near particle contact. This feature cannot be captured at all by the superposition approximation, which always yields a force between single multipole charges decaying like a power law of the separation. Finally, Lehle et al. [51] address the case of highly elongated ellipsoidal particles as employed in recent experiments [16,52,53]. Use of elliptic coordinates allows an efficient numerical computation of the interfacial profile for elliptical contact lines and the deformation is expressed as an “elliptic” multipole expansion. Corrections to the spherical quadrupole behavior predicted by the superposition approximation are found as the interparticle separation decreases. In particular, in the range of parameters and separations probed by the experiment of Loudet et al. [16], the effective capillary potential decays more slowly than 1/d4: Lehle et al. [51] suggest this as the explanation for the apparent value -3.1 of the decay exponent quoted in reference [16]. 2.3.1.2 Interface under the Effect of a Pressure Field Another case of experimental relevance occurs when P π 0 in Equation 2.8. The most studied situation is a pressure field of the form of Equation 2.16 due to gravity or a disjoining pressure. We have already mentioned how this field modifies the monopole–monopole interaction from Equation 2.15 to Equation 2.17. Quite generically, Equation 2.8 with this pressure field is formally analogous to the Debye–Hückel theory of dilute electrolytes and the capillary length l plays the role of a screening length. Therefore, when the lengths involved are much smaller than l, that is, in the submillimeter range, the effect of this pressure is negligible and one can resort to the results derived for a pressurefree interface, as has been illustrated previously with several examples. The recent theoretical investigation has addressed the case of a localized pressure field centered at the particles [35,37,54–58]. This was motivated by the suggestion [11] that the interfacial deformation by the electric field emanating from a charged colloidal particle is the origin of an apparent long-ranged attraction between electrically charged particles in 2D colloids [7–14]. One can apply the electrostatic analogy in a manner paralleling the pressure-free case. An isolated particle at an asymptotically flat interface creates a pressure field Pisol(r) and the deformation (Equation 2.12) has to be amended with the contribution by the extended “capillary charge” Pisol(r),

uisol (r , j) =

q0 L 1 ln + 2 pg r 2 pg

•

qs cos s(j - q s ) 1 + s 2 pg sr s =1

Â

Ú dA¢P

isol

(r ¢ ) ln

L . r - r¢

(2.26)

The capillary multipoles Ps of the capillary charge distribution Pisol(r) are (see the Appendix): Ps = ps eisy := S

s i sj

Ú dAr e

Pisol (r ),

(2.27)

which is defined only if P(r) has a compact support or decays with distance r from the particle faster than any power of r. Otherwise, if Pisol(r Æ •) ~ r –n, Ps is defined only for s < n - 2. One can introduce the total multipoles Ms of the combined system “particle + charge distribution Pisol(r)” as M s := Qs + Ps , if s < n - 2 M s := Qs , if s ≥ n - 2

(2.28)

47

Capillary Forces between Colloidal Particles at Fluid Interfaces

and Ms = mseisws. Then, Equation 2.26 takes the form uisol (r ) = umult (r , j; {M s }) + Duisol (r , j; Pisol ),

(2.29)

where •

ms cos s(j - w s ) sr s s =1

umult (r , j; {M s }) :=

m0 L 1 ln + 2 pg r 2 pg

1 2 pg

p L L 1 - 0 ln 2 pg r 2 pg r - r¢

Â

and

Duisol :=

Ú

dA¢Pisol (r ¢ ) ln

n-2

Â s =1

ps cos s(j - y s ) . sr s

Here, umult is the multipole expansion associated to the total multipole charges Ms, and Duisol is the difference between the full contribution of Pisol(r) and its approximation by a multipole expansion (possibly truncated at a finite order s < n - 2). By expanding the logarithm, it is shown that Duisol ~ r 2–n as r Æ • [34]. This demonstrates incidentally that the decay exponent n of the pressure field must be larger than 2; otherwise, the deformed interface does not approach the reference flat interface far from the particle and the whole perturbative scheme is invalid: what would actually happen is that the distant interface would be in general curved because of the pressure Pisol(r) and one should perturb around this curved profile. In Section 2.3.2, we address this issue with examples. In conclusion, the interface profile described by Equation 2.29 is dominated as r Æ • by the lowestorder nonvanishing total multipole and the superposition approximation holds if the order of this multipole is s < n - 2; otherwise, the interfacial deformation decays like r 2–n. Consider now a system with N particles, located at positions {ri}: The deformation has again the (i) (i) form of a superposition like Equation 2.13, with corrected capillary multipoles M s + dM s accounting for effects of “capillary polarization,” plus an amendment due to P(r): N

u(r ) =

Âu

mult

(r - ri ; {M s(i ) + dM s(i ) }) + Du(r; P).

(2.30)

i =1

N Du (r − r ; P ) + du(r; P) involves the part of the capillary The amendment Du(r; P) := Âi=1 isol i isol charge distribution P(r) not captured by the multipoles, described by the fields Duisol, and also a term du due to the possible departure of the pressure field itself P(r) from a simple superposition Âi Pisol ( |r – ri| ), since, depending on the physical origin of the pressure, this need not be a good approximation; we discuss below an example. At large separations, |ri - rj| Æ •, the induced charges (i) vanish, dMs Æ 0, and also du(r; P) Æ 0, so that Equation 2.30 approaches a truly linear superposition. The capillary force and torque can be derived from an effective interaction potential: For two particles, it has the form of a multipole–multipole interaction like Equation 2.14 plus a correction term related to Du in Equation 2.30 and collecting forces and torques contributed by the capillary charge in P(r), not captured by the multipoles. An immediate conclusion is that, similarly to the deformation by an isolated particle, the capillary force and torque are dominated at large separations by the interaction of the lowest-order nonvanishing multipoles provided their order is s < n - 2, and in such case pairwise additivity of the capillary force holds. In summary, when the asymptotic behavior is dominated by a multipole decay, the pressure field modifies the results of the pressure-free case just by the renormalization of the dominant multipole

48

Structure and Functional Properties of Colloidal Systems

from Qs to Ms = Qs + Ps. For example, if the permanent total monopole of an isolated particle m 0 π 0, the effective interaction potential between two such particles is asymptotically

U eff ª -

m02 L ln , 2 pg d

(2.31)

formally analogous to the force by buoyancy (Equation 2.15). This illustrates an important point: Note that Ueff does not give just the capillary force on the particle itself (this would correspond in the above expression to replacing m20 Æ m 0 q0 assuming q0 π 0), but it gives the capillary force on the subsystem “particle + surrounding interface”—an effective particle in this respect. This is actually the relevant force in the interpretation of experiments because they are usually sensitive to the effective force after all the capillary degrees of freedom have been integrated out, that is, the potential of mean force is given by Equation 2.31. The experimentally relevant system of an electrically charged 2D colloid has been addressed with the model of identical, electrically charged, spherical particles at the interface between a dielectric fluid and an electrolyte (e.g., air/water). The direct electrostatic interaction between the particles is given asymptotically by a dipole–dipole repulsion (see Figure 2.1), described by a potential Urep ~ d-3 [59–61]. The electric field also exerts a vertical force on the particles and an electric stress at the interface (the electric pressure field by an isolated dipole decays like Pisol(r) ~ 1/r6). Thus, the interface is deformed and a capillary force between the particles can arise. By spherical symmetry, the permanent capillary multipoles of an isolated particle vanish except for the monopole. Furthermore, since the system is mechanically isolated, the vertical electric force on the particle (q0 by Equation 2.10) is, by the principle of action–reaction, balanced by the total vertical electric force p 0 acting on the interface and m 0 = q0 + p 0 = 0. Therefore, there are no permanent capillary multipoles, uisol(r) = Duisol(r) ~ r 2–n with n = 6, and the capillary interaction arises solely due to induced capillary multipoles and departures from superposition of the pressure field. The electric field E obeys the superposition principle, and the pressure field is quadratic in E, so that there is a correction to the superposition of pressure: P(r) = Pisol(r - r1) + Pisol(r - r2) + Pm(r; d), where Pm(r; d) μ Eisol(r - r1)Eisol(r - r2) and Pm ~ 1/d3 in the neighborhood of each particle asymptotically for large interparticle separation d; therefore, Pm(r; d) dominates over Pisol(d) near the particles, which is nevertheless where the major effect by P(r) is concentrated. The deformation field can be written as 2

u(r ) =

Â ÈÎu

mult

(r - ri ; {dM s(i ) }) + Duisol (r - ri ; Pisol )˘˚ + du(r; d ),

i =1

with the “capillary potential” du(r; d) = (2pg)-1 Ú dA'Pm(r¢; d) ln(L/|r – r¢|) contributed by the additional capillary charge Pm(r; d). The typical amplitude of du(r; d) must decay also like 1/d3 with separation, and is therefore asymptotically dominant over the field Duisol(d) ~ 1/d4. Furthermore, the induced capillary charges dMs(i) must decay at least like the inducing field Pm, that is, dMs(i) ~ 1/d3; actually, no monopole can be induced since the system remains mechanically isolated and no net vertical force can arise by the presence of a second particle. Therefore, the term umult(d; {dMs(i)}) must decay at least like an (induced) dipole, that is, umult ~ 1/d4 at least, which is also subdominant with respect to du. In conclusion, in the neighborhood of the ith particle one has asymptotically for |r - ri| d, u(r) ~ Duisol(r - ri; Pisol) + [umult(r - ri; {dMs(i)}) + du(d)],

P(r) ~ Pisol(r - ri) + [Pm(r; d)],

with the terms in brackets, which decay like 1/d3, representing the dominant d-dependent contributions. Clearly, the superposition approximation is invalid to estimate the capillary force. Instead,

49

Capillary Forces between Colloidal Particles at Fluid Interfaces

one has, following the electrostatic analogy, the force that the additional capillary potential umult(r - ri; {dMs(i)}) + du(d) exerts on the permanent capillary charge Pisol(r - ri) at and near the particle, and the force that the distributed additional capillary charge Pm(r) experiences in the permanent capillary potential Duisol(r - ri; Pisol) around the particle. In both cases, the potential of mean force will decay asymptotically as Ueff ~ d-3, that is, at the same rate as the electrostatic repulsion. Whether this capillary force is attractive and can overcome the electrostatic repulsion cannot be answered within the simple analysis presented here. Detailed calculations are reported in references [35,55,56] employing different models for the electrostatic problem “charged particle at a flat interface” in order to compute the electric stresses on the particle and the interface. For spherical particles of radius R and contact angle qY (π0, p), the main result is summarized in the expression U eff (d ) ª - C e FU rep (d ),

| q0 | , 2 pgR sin q Y

e F :=

(2.32)

valid asymptotically for large separations d, in terms of the vertical force q0 that the electric field exerts on the particle. Here, C is a positive numerical factor of order unity, whose precise value depends on the electrostatic model. Therefore, the capillary force is attractive, but it is much weaker than the electrostatic repulsion because the assumption of small interfacial deformation everywhere imposes the formal requirement eF 1 (the force |q0| must be small). For an interface between air and pure water (Debye length 1 mm), and water modeled with the linear Debye–Hückel theory of electrolytes, Figure 2.6 shows the region of parameters (contact radius and electric surface charge density) where the crossover from asymptotic electric repulsion to asymptotic capillary attraction is predicted to occur [35]. The numerical values in this plot should be taken with caution and understood as a guiding estimate because the crossover happens for values of eF close to one. Furthermore, the 0.5 eF =1 0.4 Capillary attraction

s

0.3

0.2

eF = eF,crit

0.1

0 0

1

2 r0

3

4

FIGURE 2.6 Parameter space spanned by the surface density of electric charge on the particle (s, in units of e/nm2) and the contact line radius (r0 = R sin qY, in mm) of a system of spherical particles at an air/water interface, assuming the Debye–Hückel theory for water with a screening length of 1 mm. The solid line is the locus of values of eF such that above that line the capillary attraction given by Equation 2.32 dominates over the electric repulsion. The dashed line is the locus eF = 1 (the small-deformation approach corresponds formally to eF 1). (Reproduced from Domínguez, A. et al. J. Chem. Phys. 127, 204706, 2007. With permission.)

50

Structure and Functional Properties of Colloidal Systems

values of the density of electric charge in the particle’s surface may not correspond to the real charge density, but rather to an effective one following from charge renormalization by nonlinear electric effects (described by the Poisson–Boltzmann model) near the particles [62]. In reference [11], it was proposed that the capillary force was of the monopole form (Equation 2.31) due to the vertical electric force on the particle leading to a monopole q0 π 0. We have seen, however, that the relevant amplitude of the monopole force is actually m0 = q0 + p0 = 0. This was noted in reference [63], where instead a capillary attraction decaying like Ueff ~ 1/d6 was proposed. This result, however, was based on a wrong analogy with the pressure-free case (the effective renormalization Qs Æ Ms was not made consistently); after proper account of the electric pressure field P(r) π 0, a repulsive capillary interaction decaying also like Ueff ~ 1/d6 is obtained [37,54] within the superposition approximation. However, the mentioned inconsistency of this approximation was noted in reference [37]; a calculation beyond this approximation [35,55,56] led finally to Equation 2.32. Similarly, the analysis in reference [64] supporting Equation 2.31 theoretically and experimentally (electrically charged glass spheres of radius 0.3 mm) was shown to be flawed [65] and indeed no evidence of a logarithmic dependence was observed in experiments repeated later by the same group [58]. In conclusion, the attraction between charged colloidal particles reported by several groups is unlikely to be explained in terms of an electrically induced capillary force. Actually, Fernández-Toledano et al. [15] proposed a simple explanation in terms of contamination of the interface. It is still a matter of debate whether the observed effect would be an artifact and, if not, what the physical explanation would be. In a recent experiment [66], the formation of 2D structures of micrometer-sized droplets of glycerol at the interface between a nematic phase and a fluid phase was observed. The anchoring boundary conditions of the nematic director at the droplet surface deforms the director field and stresses arise which may produce a nematic-mediated interaction between the droplets and, additionally, an interfacial deformation and the corresponding capillary interaction. For small director distortions, the nematic-mediated interactions look formally like the electrostatic interaction and the scenario is actually analogous to the example just discussed. In references [66,67], it was argued by analogy with the interaction in bulk that the nematic-mediated interaction is a repulsion of dipole type, Urep(d) ~ 1/d3, while the capillary interaction is of monopole type, Ueff(d) ~ ln d, following the (wrong) reasoning in reference [11]. A detailed analysis [4] shows, however, that the presence of an interface gives actually a quadrupole-like nematic repulsion, Urep(d) ~ 1/d5, while the capillary interaction is given, due to mechanical isolation, by Equation 2.32 with q0 determined by the vertical force that the nematic field exerts on the particle. However, a permanent monopole and thus a capillary interaction like Equation 2.31 arises if mechanical isolation is violated by a substrate sustaining a nematic film of finite thickness h (but still much larger than the radius of the glycerol droplets). Solution of the director field equations [4] gives q0 μ 1/h4+w, where the exponent w > 0 depends on the details of the anchoring boundary conditions. The strong dependence of q0 on h yields the effect of this permanent monopole quickly irrelevant: For the value h ~ tens of microns appropriate to the experiment described in reference [66], Ueff is at most of the order of 10-11 kT at room temperature [4] and thus cannot be invoked as an explanation of the apparent attraction.

2.3.2

SMALL DEFORMATIONS OF A CURVED INTERFACE

If the fluid interface is already curved in the absence of particles, the approach just discussed has to be modified to consider small deformations about an unperturbed curved interface. The application of the tools of differential geometry, as discussed in, for example, reference [68], is a promising approach that still remains to be developed. So far, several particular cases have been considered like the interfacial deformations of a spherical droplet by the electric field of a particle in the limit of a large droplet radius [34,37,69–71], or of an unperturbed interface that is itself a slight deformation of a flat interface [40]. In these two cases, the key assumption is a good separation of length scales: the typical radius of curvature of the unperturbed, smooth interface is much larger than the typical

51

Capillary Forces between Colloidal Particles at Fluid Interfaces

length scales (e.g., size of the particles, relative separation) characterizing the departures from this reference interface. Under these conditions, the unperturbed interface departs only slightly from its tangent plane in the relevant region around the particles and the assumption of a small deformation about an imaginary flat interface (coinciding with the tangent plane) holds (Figure 2.7). In the language of the electrostatic analogy, the small departure uref(r||) of the unperturbed curved interface from its tangent plane plays the role of a background capillary potential. 2.3.2.1 Monopole The interface of a fluid in a macroscopic vessel will not be perfectly flat if the contact angle at the vessel walls differs from p/2. The interfacial deformation uref(r) is the solution of Equation 2.8 in the presence of gravity, —||2 uref = uref /l2, together with the appropriate boundary condition at the vessel wall. If the vessel is larger than l (a few millimeters), the deformation close to the center of the vessel has a small curvature (in the electrostatic analogy, the effect of the boundary condition at the wall gets screened beyond a separation of order l), and the small-deformation approach (about the horizontal plane tangent to the interface at the center of the vessel) holds. A submillimeter particle at the interface, characterized by a capillary monopole q0 = Fbuoy due to the buoyancy force, will couple with the deformation field uref, showing up in a capillary force field acting on the particle: Fcap(r||) = q0—||uref(r||). Thus, particles pulled downward by buoyancy (q0 < 0) will drift to the minima of the background deformation uref (r||). This case was studied experimentally in reference [40] and a very good agreement was found between the measured force and the theoretical prediction. 2.3.2.2 Quadrupole Another situation of experimental interest corresponds to nonspherical particles at a curved interface. The lowest-order nonvanishing capillary charge of these particles, in the absence of external forces and torques, is the quadrupole Q2 = q2e2iq2, which couples to the field uref (r||) so that the particle experiences a torque and a force due to the curvature of the background interface. Following the electrostatic analogy, they can be derived from an effective potential of the form (see the Appendix) U eff = - 14 D : —|| —||uref ,

(2.33)

D := q2 ÈÎ(e x e x - e y e y ) cos 2q2 + (e x e y + e y e x ) sin 2q2 ˘˚ .

(2.34)

u

r|| uref

FIGURE 2.7 Small deformation (exaggerated in the schematic drawing) of a curved interface. In the neighborhood of the tangent plane (dotted line), the undeformed interface (dashed line) is parametrized by the field uref(r||), giving the departure from the tangent plane. The deformed interface (solid line) is parametrized by u(r||).

52

Structure and Functional Properties of Colloidal Systems

Here, D is a second-rank tensor (in dyadic notation) that describes the (traceless) quadrupole and q2 is the angle that the principal axes of D form with the coordinate axes {ex, ey}. Without loss of generality, one can write at any point r|| —|| —||uref (r|| ) =

1È 1 1 ˘ (e e + e y e y ) + 2 ÍÎ L1(r|| ) L 2 (r|| ) ˙˚ x x +

1È 1 1 ˘ È(e e - e y e y ) cos 2f(r|| ) + (e x e y + e y e x ) sin 2f(r|| )˘ , Í ˚ 2 Î L1(r|| ) L 2 (r|| ) ˙˚ Î x x

in terms of the radii of curvature L1 and L2 and the angle f between the principal directions of curvature and the coordinate axes. When the background interface is a minimal surface, —2uref = 0 and L1 = -L2; taking L1 > 0 conventionally, the effective potential simplifies to U eff = -

q2 cos 2 ÈÎq2 - f(r|| )˘˚ . 2 L1 (r|| )

(2.35)

Thus, the particle will experience a torque tending to align it with the principal directions of curvature of the background interface, that is, q2 = f, f + p. When aligned like this, it will experience a force pulling it to regions of increasing absolute curvature 1/L1. With the parametrization q2 = 4pgR2 H introduced in Equation 2.25 for a particle of size R inducing a quadrupole interfacial deformation of a typical amplitude H at the contact line, one typically obtains 2

H Ê R ˆ U eff ª -107 kT Á cos 2 ÈÎq2 - f(r|| )˘˚ . ˜ Ë mm ¯ L1 (r|| ) Thus, for micrometer particles even such a small ratio as H/L1 ~ 10-7 brings about forces and torques dominating over thermal motion [34].* If the particle is spherical, all the permanent capillary multipoles vanish and any interaction with the background curved interface is a consequence of multipoles induced by uref(r||). Mechanical isolation implies that the monopole and the dipole must vanish and the lowest-order nonvanishing multipole is an induced quadrupole dD. The induced multipoles must be proportional to derivatives of uref (there would be no induced multipoles in a flat interface), and the lowest-order derivative that can give rise to a quadrupole is —||—||uref by symmetry. Since the spherical particle does not introduce privileged directions, one can conclude dD = 4C—||—||uref with a positive scalar constant C (a possible dependence of C on uref would yield a contribution in dD nonlinear in uref and thus negligible in the small-deformation limit). In conclusion, one would find an effective potential Ueff = -C(—||—||uref) : (—||—||uref) = -2C/L21(r||), and the particle will experience a force toward regions of increasing curvature. This problem has been discussed in reference [70], where detailed calculations within the energy approach give C = pgR4/12, and therefore 2

2

Ê R ˆ Ê R ˆ U eff ª -106 kT Á , Ë mm ˜¯ ÁË L1 (r|| ) ˜¯ demonstrating the relevance of this effect for micrometer particles. Corrections to this expression by higher-order induced multipoles or by quadrupoles induced by higher-order derivatives of uref are expected to be subdominant by powers of the small ratio R/L1. *

In this reference, the expressions for the force and torque were misspelled and have to be corrected by a factor 1/2.

Capillary Forces between Colloidal Particles at Fluid Interfaces

53

If the surface is not minimal (L1 π -L2), the expressions are slightly changed (see the Appendix). Instead of Equation 2.33, the effective interaction of the quadrupole with the curved interface is given by U eff = -

q2 4

1 ˘ 1 È 1 1 ˘ È 1 Í L (r ) - L (r ) ˙ cos 2 ÎÈq2 - f(r|| )˚˘ - 4 S Í L (r ) + L (r ) ˙ , 2 || ˚ 2 || ˚ Î 1 || Î 1 ||

(2.36)

where the trace S is defined in the Appendix. This additional term, being independent of the angle q2, does not affect the torque, so that the above conclusions concerning the rotation and alignment of the quadrupole with the curvature directions of uref still hold. This has been demonstrated experimentally in reference [72], where the orientation of cylindrical micrometer-sized particles at nonplanar fluid interfaces has been studied. (We note that the torque derived in that work is a factor 1/2 smaller than the one obtained with Equation 2.36 because the energy functional used in reference [72] is incomplete: Relevant contributions by the pressure field and by the wetting energy of the particle are neglected altogether without justification.) The forces and torques we have just discussed are additional to and compete with the quadrupole–quadrupole capillary interaction in the presence of more than one particle. This could conceivably lead to an interesting phenomenology and also help control the assembling of structures in 2D colloids at curved interfaces, as exemplified experimentally in reference [72]. 2.3.2.3 Spherical Charged Particle In the experiment reported in reference [11], the charged spherical particles (radius R 0.75 mm) lie actually at the surface of a water droplet of radius L 24 mm. As we have shown, a monopole–monopole capillary attraction in a flat interface is ruled out asymptotically for a mechanically isolated system. It was conjectured in reference [37] that the intrinsic curvature of the spherical interface might alter this conclusion, but this possibility has been refuted [34,57,71]. Consider an isolated spherical droplet containing a charged particle at its surface. As we have seen in the flat case, the amplitude of the logarithmic deformation in Equation 2.29 would be given by the total capillary monopole, Equations 2.27 and 2.28. In the present case, however, the integral in Equation 2.27 extends only up to a maximum distance from the particle, since departures of the spherical interface from its tangent plane must be small. The interface beyond this maximum distance would be termed a “nonlinear patch,” for which the capillary monopole is nevertheless given exactly by Equation 2.10, which is the vertical electric force in this case. Since the system is isolated, this force must balance the sum of vertical electric forces acting on the particle and the “linear patch” of interface, and consequently, the permanent capillary monopole does vanish: Asymptotically there is no logarithmic interfacial deformation around a particle. Notice that this reasoning is completely general and actually not restricted to a spherical interface. This conclusion coincides with that obtained in reference [34], where a detailed calculation within the force approach is carried out for the small deformation of a spherical droplet (i.e., not restricted to the neighborhood of the tangent plane). In reference [69], a false force balance is implemented and the wrong conclusion is reached of a capillary monopole μ (R/L)2. When corrected properly [71], the absence of a permanent monopole is obtained also within the energy approach. In the experiment reported in reference [11], the droplet is actually hanging from a plate, so that mechanical isolation is violated: The total capillary monopole is the electric vertical reaction force that the plate exerts on the system (i.e., minus the electric force acting on the plate). The electric pressure decays from the particle as P ~ r–n, takes appreciable values in a region of size ~R around it, and gives an electric force q0 on the particle, so that the electric pressure far from the particle can be written on dimensional grounds as P μ (q0/R2)(R/r)n (this simple estimate is supported by a detailed solution of the electrostatic problem in both an undeformed flat [60] or spherical [73] interface, which provides the value n = 6). The force on the plate, at a separation r L and of linear extent ~L, gives therefore a

54

Structure and Functional Properties of Colloidal Systems

total monopole m0 ~ q0(R/L)n-2. For the numerical values appropriate for the experiment described in reference [11] (n = 6, L/R = 32), the interpretation of the measurements in terms of a monopole–monopole attraction gives m0/2pgR ~ 10-3 [37], and thus an unrealistically large vertical force on the particle of order q0/2pgR ~ 103 is estimated (the maximal capillary force at the contact line is expected to be q0 ~ gR, see Equation 2.3). The explanation of the apparent attraction observed in this experiment is still an open problem. As suggested in reference [34], the presence of the plate breaks anyhow the spherical symmetry and a possible force (of whatever origin) confining the particles at the apex of the droplet could be mistaken for an effective attraction between the particles.

2.4

CONCLUSIONS

2D colloids at fluid interfaces have the peculiarity, compared to bulk colloids, that the particles are subjected to the additional effect of the capillary force caused by the deformable nature of the interface. The typical values of the surface tension of fluid interfaces make this force relevant against thermal motion in a broad range of length scales, from the millimeter down to the nanometer. The simplest theoretical model addresses small deformations of a flat interface. In this case, the effective, capillary-induced interaction between particles is formally analogous to the 2D electrostatic interaction (except that equal charges attract each other). Furthermore, in the limit of wellseparated particles, the superposition approximation holds and the force is pairwise additive. Under these conditions, the effective capillary force between two particles is, barring fine-tuning, either a monopole–monopole interaction, Equation 2.15, or a quadrupole–quadrupole one, Equation 2.25. The monopole arises whenever there is an external net force acting on the particles; the quadrupole is a generic outcome of departures from sphericity. The monopole interaction is a long-ranged attraction akin to a 2D Newtonian gravitational interaction, but the characteristic phenomenology of such a nonintegrable potential is usually masked in experiments by additional effects (e.g., electrostatic repulsion). The quadrupole interaction is anisotropic, which reflects in some features of the structures formed by the particles. The case of charged, spherical particles is also of experimental interest, whereby the interfacial deformation is a consequence of electric stresses. The theoretical analysis requires to go beyond the superposition approximation and it rules out a possible net attraction at least asymptotically for large interparticle separations, as the capillary attraction is then unable to overcome the electric repulsion. When the interface is already curved by itself, there can arise a capillary force on isolated particles. In the limit of small interfacial curvature the electrostatic analogy holds again, with the curved interface playing the role of an “external electric field”: for example, a neutral, spherical particle at a minimal surface is pulled to regions of larger curvature. Thus, a novel phenomenology is predicted adding to the effects of particle–particle interaction just mentioned. There is an increasing amount of experimental evidence confirming the relevance of the capillary force in 2D colloids and the predictions for a monopole–monopole and a quadrupole–quadrupole interaction. From a more technological point of view, the theoretical analysis provides hints on how to control and tune the capillary forces, for example, via an external force field for the monopole, the particle’s shape for the quadrupole, or the form of a curved interface. When comparing with experiments, however, one must bear in mind the limitations of the theoretical analysis carried out so far. The small-deformation approximation is likely to be valid in most cases, but the superposition approximation and the asymptotic results for large separations can be conceivably unreliable for small separations—for example, in the easily realizable high-density colloids, where many-body effects could play a role. The theoretical research of these effects, necessarily assisted by numerical approaches, is a task of great interest as a complement of the experiments in the proper interpretation of the measurements.

Capillary Forces between Colloidal Particles at Fluid Interfaces

55

REFERENCES 1. B. P. Binks and T. S. Horozov (Eds), Colloidal Particles at Liquid Interfaces, Cambridge University Press, Cambridge, 2006. 2. J. C. Fernández-Toledano, A. Moncho-Jordá, F. Martínez-López, and R. Hidalgo-Álvarez, Theory for interactions between particles in monolayers, In Colloidal Particles at Liquid Interfaces, B. P. Binks and T. S. Horozov (Eds), Cambridge University Press, Cambridge, 2006. 3. M. Oettel and S. Dietrich, Colloidal interactions at fluid interfaces, Langmuir, 24, 1425, 2008. 4. M. Oettel, A. Domínguez, M. Tasinkevych, and S. Dietrich, Effective interactions of colloids on nematic films, Eur. Phys. J. E, 28, 99–11, 2009. 5. M. M. Nicolson, The interaction between floating particles, Proc. Cambridge Philos. Soc., 45, 288, 1949. 6. P. A. Kralchevsky and K. Nagayama, Capillary interactions between particles bound to interfaces, liquid films and biomembranes, Adv. Colloid Interface Sci., 85, 145, 2000. 7. F. Ghezzi and J. C. Earnshaw, Formation of meso-structures in colloidal monolayers, J. Phys.: Condens. Matter, 9, L517, 1997. 8. J. Ruiz-García, R. Gámez-Corrales, and B. I. Ivlev, Foam and cluster structure formation by latex particles at the air/water interface, Physica A, 236, 97, 1997. 9. D. Stamou, C. Duschl, and D. Johannsmann, Long-range attraction between colloidal spheres at the air-water interface: The consequence of an irregular meniscus, Phys. Rev. E, 62, 5263, 2000. 10. M. Quesada-Pérez, A. Moncho-Jordá, F. Martínez-López, and R. Hidalgo-Álvarez, Probing interaction forces in colloidal monolayers: Inversion of structural data, J. Chem. Phys., 115, 10897, 2001. 11. M. G. Nikolaides, A. R. Bausch, M. F. Hsu, A. D. Dinsmore, M. P. Brenner, C. Gay, and D. A. Weitz, Electric-field-induced capillary attraction between like-charged particles at liquid interfaces, Nature, 420, 299, 2002. 12. G. Tolnai, A. Agod, M. Kabai-Faix, A. L. Kovács, J. J. Ramsden, and Z. Hórvölgyi, Evidence for secondary minimum flocculation of Stöber silica nanoparticles at the air-water interface: Film balance investigations and computer simulations, J. Phys. Chem. B, 107, 11109, 2003. 13. O. Gómez-Guzmán and J. Ruiz-García, Attractive interactions between like-charged colloidal particles at the air/water interface, J. Colloid Interface Sci., 291, 1, 2005. 14. W. Chen, S. Tan, T.-K. Ng, W. T. Ford, and P. Tong, Long-ranged attraction between charged polystyrene spheres at aqueous interfaces, Phys. Rev. Lett., 95, 218301, 2005. 15. J. C. Fernández-Toledano, A. Moncho-Jordá, F. Martínez-López, and R. Hidalgo-Álvarez, Spontaneous formation of mesostructures in colloidal monolayers trapped at the air-water interface: A simple explanation, Langmuir, 20, 6977, 2004. 16. J. C. Loudet, A. M. Alsayed, J. Zhang, and A. G. Yodh, Capillary interactions between anisotropic colloidal particles, Phys. Rev. Lett., 94, 018301, 2005. 17. N. Aubry and P. Singh, Physics underlying controlled self-assembly of micro- and nanoparticles at a two-fluid interface using an electric field, Phys. Rev. E, 77, 056302, 2008. 18. P. Pieranski, Two-dimensional interfacial colloidal crystals, Phys. Rev. Lett., 45, 569, 1980. 19. A. Perro, S. Reculusa, S. Ravaine, E. Bourgeat-Lami, and E. Duguet, Design and synthesis of Janus micro- and nanoparticles, J. Mater. Chem., 15, 3745, 2005. 20. J. Faraudo and F. Bresme, Stability of particles adsorbed at liquid/fluid interfaces: Shape effects induced by line tension, J. Chem. Phys., 118, 6518, 2003. 21. L. Dong and D. T. Johnson, Adsorption of acicular particles at liquid-fluid interfaces and the influence of the line tension, Langmuir, 21, 3838, 2005. 22. F. Bresme and M. Oettel, Nanoparticles at fluid interfaces, J. Phys.: Condens. Matter, 19, 413101, 2007. 23. K. Zahn, J. M. Méndez-Alcaraz, and G. Maret, Two-stage melting of paramagnetic colloidal crystals in two dimensions, Phys. Rev. Lett., 79, 175, 1997. 24. M. Oettel, Entrapment of charged, nonwetting colloids near oil-water interfaces, Phys. Rev. E, 76, 041403, 2007. 25. M. E. Leunissen, A. van Blaaderen, A. D. Hollingsworth, M. T. Sullivan, and P. M. Chaikin, Electrostatics at the oil-water interface, stability, and order in emulsions and colloids, Proc. Nat. Acad. Sci. USA, 104, 2585, 2007. 26. L. Segel, Mathematics Applied to Continuum Mechanics, New York: Dover, 1987. 27. R. Golestanian, M. Goulian, and M. Kardar, Fluctuation-induced interaction between rods on a membrane, Phys. Rev. E, 54, 6725, 1996.

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28. H. Kaïdi, T. Bickel, and M. Benhamou, Surface-mediated attraction between colloids, Europhys. Lett., 69, 15, 2005. 29. H. Lehle, M. Oettel, and S. Dietrich, Effective forces between colloids at interfaces induced by capillary wavelike fluctuations, Europhys. Lett., 75, 174, 2006. 30. H. Lehle and M. Oettel, Importance of boundary conditions for fluctuation-induced forces between colloids at interfaces, Phys. Rev. E, 75, 011602, 2007. 31. J. Guven, Laplace pressure as a surface stress in fluid vesicles, arXiv:cond-mat/0602289v1, 2006. 32. M. M. Müller, M. Deserno, and J. Guven, Geometry of surface-mediated interactions, Europhys. Lett., 69, 482, 2005. 33. R. Capovilla and J. Guven, Stresses in lipid membranes, J. Phys. A: Math. Gen., 35, 6233, 2002. 34. A. Domínguez, M. Oettel, and S. Dietrich, Force balance of particles trapped at a fluid interface, J. Chem. Phys., 128, 114904, 2008. 35. A. Domínguez, M. Oettel, and S. Dietrich, Theory of capillary-induced interactions beyond the superposition approximation, J. Chem. Phys., 127, 204706, 2007. 36. P. A. Kralchevsky, V. N. Paunov, I. B. Ivanov, and K. Nagayama, Capillary meniscus interaction between colloidal particles attached to a liquid-fluid interface, J. Colloid Interface Sci., 151, 79, 1992. 37. M. Oettel, A. Domínguez, and S. Dietrich, Effective capillary interaction of spherical particles at fluid interfaces, Phys. Rev. E, 71, 051401, 2005. 38. D. Y. C. Chan, J. D. Henry Jr., and L. R. White, The interaction of colloidal particles collected at fluid interfaces, J. Colloid Interface Sci., 79, 410, 1981. 39. O. D. Velev, N. D. Denkov, V. N. Paunov, P. A. Kralchevsky, and K. Nagayama, Direct measurement of lateral capillary forces, Langmuir, 9, 3702, 1993. 40. N. D. Vassileva, D. van den Ende, F. Mugele, and J. Mellema, Capillary forces between spherical particles floating at a liquid-liquid interface, Langmuir, 21, 11190, 2005. 41. P. A. Kralchevsky, V. N. Paunov, N. D. Denkov, I. B. Ivanov, and K. Nagayama, Energetical and force approaches to the capillary interactions between particles attached to a liquid-fluid interface, J. Colloid Interface Sci., 155, 420, 1993. 42. V. N. Paunov, P. A. Kralchevsky, N. D. Denkov, and K. Nagayama, Lateral capillary forces between floating submillimeter particles, J. Colloid Interface Sci., 157, 100, 1993. 43. N. Aubry, P. Singh, M. Janjua, and S. Nudurupati, Micro- and nanoparticles self-assembly for virtually defect-free, adjustable monolayers, Proc. Nat. Acad. Sci. USA, 105, 3711, 2008. 44. R. Di Leonardo, F. Saglimbeni, and G. Ruocco, The very long range nature of capillary interactions in liquid films, Phys. Rev. Lett., 100, 106103, 2008. 45. E. A. van Nierop, M. A. Stijnman, and S. Hilgenfeldt, Shape-induced capillary interactions of colloidal particles, Europhys. Lett., 72, 671, 2005. 46. J. Lucassen, Capillary forces between solid particles in fluid interfaces, Colloids Surf., 65, 131, 1992. 47. A. B. D. Brown, C. G. Smith, and A. R. Rennie, Fabricating colloidal particles with photolithography and their interactions at an air-water interface, Phys. Rev. E, 62, 951, 2000. 48. P. A. Kralchevsky, N. D. Denkov, and K. D. Danov, Particles with an undulated contact line at a fluid interface: Interaction between capillary quadrupoles and rheology of particulate monolayers, Langmuir, 17, 7694, 2001. 49. J.-B. Fournier and P. Galatola, Anisotropic capillary interactions and jamming of colloidal particles trapped at a liquid-fluid interface, Phys. Rev. E, 65, 031601, 2002. 50. K. D. Danov, P. A. Kralchevsky, B. N. Naydenov, and G. Brenn, Interactions between particles with an undulated contact line at a fluid interface: Capillary multipoles of arbitrary order, J. Colloid Interface Sci., 287, 121, 2005. 51. H. Lehle, E. Noruzifar, and M. Oettel, Ellipsoidal particles at fluid interfaces, Eur. Phys. J. E, 26, 151, 2008. 52. J. C. Loudet, A. G. Yodh, and B. Pouligny, Wetting and contact lines of micrometer-sized ellipsoids, Phys. Rev. Lett., 97, 018304, 2006. 53. M. G. Basavaraj, G. G. Fuller, J. Fransaer, and J. Vermant, Packing, flipping, and buckling transitions in compressed monolayers of ellipsoidal latex particles, Langmuir, 22, 6605, 2006. 54. L. Foret and A. Würger, Electric-field induced capillary interaction of charged particles at a polar interface, Phys. Rev. Lett., 92, 058302, 2004. 55. M. Oettel, A. Domínguez, and S. Dietrich, Attractions between charged colloids at water interfaces, J. Phys.: Condens. Matter, 17, L337, 2005. 56. A. Würger and L. Foret, Capillary attraction of colloidal particles at an aqueous interface, J. Phys. Chem. B, 109, 16435, 2005.

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57. A. Domínguez, M. Oettel, and S. Dietrich, Capillary-induced interactions between colloids at an interface, J. Phys.: Condens. Matter, 17, S3387, 2005. 58. K. D. Danov, P. A. Kralchevsky, and M. P. Boneva, Shape of the capillary meniscus around an electrically charged particle at a fluid interface: Comparison of theory and experiment, Langmuir, 22, 2653, 2006. 59. A. J. Hurd, The electrostatic interaction between interfacial colloidal particles, J. Phys. A: Math. Gen., 18, L1055, 1985. 60. K. D. Danov and P. A. Kralchevsky, Electric forces induced by a charged colloid particle attached to the water-nonpolar fluid interface, J. Colloid Interface Sci., 298, 213, 2006. 61. A. Domínguez, D. Frydel, and M. Oettel, Multipole expansion of the electrostatic interaction between charged colloids at interfaces, Phys. Rev. E, 77, 020401(R), 2008. 62. D. Frydel, S. Dietrich, and M. Oettel, Charge renormalization for effective interactions of colloids at water interfaces, Phys. Rev. Lett., 99, 118302, 2007. 63. M. Megens and J. Aizenberg, Like-charged particles at liquid interfaces, Nature, 424, 1014, 2003. 64. K. D. Danov, P. A. Kralchevsky, and M. P. Boneva, Electrodipping force acting on solid particles at a fluid interface, Langmuir, 20, 6139, 2004. 65. M. Oettel, A. Domínguez, and S. Dietrich, Comment on “Electrodipping force acting on solid particles at a fluid interface”, Langmuir, 22, 846, 2006. 66. I. I. Smalyukh, S. Chernyshuk, B. I. Lev, A. B. Nych, U. Ognysta, V. G. Nazarenko, and O. D. Lavrentovich, Ordered droplet structures at the liquid crystal surface and elastic-capillary colloidal interactions, Phys. Rev. Lett., 93, 117801, 2004. 67. A. B. Nych, U. M. Ognysta, V. M. Pergamenshchik, B. I. Lev, V. G. Nazarenko, I. Muševicˇ, M. Škarabot, and O. D. Lavrentovich, Coexistence of two colloidal crystals at the nematic-liquid-crystal—air interface, Phys. Rev. Lett., 98, 057801, 2007. 68. M. M. Müller, M. Deserno, and J. Guven, Interface-mediated interactions between particles: A geometrical approach, Phys. Rev. E, 72, 061407, 2005. 69. A. Würger, Capillary attraction of charged particles at a curved liquid interface, Europhys. Lett., 75, 978, 2006. 70. A. Würger, Curvature-induced capillary interaction of spherical particles at a liquid interface, Phys. Rev. E, 74, 041402, 2006. 71. A. Domínguez, M. Oettel, and S. Dietrich, Absence of logarithmic attraction between colloids trapped at the interface of droplets, Europhys. Lett., 77, 68002, 2007. 72. E. P. Lewandowski, J. A. Bernate, P. C. Searson, and K. J. Stebe, Rotation and alignment of anisotropic particles on nonplanar interfaces, Langmuir, 24, 9302, 2008. 73. A. Würger, Screened electrostatics of charged particles on a water droplet, Eur. J. Phys. E, 19, 5, 2006. 74. U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.

APPENDIX Here we recall briefly some results pertaining electrostatics in two dimensions. A distribution P(r) of “capillary charge” creates a “capillary potential” u(r) that can be written, provided P(r) vanishes fast enough at infinity, as

u(r ) = Re V (z ),

V ( z) : =

1 2 pg

L

Ú dA¢P(r¢) ln z - z¢ , S

after introducing the complex variable z = reij , and Re denotes the real part. If the support S of P(r) is compact, that is, P(r) vanishes beyond a finite distance r, one can expand the logarithm in a Taylor series in z¢/z valid in the domain |z¢| < |z|. Therefore, in the region where P(r) vanishes V(z) is an analytic function that can be written as the multipole expansion

V (z ) =

Q0 1 L ln + 2 pg z 2 pg

•

Qs

Â sz , s

s =1

(2.37)

58

Structure and Functional Properties of Colloidal Systems

with the multipole moments defined as Qs : =

Ú dA¢P(r¢)(z¢) . s

(2.38)

S

By studying how Qs transforms under a rigid rotation of the “capillary charge” P(r), one can show easily that the complex-valued multipole charge can be written as Qs = qseisqs. Inserting this into Equation 2.37 one arrives at Equation 2.12. A definition of the multipole charge alternative to Equation 2.38 is provided by application of the residue theorem to Equation 2.37: Qs = ig

Ú dz z C

s

dV , dz

where C is an arbitrary simple contour enclosing completely the support S of the distribution P(r). Application of the Cauchy–Riemann relationships verified by the analytic function V(z) allows one to rewrite this expression as Qs = - g

Ú

s

d ÈÎr ◊ (e x + ie y )˘˚ (n - it ) ◊ —u,

(2.39)

C

where n and t are defined in Figure 2.4. Since this expression involves just the derivative of the deformation field u(r) at an arbitrary contour, it allows the definition and computation of the multipoles of any region that can be surrounded by a simple contour C even when the assumption of small capillary deformation does not hold inside, in which case Equation 2.39 describes “virtual multipoles.” A less compact, but useful way of giving the same information is in terms of the (s) (e - ie )], a tensor of rank s. For real-valued multipoles (in dyadic notation) Qˆs : = Re[Qs(ex - iey) ... x y example, this gives the real-valued quadrupole as written in Equation 2.34. The multipolar expansion (Equation 2.37) can be easily transformed to an expansion for u(r) = Re V(z) involving only the real quantities Qˆs. Assume now that the distribution P(r¢) is located at a point r in an externally created deformation field uext(r). The “electrostatic” energy of the coupling with this field is

Ú

U = - dA¢P(r ¢ )uext (r + r ¢ ).

(2.40)

S

(The sign of U is changed because, as remarked, capillary forces are reversed compared to the electrostatic forces.) Since —2uext = 0, the associated complex potential Vext(z + z¢) is an analytic function in the region S and can be expanded in a Taylor series about the point z. Thus, one finds •

U = - Re

Â s=0

Qs d sVext ( z ). s ! dz s

(2.41)

If uext is given in turn by a multipole expansion, this expression becomes Equation 2.14 for the interaction energy between two different multipole distributions after inserting Equation 2.37. Iterative application of the Cauchy–Riemann relationships yields s d sVext 1 = s - 1 ÈÎ(e x - ie y ) ◊— ˘˚ uext s dz 2

(s > 0),

Capillary Forces between Colloidal Particles at Fluid Interfaces

59

so that Equation 2.41 can be written alternatively in terms of only real quantities as •

U = - Qˆ 0 uext (r ) -

1

Â s!2

s -1

(s)

Qˆ s ◊ — —uext (r ).

s =1

Equation 2.33 is the special case of this expression for a quadrupole D := Q2. This expression must be modified when the field uext(r) does not describe a minimal surface, that is, —2uext π 0. By Taylor expanding in Equation 2.40, one can show that U = - Qˆ 0 uext (r ) - Qˆ1 ◊ —uext (r ) - 14 ÈÎQˆ 2 + 1S ˘˚ ◊ ——uext (r ) , where the trace S := ÚS dA¢P(r¢) (r¢)2 is a property of the charge distribution additional to the traceless tensor Qˆ2 and 1 is the unit tensor.

Part II Structure

3

Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations A. Martín-Molina, M. Quesada-Pérez, and R. Hidalgo-Álvarez

CONTENTS 3.1 3.2

Introduction ........................................................................................................................ Theoretical Background ..................................................................................................... 3.2.1 HNC/MSA Equations for a Multicomponent System ............................................ 3.2.2 Theory–Experiment Relationship ........................................................................... 3.3 Results ................................................................................................................................. 3.3.1 Divalent Counterions .............................................................................................. 3.3.2 Trivalent Counterions ............................................................................................. 3.3.3 Mixture of Electrolytes ........................................................................................... 3.4 Conclusions ......................................................................................................................... References ....................................................................................................................................

3.1

63 65 65 66 69 69 70 72 74 74

INTRODUCTION

Most colloidal particles in aqueous solution are charged. The surface charge can arise from different charging mechanisms such as ionization of surface groups, chemical binding, or physical adsorption of ions from the liquid medium. Consequently, the presence of a charged particle in an electrolyte solution implicates a distribution of ions around its surface. This ionic distribution is not uniform and gives rise to a structure termed electric double layer (EDL). Obviously, this configuration plays an important role in the stability of colloidal dispersions as well as their electrokinetic properties. For many decades, the Gouy–Chapman (GC) model, whose cornerstone is the Poisson–Boltzmann (PB) equation, has been the traditional approach to describing the EDL. Although this mean field approximation is satisfactorily applied to systems where the concentration of (mostly monovalent) electrolyte and the surface charge are low enough, this model fails under certain conditions. For instance, errors in the GC theory are noticeable for multivalent electrolytes, for elevated monovalent salt concentrations, for ions with a large diameter, for high surface charges, and for solvents with a low dielectric constant. These deficiencies have been previously corroborated by comparing the classical theory with a large variety of simulations and sophisticated models such as the modified PB theory, the density functional theory, and those models based on integral equations theories [1–10]. According to these works, failures of the GC model in the description of the multivalent ion 63

64

Structure and Functional Properties of Colloidal Systems

distribution around the charged colloids are the cause for such disagreements. The overlooking of the phenomenon generally known as overcharging is probably one of the most representative examples of the breakdown of the classical approach. Although the correct definition of overcharging is a controversial issue [11,12], this term is widely used to refer the effects derived from the counterion condensation in the immediate vicinity of a charged surface. If the concentration of counterions is so large that the surface charge is neutralized and even overcompensated, the situation is called charge reversal. Such overcompensation causes, in turn, that the effective electrical field produced by the particle plus the counterions reverses its direction with respect to the unscreened electrical field. Then, the effective reversed electrical field produces a concomitant layer of coions, next to the first counterion layer, which is referred to as charge inversion. Then the question is why the GC model fails to describe the overcharging phenomena? Although the origin of the overcharging has caused numerous discussion among scientific community related to colloidal science, it is accepted that the ion–ion correlations can explain, at least in part, these unexpected phenomena [13]. This conclusion can be inferred by comparing the results obtained from the GC model with those resulting from theoretical approximations based on the primitive model (PM) of electrolyte. Although both models have in common the assumption that the solvent is a dielectric continuum and the surface charge is uniformly distributed at the interface, in the former, ions are assumed to be point charges, whereas ions are considered as hard spheres in the latter. In the PM formalism, each ion feels a local field that is in general different from the mean field. That is to say, the effect of the charge and finite size of ions gives rise to different sort of correlations between them. A good description about the different roles of ion–ion correlations of both electrostatic and finite size origin has been elucidated by Kjellander and Greberg by analyzing various contributions to the mean force that acts on ions [14]. From a theoretical point of view, the PM can be introduced in the theoretical description of the EDL by means of the integral equation formalism [13]. In fact, the so-called hypernetted-chain/ mean-spherical approximation (HNC/MSA) began to show the deficiencies of the classical EDL model more than two decades ago, claiming since then that the PB approach is inadequate for solutions with multivalent counterions [15,16]. In particular, under conditions propitious for charge inversion (i.e., high particle surface charge densities or/and electrolyte concentrations), the ion concentration profiles calculated with the HNC/MSA exhibit an oscillatory behavior and unexpected changes in the sign of the electrostatic potential. Conversely, monotonic functions (with respect to the distance from the surface plane) are obtained for the ion concentration profiles and the electrostatic potential by using the classical EDL model [2,17]. Although similar conclusions have been reported by using computer simulations as well as other theoretical approximations, only in recent times the PM models have started to be compared with experimental data [17–23]. Accordingly, the main purpose of this work is to study the experimental charge inversion of latex particles by means of electrophoretic mobility measurements. Moreover, the experimental results will be compared with predictions obtained with the GC model as well as HNC/MSA in order to analyze the effect of the ionic size correlations for this kind of experiments. To this end, we will first study the effect of different multivalent electrolytes on the latex particles. Afterwards, we will look into mixtures of mono- and multivalent electrolytes on the same system. The objective is therefore to draw up the boundaries of the charge inversion for a colloidal model system by putting emphasis on the importance of size correlations in the description of the colloidal EDL. The remainder of the work is organized as follows. First, we review how ion size correlations are included in the model of EDL by integral equation theories (and, more specifically, by the HNC/ MSA). In this section we also discuss the importance of the ion size in theoretical predictions. Before showing the results, a brief discussion about the theory–experiment relationship is done. Afterwards, the theoretical predictions are compared with experimental data. In order to deepen into the characteristic parameters related to the overcharging phenomena, a great variety of situations are studied. Finally, some conclusions are presented.

Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations

3.2 3.2.1

65

THEORETICAL BACKGROUND HNC/MSA EQUATIONS FOR A MULTICOMPONENT SYSTEM

As stated in the introduction, in this work we have used the PM in order to include the ion size in the EDL description. To this end, it is necessary to know the structure formed by counterions and coions around the macroions. Consequently, the Ornstein–Zernike (OZ) formalism applied to isotropic systems could be a starting point for its description 4

hik (r ) = cik (r ) +

Â r Ú c (| r - s |)h j

ij

jk

( s ) d 3 s,

(3.1)

j =0

where hik(r) = gik(r) - 1, g jk(r) is the pair distribution function of species j and k, rj is the density of species j, and Cij(|r - s|) are integrals depending on the direct correlation functions of species i and j. Accordingly, this equation expresses the fact that in a dense system correlations include direct correlations (cij functions) plus correlations induced by the propagation of interactions between one particle and the others. Given that one of the objectives of the work is to study the EDL in the presence of mixtures of two electrolytes, we will show the equations for the general case in which there will be four different types of ions. Let us denote salt A and salt B as the 1 : 1 and z : 1 electrolytes (z is the valence of a multivalent counterion), respectively. Indexes i, j, and k run over from 0 to 4, where 0 is referred to macroions, 1 and 3 for counterions corresponding to the electrolytes A and B, respectively, and finally, indexes 2 and 4 will be employed for coions associated to salts A and B, respectively. In addition, if a highly dilute dispersion is assumed, r Æ 0, from Equation 3.1, we can write 4

hi 0 (r ) = ci 0 (r ) +

Â r Ú c (|r - s |)h j

ij

j0

(s ) d 3 s.

(3.2)

j =1

The next step is to find suitable approximations for the direct correlation functions (the closure relations for the OZ equation). To this end, the particle–ion correlations have been treated by using the HNC approximation:

ci 0 (r ) = - b

zi z0 e2 + hi 0 (r ) - ln[1 + hi 0 (r )], 4 pe 0 e r r

(3.3)

where b = 1/kBT (k B is Boltzmann’s constant and T is the absolute temperature); zie and z0 e are the charges of ions and particles, respectively (e is the elementary charge); and e r e0 is the permittivity of the dielectric continuum (e0 is the vacuum permittivity). Concerning the ion–ion correlations, cij (| r - s|) is customarily approximated by its bulk value, cijbulk (| r - s|), and the following decomposition is made: cijbulk (| r - s |) =

zi z j e2 + cij0 (| r - s |), 4 pe 0 e r | r - s |

(3.4)

where cij (| r - s|) are functions that include the ionic size correlations through parameters such as its salt molar concentration, the size, and charge of ions. For the case of a restrictive PM (in which all the ionic radii are supposed to have the same value a), such functions can be calculated by means of

66

Structure and Functional Properties of Colloidal Systems

the MSA according to the premise given in reference [15] (the combined use of both approximations leads to the HNC/MSA approach). Although the HNC/HNC approximation is theoretically a more consistent procedure, the previous one is usually more employed for the planar EDL [14,24–26]. In view of the fact that the particles used in this work are almost two orders of magnitude larger than ions, they can be considered as charged planes, and therefore the application of the HNC/MSA approach is expected to be successful. Once the closure relations for the OZ equation are presented, the HNC/MSA three-dimensional equations can be deduced by performing some algebraic manipulations and angular integrations [24]: 4

ln[1 + hi 0 (r )] = - bzi ey (r ) +

Ú Â r c (|r - s |)h 0 j ij

j0

( s ) d 3 s,

(3.5)

j =1

where y(r) is the Poisson equation solution electrostatic potential. From Equation 3.5, the one-dimensional HNC/MSA equations can be straightforwardly deduced: 4

ln[1 + hi 0 (r )] = -bzi ey (r ) +

Ú Â r c (|r - s |)h 0 j ij

j0

( s ) d 3 s,

(3.6)

j =1

where hi0(x) = gi0(x) – 1 is the wall-ion total correlation function, y(x) is the electrostatic potential at a distance x > 0 from the wall, and the integrals depending on the direct correlation functions of the bulk species, cij0 (|x – y|), can be calculated from reference [16]. y(x) can be related to the correlation functions through y( x) =

e er e0

ÂzrÚ j j

j

• x

( x - t )h j (t )dt.

(3.7)

If ionic size correlations between ionic species i and j are neglected, cij0 functions vanish. Consequently, Equation 3.6 becomes the widely known Boltzmann exponential expression. In order to point out the importance of the inclusion of the ion size in the description of the EDL, we show in Figure 3.1 the wall-ion distributions functions predicted by the HNC/MSA and the PB equation, for a representative case: divalent electrolyte at high concentration (0.1 M) and moderate surface charge density (s0 = -0.15 C/m2). According to the classical approach, the counterion concentration should increase monotonically by decreasing the distance x from the surface, whereas the coion density should decrease. This contrasts strongly with the HNC/MSA predictions in which a typical size for divalent ions was chosen in the calculations (a = 0.40 nm). The gi0(x) functions obtained with this theory are not monotonic any longer. This is a clear example of the charge inversion; for distances from the wall larger than one precise value (x > 3.5a in this example), it seems that the role of counterions and coions is exchanged. In other words, for the ions further out, the surface seems to have a charge that is opposite in sign to the bare surface charge. This is the reason why, far from the surface, we find an increasing concentration of coions (going toward the particle) and a decreasing concentration of counterions (just the opposite trends foreseen by the PB approach).

3.2.2

THEORY–EXPERIMENT RELATIONSHIP

Since our experimental technique is electrophoresis (motion of dispersed particles relative to a fluid under the influence of an electric field that is space uniform), the real parameter measured in our experiments is the electrophoretic mobility (m e). This parameter is the coefficient of proportionality

67

Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations

gi0(x)

10

Counterions

1

0.1

Coions

0.01 0

1

2

3 x/a

4

5

6

FIGURE 3.1 Distribution functions, gi0(x), predicted by the HNC/MSA (solid lines) and PB (dashed lines) approaches for a planar EDL (2 : 2 electrolyte, 0.1 M, s0 = -0.15 C/m2, and an ionic radius of 0.4 nm).

between the particle speed and the electric field strength and can be related to the EDL properties by means of the z-potential (or zeta-potential). This electrokinetic property is defined as the electric potential at the shear plane which may be placed up to 2–3 water molecule diameters away from the surface [27]. As a result of this definition, it is usual to assume that the z-potential coincides with the diffuse potential, z yd = y(a), which is the potential at the closest distance of the hydrated ions to the wall [15–17,20,21]. This approximation is proved to be valid for smooth double layers in the absence of organic impurities [27–29]. In this manner, the z-potential can be calculated from equations shown above and can be easily related to the electrophoretic by means of the Helmholtz– Smoluchowski (HS) equation:

me =

e0 er z , h

(3.8)

where h is the viscosity. This expression is rigorously valid in the limit kR Æ •, where R is the radius of the colloidal particle and k is the Debye–Hückel parameter (whose reciprocal is the screening Debye length). Equation 3.8 is formally identical to that obtained from the PB approach because the entire mean electrostatic potential profile at equilibrium is not required in this limiting case. Hence, ion–ion correlations are actually included in calculating both z and m e. In addition, as our colloidal particles are considerably large and we are interested in the high electrolyte concentration regime (at which charge inversion is expected to happen), the electrophoretic mobility is accepted to remain linear in z, since polarization effects (responsible for the nonlinear behavior of m e) are also small. In fact, the HS equation has been already applied successfully in studies on the reversal of the electrophoretic mobility [17,25,26,30,31]. In order to overcome the limitations of the HS equation, Lozada Cassou and González Tovar developed an electrokinetic conversion theory consistent with the HNC/MSA that allows one to convert z-potential into m e for the whole range of salt concentrations [18]. Anyway, this matter goes beyond the objectives of this work, so readers interested in this topic are referred to the articles cited previously.

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Structure and Functional Properties of Colloidal Systems

As a conclusion, the diffusion potential appears as the link between theory and electrophoresis experiments. In the case of the HNC/MSA predictions, Equations 3.6 through 3.8 will be used to estimate theoretical electrophoretic mobilities, whereas for the case of the classical GC model, theoretical values are calculated from Equation 3.8 along with the following expressions that can be deduced from the PB equation [32]:

s0 =

e 0 e r kkT Ê ey d ˆ fÁ e Ë kT ˜¯

(3.9)

with

y

f ( y) = sgn(1 - e )

2

Â

2

r [exp(- zi y) - 1]

i =1 i

Â

2 i =1

(3.10)

,

ri zi2

where sgn(x) denotes the sign of x. In Figure 3.2, we show the diffuse potential as a function of the surface charge density of a planar EDL, calculated by means of the HNC/MSA and the PB equation. In this case, we have chosen different electrolyte concentrations in order to illustrate the influence of the surface charge and the ionic strength in the diffuse potential. As in the previous example, the ion radii used in the integral equations formalism is 0.4 nm. This figure shows again certain discrepancies between the predictions of the PB equation and the HNC/MSA, namely, the diffuse potential predicted by the latter can be a nonmonotonic function of the surface charge density. What is more, a reversal in yd may be foreseen. This feature will be illustrated later with electrophoresis data of real colloids.

0.08

0.05 M

0.06

0.2 M

–yd (V)

0.5 M 0.04

0.02

0.05 M

0.00

0.2 M 0.5 M –0.02 0.00

0.05

0.10

0.15

0.20

0.25

–s0 (C/m2)

FIGURE 3.2 Diffuse potential as a function of the surface charge density predicted by the HNC/MSA (solid lines) and PB (dashed lines) approaches for several salt concentrations of a 2 : 2 electrolyte (indicated in the graph) and an ionic radius of 0.4 nm.

Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations

3.3

69

RESULTS

As mentioned in the introduction, in this section we will show some experimental results of electrophoresis that will be compared with classical and integral equations approximations. In particular, we will focus on the effect of multivalent electrolytes on two similar systems but with different surface charge density. We used two monodisperse polystyrene latexes (JL1 and SN10) with a large particle size (186 and 196 nm, respectively) so that we could apply the HS approximation. They were prepared by a two-stage “shot growth” emulsion polymerization process in the absence of emulsifiers, and subsequently, two styrene/sodium styrene sulfonate copolymers were obtained. The details of synthesis method are described in reference [20]. The surface charge densities were determined by conductometric automatic titration and the results were (-0.040 ± 0.002) and (-0.115 ± 0.017) C/m2 for latexes JL1 and SN10, respectively. As can be seen, both latexes have similar size and surface charged groups, but the absolute value of s0 for SN10 almost quadruplicates than that for JL1. This aspect will allow us to study the surface charge effect in the electrophoretic behavior when different ions are present in the solution. In addition, since the particles are negatively charged in both cases, the role of the counterions will be performed by cations. Electrophoretic mobility measurements were performed by using a ZetaPALS instrument (Brookhaven, USA), which is based on the principles of phase analysis light scattering (PALS). The setup is especially useful at high ionic strengths and nonpolar media, where mobilities are usually low [33,34]. Accordingly, the results are presented as follows: Firstly, we analyze the case of a salt with divalent counterions. Secondly, the case of trivalent counterions is studied. Finally, we will focus on the case of electrolyte mixtures made of mono- and multivalent electrolytes.

3.3.1 DIVALENT COUNTERIONS We start displaying the electrophoretic mobility for latexes JL1 (Figure 3.3a) and SN10 (Figure 3.3b) as a function of the Mg(NO3)2 (Mg++ is the counterion). In both cases, experimental data are represented by symbols with their experimental error bars (estimated from the standard deviation of a set of measurements), whereas theoretical predictions are plotted by lines (solid and dashed lines for the HNC/MSA and PB approaches). To begin with the discussion, we center on the experimental results. In order to simplify the discussion, we will refer to the electrophoretic mobility in absolute value hereafter. Accordingly, the electrophoretic mobilities of the more charged latex (SN10) are logically

(a)

1

(b)

0 me (10–8 m2 V–1s–1)

0 me (10–8 m2/ Vs)

1

–1 –2 –3

–1 –2 –3 –4

–4 –5 0.1 CMg(NO3)2 (M)

0.1

1 CMg(NO3)2 (M)

FIGURE 3.3 Electrophoretic mobility as a function of the Mg(NO3)2 concentration for latexes JL1 (Figure 3.3a) and SN10 (Figure 3.3b). Symbols stand for experimental data and lines are the predictions obtained from the HNC/MSA (solid lines) and PB–GC (dashed lines) approaches. The hydrated ion radius used in the former theory is a = 0.4 nm.

70

Structure and Functional Properties of Colloidal Systems

larger than those measured for the less charged latex (JL1). Apart from the difference in the magnitude of the mobilities, in both cases, m e decreases with the salt concentration. However, a reversal of the mobility sign is never reached. Anyhow, it is worth comparing these results with the predictions obtained from the PB and HNC/MSA approaches. In general, the results obtained with the classical model are markedly larger than the experimental ones. In order to find better agreement between experimental results and PB predictions, we could imagine that the Stern layer contains adsorbed ions. In such a case, Equation 3.11 must be applied carefully since s0 should be replaced by a diffuse charge density that is not identical to the surface charge density. This analysis is feasible but calls for further information that a priori is not usually available. Consequently, the diffuse charge should be used as phenomenological parameter. The reader interested in additional details about this discussion is referred to our previous work [17]. In contrast, the HNC/MSA predictions are much closer to experimental data. According to this model, for lower salt concentrations, the curves match fairly well with the experimental results, whereas for higher ionic strengths, the modest theoretical reversals in the mobility expected to come out are not experimentally corroborated. Anyway, the integral equations theory provides better results, especially for latex SN10 (Figure 3.3b). The hydrated ion radius chosen for the computation of these predictions was the same as in the examples shown in Figures 3.1 and 3.2 (0.4 nm) [30]. According to these results, some general conclusions can be deduced. First, the PB–GC theory seems to be applicable just for systems with low surface charge. For the highly charged latex, this approach presents overestimated mobilities with respect to both experimental and HNC/MSA data. The fact that the PB model works relatively well for systems with smaller s0 is essentially the same as that obtained by Quesada et al. in their study on renormalization processes in charged colloids [35]. According to this work, the electrokinetic charge density, estimated from electrophoresis, tends to be close to the actual surface charge density for low values of s0 and elevated ionic strengths. In other words, under these conditions, the electrophoretic mobilities calculated from GC model are comparable to the experimental ones. On the other hand, the fact that the HNC/MSA matches quite better with the mobility measurements than the previous case, in particular for the highest charged latex, gives an idea about the importance of ionic size correlations for this kind of systems. In the same way, this conclusion can be drawn from reference [17] where Attard et al. compare the zeta and surface potentials for several charged systems by using the HNC theory and experimental data. In this work, the importance of considering diverse aspects, such as ion–ion correlations that are not completely included in the classical model, are underlined.

3.3.2 TRIVALENT COUNTERIONS As in the previous case, we analyze the effect of a 3 : 1 electrolyte on the electrophoretic mobility for our colloidal systems. In particular, in Figure 3.4 the mobility measurements (symbols) and their corresponding theoretical predictions (lines) are plotted as a function of La(NO3)3 for the latexes JL1 (Figure 3.4a) and SN10 (Figure 3.4b). As can be seen, the most remarkable feature of these results is the existence of a reversal in the experimental mobility for latex SN10. Before commenting this fact, we will focus on latex JL1. For this system, the experimental reversal in the mobility is not reached. Unlike the previously analyzed electrolyte case, the PB forecast (dashed line) provides absolute values of mobility quite larger than the experimental ones. It seems evident that the counterion valence plays an important role in electrophoresis. This may be the reason why the classical model cannot reproduce the experimental data even for systems of low surface charge density (such as JL1). With regard to the HNC/MSA theory (solid line), predictions are better as the electrolyte concentration increases, whereas the tendency becomes worse at lower ionic strengths. The ionic radius used here was 0.48 nm, which is the typical value for a trivalent hydrated cation [30]. Although the results are an improvement on the PB ones, some discrepancies can be observed at low electrolyte concentrations. Such disagreements can be explained in terms of the HS approximation in the conversion of mobility into z-potential (Equation 3.8). As mentioned before, this approach is valid

71

(a)

(b)

1

me 108 (m2 V–1s–1)

Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations

0

me 108 (m2 V–1s–1)

0 –1 –2 –3

–4

–4 –5

–3

1E-3

0.01 CLa(NO3)3 (M)

0.1

–5 1E-3

0.01

0.1

CLa(NO3)3 (M)

FIGURE 3.4 Electrophoretic mobility as a function of the La(NO3)3 concentration for latexes JL1 (Figure 3.4a) and SN10 (Figure 3.4b). Symbols stand for experimental data and lines are the predictions obtained from the HNC/MSA (solid lines) and PB–GC (dashed lines) approaches. The hydrated ion radius used in the former theory is a = 0.48 nm.

for elevated electrokinetic radii and it should be noted that now we are dealing with salt concentrations lower than that in the 2 : 1 case. Although kR is still larger than 1 for all ionic concentrations, it is important to keep in mind that as the electrolyte concentration decreases, this approximation is expected to provide worse results. In fact, a correction on this approximation has been already done by means of the electrokinetic conversion theory developed by Lozada-Cassou and González-Tovar [36]. Therein, it can be inferred that HS corrections only affect to low concentrations of multivalent electrolytes, and therefore it is proved that Equation 3.8 is appropriate to analyze the results concerning charge inversion (that takes place at high salt concentrations). In the case of latex SN10, a reversal in the sign of the mobility is clearly attained (as has been previously remarked). This feature was already observed by Ottewill and Shaw some decades ago [37]. Recently, other authors have also reported this phenomenon with trivalent counterions [38–40]. In our case, the reversal is observed for a salt concentration of about 15 mM (to which we will refer as the reversal concentration). It should be stressed that values of the same order of magnitude have also been found for other polymeric latexes with strong acid groups (sulfate [38] and sulfonate [40]). In contrast, much lower reversal concentrations (of the order of 10-5 M) have been found for carboxylic latexes [37]. This seems to suggest that the chemical nature of the surface groups has certain influence on mobility reversal in 3 : 1 electrolytes. There also exist differences in the magnitude of mobility for electrolyte concentrations under or above the reversal concentration. In our work, the electrophoretic mobility values are smaller than those reported by Ottewill, but resemble the results reported for similar sulfonated latexes [40]. Regarding the theoretical PB–GC prediction, it is obvious that this approach cannot explain the experimental mobility unless a mechanism of ion-specific adsorption is assumed. Ottewill and Shaw argued that La3+ ions could be electrostatically bound to carboxylic groups. However, these groups are not present on the surface of these latexes. Accordingly, this theory is not able to reproduce the previously cited reversal and the mobility values are very different to the experimental and the HNC/MSA data (especially for the more charged system). Concerning the HNC/MSA theory, one of the most relevant aspects is its capability to reproduce the electrolyte concentration for which the mobility reversal occurs. In our case, this reversal concentration is about 15 mM of La(NO3)3. Apart from the reversal concentrations, there are more aspects to mention about the HNC/MSA prediction. For instance, the model matches fairly well with the experimental data for lower salt concentrations. This agreement is notorious for the case of the more charged latex. This feature can be understood taking into account that, according to the PB model, the more charged system, the higher its mobility. However, many colloidal systems do not

72

Structure and Functional Properties of Colloidal Systems

obey this rule [20,35,41–43]. In particular, this discrepancy takes place in the present work. For instance, the mobilities for both latexes are very similar when the electrolyte is 2 : 1. Moreover, for the 3 : 1 case, the most charged latex exhibits smaller mobilities (in absolute value). To understand this discrepancy, we must turn back our attention to Figure 3.2. According to the HNC/MSA predictions, at electrolyte concentrations of 0.05 and 0.2 M, almost the same theoretical mobilities can be obtained for the surface charge densities corresponding to these latexes (0.04 and 0.115 C/m2 for JL1 and SN10, respectively). Hence, it is reasonable to suppose that lower mobilities can be found for latex SN10 when moderate ionic strengths of a 3 : 1 electrolyte are used, which confirms the importance of the ionic size correlations in this sort of studies. For the highest ionic strengths (above the reversal concentration), the HNC/MSA theory predicts that the reversal mobility rises with increasing salt concentration. As can be observed, this behavior does not agree with the experimental results. The occurrence of a plateau in the experimental mobility for high electrolyte concentrations seems to suggest the idea of saturation in the charge reversal. In relation to this, we should mention that recent simulations for trivalent counterions seem to corroborate this idea [44]. In fact, this saturation in the inverted mobility values was also reported by Tanaka and Grosberg in their molecular-dynamics study [45]. The reason why simulation predictions are closer to the experimental data than the theoretical ones could be understood in terms of a numerical breakdown of the HNC/MSA at high electrolyte concentrations. In references [44] and [45], an exhaustive comparison between the HNC/MSA approach and Monte Carlo (MC) simulations is carried out. Therein, one can observe how the theory overestimates the charge oscillation in the second layer of ions when a highly charged planar wall is analyzed in the presence of an elevated 3 : 1 salt concentration (see Figure 12 of the cited reference). This overestimation is preceded by slight differences in the concentration of coions in the region adjacent to the surface, in which the theory predicts less presence of these ions. Probably, the simulations consider the fact that the coions are attached to counterions in that region in spite of the electrostatic repulsion due to the wall charge. In other words, coions manage to approach the likely charged surface condensed onto the counterions. This effect can be clearly observed in the snapshots of the simulation performed by Tanaka and Grosberg [45]. At this point, we should emphasize the fact that, in the HNC/MSA, ion–ion correlations are treated within the MSA (see Section 3.2). Such an approximation is essentially linear since the ion–ion interaction energy cannot be larger than the thermal energy. In the strong coupling limit (ions with large valences, high salt concentrations, and low temperature), this condition would be violated to a great extent, which might account for the breakdown of the MSA. Although improved integral approximation can then be applied [24], we should point out that there are Wigner-crystal-like formalisms that could work properly (at least from a qualitative viewpoint) in that limit, such as the model put forward by Levin on the one hand, and by Shklovskii on the other, with their respective coworkers [46–50], which assumes a layer of condensed counterions on the colloidal particle surface.

3.3.3 MIXTURE OF ELECTROLYTES As we have shown in the previous cases, the inclusion of the short-range correlations between ions in the study of the ionic distribution leads to quite different results with respect to the classical treatment. In particular, these discrepancies become more noticeable for highly charged particles in the presence of moderate or highly multivalent counterions concentrations. Accordingly, it is expected that the effect of ion correlations in electrolyte mixture will be also important. In particular, it can be shown that the existence of an electrolyte mixture consisting of multi- and monovalent counterions may cause charge inversion in a colloidal system. Precisely, this aspect has recently originated an enormous interest in the field of biophysics. For instance, DNA forms a tightly packed torus in aqueous solutions containing spermine (Z = +3) or spermidine (Z = +4) [49]. In order to find out the relevance of ion correlations in electrolyte mixtures, in this section we study the electrophoretic mobility of the more charged latex previously studied (SN10), in the presence of different 1 : 1 and 3 : 1 salt mixtures. Likewise, the results are analyzed within the HNC/MSA.

Ionic Structures in Colloidal Electric Double Layers: Ion Size Correlations

73

exp. [NaNO3] = 0.00 M 1.0

exp. [NaNO3] = 0.01 M exp. [NaNO3] = 0.10 M

me 108 (m2 V–1s–1)

0.5 0.0 –0.5 –1.0 –1.5 HNC/MSA [NaNO3] = 0.00 M, a = 0.48nm –2.0

HNC/MSA [NaNO3] = 0.01 M, a = 0.43nm HNC/MSA [NaNO3] = 0.10 M, a = 0.40nm

–2.5 1E-3

0.1

0.01 CLa(NO3)3 (M)

FIGURE 3.5 Electrophoretic mobility as a function of the La(NO3)3 concentration for 0, 0.01, and 0.1 M of NaNO3 (squares, circles, and triangles, respectively). The lines denote the HNC/MSA predictions (for the ionic radii indicated in the figure).

In Figure 3.5, electrophoretic mobility measurements are shown as a function of the La(NO3)3 concentration in the presence of NaNO3 (0, 0.01 and 0.1 M). As can be seen, there are no significant differences between 0 and 0.01 M of monovalent salt but the results for 0.1 M present the following noticeable features: (i) for La(NO3)3 concentrations lower than 0.02 M, the magnitude of me is higher in the presence of 0.1 M of NaNO3 (a similar trend was reported by other authors); (ii) the concentration of zero mobility (to which we will refer as inversion point) is shifted toward higher lanthanum concentrations; (iii) beyond the inversion point, the mobility is lower for the mixture with 0.1 M of monovalent salt. Regarding the situations pointed out in (i) and (iii), they result counterintuitive from a classical point of view. Although PB–GC predictions are not shown, the magnitude of me calculated with this approach is not expected to increase adding NaNO3. Concerning the HNC/ MSA, different ion radii were used in the calculations according to the monovalent salt concentrations. When there is only trivalent electrolyte, the ion size was a = 0.48 nm (as in the previous case). As the 1 : 1 salt concentrations increases, lower values were employed in order to average this magnitude. Anyway, values close to the trivalent one were chosen because it is assumed that this ionic specie is predominant in the immediate vicinity of the charged surface. As can be seen in the figure, mobility reversal is clearly foreseen by the HNC/MSA. Moreover, the behavior below the inversion point (i.e., for La3+ concentrations lower than 0.02 M) is qualitatively captured. More specifically, the theoretical results for 0 and 0.01 M of NaNO3 are quite similar and lower than those predicted for 0.1 M of this salt. This increase in mobility when adding monovalent electrolyte cannot be justified easily if the HNC/MSA is replaced by a PB–GC approach. In this sense, mobilities predicted by the PB–GC theory under similar conditions are usually much greater [19]. Consequently, the HNC/ MSA is able to reproduce (at least) the order of magnitude obtained in experiments. Regarding the inversion points predicted, they are of the order of 0.01–0.02 M of La(NO3)3. The shift of the inversion point found experimentally is also qualitatively corroborated by the theory. However, this agreement is due to, at least partially, changes in the ionic radii used in the calculations. If a fixed value is employed in all cases, differences in the inversion point are less significant. Finally, beyond

74

Structure and Functional Properties of Colloidal Systems

the inversion point, the theoretical results do not fit the experimental data over this range of La3+ concentrations (as in the previous case). More specifically, me should increase markedly according to the HNC/MSA, which contrasts strongly with the plateau found in experiments. The cause of this disagreement can be explained once more in terms of overestimation of the charge predicted by the theory (see discussion of the previous section). This feature is deeply analyzed for the case of mixture of electrolytes by means of MC simulations in reference [51].

3.4

CONCLUSIONS

Although diverse findings have been drawn along the chapter, we would like to conclude this work by summing up concisely some conclusions. (i) In general, the experimental mobilities for charged colloids can be justified moderately well by using equation integral theories that take the ion size into account. In contrast, the PB–GC approach gives worse results than the previous one and its applicability is limited for low charged systems. (ii) The correlation effects due to ionic size become especially important for highly charged systems in the presence of significant concentrations of multivalent counterions. This feature is especially relevant in the case of electrolyte mixtures. (iii) Although the integral equation formalism can quantitatively justify the mobility reversal and the behavior below the inversion point, neither the HNC/MSA nor the PB–GC approaches manage to predict properly the behavior of the electrophoretic mobility beyond the inversion point. The existence of some numerical discrepancies is not surprising since there are several approximations involved in these calculations, for instance, z yd, the HS equation, or assuming the same size for all the ionic species. Therefore, in our opinion, they might explain the numerical differences between theory and experiment found below the inversion point, but they would hardly justify major discrepancies, such as those observed beyond this point. In summary, if diverse parameters have to be included to improve the EDL description, it is worth using a sophisticated model to consider them instead of a classical one that ignores basic features such as the ionic size.

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33. Miller, J. F., K. Schätzel, and B. Vincent. 1991. The determination of very small electrophoretic mobilities in polar and nonpolar colloidal dispersions using phase analysis light scattering. Journal of Colloid and Interface Science 143 (2): 532–554. 34. McNeil-Watson, F., W. Tscharnuter, and J. Miller. 1998. A new instrument for the measurement of very small electrophoretic mobilities using phase analysis light scattering (PALS). Colloids and Surfaces A: Physicochemical and Engineering Aspects 140 (1–3): 53–57. 35. Quesada-Pérez, M., J. Callejas-Fernández, and R. Hidalgo-Álvarez. 1999. Renormalization processes in the charge density of polymer colloids. Colloids and Surfaces A: Physicochemical and Engineering Aspects 159 (2–3): 239–252. 36. Quesada-Pérez, M., E. González-Tovar, A. Martín-Molina, M. Lozada-Cassou, and R. Hidalgo-Álvarez. 2005. Ion size correlations and charge reversal in real colloids. Colloids and Surfaces A: Physicochemical and Engineering Aspects 267 (1–3): 24–30. 37. Ottewill, R. H. and J. N. Shaw. 1968. An electrophoretic investigation of the behavior of monodisperse polystyrene latices in solutions of lanthanum, neodymium, and thorium nitrates. Journal of Colloid and Interface Science 26 (1): 110–119. 38. Elimelech, M. and C. R. O’Melia. 1990. Effect of electrolyte type on the electrophoretic mobility of polystyrene latex colloids. Colloids and Surfaces 44 (C): 165–178. 39. Galisteo F., F. J. De las Nieves, M. Cabrerizo, and R. Hidalgo-Álvarez. 1990. Effects of particle concentration, ionic strength, pH and temperature on the microelectrophoretic mobility of cationic polysterene latex. Progress in Colloid & Polymer Science 82: 313–320. 40. Bastos, D., F. J. De las Nieves, P. Laggner, and O. Glatter. 1993. On the zeta-potential of sulfonated polystyrene model colloids. Progress in Colloid & Polymer Science. Graz, Austria: Springer-Verlag GmbH & Company KG. 41. Huang, Q. R., P. L. Dubin, C. N. Moorefield, and G. R. Newkome. 2000. Counterion binding on charged spheres: Effect of pH and ionic strength on the mobility of carboxyl-terminated dendrimers. Journal of Physical Chemistry B 104 (5): 898–904. 42. Haydon, D. A. 1960. A study of the relation between electrokinetic potential and surface charge density. Proceedings of the Royal Society of London Series A—Mathematical and Physical Sciences 258 (1294): 319–328. 43. Brendan J. C. and D. A. Haydon. 1975. Electrokinetic and surface potentials at liquid interfaces. Journal of the Chemical Society, Faraday Transactions 1 71 (10): 361–377. 44. Martín-Molina, A., J. A. Maroto-Centeno, R. Hidalgo-Álvarez, and M. Quesada-Pérez. 2008. Charge reversal in real colloids: Experiments, theory and simulations. Colloids and Surfaces A: Physicochemical and Engineering Aspects 319 (1–3): 103–108. 45. Tanaka, M. and A. Yu. Grosberg. 2002. Electrophoresis of a charge-inverted macroion complex: Molecular-dynamics study. European Physical Journal E 7 (4): 371–379. 46. Perel, V. I. and B. I. Shklovskii. 1999. Screening of a macroion by multivalent ions: A new boundary condition for the Poisson–Boltzmann equation and charge inversion. Physica A: Statistical Mechanics and its Applications 274 (3): 446–453. 47. Nguyen, T. T., A. Yu. Grosberg, and B. I. Shklovskii. 2000. Screening of a charged particle by multivalent counterions in salty water: Strong charge inversion. Journal of Chemical Physics 113 (3): 1110–1125. 48. Levin, Y. 2004. Introduction to statistical mechanics of charged systems. Brazilian Journal of Physics 34 (3 B): 1158–1176. 49. Pianegonda, S., M. C. Barbosa, and Y. Levin. 2005. Charge reversal of colloidal particles. Europhysics Letters 71 (5): 831–837. 50. Martín-Molina, A., J. A. Maroto-Centeno, R. Hidalgo-Álvarez, and M. Quesada-Pérez. 2006. Testing one component plasma models on colloidal overcharging phenomena. Journal of Chemical Physics 125 (14): 144906-1–144906-9. 51. Martín-Molina, A., M. Quesada-Pérez, and R. Hidalgo-Álvarez. 2006. Electric double layers with electrolyte mixtures: Integral equations theories and simulations. Journal of Physical Chemistry B 110 (3): 1326–1331.

4

Effective Interactions of Charged Vesicles in Aqueous Suspensions C. Haro-Pérez, L.F. Rojas-Ochoa, V. Trappe, R. Castañeda-Priego, J. Estelrich, M. Quesada-Pérez, José Callejas-Fernández, and R. Hidalgo-Álvarez

CONTENTS 4.1 4.2 4.3

Introduction ........................................................................................................................ Experimental System and Methods .................................................................................... Results and Discussions ...................................................................................................... 4.3.1 Structural Hallmarks .............................................................................................. 4.3.2 Effective Interactions Potentials ............................................................................. 4.3.3 Dynamics ................................................................................................................ 4.3.4 Energy Content: Link between Static and Dynamic Properties ............................. 4.3.5 Discussion ............................................................................................................... 4.4 Summary ............................................................................................................................ Acknowledgments ........................................................................................................................ References ....................................................................................................................................

4.1

77 78 79 79 82 86 89 90 91 91 91

INTRODUCTION

Liposomes are vesicular structures consisting of one or more lipid bilayers enclosing an aqueous volume. Since the first publications in the 1960s, liposomes have become a major area of research, which to date keeps expanding. Indeed, the interest in studying them is manifold [1]. Their biocompatibility makes them ideal candidates for the delivery of drugs and other agents of therapeutic value [2,3]. Moreover, they can be used as valuable models for the study of more complex biological systems such as cellular membranes. Irrespective of their application, lipid vesicles can be considered as colloids whose stability is theoretically described in terms of classical colloidal forces [4]. Any progress in the knowledge of these forces can potentially contribute to the better understanding and control of the processes in which liposomes are involved. A large number of lipids have electrically charged head groups, such that electrostatic interactions play an important role in defining the thermodynamics and dynamics of liposome dispersions [5]. The theory used to describe the stability of liposome and, in general, charged colloidal dispersions is the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory that considers charged particles to interact through repulsive screened Coulomb interactions. The validity of this theory, however, has been recently challenged by experimental and theoretical work investigating the behavior of 77

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highly charged systems at quasi-deionized conditions. Indeed, these investigations showed evidence for nontrivial phase behavior, such as void formation [6], liquid–solid re-entrance [7], and gas– liquid spinodal instabilities [8], suggesting the existence of effective long-range attractive forces between like-charged particles in contradiction to the well-established DLVO potential (for a review see reference [9]). In this contribution we present experimental results, indicating that such effective attraction also governs the behavior of charged liposome dispersions at quasi-deionized conditions. The chapter is organized as follows. In Section 4.2 we introduce the experimental system and methods. In Section 4.3 we present the volume fraction-dependent structural and dynamical properties of our liposomes, which we investigated by using static light scattering (SLS) and dynamic light scattering (DLS). Subsequently, we show the results obtained from the modeling of the structure factor, for which we also perform Monte Carlo (MC) simulations. We close Section 4.3 by a discussion and finally present a brief summary in Section 4.4.

4.2

EXPERIMENTAL SYSTEM AND METHODS

The monolamellar vesicles used in our experiments are composed of bovine spinal cord phosphatidylserine (PS) and egg phosphatidylcholine (PC) at a ratio of PS/PC 1. They are produced by using the lipid-film method [10], which consists of placing a solution of lipids in chloroform in a round-bottom flask and drying it under reduced pressure at 50°C in a rotary evaporator. The thin film formed on the walls of the flask is subsequently hydrated with a given quantity of ultrapure water. Applying constant vortexing and sonication leads to the formation of large multilamellar liposomes; subsequent extrusion through polycarbonate membrane filters with 100 nm pore sizes finally yields small unilamellar vesicles that are moderately polydisperse in size [5]. The form factor determined in SLS experiments using dilute dispersions of these liposomes is well approximated by the polydisperse model for vesicles assuming a discrete Schulz distribution composed of three species of different size [11]. The mean diameter of the vesicles used in the present study is d = 120 ± 18 nm as determined by both SLS and DLS. The pH is set to ~7, where we presume the zwitterionic phospholipid PC to be uncharged, such that the total surface charge of the liposome is determined by the monovalent ionic phospholipid PS. Assuming an area per lipid of 0.7 nm2 [12], we estimate the number of ionizable groups per particle to be 30,000. Following the argumentations of reference [13], this would correspond to 2000 electrons/particles or equivalently to 0.7 μC/cm2. This charge reduction is due to the fact that the pH in the vicinity of the particles can be lower than in the bulk, which influences the degree of ionization of the surface chemical groups. The volume fraction of the initial stock solution, f = (c/rs)[a3/(a3 - b3)] 0.12, is estimated from the lipid concentration c used in the preparation by assuming that the thickness and the density of the phospholipid shell are respectively a - b = 4 nm and rs = 1.015 g/cm3 [14–16]; a and b are respectively the outer and inner radius of the vesicle. Additional suspensions with volume fractions ranging from f 0.009 to f 0.12 are prepared by diluting the stock solution with ultrapure water. To ensure the lowest possible ionic strength, the suspensions are sealed in contact with a mixed bed of ion exchanger resins in cylindrical quartz cells 5 days prior to the experiments. Before use the ion exchange resin is washed several times with ultrapure water till we find that a small amount of water that has been in contact with the resin for several days does not contain any contamination, such as polyelectrolyte, that absorb in the UV range [17]. All our liposome suspensions are remarkably transparent indicating that the refractive index of the liposome nl is similar to the one of water nw. We determine nl by measuring the concentration dependence of the effective refraction index of the liposome dispersions, neff, at different wavelengths l and T = 23°C using an Abbe refractometer. Assuming n2eff = (n2l - n2w)f + nw2 [18], the refractive indices nl(l) are obtained from the linear fits to the data plotted as n2eff versus f, as shown in Figure 4.1. Using the dispersion law nl(l) = A0 + Al/l2 to describe the wavelength dependence of

79

Effective Interactions of Charged Vesicles in Aqueous Suspensions

nlipos(l = 435.8 nm) = 1.362 1.80

n2eff

1.79 nlipos(l = 546.1nm) = 1.360

1.78

nlipos(l = 579.1nm) = 1.359 1.77 0.00

0.02

0.04

0.06

0.08

0.10

φ

FIGURE 4.1 Squared effective refraction index versus volume fraction for different wavelengths: l = 435.8 nm (squares), l = 546.1 nm (circles), and l = 579.1 nm (triangles). The lines are linear fits to the data.

nl(l), we determine nl = 1.3602 for l0 = 514.5 nm, the wavelength used in our light scattering experiments. The optical contrast between particles (nl = 1.36) and suspending medium (nw = 1.33) is sufficiently small to prevent significant multiple scattering. This makes these systems ideal model systems for light scattering studies, allowing for the use of standard light scattering techniques. We use a commercial goniometer from ALV-GmbH equipped with an Argon-ion laser operating at a wavelength of l0 = 514.5 nm, which allows for the measurement of the scattered intensity in a scattering vector range of 0.0042 £ q £ 0.031 nm-1. To assure that the measurements are indeed not corrupted by multiple scattering, we additionally perform several cross-check experiments using a home-made 3D cross-correlation device that allows for the measurement of the single scattering contributions in the presence of multiple scattered light [19,20]. The results do not differ by more than 3% from those measured with the commercial device, indicating that the use of the standard device does not lead to significant errors.

4.3

RESULTS AND DISCUSSIONS

The deionization of our dispersions leads to remarkably different effects depending on whether we deionize a sample with f £ 0.06 or f ≥ 0.09. Indeed, for f up to 0.06, the viscosity gradually increases during few hours after the liposome dispersion has been put into contact with the ion exchange resin. By contrast, for the two largest f investigated, f = 0.09 and 0.12, shaking up the just prepared mixtures of dispersion and resin transform the samples in elastic solids that withstand even rigorous shaking. Upon addition of a small quantity of a concentrated electrolyte solution, these systems revert back to a fluid state that again transform into solids as the ions are removed by the resin. To explore the origin of this remarkable behavior, we investigate the structure and dynamics of our dispersions by SLS and DLS.

4.3.1 STRUCTURAL HALLMARKS From the SLS data we determine the static structure factor S(q) = (r0 /r)[I(q)/I0(q)], where I0(q) is the light intensity scattered by a suspension of noninteracting particles with number density r0 [21] and

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I(q) is the ensemble-averaged scattered intensity of the suspension of interacting particles with number density r. Contrary to the fluid-like systems, the time-averaged intensity is not equal to the ensemble-averaged one in the solid-like systems, even if the intensity is averaged over long-time periods. In these cases, the ensemble-averaged intensity is determined by steadily rotating the sample during the experiments [22]. For the samples with f ranging from 0.009 to 0.06, S(q) exhibits a pronounced particle–particle peak whose position strongly depends on liposome concentration, as shown in Figure 4.2a. The peak position, qmax, shifts to larger q-values as f increases exhibiting a power-law behavior, qmax ~ f1/3, characteristic of salt-free charged suspensions. In Figure 4.2b we display the related mean interparticle distance, rm = 2p/qmax, that consequently scales as f -1/3 [23]. Unfortunately, although we clearly observe an increase in S(q) at large q, for the solid-like samples (f = 0.09 and f = 0.12), the peak positions are shifted to q-values that are beyond the q-range accessible in our experiments. To test whether the peak position of these systems follows the same trend as the one observed at lower f, the rm-values obtained by extrapolation (open circles in Figure 4.2b) are used to normalize the q-axis in the S(q) versus q plot. As shown in Figure 4.3a, this scaling procedure leads to a reasonable coincidence of all S(q)-data around the peak, which indicates that the monotonic decrease of rm with f -1/3 persists for the largest volume fractions investigated. Further evidence for a continuous development of S(q) with f is shown in Figure 4.3b, where we display the derivative of the static structure factor at S(qrm) 1, as determined by the slope in the q-range comprised by the horizontal lines shown in the Figure 4.3a. This slope develops continuously with f in the volume fraction range investigated; it does not exhibit an abrupt change at the fluid–solid transition, which would indicate a thermodynamic phase transition. Indeed, the continuous change of the structure reveals that the transition is not related to a liquid–crystal transition, where we would expect this slope to be significantly higher for a crystal phase than for the liquid disordered phase. The apparent total absence of crystals in our system is most likely due to charge polydispersity, which suppresses crystallization in charged colloids [24] as size polydispersity does for hard spheres [21]. The f -1/3 scaling of rm is indicative of the repulsive nature of our charged systems, where particles configure such that to achieve a maximal distance between them; this indirectly implies that space is homogeneously filled with particles. In seeming contradiction to this, S(q) exhibits an upturn at low q for all f investigated, which suggests that the dispersions are structurally heterogeneous at large (a)

(b) 3 4

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2 S (q)

rm/d

–1/3

2

1

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FIGURE 4.2 (a) Wave vector dependence of the static structure factor: from left to right f = 0.009, 0.016, 0.020, 0.027, 0.040, 0.060, 0.090, 0.120. (b) Mean particle distance rm as a function of f. Solid circles: data obtained from the peak position in S(q); open circles: rm for f = 0.090 and f = 0.120 as expected from extrapolation. Line corresponds to a power-law fit to the data, as expected for repulsive systems rm ~ f -1/3. (Reprinted from Haro-Pérez C., et al. 2009. Phys. Rev. Lett. 102: 081301-1–4. With permission.)

81

Effective Interactions of Charged Vesicles in Aqueous Suspensions (a) 3

(b) 1.5

S (q)

Slope

2 1.0

1 0.5 0

0

4

8 qrm

12

16

0.01

0.1 f

FIGURE 4.3 (a) Normalization of q with rm leads to a reasonable coincidence of all data around the peak. Vertical line indicates the q-value where S(q) 1, the q-value at which we determine the intensity correlation function. (Reprinted from Haro-Pérez C., et al. 2009. Phys. Rev. Lett. 102: 081301-1–4. With permission.) (b) Slope of S(q) calculated in the interval marked by horizontal lines in (a).

length scales. To ensure that this low q-rise in S(q) is not simply due to polydispersity [24], we use MC simulations to calculate S(q) for a repulsive Yukawa pair potential, considering polydispersity in two different ways: (i) size polydispersity only and (ii) size and charge polydispersity combined. In both cases we implement a discrete Schulz distribution with a polydispersity of 10%, corresponding to the one obtained from the fits to the particle form factor. The charge polydispersity is obtained by fixing the surface charge and calculating the number of charged groups associated to the particle size distribution as described in reference [23,25]. Both calculations lead to an increase in S(q) at low q as compared to the behavior expected for a monodisperse system; this is shown in Figure 4.4, where we display the numerical results for the monodisperse case (black line), the size (dark gray line), and 3

S(q)

2

1

0 0

1

2

3 qd

4

5

6

FIGURE 4.4 Scattering vector dependence of static structure factor. Black circles: measured S(q) for the system at f = 0.027. Black line: calculated S(q) for dispersion of monodisperse repulsive spheres. Dark gray line: calculated S(q) for a repulsive system with size polydispersity only. Gray line: calculated S(q) for repulsive system with size and charge polydispersity.

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charge-size (gray line) polydispersities, together with the experimental results at f = 0.027 (black symbols). Clearly, although polydispersity increases the value of S(q = 0), it does not lead to an uprise in S(q), as the one characterizing the measured S(q). This strongly suggests that the low q-rise has indeed to be attributed to structural heterogeneities rather than to polydispersity. The formation of larger length-scale structures is reminiscent of attractive colloidal systems, where aggregation of the particles naturally leads to the formation of space-spanning networks. However, in a short-range attractive system the particles touch each other and generally exhibit a mean particle distance, which is independent of volume fraction. By contrast, in our system the particles are a considerable distance apart from each other and the mean particle distance exhibits a volume fraction dependence that is typical for repulsive systems. This indicates that the effective interactions between the particles comprise a long-range repulsion that governs the local structure and an even longer range attraction that leads to heterogeneities.

4.3.2 EFFECTIVE INTERACTIONS POTENTIALS In order to progress we thus aim to expose the effective interparticle interaction potential leading to the peculiar particle configuration in our systems. To do this we use an inversion protocol, iteratively approximating the experimental S(q) as follows. (i) The peak height and position of the measured S(q) is fitted assuming a pure repulsive Yukawa potential to solve the Ornstein–Zernike equation with an appropriate closure relation, for example, Hypernetted Chain (HNC) and/or Rogers–Young. The resultant pair correlation function g0(r) is then used to calculate the potential of mean force bUMF(r) = -ln[g0(r)], which is used as first trial effective potential. (ii) This potential is used in an MC simulation to obtain a new set of Sk(q) and gk(r). (iii) The new set is then used to correct the effective potential, bUk(r), as described in reference [26,27]. The procedure is repeated from (ii) until the best possible agreement of Sk(q) with the experimentally determined structure factor is obtained, as shown for f = 0.009, 0.027, and 0.09 in Figure 4.5a, b, and c, respectively; the corresponding

3

(a)

(b)

3

(c)

2

1

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S(q)

2

0

0

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3

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50

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(d)

3 U(r)/kBT

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(e)

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(f )

3

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2

3

4 r/d

5

61

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61

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3

4

5

–2 6

r/d

FIGURE 4.5 Structure factor resulting from the inversion protocol (solid gray symbols) in comparison to the experimental S(q) (open symbols) for (a) f = 0.009, (b) f = 0.027, and (c) f = 0.090. Corresponding effective interactions as a function of the normalized particle distance for (d) f = 0.009, (e) f = 0.027, and (f) f = 0.090. (Reprinted from Haro-Pérez C., et al. 2009. Phys. Rev. Lett. 102: 081301-1–4. With permission.)

Effective Interactions of Charged Vesicles in Aqueous Suspensions

83

effective potentials are shown in Figure 4.5d, e, and f. As anticipated, the potentials are characterized by a long-range attractive component and a shorter-ranged repulsive component. As a self-consistency test, we also apply the inversion protocol described above to integral equations, where the effective potentials obtained at each step are used as an input to solve the Ornstein–Zernike equation with the soft mean spherical approximation (SMSA) as closure relation [28] instead of using them as input for MC simulations. The SMSA closure has been previously shown to be suitable for the description of scattering data of dense systems with both repulsive and attractive interactions [29,30]. This calculation yields structure factors that agree with our experimental findings as well, which provides additional evidence for the validity of our effective potentials. The MC simulations are performed at constant number of particles (N) volume (V), and temperature (T), (NVT ensemble) with N = 5832 particles placed in a cubic box of volume L3 = N/r and periodic boundary conditions in each direction, where r is the number density. We partly account for polydispersity by considering the effect of polydispersity on the optical properties of the system. The three particle sizes that define the discrete Schulz distribution best describing the form factor of our liposomes are implemented in the simulations, where we assume that the same effective potential governs the interaction among the three species. For the calculation of Sk(q), we then account for the differences in optical properties by using a weighted size-average of the partial structure factors of the three components. This rather crude way to deal with polydispersity is dictated by our limited knowledge of the “real” interaction potential. For the two most concentrated samples, f = 0.09 and f = 0.12, for which we do not access the particle–particle peak experimentally, we additionally constrain the parameters leading to the best agreement with the experimental data to also reproduce the peak position as determined by extrapolation in Figure 4.2b. As a typical example for the particle configuration obtained in our MC simulations after optimizing the agreement between simulation and experimental S(q), we show in Figure 4.6c a result for f = 0.027; the corresponding effective potential and structure factor are shown in Figure 4.6a and the inset, respectively. For the sake of clarity, we do not show the entire simulation box but a slice of thickness 3rm, with rm being the mean interparticle distance. For comparison, we also perform MC simulations for equivalent purely repulsive systems, where we implement Yukawa potentials with parameters chosen so as to reproduce the peak height of the S(q) of our experimental systems; the corresponding particle configuration for f = 0.027 is shown in Figure 4.6d, the potential and structure factor are shown in Figure 4.6b and the inset, respectively. To appreciate the difference between the two systems, we divide the entire simulation box into cubic boxes of size 3rm and determine the local volume fraction within these boxes for several equilibrium particle configurations. In Figure 4.6e and f we report the histograms of the local volume fraction fl normalized by the average particle volume fraction f. In the case of the purely repulsive system shown in Figure 4.6f, the distribution is rather narrow. By contrast, for our system the distribution is wider, as shown in Figure 4.6e; this indicates that the probability to find both space with lower and higher local volume fraction than the bulk volume fraction is higher in this system than in the purely repulsive one. The distribution function remains unimodal, which reveals that the heterogeneities are not well defined. This is in contrast to the findings of Tata et al. who previously reported the formation of heterogeneities in charged colloidal systems [6]. These authors observed the existence of voids with welldefined shapes and sharp interfaces, which is not the case in the present system. The effective interaction potentials U(r) determining the particle configuration in our systems exhibit a peculiar volume fraction dependence, as shown in Figure 4.7. The magnitude of the maximal attractive interaction characterized by the minimum in U(r) exhibits a nonmonotonous development; it initially increases with increasing volume fraction, reaches a maximal value at f = 0.02 to then decrease again at higher volume fractions. The range of both the attraction and the repulsion systematically decreases with increasing volume fraction in the lower volume fraction range to then remain approximately the same for f ≥ 0.06. In fact, the position of the repulsive barrier r b, as determined at U(r) = 0, exhibits a f -1/3-dependence at low volume fractions to then remain constant for f ≥ 0.06, as shown in Figure 4.8a. A direct comparison with the volume fraction-dependent mean

84

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Structure and Functional Properties of Colloidal Systems

(b) 1

2

3

4 r/d

5

6

(d)

(f ) 0.0 0.5

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1.5

FIGURE 4.6 Graphs comparing the simulation data obtained for systems with the mixed potential describing the behavior of our system (top row) and with a purely repulsive potential, where the parameters are fi xed so that the peak height in S(q) reproduces the one of our experimental system (bottom row). For both systems f = 0.027. Displayed in (a) and (b) are the effective potentials and in the inset the resulting structure factors. Typical equilibrium configurations are shown in (c) and (d); the depth of the image corresponds to 3rm with rm being the mean interparticle distance. Histograms of the local particle volume fraction determined in cubic cells of size 3rm are shown in (e) and (f).

4 3

U(r)/kBT

2 1 0 –1 –2

1

2

3

4

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r/d

FIGURE 4.7 Effective potential estimated from S(q) and used in the MC-NVT simulations: from right to left f = 0.009, 0.016, 0.020, 0.027, 0.040, 0.060, 0.090, and 0.120.

85

Effective Interactions of Charged Vesicles in Aqueous Suspensions (b)

4

12

3

8

rb/d

U(rm)/kB T

(a)

–1/3

4

2 0 0.01

0.1

0.01

0.1

f

f

FIGURE 4.8 (a) Volume fraction dependence of the position of the repulsive barrier, r b, as determined at U(r) = 0 from the effective mean interaction potential shown in Figure 4.7. (b) Effective mean interaction potential at the mean interparticle distance as a function of f. Dashed line indicates U(rm)/k BT = 1.

interparticle distances rm shown in Figure 4.2b reveals that r b ~ rm for f £ 0.06, while r b > rm for f = 0.09 and f = 0.12. This actually suggests that in the solid-like systems the particles are pushed together, so as to sense the strong repulsion between them. This is most clearly seen in Figure 4.8b, where we report the energy felt by particles that are placed at the mean interparticle distance from each other, U(rm). This energy is around 1 k BT for volume fractions up to 0.06, but it increases to 8 and 13 k BT for f = 0.09 and f = 0.12, respectively. This result suggests that the macroscopically observed transition from fluid- to solid-like behavior is repulsive in origin. Note, however, that we expect the position of the repulsive barrier to continuously decrease with increasing f for equivalent purely repulsive systems. This is shown in Figure 4.9a, where we display the Yukawa potentials calculated for systems with volume fractions as the one investigated and for which we impose that the peak heights in S(q) correspond to the one of our experimental S(q)s. The position of the repulsive barrier decreases here proportionally to rm over the entire volume fraction range investigated. This is shown in Figure 4.9b, where we report r b determined at U(r)/k BT = 1 as a function of f; as the mean interparticle distance, the position of the repulsive barrier exhibits the

(a) 4

(b)

8

3

rb/d

U(r)/kBT

6 2 1

4

–1/3

0

–1 –2

1

2

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4 r/d

5

6

2

0.01

0.1 f

FIGURE 4.9 (a) Yukawa potentials for systems equivalent to those investigated. (b) Volume fraction dependence of the position of the repulsive barrier, r b, as determined at U(r b)/k BT = 1 from the Yukawa potentials shown in (a).

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Structure and Functional Properties of Colloidal Systems

volume fraction dependence characteristic for repulsive systems, r b ~ f -1/3, over the entire f range. This is in contrast to the behavior exhibited by our experimental system, where we find that the barrier is arrested for f > 0.06. This difference indicates that although the repulsive energy determines the observed fluid–solid transition, it is the fact that r b remains constant at large volume fractions, which imposes the arrest condition. The arrest of r b, however, seems directly correlated to the presence of effective attractions, as we do not find such arrest in equivalent purely repulsive systems.

4.3.3 DYNAMICS To obtain quantitative information about the dynamical hallmarks of the fluid–solid transition, we measure the intensity autocorrelation function g2(q, t) = ·I(t¢)I(t¢ + t)Ò/·I(t¢)Ò2 at qrm 5, where S(q) 1, as indicated by the vertical line in Figure 4.3a. This particular choice of the scattering vector is motivated by the fact that the autocorrelation function is mainly determined by the self-motion of the particles for S(q) = 1 [31]. For the fluid-like ergodic systems g2(q, t) relates to the autocorrelation function of the electric field, g1(q, t), via the Siegert relation gl(q, t)2 = [g2(q, t) - 1]/b, where b is an experimental constant of order one. In the case of nonergodic, solid-like systems, we obtain g1(q, t) following the procedure described in references [32,33]. The resulting field correlation functions are shown in Figure 4.10, where we normalize the time axis with q2 to account for the variation in q at which we perform our measurements. Our DLS data exhibit the behavior anticipated from the macroscopically observed transition from viscous, fluid-like behavior for all samples with f £ 0.06 to elastic, solid-like behavior for the samples with f ≥ 0.09. The autocorrelation functions fully decay for all fluid-like samples (f £ 0.06), while they are characterized by a nondecaying component at long times for the two solid-like samples (f ≥ 0.09). This clearly indicates that our system undergoes an ergodic to nonergodic transition, that is, a fluid to solid transition as the volume fraction increases beyond a critical value f c, where 0.06 < f c < 0.09. The dynamics of all our fluid-like, ergodic systems is characterized by a two-step decay, which is a typical characteristic of the dynamics of colloidal suspensions at larger effective volume fractions; the short-time dynamics is associated to the diffusion of the spheres within the cages of nearest neighbors, while the long-time dynamics is due to the structural rearrangement of the cages themselves.

g1(q, t)

1.0

f

0.5

0.0

10–9

10–7 tq2(s nm–2)

10–5

FIGURE 4.10 Field autocorrelation function determined at q 5/rm for f = 0.009, 0.016, 0.020, 0.027, 0.040, 0.060, 0.090, and 0.120. To account for the change in q at which we perform our measurements the time axis is normalized by q2. (Reprinted from Haro-Pérez C., et al. 2009. Phys. Rev. Lett. 102: 081301-1–4. With permission.)

87

(a) 1.000

(b) 1.000

g1(q, t)

g1(q, t)

Effective Interactions of Charged Vesicles in Aqueous Suspensions

0.368

0.135 0.900 0

3 × 1010 6 × 1010 2 tq (s m–2)

9 × 1010

0

2 × 1012 4 × 1012 2 –2 tq (s m )

6 × 1012

FIGURE 4.11 Half-logarithmic representations of the field autocorrelation functions shown in Figure 4.10 to demonstrate that both short- and long-time behavior can be approximated by simple exponential functions: (a) reduced graph to assess the short-time behavior; (b) full graph to assess the long-time behavior.

The nondecaying behavior exhibited by the solid-like systems is characteristic for dynamically arrested systems, for which the structural relaxation process is suppressed. To quantify both short- and long-time dynamics, we analyze the decay rates describing the short- and long-time behavior of g1(q, t). The half-logarithmic representations of the field correlation functions shown in Figure 4.11 reveals that both short- and long-time behavior can be reasonably described by single exponential functions; since the time axis are normalized with q2, we obtain the short- and long-time diffusion coefficients, Dshort and Dlong, directly from the decay rates as g1(q, t) = exp(-q2Dshort t) and g1(q, t) μ exp(-q2Dlong t). In Figure 4.12 we display the volume fractiondependent behavior of both diffusion coefficients normalized by the diffusion coefficient of the freely diffusing noninteracting vesicles at low volume fractions D 0. The short-time diffusion coefficient (squares) essentially corresponds to the free diffusion coefficient over the entire volume fraction range investigated. This indicates that hydrodynamic effects are not significant; the particle diffuses within the cage of nearest neighbors just like in a very dilute suspension. By contrast, the

Dself/D0

1

0.1 Dshort/D0 Dlong/D0 0.01

0.1 f

FIGURE 4.12 Volume fraction dependence of the short- (squares) and long-time (circles) self-diffusion coefficients. Both coefficients are normalized by the free diffusion coefficient.

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Structure and Functional Properties of Colloidal Systems

presence of the neighboring particles is clearly felt at longer times. The long-time diffusion coefficient (circles) is significantly lower than D 0 and decreases in the volume fraction range of 0.009–0.04 by about one order of magnitude as f increases. For systems approaching fluid–solid transitions, which are solely defined by the dynamical arrest of the constituents of the system, one in principle expect that the long-time diffusion coefficient exhibits a critical-like divergence. Instead, Dlong is slightly larger at f = 0.06 as compared to D long at f = 0.04. This unexpected increase in the mobility of the particles near the fluid–solid transition is correlated to a decrease in the peak height of S(q) (see Figure 4.2a) and prevents the characterization of the fluid–solid transition in terms of a critical transition at which we would expect the long-time relaxation to diverge. To gain a better understanding of the constraints set by the caging of particles, we convert the correlation functions into mean-squared displacement ·Dr 2(t)Ò according to gl(q, t) = exp{-1/2 q2·Dr 2(t)Ò/3}, with results shown in Figure 4.13a. Although we determine g1(q, t) in the scattering vector range below the peak, where we generally expect collective effects to become important, such conversion is justified for our particular choice of scattering vector, where we ensure that S(q) 1. Indeed, as mentioned above, we expect that g1(q, t) is mainly determined by self-diffusion at S(q) = 1 [31]. To ascertain that this is really the case, we verify for a sample where S(q) 1 is experimentally accessible at two different q-values that the conversion from g1(q, t) to ·Dr 2(t)Ò yields the same results for both q-values. In Figure 4.13b we show the conversion results obtained for f = 0.04 at q1, a scattering vector below the peak (qrm 5), and q2, a scattering vector above the peak (qrm 7.5); for both scattering vectors S(q) 1, as shown in the inset of Figure 4.13b. The meansquared displacements obtained for both q-values are almost identical, validating that the assumption for g1(q, t) is mainly determined by self-diffusion when S(q) = 1, regardless whether the probed q-value is below or above the particle–particle peak. As discussed above, the dynamics of the fluid-like samples is characterized by two time scales. At short times, the mean-squared displacement increases linearly with time reflecting the particle diffusion within the cage, where the frictional drag is entirely set by the solvent viscosity. At intermediate times, ·Dr 2(t)Ò starts to exhibit a deviation from the linear increase at some characteristic ·Dr 2(t)Ò-value, which is related to the spatial extent of the cage. At long times, ·Dr 2(t)Ò increases again nearly linearly reflecting the diffusion of particle past their nearest neighbors, which requires cooperative rearrangements of the particle configuration. For the two solid-like systems, this final increase in the mean-squared displacement is missing, the particle motion being clearly constrained to a maximum mean-squared displacement ·Dr 2(t)Òmax, which is a measure of the spatial extent of the nonrelaxing cage of nearest neighbors. (a)

(b) q1 q2

103

101

=1

103 3 2 S(q)

(nm2)

(nm2)

f

101

0 0

10–7

10–5

10–3 t(s)

10–1

10–7

10–5

q1

q2

1 4

10–3 t(s)

8 qrm

12

16

10–1

FIGURE 4.13 (a) Time dependence of the mean-squared displacement. For the sake of clarity, we omit to show the data obtained at f = 0.016, 0.02, and 0.06. From left to right f = 0.009, 0.027, 0.04, 0.09, and 0.12. (Reprinted from Haro-Pérez C., et al. 2009. Phys. Rev. Lett. 102: 081301-1–4. With permission.) (b) Comparison of the conversion of g1(q, t) obtained at q1 and q2 into ·Dr 2(t)Ò for f = 0.04, as denoted in the inset S(q) 1 for both q-values.

89

Effective Interactions of Charged Vesicles in Aqueous Suspensions (b) 0.3

(a)

0.2 rloc/rm

/yo

103

101

10–1 –9 10

10–7

10–5 t/t0

10–3

10–1

0.1

0.01

0.1 f

FIGURE 4.14 (a) Result of the scaling procedure described in the text. (b) Volume fraction dependence of the localization length in units of mean interparticle distances. Dashed horizontal line corresponds to the Lindemann criterion, r loc/rm = 0.1.

To characterize the confinement set by the cage of nearest neighbors for both fluid- and solid-like samples, we adopt a scaling technique. The time dependences of the mean-squared displacements obtained for the various volume fractions are shifted onto the one obtained for f = 0.12, so as to get the best overlap of the data in the short-time and transition regime, as shown in Figure 4.14a. The shift parameters used to scale the y-axis, y0, are then used to estimate the maximum mean-squared displacement in reference to the one obtained at f = 0.12, where ·Dr 2(t)Òmax is clearly defined at long ___________ times. We convert ·Dr 2(t)Òmax into “localization lengths” according to r loc = √·Dr 2(t)Ò max/3 [34] and normalize r loc with the mean interparticle distance rm in order to evaluate the arrest conditions in terms of the Lindemann criterion as applied to an amorphous phase [35]. The Lindemann criterion is an empirical rule used to define the melting point of solids; it demands that r loc/rm 0.1 at the liquid–solid transition [36]. Consistent with a transition between f = 0.06 and f = 0.09, r loc/rm > 0.15 for f £ 0.06 while r loc/rm < 0.1 for f ≥ 0.09, as shown in Figure 4.14b. Clearly, the macroscopically observed differences in the mechanical properties of the samples at f £ 0.06, fluid-like, and f ≥ 0.09, solid-like, are reflected in their dynamics. Interestingly, r loc/rm remains almost constant at r loc/ rm ~ 0.2 for all fluid-like samples investigated, while dropping significantly above the fluid–solid transition to r loc/rm = 0.09 and r loc/rm = 0.07 for f = 0.09 and f = 0.12, respectively. This behavior is strongly reminiscent of the abrupt change observed in U(rm) (see Figure 4.8b), suggesting that the observed fluid–solid transition is rather sharp and does not exhibit the typical critical-like behavior of other colloidal systems undergoing a glass transition, [32,33]. However, to unambiguously prove this would require a finer scan of the volume fraction dependence in the transitional range, f = 0.06– 0.09, than the one performed in the present investigation.

4.3.4

ENERGY CONTENT: LINK BETWEEN STATIC AND DYNAMIC PROPERTIES

To further explore the correlation between dynamical and structural hallmarks of the arrest conditions, both the localization length and the mean interparticle energy are converted into elastic moduli, G. For the localization length we use the microrheology approach Gd = k BT/[rm(3r loc2)] [37]; for the mean interparticle energy we use the Zwanzig and Mountain model Gs = nk BT + 2p/15n2 • Ú 0 g(r) d/dr(r4 dU(r)/dr)dr [38]. The latter model was originally developed to describe the high-frequency elastic modulus of molecular liquids composed of particles interacting by a Lennard–Jones potential and was later successfully applied to colloidal systems [39]. Both approaches yield elastic moduli that compare remarkably well, as shown in Figure 4.15, where we report Gd obtained from the dynamic data as open symbols and Gs obtained from the static data as solid symbols. For the samples with f £ 0.06, the elastic modulus varies insignificantly around G ~ 1.5 Pa. As these samples are fluid-like at long times, the elastic modulus reported here corresponds to the

90

Structure and Functional Properties of Colloidal Systems

G (Pa)

40

20

0 0.01

f

0.1

FIGURE 4.15 Comparison of the elastic modulus as estimated using the microrheology approach, Gd (open symbols), and the elastic modulus as calculated from the interparticle potential shown in Figure 4.7, Gd (solid symbols).

elasticity probed when subjecting the sample to a strain or stress at a frequency higher than the relaxation frequency. For the two solid-like samples, the elastic modulus is significantly higher and corresponds to the plateau modulus that characterizes the zero frequency elasticity of solid systems. Although the agreement between Gd and Gs is less convincing at the largest volume fraction, the basic feature of the transition is captured in both cases: G significantly increases above the fluid–solid transition. This result is remarkable as it demonstrates that the static data can be used to predict the dynamics and in particular the dynamical arrest of the system. Note that we choose to show G as a parameter to demonstrate the conversion of static data into dynamic data, but we could equally convert Gs into r loc, thereby directly extracting a measure of the so-called Debye–Waller factor [40] from information obtained from the static data; this despite the fact that the systems do not exhibit any particular structural change at the transition.

4.3.5

DISCUSSION

Clearly, the fluid–solid transition of our systems is not just due to an overcrowding effect, as this is the case for hard-sphere colloids at the glass transition. It appears to be due to the arrest of the repulsive barrier at high volume fraction, which leads to the sudden increase in the energy content and consequently to dynamic arrest. As we do not expect such peculiar development of the repulsive barrier for a purely repulsive system in the range of volume fractions investigated here, the cessation of the repulsive barrier must be directly linked to the low q-rise in S(q) indicative of effective attractions. Our results in fact indicate that the mechanism leading to the appearance of effective attractions is at the origin of both structural heterogeneities and dynamic arrest. The fact that the repulsive barrier does not evolve as we increase the volume fraction beyond a certain point seemingly indicates that at this point the addition of counterions does not lead to a decrease in the screening length. This evokes that the added counterions are located in space where there are no particles. This interpretation is somewhat reminiscent of the one implied by Sogami and Ise [41], who report the existence of an effective attraction between two macroions, because the counterions can be attracted simultaneously by neighboring particles. Such counterionmediated attraction is only possible if the space between two particles is somewhat depleted of counterions. Our experimentally determined potentials are in fact very similar to the one reported by Sogami and Ise, such that we could conceive that both the effective attraction and the cessation of the evolution of the repulsive barrier are caused by the fact that counterions migrate to the space with a low particle concentration. This interpretation is obviously very speculative and our data do not exclude the possibility that the appearance of effective attractions is due to many body effects

Effective Interactions of Charged Vesicles in Aqueous Suspensions

91

as evoked by other authors [42]. Indeed, we believe that our findings could serve as an excellent benchmark for theories accounting for the appearance of effective attractions in highly charged colloidal systems.

4.4

SUMMARY

We have investigated the structure and dynamics of aqueous dispersions of charged liposomes as a function of volume fraction. The structural investigations reveal that these systems exhibit structural heterogeneities that are characteristic of attractive systems, while exhibiting a volume fraction dependence of the mean particle spacing that is typical of strongly repulsive systems. Modeling the structure factor assuming an effective particle interaction potential composed of a long-range attraction and a shorter-range repulsion reveals a peculiar development of the effective particle interactions with volume fraction. In particular, we find that the position of the repulsive barrier ceases to decrease at large volume fractions, while the mean interparticle distance continues to decrease. This indicates that at high volume fraction the particles are squeezed together, so as to sense the strong repulsion between them. Consequently, the energy content is markedly increased, which leads to the dynamic arrest of the system observed in our dynamical investigation.

ACKNOWLEDGMENTS We thank the European Regional Development Fund (ERDF), the “Ministerio de Ciencia y Tecnología, Plan Nacional de Investigación, Desarrollo e Innovación Tecnológica (I + D + I)” project MAT2006-12918-CO5-01-02-05 (C.H.P., J.C.F., M.Q.P., R.H.A., and J.E.), the Swiss National Science Foundation (L.F.R.O. and V.T.), and Conacyt-México grants 51669 & 46373 (L.F.R.O. and R.C.P.) for financial support. We are indebted to Professor P. Schurtenberger for the use of his laboratory.

REFERENCES 1. Haro-Pérez C., Rojas-Ochoa L. F., Castañeda-Priego R., Quesada-Pérez M., Callejas-Fernández J., Hidalgo-Álvarez R., and Trappe V. 2009. Phys. Rev. Lett. 102: 081301-1–4. 2. Gregoriadis G. and Florence A. T. 1993. Drugs 45: 15–28. 3. Barenholz Y. 2001. Curr. Opin. Colloid Interface Sci. 6: 66–77. 4. Ohki S. and Ohshima H. 1999. Colloids Surf. B 14: 27–45. 5. Haro-Perez C., Quesada-Pérez M., Callejas-Fernández J., Sabaté R., Estelrich J., and Hidalgo-Álvarez R. 2005. Colloids Surf. A 270, 271: 352–356. 6. Tata B. V. R., Yamahara E., Rajamani P. V., and Ise N. 1997. Phys. Rev. Lett. 78: 2660–2663. 7. Yamanaka J., Yoshida H., Koga T., Ise N., and Hashimoto T. 1998. Phys. Rev. Lett. 80: 5806–5809. 8. Zoetekouw B. and van Roij R. 2006. Phys. Rev. Lett. 97: 258–302. 9. Tata B. V. R. and Jena S. S. 2006. Solid State Commun. 139: 562–580. 10. Deamer D. and Bangham A. D. 1975. Biochem. Biophys. Acta 443: 629–634. 11. Chong C. S. and Colbow K. 1976. Biochim. Biophys. Acta 436: 260–282. 12. Eisenberg M., Gresalfi T., Ricio T., and McLaughlin S. 1979. Biochemistry 18: 5213–5223. 13. Gisler T., Schulz S. F., Borkovec M., Sticher H., Schurtenberger P., D’Aguanno B., and Klein R. 1994. J. Chem. Phys. 101: 9924–9936. 14. Stuchly M. A., Stuchly S. S., Liburdy R. P., and Rousseau D. A. 1988. Phys. Med. Biol. 33: 1309–1324. 15. Huang C. and Mason J. F. 1978. Proc. Natl. Acad. Sci. USA 75: 308–310. 16. Lasic D. D. 1993. Liposomes: From Physics to Applications. Amsterdam: Elsevier. 17. van den Hul H. J. and Vanderhoff J. W. 1968. J. Colloid Interface Sci. 28: 336–337. 18. Alexander M., Rojas-Ochoa L. F., Leser M., and Schurtenberger P. 2002. J. Colloid Interface Sci. 24: 35–46. 19. Schaetzel K. 1991. J. Mod. Opt. 38: 1849–1865. 20. Urban C. and Schurtenberger P. 1998. J. Colloid Interface Sci. 207: 150–158.

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21. Pusey P. N. 1991. Colloidal Dispersions in les Houches Session L1: Liquids, Freezing and Glass Transition, J. P. Hansen, D. Levesque, and J. Zinn-Justin (Eds), Chapter 10. Amsterdam: Elsevier. 22. Joosten J. G. H., Geladé E. T. F., and Pusey P. N. 1990. Phys. Rev. A 42: 2161–2175. 23. Nägele G. 1996. Phys. Rep. 272: 215–372. 24. Tata B. V. R. and Arora A. K. 1991. J. Phys.: Condens. Matter 3: 7983–7993. 25. D’Aguanno B. and Klein R. 1992. Phys. Rev. A 46: 7652–7656. 26. Lyubartsev A. P. and Laaksonen A. 1995. Phys. Rev. E 52: 3730–3737. 27. Tóth G. 2001. J. Chem. Phys. 115: 4770–4775. 28. Madden W. G. and Rice S. A. 1980. J. Chem. Phys. 72: 4208–4215. 29. Katsov K. and Weeks J. 2000. J. Stat. Phys. 100: 107–134. 30. Zerah G. and Hansen J. P. 1980. J. Chem. Phys. 84: 2336–2343. 31. Pusey P. N. 1978. J. Phys. A: Math. Gen. 11: 119–135. 32. van Megen W. and Pusey P. N. 1991. Phys. Rev. A 43: 5429–5441. 33. van Megen W., Underwood S. M., and Pusey P. N. 1991. Phys. Rev. Lett. 67: 1586–1589. 34. Wilke S. D. and Bosse J. 1999. Phys. Rev. E 59: 1968–1975. 35. Xia X. and Wolynes P. G. 2000. Proc. Natl. Acad. Sci. USA 97: 2990–2994. 36. Lindemann F. A. 1910. Z. Phys. 11: 609–612. 37. Mason T. G., Gang H., and Weitz D. A. 1997. J. Opt. Soc. Am. A 14: 139–149. 38. Zwanzig R. and Mountain R. D. 1965. J. Chem. Phys. 43: 4464–4471. 39. Wagner N. J. 1993. J. Colloid Interface Sci. 161: 169–181. 40. Gotze W. and Sjogren L. 1992. Rep. Prog. Phys. 55: 241–376. 41. Sogami I. and Ise N. 1984. J. Chem. Phys. 81: 6320–6332. 42. Dobnikar J., Chen W., Rzehak R., and Von Grünberg H. H. 2003. J. Chem. Phys. 119: 4971–4985.

5

Structure and Colloidal Properties of Extremely Bimodal Suspensions A.V. Delgado, M.L. Jiménez, J.L. Viota, R. Rica, M.T. López-López, and S. Ahualli

CONTENTS 5.1 5.2

Introduction ...................................................................................................................... Permittivity of a Bidisperse Suspension ........................................................................... 5.2.1 Generalities ........................................................................................................... 5.2.2 Dielectric Relaxation of Bimodal Suspensions .................................................... 5.2.3 Experimental Data on the Permittivity of Bimodal Systems ............................... 5.3 Dynamic Mobility of Bimodal Suspensions ..................................................................... 5.3.1 General Behavior and Effect of Particle Size ....................................................... 5.3.2 Case of Bidisperse Systems .................................................................................. 5.3.3 Experimental Data on the Dynamic Mobility ...................................................... 5.4 Case of Magnetic Particles ............................................................................................... 5.4.1 Introduction .......................................................................................................... 5.4.2 Magnetic Properties of Bimodal Suspensions ...................................................... 5.4.3 Colloidal Stability of Concentrated Bimodal MFs ............................................... 5.5 Conclusions ....................................................................................................................... Acknowledgments ...................................................................................................................... References ..................................................................................................................................

5.1

93 95 95 97 98 102 102 103 105 107 107 109 110 112 113 113

INTRODUCTION

A number of phenomena have been recently described that take place in concentrated colloidal dispersions and are a consequence of the interactions (both hydrodynamic and electrostatic) between the particles in suspension. Some remarkable examples are phase separation in bidisperse systems [1–3] and field-induced structures [4–12]. Less intuitive are phenomena in which structures induced by applied electric fields give rise to negative electric birefringence in suspensions of elongated particles: these appear to get oriented with their long axis preferably perpendicular (not parallel as in ordinary electric birefringence) to the electric field [13–17]. Complex as they may be, these phenomena can be related to the behavior of suspensions in which the solid concentration is high (this often means volume fractions f in the order of, say, 10% and above). There are other phenomena that appear even in dilute suspensions and are subtle examples of how particles of populations widely differing in size can produce significant interactions that manifest in changes

93

94

Structure and Functional Properties of Colloidal Systems

of such macroscopic properties as electrophoretic mobility (both dc and ac), dielectric relaxation, colloidal stability, and so on. This is the situation on which the present chapter is focused. We will consider how electrokinetic and related phenomena in moderately dilute systems composed by two populations of particles greatly different in size can be largely affected by the composition, and in fact the results can be very different from what one would expect by simple consideration of linear mixture formulae. Let us mention, by way of example, the intriguing results recently reported in reference [18]. In that work, Mantegazza et al. found experimentally that the above-mentioned parallel orientation of elongated particles under the action of an electric field is strongly affected by the addition of small amounts of nanoparticles of very diverse characteristics. This is demonstrated by electric birefringence spectroscopy of the bimodal suspensions: The results indicate that the specific arrangement of the small particles (SPs) induced by the field around the larger ones (LPs), together with the liquid flows of electroosmotic origin at the double layers of the LPs, provoke a tendency of the elongated ones toward a perpendicular orientation. This is far from expected, as the larger dipole moment in the direction of the long axis should bring about a torque tending to parallel orientation. Because of the nonlinear (with respect to the applied field) nature of the electro-orientation effects, with the added difficulty of the nonspherical shape of the LPs, a quantitative explanation of the phenomenon is still lacking. However, these results have stimulated the investigation of electrokinetic phenomena in bidisperse systems for cases where the quantitative evaluation and comparison with models are easier because of the sphericity and monodispersity of both the LPs and the SPs. One such phenomenon is the low-frequency dielectric dispersion (LFDD) of suspensions. This is the denomination given to the frequency dependence of the permittivity of dispersed systems for applied electric field frequencies close to characteristic frequencies of relaxation (typically in the kHz to MHz range) in the electrical double layer. A significant effect has been reported of the properties of the medium, of the particle, and of their interface on the relaxation pattern of the permittivity, in particular: number of relaxations observed, natural frequencies, and amplitudes of those relaxations. This explains the increased interest in the determination of the permittivity of colloidal systems during the last decade or so [19–26]. In this contribution, we will show results that clearly demonstrate that the presence of the SPs affects the amplitude and characteristic frequency of the LFDD far more than could be explained by simple considerations of accumulation of effects. Similarly to LFDD, there is a set of electrokinetic techniques that involves ac fields and that can be applied to suspensions of arbitrary particle concentration, as they do not rely on optical techniques of evaluation. These are the so-called electroacoustic techniques, which enable the determination of the dynamic or ac mobility, ue, of colloidal particles (the ac counterpart of the dc or classical electrophoretic mobility) as a function of frequency. There are basically two such techniques. One is based on the determination of the electric potential difference induced by the passage of a sound wave through the system: it is called colloid vibration potential (CVP) or colloid vibration current (CVI), depending on the quantity measured. In the second technique, reciprocal of CVP or CVI, the basic process is the generation of a pressure wave when an ac electric field is applied to the suspension: the amplitude of the sound wave, AESA, is known as electrokinetic sonic amplitude, and so we speak of the ESA effect. After the very early works in the subject, O’Brien [27,28] was the first author to perform a rigorous investigation on the physical foundations of electroacoustic techniques, and he found that ue is in fact proportional to AESA [28]: rm f ue μ AESA ____ , Dr

(5.1)

where f is the volume fraction of solids and Dr = rp – rm is the density contrast, rp (rm) being the density of the particles (the dispersion medium). Turning again to the problem of bidisperse suspensions, we will show below that the presence of even minute amounts of tiny particles can greatly affect the dynamic mobility of large latex particles. Comparison between experimental data and theoretical approaches to the calculation of ue in the

95

Structure and Colloidal Properties of Extremely Bimodal Suspensions

case of these extremely bimodal suspensions will demonstrate that such an important influence is a consequence of the perturbation of the ionic clouds due to the presence of the SPs around the LPs, showing the relationship between structure at the nanoscale and macroscopic functionalities of these systems. In this chapter we will equally consider a more applied, so to say, set of systems: in these, the particles (both SP and LP) are ferro- or ferri-magnetic. This means that the application of an external magnetic field will induce complex structures that substantially modify many properties (noticeably, rheological and optical) of the systems. We will not focus on these effects in the present chapter (see, however, references [8,29–31]), but we will rather consider the effect of the presence of the SPs in two particular aspects, namely, the magnetic and stability properties of the bidisperse suspensions [32,33]. We note that, although the remanence of iron and magnetite is small (they are typical soft magnetic materials), the small remnant magnetization of the LPs will give rise to the existence of a magnetic moment in each particle that will hence interact, via van der Waals and magnetic forces, with the small ones. As a result, the latter arrange around the LPs producing microstructures that manifest themselves in such important aspects as the magnetic response of the suspension and its colloidal stability. These special situations will also be briefly reported in order to show the specific properties of the bimodal magnetic fluids (MFs).

5.2 PERMITTIVITY OF A BIDISPERSE SUSPENSION 5.2.1

GENERALITIES

It may be recalled here that this technique is best suited for the evaluation of the electric double-layer dynamics. Depending on the system parameters, a plot of the real part of the relative permittivity (e* = e¢ – ie≤) of a suspension, in the frequency range mentioned, will display two welldefined relaxations, as schematically shown in Figure 5.1. The first and most important one (typically

100

95 1 e≤(n)

e¢(n)

90

85 0.1 80

0.01

75 101

102

103

104

105

106

107

108

109

n (Hz)

FIGURE 5.1 A plot of the theoretical relative permittivity e* of a suspension of spherical particles 100 nm in radius in a 1 mM KCl solution at 25°C, as a function of the frequency n of the applied field. The zeta potential of the particles is -100 mV. The left and right axes correspond, respectively, to the real (e¢ = Re{e*}) and imaginary (e≤ = Im{e*}) components of the relative permittivity. Volume fraction of solids f = 1%. The vertical arrows indicate the approximate positions of the a- (left) and MWO- (right) relaxations.

96

Structure and Functional Properties of Colloidal Systems

occurring at the kHz frequency range) is the so-called a-relaxation, and it takes place with characteristic frequency na or time t a: na @

D 2 p(a + k -1 )2

(5.2)

(a + k -1 )2 , ta @ D

where D is the effective diffusion coefficient of the ions in solution, a is the particle radius, and k -1 is the double-layer thickness (Debye length). In the case of a solution containing just two kinds of ions (cations and anions), D=

2D+ D. D+ + D-

(5.3)

The a-relaxation is a consequence of the concentration polarization effect. Assume, as shown in Figure 5.2, that an electric field E0, directed from left to right is acting on a negatively charged (a)

E0 j–D

j–em

j+D

j+em

+ + + + +–+ – – – + + + + – + +jem j D+ – – – – ++ + +– – ++ – – – – –+ –++ – – ++++ + – jD – + + – + jem + – – – + ++ + + ++ + + c(r)

– jem

jD –

j+em

j+D

c•

c• c(r) +

– d

(b)

E0 – jem

j+em

+ + +– + –– + +– – – – –+ – +– + – ++ + +

+ – + –+ + – – + j+ – + + em + ++ – + –– – ++++ – + + – + jem – – – + + ++ +

j–em j+em

+

– d

FIGURE 5.2 Schematics of the fluxes of counterions (+) and coions (-) around a negatively charged colloidal particle in the presence of an electric field (E0). In (a) the concentration polarization (electrolyte concentration, ± c(r), increased on the right and depleted on the left) provokes diffusion fluxes ( j D) that superimpose to the electromigration fluxes ( j ±em). In panel (b) the frequency is above the a-relaxation, and only electromigration can take place. (Reprinted from Jiménez, M.L. et al. 2007. J. Colloid Interface Sci. 309: 296–302. With permission from Elsevier.)

Structure and Colloidal Properties of Extremely Bimodal Suspensions

97

particle. Counterions (cations) will accumulate on the right-hand side of the particle and so will coions, brought by the field from the bulk solution; the overall result is that both counterions and coions get accumulated on the right and depleted on the left of the particle. If the field frequency is above na (Equation 5.2), there is not enough time between successive field oscillations for the electrolyte gradient to be established, and the concentration polarization is greatly reduced. This brings about a decrease in the diffusion currents ( j D± in Figure 5.2a); such a decline is macroscopically equivalent to a reduction in permittivity. Such a reduction is in fact the a-relaxation. For n > na the mismatch between particle and medium conductivities still may allow for counterion and coion accumulation on opposite sides of the particle, meaning another source of double-layer polarization [34–36]. Following the same reasoning as above, it is easy to understand that again a characteristic frequency will be reached where this mechanism will also be frozen. This is the Maxwell–Wagner– O’Konski (MWO) relaxation frequency, dependent on the electric permittivities ep e0, eme0, and the conductivities Kp, Km, of the particle and the dispersion medium, respectively: nMWO =

1 (1 - f)K p + (2 + f)K m , 2 p (1 - f)e p e 0 + (2 + f)e m e 0

(5.4)

where e0 is the vacuum permittivity. Note that even if the particle is intrinsically insulating, the presence of its double layer will manifest in an effective, nonzero conductivity: 2Ks Kp = ____ a

(5.5)

with Ks being the surface or double-layer conductivity, produced by the excess of counterions in the electric double layer [37]. The value of nWMO is typically around 10 MHz, that is, some 2–3 orders of magnitude above na (the MWO relaxation is barely observed in Figure 5.2). These processes influence to a large extent the strength and direction of the dipole moment induced by the field. At high frequencies, above nMWO, the dipole will be a consequence of the permittivity mismatch, and will point opposite to the field if ep < em; for na < n < nMWO, the dipole can change direction if the double-layer conductivity is high enough (as depicted in Figure 5.2b); finally, the presence of the electrolyte gradient for n < n a; reduces this positive dipole because of the diffusion associated to the concentration gradient. All these changes manifest in the permittivity of the suspension, as this can be expressed as a function of the real and imaginary components of the induced dipole coefficient (C* = C1 – iC2), in turn relating the dipole moment d to the applied field [19–23,38,39]. For a spherical particle of radius a: d = 4 pe 0 e m a 3 (C1 - iC2 ) E0 Km È ˘ e ¢( n) = e m + 3fe m ÍC1 ( n) C ( n) = e m + de( n) = e m + fDe( n), 2pne m e 0 2 ˙˚ Î

(5.6)

where de(n) is the so-called dielectric increment and De(n) is its specific (per unit volume fraction) counterpart. The variety of mechanisms mentioned for dipole moment modification explains the sensitivity of permittivity determinations to the double-layer characteristics and, more important for the purposes of this contribution, to the existence of mesoscopic structures in the bulk suspension. In particular, we will consider in the next section the effect of the composition of our bidisperse system on the expected permittivity–frequency pattern.

5.2.2

DIELECTRIC RELAXATION OF BIMODAL SUSPENSIONS

Following arguments similar to those reported by Carrique et al. [40], it is possible to estimate the permittivity of a suspension containing two kinds of spherical particles with radii aS and aL and

98

Structure and Functional Properties of Colloidal Systems

respective volume fractions fS and fL. Assuming that the contributions of each particle population are additive, Equation 5.6 can be modified to e¢(n) = em + de(n) = em + fSDeS(n) + fLDeL(n),

(5.7)

with DeS, DeL being the specific dielectric increments corresponding to SPs and LPs, respectively. Note that, according to Equation 5.7, and considering that in reference [40] it was shown that the permittivity increment scales roughly with the third power of the particle radius, there is an appreciable weight of the LPs in the overall dielectric response, with a modest effect of the SPs. Figure 5.3 illustrates this fact.

5.2.3

EXPERIMENTAL DATA ON THE PERMITTIVITY OF BIMODAL SYSTEMS

As LPs for these experiments, two negatively charged polystyrene latexes were used. They will be denoted as L-530 and L-336 (see reference [41] for a more detailed description). The former was purchased from Interfacial Dynamics Corporation, USA; according to the manufacturer, the surface charge density of these particles comes from sulfonate groups. L-336 was a gift of Professor E. Enciso, Universidad Complutense de Madrid, Spain, and its surface charge is generated by dissociation of carboxylic groups. Their respective particle radii are 265 and 168 nm, as deduced from electron microscope pictures. The SPs were negatively charged silica nanoparticles (Aerosil A300, 7 nm in radius, manufactured by Degussa-Hüls, Germany). All the suspensions were prepared in 0.5 mM KCl, at a pH of 4.5. In such conditions, the electrophoretic mobilities of the particles were measured in a Malvern Zetasizer 2000 (Malvern Instruments, England). The values obtained (in units of 10-8 m2/s/V) were -5.7 (L-530), -4.3 (L-336), and -2.5 (A300). Since all particles are negatively charged, electrostatic attraction between SPs and LPs can be ruled out in all cases. The electric permittivity of the suspensions was determined, for the frequency interval 1 kHz to 1 MHz, by measuring the complex impedance of a conductivity cell with variable electrode separation

fs = 0%

20

fs = 0.5%

15

de (n)

fs = 1% 10

fs = 1.5% fs = 2%

5

0 –5 101

102

103

104 n (Hz)

105

106

107

FIGURE 5.3 Dielectric increment as a function of frequency for suspensions containing the indicated volume fractions fS of SPs and a constant volume fraction of LPs fL = 1.9% (cf Equation 5.7). The zeta potentials of the two kinds of particles are -50 mV (SP) and -80 mV (LP), and the ionic strength is 1 mM KCl. Radii: small, aS = 7 nm; large, aL = 168 nm.

Structure and Colloidal Properties of Extremely Bimodal Suspensions

99

filled with the suspension, using an HP-4284A impedance meter [42]. The logarithmic derivative method [43] was used to reduce the perturbing effect of electrode polarization at low frequencies (below 10 kHz, say). Without going into details, let us mention here that the procedure is based on the evaluation of a quantity, the logarithmic derivative e≤D(n), obtained from the real part of the permittivity e¢(n) as follows: e ¢¢D (n) = -

p de ¢(n) . 2 d ln n

(5.8)

The advantage of using this quantity has been thoroughly discussed in reference [43]; suffice it to say here that the undesired contribution of electrode polarization to the apparent permittivity can be greatly reduced in many cases by using e≤(n) instead of e≤D(n) or e¢(n). Even more, the tiny MWO relaxation, often unobservable in comparison with the a process, can be found using the proposed method. Nevertheless, since the quantity with the real physical meaning is e¢(n) [or e≤(n); they are Kramers–Krönig related and bear the same physical information], it is convenient to consider how to obtain it from the logarithmic derivative. The procedure that was used in the presented data involved first integrating e≤D(n) in the whole frequency spectrum; from this, e¢(n) was obtained except for an integration constant, and hence after performing the integration we obtain in fact e¢(n) – e¢(•), where of course infinite means ( nMWO); in consequence, according to Equation 5.6, such difference is in fact the dielectric increment de(n). These data are fitted to a relaxation function like Havriliak–Negami’s [44]: Ï ¸ de(0) Ô Ô de(n) = fDe( n) = Re Ì , b ˝ a ÔÓ ÈÎ1 + (2 pint) ˘˚ Ô˛

(5.9)

where the parameters de(0) and t correspond, respectively, to the dielectric increment at zero frequency (the amplitude of the relaxation, roughly speaking) and the characteristic relaxation time. The constants a and b refer to the shape of the curve. Using this procedure, it was possible to demonstrate that no relaxation is observed if a simple solution is measured, as expected from the absence of interfacial polarization mechanisms; in addition, it was found that the silica SP do not show relaxations either, since their small size would lead to na values beyond the maximum frequency attainable in these experiments. Finally, both L-530 and L-336 display well-defined a-relaxations in the 10 kHz frequency region. Figure 5.4 illustrates these results, showing for comparison the LFDD of L-336, A300, and KCl solution. The point now is to consider how the permittivity of the LP suspension changes when different concentrations of SP are added to the latex suspensions. Figure 5.5 shows the results in the case L-336 and L-530 for different A300 concentrations and the same ionic strength of 0.5 mM KCl. Note that in all cases the addition of the SP brings about a reduction of the dielectric increment and an increase in the a-relaxation frequency. After fitting the data to the Havriliak–Negami function (Equation 5.9), it is possible to evaluate more quantitatively those effects, by estimating the dependence of both de(0) and t on the concentration of A300. The results are plotted in Figure 5.6. It is immediately clear that for a given ionic strength (similar results were reported in reference [41] for other ionic contents, except for the most dilute solutions), the addition of SP decreases both the dielectric increment and the characteristic time. This is not a result of the simple linear combination of properties obtained from the use of Equation 5.7 for the permittivity increment or from the evaluation of the characteristic time for the a-relaxation using Equation 5.2. The comparison of Figures 5.6 (experimental) and 5.7 (calculated data) confirms that, as previously described (see also references [45,46]), the overall permittivity response of the suspensions must be the result of the contributions of individual particles and their ionic atmospheres. In the systems studied above,

100

Structure and Functional Properties of Colloidal Systems 250 KCl solution 0.43% A300 1.9% L-336

200

e˝D(n)

150 100 50 0 –50 10000 n (Hz)

1000

100000

FIGURE 5.4 Logarithmic derivative of the relative permittivity of a solution 1 mM KCl (䊏), a suspension of A300 silica nanoparticles with volume fraction 0.43% (䊊), and a suspension of L-336 1.9% (䉱). Ionic strength: 1 mM KCl.

considering that the SPs do not present any relaxations in the frequency range investigated, their contribution to the permittivity of the bimodal suspensions must be frequency independent and very small. In addition, because of their relative small size, their concentration polarization clouds will form much faster than those of the LP. This means that whatever the frequency, the nonequilibrium (a)

60

L336-A300

120 90 de (w)

e˝D (w)

40

20

0

30

1.9–0 1.9–0.23 1.9–0.45 1.9–0.90 104

5

0 6

400

L530–A300

2–0.23 80

2–0.45

105

106

w (rad/s)

2–0.011 2–0.23 2–0.45

200 100 0

0–0.90 104

de (w)

e˝D (w)

2–0.011

105 106 w (rad/s) 2–0

300

2–0 160

1.9–0 1.9–0.23 1.9–0.45 1.9–0.90 104

10 10 w (rad/s)

(b) 240

0

60

0–0.90 104

105 106 w (rad/s)

107

FIGURE 5.5 Left: logarithmic derivative of the relative permittivity of 1.9% L-336 (a) and 2% L-530 (b) suspensions as a function of the angular frequency (w = 2pn) of the field, in 0.5 mM KCl and the SP concentrations indicated (key: % volume fraction of LP-% volume fraction of SP). Right: dielectric increment for the same suspensions. The lines are the best fits to the Havriliak–Negami relaxation function.

101

de(0)/de(0)(A300 = 0), ta/ta(A300 = 0)

Structure and Colloidal Properties of Extremely Bimodal Suspensions

1.0 0.9 0.8 0.7 0.6 0.5 0.01

0.1 A300 (% v/v)

1

FIGURE 5.6 Dielectric increment at zero frequency, de(0), and characteristic a-relaxation time, t a , as a function of the concentration of A300 SP for L-336 (1.9%) suspensions in 0.5 mM KCl. Symbols: (䊐): de(0); (䊏): ta. The values are relative to those measured in the absence of Aerosil.

ionic clouds around the SPs have reached their steady configuration long before those around the LPs. This behavior reminds the particle concentration effects in LFDD: in reference [24] it was shown that when f is large, the neutral electrolyte gradients of neighbor particles overlap, leading to a decrease in the concentration polarization of each particle. However, in the present case, the particle concentrations are not high enough to suggest any electric double-layer overlap as a justification for the results. All these arguments point to an explanation based on the arrangement of SPs around each LP. A proposal was given in reference [41], as schematically shown in Figure 5.8: each SP in the proximity of an LP will bear its own concentration polarization gradient. This may alter that of the LP by increasing the ionic concentration on the left and decreasing it on the

de(0)/de(0)(A300 = 0), ta/ ta(A300 = 0)

1.025 1.020 1.015 1.010 1.005 1.000 0.995 0.0

0.5

1.0 1.5 A300 (% v/v)

2.0

FIGURE 5.7 Calculated values of the dielectric increment at zero frequency, de(0), and the characteristic time, t a , as a function of the concentration of A300 SP for suspensions of L-336 (1.9%) in 0.5 mM KCl. Symbols: (䊐): de(0); (䊏): ta. For the calculation, use was made of Equation 5.7. The values are relative to those calculated in the absence of Aerosil.

102

Structure and Functional Properties of Colloidal Systems (a) E0

(b) CLP(r)

(i) CLP(r)

CLP+SP(r) (ii) CLP+SP(r)

FIGURE 5.8 (a) Schematic representation of a LP surrounded by SPs. The darkest regions of the ionic clouds represent the highest electrolyte concentration. (b) The distribution of neutral electrolyte concentration for the LP alone (i) and the LP surrounded by SPs (ii). (Reprinted from Jiménez, M.L. et al. 2007. J. Colloid Interface Sci. 309: 296–302. With permission from Elsevier.)

right of the particle: as a result, the gradient is reduced, and this lowers the amplitude of the a-relaxation. Furthermore, the average distance to the surface of the cloud maximum and minimum is reduced, and thus the characteristic time that counterions need to establish the polarization is shorter. Both aspects are in agreement with the experimental observations, and confirm the sought relationship between microstructure and macroscopic properties.

5.3 DYNAMIC MOBILITY OF BIMODAL SUSPENSIONS 5.3.1

GENERAL BEHAVIOR AND EFFECT OF PARTICLE SIZE

Charged colloidal particles in aqueous suspensions oscillate under the action of an alternating electric field with a velocity that is proportional to the field. Because the different mechanisms responsible for the steady velocity need finite times to be established, such velocity will have a phase difference with respect to the field, and hence it will be a complex quantity. The forces acting on each particle are of electric and viscous origin, and come, basically, from the action of the external field on the particle charge, the viscous friction with the fluid (also in motion because of the effect of the field on the charged double layer), and, eventually, the local field generated by the polarized double layer. To this we must add the hydrodynamic and electrical interactions between particles, if their concentration is large enough. An approximate formula (valid for ka 1) accounting for these effects is, for dilute systems [47–49]: ue( n) =

2 e 0 e mz È1 - (C1 ( n) - iC2 ( n) )˘˚ G( n), 3 h Î

(5.10)

Structure and Colloidal Properties of Extremely Bimodal Suspensions

103

where h is the viscosity of the medium and G is the complex function that accounts for the particle and fluid inertia in the presence of the applied field. According to O’Brien [27,28], this function depends on frequency as follows: G=

1+ l . 1 + l + (l / 9)(3 + 2(3 + 2 Dr / rm )) 2

(5.11)

In this expression, the frequency and size dependences are included in l, which is equal to the product ka, where k is the complex wave vector of the fluid velocity wave generated by the oscillations of the particles in the liquid medium [50]:

l = (1 + i)

pna 2rm . h

(5.12)

Hence, to a first approximation, the dynamic electrophoretic mobility will be determined by the zeta potential, and both the dipole coefficient and inertia factor at each frequency. It is precisely the dependence on the dipole coefficient that brings about a clear connection between permittivity and dynamic mobility determinations: the information provided by each of them has some common points and is complementary in some other aspects. The dynamic mobility will exhibit one or more relaxation phenomena, in addition to the high-frequency decline, always present because of inertia. This is illustrated in Figure 5.9, for different particle radii. Clear increases (at low frequencies) and decreases (high frequencies) are observed in Re(ue), with characteristic frequencies particularly well defined in the imaginary component, and strongly size-dependent. The low-frequency raise is associated to the MWO relaxation of the induced dipole. Note that, unlike permittivity, the mobility of the suspension is almost insensitive to the a-relaxation, and only the MWO process is observed in ue. In addition, for high enough frequencies the mobility decreases due to the inertia of the particle; such a decrease will be clearly separated from the MWO for the smallest particles (Figure 5.9), but for colloids a few hundred nanometers in size (dotted lines in the figure), the Maxwell–Wagner relaxation will be hidden by the inertial decay. If the particle concentration is sufficiently high (as mentioned, above a few percent), the theoretical model above described must be modified in order to take into consideration the hydrodynamic interactions between particles, and, if the double layers overlap, also their electrical interactions. The problem has been faced by using the so-called cell models [47,51] based on the hypothesis that such interactions can be modeled by assuming that a given particle is surrounded by a shell of electrolyte (the cell), and the interactions mentioned can be evaluated by proper choice of the boundary conditions on the shell surface.

5.3.2

CASE OF BIDISPERSE SYSTEMS

If we consider a system consisting of a distribution of volume fractions f(a) of particles with radius a, a superposition approach will allow us to assume that the ESA signal is the result of the individual sonic waves generated by every kind of particles, so that the total signal will be given by the following modification of Equation 5.1: •

Ú

AESA μ ue (a )f(a ) 0

Dr(a ) da, rm

(5.13)

104

Structure and Functional Properties of Colloidal Systems (a)

300 nm

100 nm

4

–Re[ue]

3 5 nm 2 Maxwell-Wagner relaxation

1

Inertial decrease

0 10–1

101

103 n (kHz)

105

107

(b) 2 Inertial peak

Im[ue]

300 nm 1

100 nm

5 nm

Maxwell-Wagner relaxation

0

10–1

101

103 n (kHz)

105

107

FIGURE 5.9 Real (a) and imaginary (b) parts of the dynamic mobility of suspensions of spherical particles in 0.5 mM KCl, with f = 1%, z = -100 mV and the particle radii indicated. (Reprinted from Jiménez, M.L. et al. 2007. Croat. Chem. Acta 80: 453–459. With permission from the Croatian Chemical Society.)

where ue(a) is the dynamic mobility of particles with radius a and Dr(a) is their corresponding density contrast. The instrument software converts this macroscopic quantity into an equivalent or average dynamic mobility, ·ueÒ; this requires that the user provides effective values for the volume fraction and the density contrast, f eff and Dreff, such that AESA μ ·ue Ò

Dreff f . rm eff

(5.14)

If we restrict ourselves to the case of a bidisperse suspension (with SPs and LPs), then Equation 5.13 can be written as AESA μ

uS fS DrS + uL f L Dr L . rm

(5.15)

Now considering that the density reff of the mixed suspension will be given by reff = rm + feff Dreff = rm + fSDrS + fLDrL, we can use the following value for the product of the effective quantities: feff Dreff = fSDrS + fLDrL.

(5.16)

105

Structure and Colloidal Properties of Extremely Bimodal Suspensions

Finally, combining Equations 5.14 through 5.16 leads to the connection between equivalent and individual dynamic mobilities: ·ue Ò =

uSfS DrS + uL f L DrL uSfS DrS + uL f L Dr L = . Dreff feff DrSfS + Dr Lf L

(5.17)

We will later return to the application of this equation to the experimental evaluation of the dynamic mobility of bimodal suspensions.

5.3.3

EXPERIMENTAL DATA ON THE DYNAMIC MOBILITY

Experiments have been reported in reference [52] on suspensions containing L-336 latex spheres as LP together with Aerosil A300 as SP. The dynamic mobility was determinated from the electrokinetic sonic amplitude (ESA) technique, using an AcoustoSizer II (Colloidal Dynamics, USA). In this device, the particles vibrate under the action of an oscillating electric field (in the frequency range 1–18 MHz) generating a sonic wave with amplitude AESA. In the case of our bidisperse systems, we used Equation 5.16 for the estimation of the effective or average volume fraction and density contrast product. The accuracy of the theoretical model in describing the dynamic mobility can be made clear from the plots in Figure 5.10. Note that in the case of A300 no Maxwell–Wagner effect is observed, since the particles are very small, and, in addition, the characteristic frequency of the inertial decrease is beyond the frequency range available. Because of this (and also because of the high probability of aggregation between such SPs) the cell model does not give an accurate description of the data as in the case of the LPs. The zeta potentials that best fitted the results were -150 mV (L-336) and -16 mV (Aerosil). The results corresponding to mixtures are plotted in Figure 5.11. As observed, the addition of even small amounts of SPs provokes a sharp mobility decrease, suggesting that u e approaches the values corresponding to pure Aerosil suspensions. The MWO relaxation reported in Figure 5.10a

lm[ue] (10–8m2V–1s–1)

–Re[ue] (10–8m2V–1s–1)

(a)

(b) 0.34

10

0.32

8

0.30

6

0.28

4

0.26

2

0.24

0 3

0.22 –2

10

100

102

104

100

101

102

103

100

101

102

103

0.2

2 0.1

1 0

0.0

–1 –0.1

–2 10–2

100

102

104 n (MHz)

FIGURE 5.10 Real and imaginary components of the dynamic mobility of (a) L-336 latex with 4.77% volume fraction, and (b) Aerosil A300 with 5.22% volume concentration, in 0.5 mM KCl as dispersion medium. The lines correspond to theoretical predictions according to the model in reference [51].

106

Structure and Functional Properties of Colloidal Systems (a)

–Re[ue] (10–8 m2 V–1 s–1)

8

0

6

0.036 0.091

4

0.223 0.453

2

0.91 0.8 1

2

1m[ue] (10–8 m2 V–1 s–1)

(b)

4 6 n (MHz)

8

10

20

3

0

2

0.036 0.091 0.223

1

0.453 0.91

0

–1 0.8 1

2

4

6

8

10

20

n (MHz)

FIGURE 5.11 Real (a) and imaginary (b) components of the dynamic mobility of suspensions containing 4.77% L-336 and the concentrations (% volume fraction) of Aerosil A300 indicated. Dispersion medium: 0.5 mM KCl. (Reprinted from Jiménez, M.L. et al. 2007. Croat. Chem. Acta 80: 453–459. With permission from the Croatian Chemical Society.)

for L-336 disappears, and the inertial decline is greatly reduced. In order to understand these effects it is first of all necessary to discard the possibility that the explanation is based simply on the predominance of one type of particles over the other. With that aim, one can compare the mobility determinations in mixed suspensions containing (x% LP + y% SP) with systems containing y% SP (y between 0.452 and 1.79). The results are shown in Figure 5.12 and demonstrate that the LPs in the mixtures are almost completely masked by the silica SPs, in spite of the lower volume fraction and zeta potential of the latter. In addition, the MWO relaxation disappears when SPs are added. As an alternative, it can be useful to consider the mixture formula deduced above, Equation 5.15. A comparison can be carried out between the frequency dependence of the dynamic mobility in the mixture 4.77% L-336 and 0.452% A300 and the calculations based on Equation 5.17, using as input data the mobilities independently determined of the latex and the SPs in the same conditions (volume fractions and ionic strength, 0.5 mM KCl, as in the mixture). Both series of data are plotted in Figure 5.13. It is immediately clear that, although the average mobility is qualitatively similar to that experimentally obtained, the latter is significantly larger and the weight of the larger particles on the average mobility is greater than that of silica. We can conclude, in agreement with

–Re[ue] (10–8 m2 V–1 s–1)

Structure and Colloidal Properties of Extremely Bimodal Suspensions

107

2.0

1.5

1.0

1

2

4

6

8 10

20

1

2

4 6 n (MHz)

8 10

20

1m[ue] (10–8 m2 V–1 s–1)

0.8

0.4

0.0

FIGURE 5.12 Frequency dependence of the real and imaginary components of the dynamic mobility of the mixtures of L-336 4.77% and different concentrations of A300 (full symbols) and of the suspensions of A300 alone at the same concentrations (open symbols) always in 0.5 mM KCl. Squares: 0.452% A300; circles: 0.910% A300; triangles: 1.79% A300. (Reprinted from Jiménez, M.L. et al. 2007. Croat. Chem. Acta 80: 453–459. With permission from the Croatian Chemical Society.)

the results reported above on the permittivity of similar mixtures, that the bimodal system does not behave as the superposition of the two components; it appears that the overall behavior is controlled by the LPs, and that the role of the SPs is to perturb the fluid and ionic flows around the larger ones. This conclusion reinforces our LFDD results.

5.4 CASE OF MAGNETIC PARTICLES 5.4.1

INTRODUCTION

A special situation that we will also consider is that of magnetizable particles: Because of either remnant or field-induced magnetizations (or both), these particles can bear magnetic moments that bring about additional particle–particle interactions not achievable in the systems described so far in this contribution. These are particularly important when one deals with the use of the particles in suspension in aqueous or nonaqueous liquid carriers, forming magnetic fluids (MFs). If the particles are in the nanometer size range, one speaks of a ferrofluid (FF), whereas if the size is rather around one micrometer, the fluid is called a magnetorheological fluid (MRF) [8,53–56]. These MFs have received most attention in recent years because of their actual and potential applications as active vibration dampers or torque transducers, for instance.

108

Structure and Functional Properties of Colloidal Systems

–Re[ue] (10–8 m2 V–1 s–1)

2.2 2.0 1.8 1.6 1.4

1m[ue] (10–8 m2 V–1 s–1)

1.2 0.8 1

2

0.8 1

2

4

6

8 10

20

4 6 n (MHz)

8 10

20

0.7

0.0

FIGURE 5.13 Frequency dependence of the real and imaginary components of the dynamic mobility of the mixture of L-336 4.77% and 0.452% A300 in 0.5 mM KCl (䊏), together with the predictions of Equation 5.18 (䊐).

Such applications are based on the substantial, reversible, and rapid change in the rheological behavior of these systems when a uniform magnetic field is applied to them: the interactions between magnetized particles produce columnar or fibrillar aggregates in the field direction (the magnetorheological effect). These structures are an outstanding example of the connection between microstructure and macroscopic properties, as they provoke the transformation of the initially Newtonian suspension into a viscoelastic fluid. This change is the basis for the applications mentioned. A more generalized use of these fluids is limited by the essential issue of their stability against settling, considering the large mass density of the magnetic particles (typically iron or ferrites). For this reason, a number of methods have been devised to reduce the settling rate and/or to increase the porosity of the sediments in order to make the MRF less prone to settling and, eventually, more easily redispersible. A group of methods are based on the use of mixed MRFs whereby the micron-sized particles (responsible for the magnetorheological response) are suspended in a carrier liquid containing magnetic nanoparticles, mostly magnetite [30,32,33,57–60]. The carrier is hence a FF, and it has been proposed that the nanoparticles in the FF stabilize the micron-sized particles through the formation of a halo (as described in reference [61] for nonmagnetic particles) around the latter, a consequence of magnetic interactions between both types of particles. These occur because the magnetic LP will have some (although small) remnant magnetization, while the nanoparticles are single domain and will thus have a permanent magnetic moment. We will briefly describe the magnetic properties and sedimentation behavior of mixed suspensions of micron- and nanometer-sized magnetite, although a wider investigation can be found in references [32,33].

109

Structure and Colloidal Properties of Extremely Bimodal Suspensions

5.4.2

MAGNETIC PROPERTIES OF BIMODAL SUSPENSIONS

In order to gain information on the microstructure of these mixed systems, it seems first of all interesting to pay attention to their magnetization properties (Figure 5.14) in relation to those of individual SPs and LPs [32]. Their average particle diameters were 8 ± 3 nm and 1450 ± 190 nm, respectively. Figure 5.14 clearly shows that the magnetic properties of the two kinds of particles are quite similar, as both show similar saturation magnetizations (about 450 kA/m, close in fact to the bibliographical data for bulk magnetite, 510 kA/m) and very narrow hysteresis cycles. They differ, however, in two aspects: one is that nanoparticles have a larger initial susceptibility, and the other is that the coercive field is also larger. These aspects can be related to the fact that the size distribution of the nanomagnetite includes a large fraction of particles in the 3–7 nm size range, these contributing a superparamagnetic behavior to the sample. The larger size fraction, although still in the singledomain region, is responsible for the increased coercivity, as for such particles, contrary to the multidomain case, the coercivity increases rapidly with size [62]. The behavior of the bimodal suspensions is illustrated in Figure 5.15, for suspensions containing a constant volume fraction of LPs, fL = 10%, and different concentrations of SPs, fS = 0.5–7%, always in water. It is worth pointing out that all the suspensions investigated show very small remanence and coercivity, similar to the dry powder. This suggests that any field-induced structure disappears upon reducing or eliminating the field. The important aspect of this figure is, however, that the addition of the SPs introduces again significant changes in the behavior of the suspension of LPs alone. Note that the saturation magnetization increases with the fraction of SPs, at least up to 5% volume fraction, and also that the initial susceptibility is higher, the larger the fS. The latter fact can be explained, as before, by considering that we are adding increasing amounts of superparamagnetic particles. But, taking into account the results at high fields, an additional mechanism must be brought into consideration: it has been suggested that this is the heteroaggregation, above mentioned, between the magnetic LPs and SPs. Indeed, classical mean-field models consider that the particle–particle interactions can be effectively taken into account if it is assumed that each particle is subjected to the external field plus an additional field created by the magnetization of the suspension. If these ideas are applied to our bimodal systems, we could imagine that the halo of nanoparticles

400

300 200 100 0

200

–100

M (kA/m)

–200 –300 –100

0

100

0

Micro↓ Micro↑ Nano↓ Nano↑

–200

–400 –1500

–1000

–500

0

500

1000

1500

H (kA/m)

FIGURE 5.14 Magnetization curves of micrometer- (䊏, 䊐) and nanometer- (∑, ) sized magnetite particles. Inset: detail of the low-field region. (Reprinted from Viota, J.L. et al. 2007. J. Magn. Magn. Mater. 314: 80–86. With permission from Elsevier.)

110

Structure and Functional Properties of Colloidal Systems

400

300

5, 7

200

300

100

200

–100

3

0

1

M (kA/m)

–200

100

–300 –50

0.5 –25

0

25

50

0 –100 –200 –300 –400 –1000

–500

0

500

1000

H (kA/m)

FIGURE 5.15 Magnetization M of suspensions containing fL = 10% volume fraction of large magnetite particles, and the indicated volume fraction of nanoparticles. The inset is a magnification of the low-field region. In this, the increasing and decreasing branches of the magnetization can be distinguished for the cases fS = 0.5% and 7%. (Reprinted from Viota, J.L. et al. 2007. J. Magn. Magn. Mater. 314: 80–86. With permission from Elsevier.)

keeps the LPs beyond the range of electric double-layer repulsions leaving only magnetic attractions between them; they will form oriented structures with the SPs as bridges between LPs. This chain formation can contribute to the increased initial susceptibility experimentally found. The mechanism described is also in agreement with the increased saturation magnetization when SPs are added: the additional field provoked by the presence of neighbor particles will facilitate the domain orientation of the LPs in the direction of the external field, increasing the sample magnetization. The phenomenon appears to be less effective for sufficiently high concentrations of SPs: these would correspond to a full coverage by the nanoparticles, with no further increase in the heteroaggregation.

5.4.3

COLLOIDAL STABILITY OF CONCENTRATED BIMODAL MFS

As mentioned, a key condition for the application of MRFs is the avoidance of compact sediments and the reduction of the velocity of settling. This task is not easy considering the density (5.2 × 103 kg/m3 for magnetite; 7.5 × 103 kg/m3 for iron) of the particles and the need of high volume fractions (in excess of 20%) for achieving sufficient magnetorheological response. As mentioned, it has been suggested that the problem may be partially solved by using thyxotropic agents dissolved in the carrier fluid to reduce settling, or polymers adsorbed on the particles to avoid their aggregation [63–66]. In this contribution we will report on the investigations performed on the effect of the addition of magnetic nanoparticles to the base fluid, in aqueous suspensions containing the same LPs and SPs previously characterized from the magnetic point of view. Although different methods have been described for the evaluation of the settling behavior of concentrated magnetic suspensions, we will focus on a procedure described in references [33,67,68]. It is based on the determination of the inductance L of a sensing coil surrounding the sample, and it has proved very versatile and sensitive, as long as the particles are magnetizable. The evaluation of L is carried out by measuring the resonance frequency of a parallel LC circuit, where C is chosen in ___ such a way that the resonance frequency (2p/√ LC ) is within the range of the instrument used to determine the frequency (a Keythley 2700 multimeter, USA, in reference [68]). In our experiments the frequency is typically around 450 kHz.

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Structure and Colloidal Properties of Extremely Bimodal Suspensions

Neglecting edge effects, use can be made of the relationship between L and the relative magnetic permeability of the suspension at the measurement point, m r (L = m r m0 N 2A/, if the coil has N turns and total length ; m0 is the permeability of vacuum) for obtaining m r. In order to determine the relation between this quantity and the particle concentration, it suffices to use the Maxwell–Garnett equation (see references [34,68]), which, for the case mrp m rm (the permeability of the particles much larger than that of the medium) reads as

m r = m rm

1 + 2f , 1- f

(5.18)

where f being the volume fraction of magnetic particles. A possible method to record the changes in the particle concentration with time (indirectly, this means to follow the settling rate of the suspensions) is to evaluate the relative resonant frequency [33,68]:

f r (t ) =

f (t = 0) Ê 1 + 2f(t ) ˆ μÁ f (t ) Ë 1 - f(t ) ˜¯

1/2

(5.19)

.

Some results are shown in Figure 5.16 [33]. Two mechanisms can be proposed to explain them, based again on the formation of structures at the nanoscale manifesting in the settling behavior. One of these mechanisms has already been suggested by the magnetization hysteresis discussed above: clouds of nanoparticles are magnetically attracted to the large magnetite particles, and such clouds will limit the maximum distance of approach between the LPs, avoiding the formation of compact, rapidly settling structures. In addition, as suggested by Klokkenburg et al. [69], the magnetic SPs can form (in the absence of any magnetic field) chain-like structures through magnetic interactions between their permanent dipoles. Such structures may not only join the SPs increasing the volume of the aggregates (and reducing their average density), but also will have a thickening (viscosity increasing) effect on the dispersion medium, additionally reducing the settling rate.

fS: 7%

1.000

fS: 3%

fr

0.998

0.996

fS: 2% fS: 5%

0.994

0.992

fS: 1% 0

20

40

60 Time (s)

80

100

120

FIGURE 5.16 Time dependence of the relative resonance frequency fr (Equation 5.19) of suspensions containing fL = 10% large magnetite particles, and the indicated volume fractions fS of magnetite nanoparticles. The base fluid is water at pH 4. (Reprinted from Viota, J.L. et al. 2007. J. Colloid Interface Sci. 309: 135–139. With permission from Elsevier.)

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Structure and Functional Properties of Colloidal Systems

FIGURE 5.17 Photographs of the suspensions corresponding to Figure 5.16 after 24 h sedimentation. The arrows indicate the sediment height. The volume fractions are fL = 10% (in all samples) and fS = 0%, 1%, 2%, 3%, 5%, 7% (from left to right). (Reprinted from Viota, J.L. et al. 2007. J. Colloid Interface Sci. 309: 135–39. With permission from Elsevier.)

Both mechanisms appear to have different characteristic times: at short times (below about 30 s) the relative frequency (hence the volume fraction) falls faster the lower fS, that is, the suspensions settle more rapidly in such conditions. This fact points to a predominance of the mechanism regarding viscosity changes in the medium: lower concentrations of SP should lead to lower viscosities and hence faster settling. It is likely that at longer times the SP clouds have acquired their final arrangement, and the process of heteroaggregation must be noticeable in stability evaluation; it must be noted that the cloud formation will provoke the SP dilution of the carrier liquid and a subsequent viscosity decrease. In this new situation, the aggregates grow bigger and will settle rapidly in a less viscous fluid. This change of tendency is observed in Figure 5.16 for fS = 1%, 2%, and 3%; for fS = 7% the SP concentration is high enough for maintaining the MF essentially stable during the whole experiment duration. It appears that at fS = 5%, the two mechanisms partially balance each other and the larger weight of the aggregates is the dominant factor of the sedimentation. That the LPs fall under gravity carrying with them the SPs (the nanoparticle halo) is well evident in the photographs shown in Figure 5.17: The nanoparticle clouds yield large, open aggregates, occupying large sediment volumes. The effect is more significant the larger the volume fraction of SPs, thus confirming the long-time importance of the heteroaggregate structure.

5.5

CONCLUSIONS

In this chapter we have revised some existing information on the colloidal behavior of suspensions consisting of two particle populations widely different in size (two or more orders of magnitude difference), the so-called extremely bimodal or bidisperse systems. The focus has been on how the nanoscale structure manifests in certain aspects of the macroscopic behavior of the suspensions. We have considered four kinds of experiments that lead to coherent observations; they included electric permittivity determinations, dynamic electrophoretic mobility, and for the case of magnetizable particles, magnetic hysteresis, and colloidal stability evaluations. The whole set of data indicate that the mixture does not behave as expected if a simple superposition of SP and LP contributions to the phenomenon is investigated. Thus, the permittivity and the mobility are higher (for otherwise identical conditions) than expected from the combination of SP and LP properties, and tend to be dominated by LPs, although with a significant effect of the SPs on the nonequilibrium electric double-layer structure of the large particles. A very noticeable effect has also been reported of the SP on the magnetic properties and settling rate of the mixed suspensions: the results are consistent again with a special arrangement of the nanoparticles around the magnetite micron-sized particles, in this case mediated by magnetic interactions between the remnant magnetic moment of the LPs and the permanent one in the single-domain SPs.

Structure and Colloidal Properties of Extremely Bimodal Suspensions

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ACKNOWLEDGMENTS Financial support for this work by MEC (Spain) (projects FIS2005-06860-C02-01, 02, and MAT2005-07746-C02-01) and Junta de Andalucía (Spain) (project 2005-FQM410) are gratefully acknowledged.

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26. Delgado, A.V. (Ed.) 2002. Interfacial Electrokinetics and Electrophoresis. New York: Marcel Dekker. 27. O’Brien, R.W. 1988. Electro-acoustic effects in a dilute suspension of spherical particles. J. Fluid Mech. 190: 71–86. 28. O’Brien, R.W. 1990. The electroacoustic equations for a colloidal suspension. J. Fluid Mech. 212: 81–93. 29. López-López, M.T., Kuzhir, P., Lacis, S., Bossis, G., González-Caballero, F., and Durán, J.D.G. 2006. Magnetorheology for suspensions of solid particles dispersed in ferrofluids. J. Phys.: Condens. Matter 18: S2803–S2813. 30. López-López, M.T., de Vicente, J., Bossis, G., González-Caballero, F., and Durán, J.D.G. 2005. Preparation of stable magnetorheological fluids based on extremely bimodal iron–magnetite suspensions. J. Mater. Res. 20: 874–881. 31. Viota, J.L., Durán, J.D.G., and Delgado, A.V. 2009. Study of the magnetorheology of aqueous suspensions of extremely bimodal magnetite particles. Eur. Phys. J. E. in press. 32. Viota, J.L., Durán, J.D.G., González-Caballero, F., and Delgado, A.V. 2007. Magnetic properties of extremely bimodal magnetite suspensions. J. Magn. Magn. Mater. 314: 80–86. 33. Viota, J.L., González-Caballero, F., Durán, J.D.G., and Delgado, A.V. 2007. Study of the colloidal stability of concentrated bimodal magnetic fluids. J. Colloid Interface Sci. 309: 135–139. 34. Maxwell, J.C. 1954. Electricity and Magnetism, Vol. 1. New York: Dover. 35. Wagner, K.W. 1914. Explanation of the dielectric behavior on the basis of the Maxwell theory. Arch. Elektrotech. 2: 371–389. 36. O’Konski, C.T. 1960. Electric properties of macromolecules. V. Theory of ionic polarization in polyelectrolytes. J. Phys. Chem. 64: 605–619. 37. Lyklema, J. 1995. Fundamentals of Interface and Colloid Science, Vol. II, pp. 4.31–4.37. London: Academic Press. 38. DeLacey, E.H.B. and White, L.R. 1981. Dielectric response and conductivity of dilute suspensions of colloidal particles. J. Chem. Soc. Faraday Trans. II 77: 2007–2039. 39. Mangelsdorf, C.S. and White, L.R. 1998. The dynamic double layer. Part 1. Theory of a mobile Stern layer. J. Chem. Soc. Faraday Trans. II 94: 2441–2452. 40. Carrique, F., Arroyo, F.J., and Delgado, A.V. 1998. Effect of size polydispersity on the dielectric relaxation of colloidal suspensions: A numerical study in the frequency and time domains. J. Colloid Interface Sci. 206: 569–576. 41. Jiménez, M.L., Arroyo, F.J., Delgado, A.V., Mantegazza, F., Bellini, T., and Rica, R. 2007. Electrokinetics in extremely bimodal suspensions. J. Colloid Interface Sci. 309: 296–302. 42. Tirado, M.C., Arroyo, F.J., Delgado, A.V., and Grosse, C. 2000. Measurement of the low-frequency dielectric properties of colloidal suspensions: Comparison between different methods. J. Colloid Interface Sci. 227: 141–146. 43. Jiménez, M.L., Arroyo, F.J., van Turnhout, J., and Delgado, A.V. 2002. Analysis of the dielectric permittivity of suspensions by means of the logarithmic derivative of its real part. J. Colloid Interface Sci. 249: 327–335. 44. Havriliak, S. and Negami, S. 1967. A complex plane representation of dielectric and mechanical relaxation processes in some polymers. Polymer 8: 161–210. 45. Arroyo, F.J. and Delgado, A.V. 2002. Electrokinetic phenomena and their experimental determination. In: Interfacial Electrokinetics and Electrophoresis (Delgado, A.V., Ed.), Chapter 1. New York: Marcel Dekker. 46. Grosse, C. and Pedrosa, S. 2002. Relaxation mechanisms of homogeneous particles and cells suspended in aqueous electrolyte solutions. In: Interfacial Electrokinetics and Electrophoresis (Delgado, A.V., Ed.), Chapter 11. New York: Marcel Dekker. 47. Arroyo, F.J., Carrique, F., Ahualli, S., and Delgado, A.V. 2004. Dynamic mobility of concentrated suspensions. Comparison between different calculations. Phys. Chem. Chem. Phys. 6: 1446–1452. 48. Dukhin, A.S., Ohshima, H., Shilov, V.N., and Goetz, P.J. 1999. Electroacoustics for concentrated dispersions. Langmuir 15: 3435–3445. 49. Dukhin, A.S., Shilov, V.N., Ohshima, H., and Goetz, P.J. 1999. Electroacoustic phenomena in concentrated dispersions: New theory and CVI experiment. Langmuir 15: 6692–6706. 50. Landau, L.D. and Lifshitz, E.M. 1966. Fluid Mechanics, Chapter II. Oxford: Pergamon. 51. Ahualli, S., Delgado, A.V., Miklavcic S.J., and White, L.R. 2006. Dynamic electrophoretic mobility of concentrated dispersions of spherical colloidal particles. On the consistent use of the cell model. Langmuir 22: 7041–7051. 52. Jiménez, M.L., Arroyo, F.J., Ahualli, S., Rica, R., and Delgado, A.V. 2007. Electroacoustic characterization of bidisperse suspensions. Croat. Chem. Acta 80: 453–459.

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53. Rosensweig, R.E. 1995. Ferrohydrodynamics. Cambridge: Cambridge University Press. 54. Ginder, J.M. 1998. Behavior of the magnetorheological fluids. MRS Bull. 23: 26–29. 55. Charles, S.W. 2002. The preparation of magnetic fluids. In: Ferrofluids (Odenbach, S., Ed.), p. 3. Berlin: Springer. 56. Bossis, G., Volkova, O., Lacis, S., and Meunier, A. 2002. Magnetorheology: Fluids, structures and rheology. In: Ferrofluids (Odenbach, S., Ed.), p. 202. Berlin: Springer. 57. Poddar, P., Wilson, J.L., Srikanth, H., Yoo, J.H., Wereley, N.M., Kotha, S., Barghouty, L., and Radhakrishnan, R. 2004. Nanocomposite magneto-rheological fluids with uniformly dispersed Fe nanoparticles. J. Nanosci. Nanotechnol. 4: 1926. 58. Ginder, J.M., Larry, D.E., and Davis, L.C. 1996. Magnetic fluid-based magnetorheological fluids. US Patent No. 5,549,837. 59. Chin, B.D., Park, J.H., Kwon, M.H., and Park, O.O. 2001. Rheological properties and dispersion stability of magnetorheological (MR) suspensions. Rheol. Acta 40: 211–219. 60. Park, B.J., Jang, I.B., Choi, H.J., Pich, A., Bhattacharya, S., and Adler, H.J. 2006. Magnetorheological characteristics of nanoparticle-added carbonyl iron system. J. Magn. Magn. Mater. 303: 290–293. 61. Tohver, V., Smay, J.E., Braem, A., Braun, P.V., and Lewis, J.E. 2001. Nanoparticle halos: A new colloidal stabilization mechanism. Proc. Nat. Acad. Sci. 98: 8950–8954. 62. Dunlop, D.J. 1990. Developments in rock magnetism. Rep. Prog. Phys. 53: 707–792. 63. Pugh, R.J. 1994. Dispersion and stability of ceramic powders in liquids. In: Surface and Colloid Chemistry in Advanced Ceramics Processing (Pugh, R.J., Bergström, L., Eds), Chapter 4. New York: Marcel Dekker. 64. Bossis, G., Lemaire, E., Volkova, O., and Clercx, H. 1997. Yield stress in magnetorheological and electrorheological fluids: A comparison between microscopic and macroscopic structural models. J. Rheol. 41: 687–704. 65. Winslow, W.M. 1949. Induced fibration of suspensions. J. Appl. Phys. 20: 1137–1140. 66. Viota, J.L., de Vicente, J., Durán, J.D.G., and Delgado, A.V. 2005. Stabilization of magnetorheological suspensions by polyacrylic acid polymers. J. Colloid Interface Sci. 284: 527–541. 67. Durán, J.D.G., González-Caballero, F., Delgado, A.V., and Iglesias, G. 2005. Fluido magnetoreológico. Spanish Patent P200502282. 68. Iglesias, G.R., Ruiz-Morón, L.F., Insa, J., Durán, J.D.G., and Delgado, A.V. 2007. An experimental method for the measurement of the stability of concentrated magnetic fluids. J. Colloid Interface Sci. 311: 475–480. 69. Klokkenburg, M., Dullens, R., Kegel, W.K., Erné, B.H., and Philipse, A.P. 2006. Quantitative real-space analysis of self-assembled structures of magnetic dipolar colloids. Phys. Rev. Lett. 96: 037203.

6

Structure and Stability of Filaments Made up of Microsized Magnetic Particles F. Martínez-Pedrero, A. El-Harrak, María Tirado-Miranda, J. Baudry, Artur Schmitt, J. Bibette, and José Callejas-Fernández

CONTENTS 6.1 6.2 6.3

Introduction ...................................................................................................................... Microsized Magnetic Particles ......................................................................................... Preparation of Magnetic Filaments .................................................................................. 6.3.1 Magnetic Properties of the Filaments ................................................................... 6.3.2 Diffusive Filament Motion ................................................................................... 6.4 Structure of Magnetic Filaments ...................................................................................... 6.5 Stability of Magnetic Filaments ....................................................................................... 6.6 Aggregates of Filaments ................................................................................................... 6.7 Conclusions ....................................................................................................................... Acknowledgments ...................................................................................................................... References ..................................................................................................................................

6.1

117 118 120 121 122 125 128 131 132 132 132

INTRODUCTION

When an external magnetic field is applied to an initially stable suspension of magnetic particles, a dipolar interaction arises that leads to the formation of anisotropic aggregates aligned along the field direction. The final morphology of the structures mainly depends on the particle volume fraction of the colloidal suspension:1 1. At high particle concentrations, long chain-like aggregates of dipolar particles form aligned along the field direction. Due to dipolar interactions perpendicular to the filament axes, the magnetic filaments aggregate laterally among themselves, forming columns or networks. The aligned structures give rise to changes in the optical and rheological properties of the colloidal suspensions.2,3 Since the linear aggregates have a preferential orientation, they restrict the movement of the fluid perpendicular to the direction of the magnetic field, increasing its viscosity up to the point of becoming a viscoelastic solid. Hence, reversible changes in the medium viscosity are quickly achieved. On the other hand, colloidal suspensions of aligned magnetic particles present optical anisotropies. In particular, birefringence due to different phase velocities and dichroism effects due to different attenuation for each polarization component of the scattered waves are described in terms of anisotropies in the real and imaginary parts of the refractive index, respectively.4,5 117

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Structure and Functional Properties of Colloidal Systems

2. At low particle concentrations, particles aggregate in regular one-particle-thick chain-like filaments. If the bonds established between the chain-forming particles are stable, magnetic filaments able to survive in the absence of the applied magnetic field are obtained.6 Such linear aggregates may be employed as a system for determining the force–distance profile between tiny colloidal particles,7 as microscopic stirrers when they are subjected to an external rotating magnetic fields,8 or even as artificial swimmers.9 Owing to their response to external fields, magnetic filaments show a growing number of applications in different fields such as rheology, microfluidics, light-transmission devices, and so on. In this chapter, we describe how to determine the structure and stability of the field-induced filaments by means of light-scattering and videomicroscopy techniques and focus our attention on the role played by the electrolyte concentration with regard to their stability. An improved understanding of the structural properties of magnetic filaments will undoubtedly help scientists to improve industrial processes and devices that are based on such systems.

6.2

MICROSIZED MAGNETIC PARTICLES

Microsized magnetic filaments are often made up of composite colloidal particles, such as silica or polystyrene microparticles, emulsion droplets, or liposomes, that contain small magnetic grains of roughly 10 nm size. Some examples can be seen in Figure 6.1. Since magnetic materials usually oxidize quite easily, most of the magnetic grains employed consist of iron oxides. The grains increase the relative density of the particles with regard to the dispersion medium. This favors particle sedimentation10 and thus becomes a serious problem for many manufacturers of technological applications. The superparamagnetic character of the grains, however, is not affected by embedding them in composite particles. Consequently, dipolar magnetic interactions between the particles only appear in the presence of an applied magnetic field. In aqueous

(a)

(b)

0.5 mm

0.5 mm (c)

(d)

50 mm

10 mm

FIGURE 6.1 (a) TEM micrographs of magnetic polystyrene particles (R0039, Merck). The small magnetic grains randomly distributed within polystyrene spheres appear as dark spots in the images. (b) Linear aggregates that have been formed under the presence of a constant magnetic field. In this particular case, the linear aggregates formed are stable enough to withstand the absence of the magnetic field. (c) and (d) Scanning electron microscopy images of magnetic silica particles, synthesized by Dr. El-Harrak. In this case, particles assemble in chains orientated along the field direction. The latter set of images was taken at different magnifications.

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119

suspensions, the surface of the particles may carry dissociated groups or charged polymers that are chemically attached to it. Therefore, a charge distribution referred to as electric double layer arises in the vicinity of the particle surface. The thickness of the electric double layer and hence the range of the corresponding electrostatic interactions depend on the electrolyte concentration of the aqueous phase.11 When a magnetic field is applied, the magnetic particles experience a dipolar interaction among them. This interaction is tunable through the strength of an external magnetic field. Analogous behaviors are observed in a suspension of dielectric spheres suspended in a dielectric medium, that is, the so-called electrorheological (ER) fluids. In fact, most of the concepts and fundamental ideas developed for magnetic colloids can be directly applied to ER suspensions.12,13 There are, however, some biasing factors, such as the surface charge or electrode polarization, which have limited the number of practical applications up to now. Unlike electric properties, magnetic interactions are generally not affected by parameters such as pH, surface charge, or ionic concentrations. A brief discussion of the limitations of ER fluids can be found in an interesting paper by Promislow et al.14 Using magnetic colloids, some authors have prepared stabilized pickering emulsions that undergo phase separation when an external magnetic field is applied.15 Aqueous dispersions of paramagnetic colloidal particles, deposited on magnetic bubble domains of a uniaxial ferrimagnetic garnet film, serve as microscopic stirrers when they are subjected to rotating external magnetic fields.16 Microsized magnetic colloidal particles are also used in medicine for diagnostic and therapeutic applications such as17,18 • Hyperthermia: The magnetic particles are heated selectively by a high-frequency magnetic field. Hyperthermia is proposed as a cancer treatment for killing or weakening of tumor cells. It is expected to have little or negligible side effects on healthy cells. • Magnetic carriers for drug vectorization: The particles are loaded with the active pharmaceutical substance and directed by means of a magnetic field gradient toward the desired location in the body. • Contrast agents for magnetic resonance imaging (MRI): The design of tissue-specific contrast agents is now a well-established field of biomedical research. Further applications, such as isolation and purification of biomolecules, separation of biochemical products, or cell labeling and sorting, have been performed using magnetic microparticles. The stability of colloidal suspensions of magnetic particles is determined by the particle–particle interactions. In aqueous media, the net interaction is the sum of a repulsive electrostatic term, an attractive London–van der Waals force, and an additional magnetic dipole–dipole interaction term that appears as soon as the external field is applied.19 Since one-particle-thick chain-like aggregates form in diluted conditions, hereafter we will neglect the effects of hydrodynamic interactions on the formation of linear aggregates. The electrostatic repulsion between the particles arises when their electrical double layers overlap. The repulsion between charged colloidal particles suspended in aqueous media can be controlled by changing the electrolyte concentration in the medium. Additional electrolyte ions in the medium compress the electric double layers around the particles and so vary mainly the range of the electrostatic interaction. The destabilizing London–van der Waals forces arise from molecular interactions between the particles. They can lead to irreversible aggregation since they are usually strong enough to keep the particles together. Nevertheless, these attractive forces are of a relatively short range and thus are frequently masked by interactions of longer range. In order to estimate the magnetic interactions, the magnetized spheres are usually modeled as point dipoles of a well-defined magnetic moment m that is oriented along the field direction, and are 14 placed at the particle centers. The magnetic dipole–dipole interaction is anisotropic and depends on the relative orientation of the external field and the position vector that connects the particle centers. A schematic plot of the total potential energy versus the distance between the particle surfaces is shown in Figure 6.2.

120

Structure and Functional Properties of Colloidal Systems (a) Energy/kBT

(b) Energy/kBT 10

10 0

5

–10

0

10

8 Y(

6

6

nm

4

4 )

nm

10

–5

ns(t)

–10

)

X(nm)

X(

2

2

8

nf (t)

H

0

(c) Energy/kBT

np(t)

10

20

30

40

50

(d) Energy/kBT

20 15

10 0 –10 –10

10

H –10

–5 Y(

5

nm

0

Y(nm) 10

20

30

40

50

–5 0

)

0 5

5

nm

)

X(

10 10

FIGURE 6.2 Typical net potential energy profiles corresponding to two EDLMP in the presence of an external magnetic field. Figures (a) and (d) show the anisotropic potential energy profile caused by a magnetic particle located at the origin. The magnetic field is oriented along the X axis. The interaction is repulsive in directions that are perpendicular to the field and attractive along the field direction. (b) Potential energy obtained for two EDLMP approaching along the field direction. The total potential energy curve (dark curve) is the sum of the van der Waals (light gray), electrostatic (upper gray curve), and magnetic interactions (lower gray curve). (c) Potential energy obtained for two EDLMP approaching perpendicularly to the field direction. Here, the potential energy was calculated for two different electrolyte concentrations, (upper curve) 10 and (lower curve) 50 mM. The X and Y axes represent the distance between the particle surfaces.

The interaction is attractive if the particles approach along the field direction, and repulsive if the particles approach perpendicularly to the field direction. Electric double-layered magnetic particles (EDLMP) approaching along the field direction, in general, give rise to a potential energy curve with a deep primary minimum at short distances and a shallower secondary minimum separated by an energy barrier (Figure 6.2b).20,21 The primary energy minimum is due to the attractive short-range van der Waals interactions, whereas the secondary minimum is mainly a result of the interplay between the electrostatic repulsions and longrange dipolar interactions. The height of the energy barrier is mostly determined by the electrostatic repulsion between the particles (Figure 6.2c).

6.3

PREPARATION OF MAGNETIC FILAMENTS

When a uniaxial magnetic field is applied to a colloidal suspension of magnetic particles, a magnetic moment is induced in each bead and an anisotropic dipolar interaction arises. In diluted suspensions, the particles self-organize due to the action of the field, forming one-particle-thick chains aligned along the field direction. Depending on the interplay between the repulsive and the attractive interactions, primary or secondary minimum aggregation or stable particle suspensions may arise (see Figure 6.2b). Aggregation takes place usually in a secondary energy minimum and thus chain rupture is observed as soon as the external magnetic field is removed, that is, the magnetic interaction vanishes and the electrostatic repulsion, which controls the stability of electric doublelayered particles suspended in aqueous media, pushes the particles away from each other. Hence, the chains break into their constituent particles and the system returns to its nonaggregated state (Figure 6.3). On the other hand, particles aggregated in a primary energy minimum form permanent filaments. Such aggregates do not break and preserve their linear geometry even when the magnetic

Structure and Stability of Filaments Made up of Microsized Magnetic Particles

121

FIGURE 6.3 (a) Videomicroscopy images of silica particles aggregated at 0 mM due to an applied magnetic field. (b) The aggregation process is reversible. As soon as the magnetic field is turned off, the electrostatic repulsion controls the stability of the system and pushes the particles away from each other, giving rise to a complete break up of the linear aggregates.

field is turned off. Adhesion of neighboring chain-forming particles may be induced by van der Waals attraction,22,23 or by adsorbed molecules that form a strong “bridge” between the particles. There is a variety of linking molecules that have been used to create permanently linked chains, including absorbing polymers,24 poly(ethylene glycol),25 polyvinylpyrrolidone,26 or caseinate.27 The linear aggregates fabricated in this way have an almost infinite lifetime, and the stable bonds are strong enough to withstand not only the absence of the magnetic field but also the drying process that is necessary for taking transmission electron microscopy (TEM) images, as can be observed in Figure 6.1b. The final length of the linear aggregates can be controlled by tuning the exposure time to the magnetic field and the relative strength of the different interparticle interactions,28 or even more accurately by forming the chains in microchannels of a given height.6 Magnetic colloids patterned on a surface can serve as templates for chain growth starting from the fixed particles, anchored on surfaces to prevent migration in flow processes.29,30 Using alternative experimental protocols, more rigid magnetic chains were obtained by other authors. If the synthesis of the magnetic nanoparticles is dominated by aggregation of primary units, an applied magnetic field has been shown to have a dramatic effect on the morphology. In this case, extremely rigid rod-like particles may be formed.31 One-dimensional structures have also been obtained for cobalt nanoparticles that undergo a superparamagnetic to ferromagnetic transition as their size increases during the synthesis process.32

6.3.1

MAGNETIC PROPERTIES OF THE FILAMENTS

The magnetic character of microsized magnetic particles is usually due to the presence of small magnetic grains of roughly 10 nm size. Since the size of the magnetic grains is often smaller than a magnetic domain, each one is considered as a magnetic monodomain. Hence, each grain has a magnetic moment that depends on the grain’s size and the magnetic material. Moving the dipoles away from an easy axis of magnetization costs a given amount of energy known as energy of anisotropy. This energy, however, decreases when the grain size decreases. If the magnetic grain size reaches a minimum value then the energy of anisotropy becomes of the same order of magnitude as the thermal energy k BT. Consequently, even when the temperature drops below the Curie temperature, the thermal energy is sufficient to change the direction of magnetization of the entire grain, and the magnetization vector fluctuates around the easy axis of magnetization with a characteristic relaxation time.11 Hence, average grain magnetization in zero field is zero. When an external field is applied, the dipole moment of each grain still align on average along the direction parallel to the easy axis of magnetization, although it will point for a longer time in the direction that maximizes the dipole’s projection on the external field direction. As a result, a net magnetization will be induced

122

Structure and Functional Properties of Colloidal Systems

in the magnetic grain.33 In composite particles, the easy axis of magnetization of the grains are often randomly oriented in the absence of an external magnetic field. Consequently, the randomly distributed oscillating dipoles do not contribute to a net moment of the entire particle. By applying a magnetic field, however, one of the two easy axis directions is favored in each magnetic grain, and the particles acquire a net magnetic moment m. The magnetization process is usually completely reversible and the magnetization curve does not show any hysteresis. This means that the material behaves on a macroscopic scale like a paramagnetic matter. Such behavior is known as superparamagnetism. At room temperature, experiments confirm that linear structures made up of superparamagnetic particles are also superparamagnetic.26 As expected, filaments are more magnetizable than the isolated magnetic nanoparticles, since the cooperative response of the magnetic particles is more sensitive to the magnetic field.34 When two magnetic particles are under the influence of an external magnetic field, they generate an additional magnetic field at the position of the other particle. Zhang and Widom studied the magnetic linear aggregate as a function forces acting within a field-induced of the applied external field H and the particle separation r .35 According to these authors, mutual induction increases the magnetic moment of the particles dramatically. If the particles are in contact and their magnetic susceptibility is c ª 1, then a particle in an infinite linear aggregate is about 20% more magnetized than an isolated particle. Mutual induction also increases the aperture angle of the zone where the dipolar interaction is attractive. Considering c ª 1, the aperture angle increases by roughly 8%.33

6.3.2

DIFFUSIVE FILAMENT MOTION

The diffusion coefficient of a Brownian aggregate is given by the Einstein relation D 0 = k BTg -1, where the friction coefficient g is a ratio between the force that the fluid exerts on the aggregate and the aggregate’s velocity. Since the fluid flow velocity induced by the motion of a particle affects the motion of the remaining particles in the aggregate, the friction coefficient of each particle depends on the positions and velocities of the others. In case of rigid arrays of N connected spherical particles, the hydrodynamic forces are approximately equal for each bead. Only the beads near the ends of the rod experience differing hydrodynamic forces. For very long rods the end effects can be neglected, and the Navier–Stokes equation has an analytical solution given by36 4phNa g = ______ , ln(N)

(6.1a)

8phNa g^ = ______ , ln(N)

(6.1b)

8ph(Na)3 gr = ________ , ln(N)

(6.1c)

where a is the particle radius and h is the shear viscosity. The rotational friction coefficient gr and the friction coefficients parallel g and perpendicular g^ to the rod axis include hydrodynamic effects. As a result, the friction coefficient in the direction parallel to the rod axis is about twice as large as the one for perpendicular coefficient. Tirado et al. included end effects to assess the friction coefficient of finite rigid cylinders.37,38 These authors solved the hydrodynamic equations in cylindrical coordinates and found that the translational and rotational friction coefficients are given by 4phNa , g = ____________ ln(N)+ g end (N)

(6.2a)

Structure and Stability of Filaments Made up of Microsized Magnetic Particles

123

8phNa , g^ = ____________ ln(N)+ g end ^ (N)

(6.2b)

8ph(Na)3 , gr = _____________ 3 ln(N)+ g end r (N)

(6.2c)

end end where the cylinder length funcitons g end r (N), g ^ (N), and g (N) account for the so-called end of chain effects. In their theoretical approach, the end functions are given by

0.90 ____ g end = -0.21 + N ,

(6.3a)

0.18 ____ 0.24 ____ g end ^ = 0.84 + N + N 2 ,

(6.3b)

0.92 - ____ 0.05 , ____ g end r = -0.66 + N N2

(6.3c)

and the diffusion coefficients are finally given by k BT D = ______ (ln(N) + g end (N)), 4phNa

(6.4a)

k BT D^ = ______ (ln(N) + g end ^ (N)), 8phNa

(6.4b)

3k BT Dr = ________ (ln(N) + g end r (N)), 8ph(Na)3

(6.4c)

For a given temperature, the diffusion coefficients are determined by the length of the filaments and the size of the chain-forming particles. Other treatments have been proposed in order to obtain the friction coefficients of finite rigid cylinders. All of them consider different approximations and lead to a variety of values for the end of chain-effect corrections.39,40 We are interested in the diffusive motion of relatively stiff chains made of magnetic particles either when the filaments are under the presence of a constant field or when the magnetic field is turned off. In the presence of an external magnetic field, the linear aggregates are forced to align along the field direction, so rotational chain diffusion is forbidden. Consequently, only the translational diffusion coefficients D and D^ have to be considered for the theoretical analysis. There are several ways to determine dynamic information about the Brownian particle movement in solution. One of these methods is dynamic light scattering (DLS). In DLS, the fluctuations of the scattered light intensity are analyzed at a fixed scattering angle and the dynamic information about the particle movement is derived from the intensity autocorrelation function recorded during the experiment. Hence, DLS assesses directly the effective diffusion coefficient Deff of the scatters present within the scattering volume. The average diffusion coefficient depends on the different diffusive modes that the aggregates may undergo. Since this quantity is related to the average aggregate size, it will allow the state of aggregation to be monitored. In our experimental setup, an external magnetic field is applied perpendicularly to the scattering plane, forcing the filaments to align in the same direction. Due to this geometry, the measurements are only sensitive to the transversal motion of the linear aggregates, and the parallel diffusion coefficient D may also be neglected.41 On the other hand, relative positional particle fluctuations inside the linear aggregates may take place due to the competition between Brownian motion and the magnetic dipole–dipole interactions

124

Structure and Functional Properties of Colloidal Systems (a)

(b)

(c)

(d)

FIGURE 6.4 Videomicroscopy images of magnetic particles aggregated in an electrolyte solution under the presence of a constant magnetic field. (a) At high field strengths, the linear aggregates behave like rigid filaments. (b) At low magnetic field strengths, however, relative positional fluctuations inside the linear aggregates may take place. Detailed views of an individual filament are depicted in the respective inset. The images prove that (a) the filaments have a relatively straight and linear form at high field strengths, whereas at lower field strengths (b) they look more bended and twisted. The added electrolyte allows the particles to come close enough so that van der Waals interactions dominate and permanent aggregates are formed. Once the magnetic field is turned off, these aggregates start to rotate freely and lose part of their linear character (see (c) and (d)).

arising in the presence of the field (see insets in Figure 6.4). Linear aggregates of magnetic particles may bend and twist slightly due to Brownian motion. Such internal fluctuations may give rise to an additional contribution to the measured average diffusion coefficient. DLS is sensitive to movement at different spatial scales. When the characteristic length q-1 is higher than the particle radius a, the center of mass diffusion of the filaments is detected. For q-1 values close to the particle radius a, however, DLS reflects mainly on the internal movement of the particles contained within the aggregates. In an experimental study, Cutillas and Liu have shown experimentally that such positional fluctuations of the particles within the chains cause a linear dependency of the average diffusion coefficient measured by light scattering on q. These authors found 2__ qaD , Deff (q) μ ___ ^ √l

( )

(6.5)

where a is the magnetic particle radius. The slope in this relationship depends on the magnetic field strength through the dimensionless parameter l.41 Using surfactant-coated kerosene-based ferrofluid droplet particles, their experiments verified that the chains become more rigid and the fluctuations disappear as the field strength increases. If the magnetic field is strong enough, significant relative motions within the filaments are suppressed and the filament diffusion coefficient should not depend on q.42 Chain diffusion may be reliably assessed by light-scattering techniques when the filaments are either aligned due to the action of the magnetic field or freely diffusing when the magnetic field has been removed. The movement of freely diffusing rods is difficult to describe because the rods will have two independent components in their translational diffusion tensor: one component stands for translations parallel to the long rod axis D and the other perpendicular to it, D^ . Furthermore, the coupling of translational and rotational diffusive modes also has to be taken into account. This general case yields rather complicated results. Moreover, there is not only a translational and rotational displacement of the entity as a whole when a chain moves. As we have observed previously, there may also be small internal vibrational modes that may affect the dynamics of the system. Therefore, an accurate description of the diffusive motion by light-scattering techniques is not

Structure and Stability of Filaments Made up of Microsized Magnetic Particles

125

straightforward at all. Nevertheless, these techniques allow for much better statistics, and shorter measuring times if compared with most of the well-established imaging methods. These advantages can make light scattering a highly valuable tool for the development and standardization of materials made of magnetic filaments. Hereafter, we will simplify the problem by treating the diffusion of linear aggregates as the diffusion of a simple but important model: the rigid rod. If the average length of a linear scatterer is short when compared with the light wavelength, then its diffusive motion is separable in rotational and translational contributions. When the filaments are sufficiently large, coupling between translational and rotational diffusive modes are expected. Maeda and Fujime included in their calculations the coupling of translational and rotational motion implicitly.43 These authors showed that a complete expression for the autocorrelation function is only possible numerically. Without solving the diffusion equation explicitly, Maeda and Fujime derived an analytical expression for the effective diffusion coefficient assessed through the field autocorrelation function. They obtained __ L2 D f (qNa) 1 - f (qNa) + ___ Deff = D -DD __ 12 r 1 3 2

[

]

(6.6)

__

where D ⬅ (1/3)D + (2/3)D^ , DD ⬅ D - D^· f1(qNa) and f 2(qNa) are numerical functions given by Maeda and Fujime.43 All these parameters can be used together with hydrodynamic theories to obtain information about the size of the freely diffusing filaments in solution.22,44 This model is quantitatively correct only when 2qN is large. Furthermore, their approach is restricted to thin rods, that is, small beads, since the Rayleigh–Gans–Debye (RGD) condition was assumed to be valid.45 Even if linear aggregates formed by magnetic particles seem to be rigid, they might still be somewhat flexible when they become very long. Thus, one would also have to take into account the effect of filament flexibility on the correlation function.46 In this chapter, however, flexibility effects will not be considered. Since all the assumptions mentioned before are required for the derivation of the previous equations, the results obtained might be influenced if one of them does not hold. The characteristic features of their model, however, may always serve as a guide for the analysis of experimental data for any pair of L = 2Na and q values.

6.4

STRUCTURE OF MAGNETIC FILAMENTS

When the chain-forming particles aggregate in the primary minimum, the filaments or at least parts of them will remain assembled even in the absence of the magnetic field. In this section, we study the three-dimensional structure of the magnetic filaments formed by means of aggregation. The deformation and magnetorheological properties of these permanent chains have been the subject of several works during the last years.8,47–49 Light-scattering techniques, widely used for studying the structure of colloidal aggregates, can also be employed for obtaining valuable properties of the magnetic filaments. Light scattering is used to measure the fractal dimension, df, which is understood as a measure on how the particles fill the three-dimensional space, quantifying how mass varies with length scale. It provides a convincing measure for the compactness of fractal aggregates and gives information about the internal structure of the magnetic filaments. When the average scattered intensity I(q) is measured as a function of the scattering wave vector q, the experiment is usually called static light scattering (SLS). The collected light intensity I(q) depends mainly on the particle form factor P(q) and the aggregate structure factor S(q). In these terms, I(q) reads50 as I(q) = KP(q)S(q).

(6.7)

126

Structure and Functional Properties of Colloidal Systems

The proportionality constant K depends on the optical properties of the light-scattering device. The particle form factor P(q) contains information on the optical properties of the individual particles, whereas the structure factor S(q) quantifies mass correlations within the aggregates. In stable systems, without any spatial correlation between the monomer positions, the structure factor becomes S(q) ⬇ 1. In this case, the particle form factor can be determined easily from direct measurements of the scattered light intensity, since then I(q) μ P(q). Our analysis of the structure of the field-induced aggregates starts checking to which extent the mean intensity of the light scattered by our colloidal particles is affected by the increased magnetic permeability and the light adsorption caused by the magnetic grains distributed within the particle. For this purpose, the I(q) curve was measured for a stable sample of magnetic polystyrene particles of 170 nm size. When the magnetic particles are distributed at random, this curve gives simply the form factor P(q) of the individual particles. Figure 6.5 shows the obtained results. Mie theory, which is a complete analytical solution of Maxwell’s equations for the scattering of electromagnetic radiation by spherical particles of any size,51 has been employed for calculating the theoretical form factor for pure polystyrene latex particles without any magnetic content. A sphere diameter of 165 nm, a relative refractive index of n = 1.12 corresponding to polystyrene in water, and a relative permeability of m = 1.0 were used for the calculations. Although neither the magnetic character of the particles nor light absorption was considered in the Mie equations, the obtained curve agrees quite well with the experimental data for the magnetic polystyrene particles (see Figure 6.5). This suggests that the polydispersity of this sample is low and no evidence of aggregation is found. In this particular case, it is possible to assume, at least from an experimental point of view, that light absorption and an increased magnetic permeability do not significantly alter the light scattered by the sample at the employed wavelength and particle concentration. RGD theory is of quite limited validity, since it assumes that each segment of scattering particles sees the same (or nearly the same) incident light wave. RGD, however, also fits the scattering data of the polystyrene magnetic particles (see Figure 6.5). Here, we used the particle diameter as fitting parameter, obtaining d = 170 nm. The observed result is quite significant, since the previously described Maeda’s model implicitly assumes the RGD theory to be valid.43 For an aggregated system, the structure factor S(q) is directly related to the fractal dimension of the formed clusters. For structures having a fractal dimension df, S(q) becomes I(q) S(q) = ____ μ q-df P(q)

(6.8)

I (a.u.)

1.0

0.8

0.6

0.2

0.4

0.6

1.0

0.8 q

1.2

1.4

1.6

1.8

(107 m–1)

FIGURE 6.5 Normalized I(q) curve for a stable sample of a superparamagnetic polystyrene particle suspension. The experimental data are shown as points (䊊). The continuous curve (-) was calculated according to Mie’s theory for polystyrene particles of the same size without any magnetic content. The dashed line (- -) was calculated according to the RGD theory.

Structure and Stability of Filaments Made up of Microsized Magnetic Particles

127

making clear how SLS characterizes the aggregate morphology in terms of the cluster fractal dimensions. This relationship is valid for Rh-1 < q < a-1, where Rh is the average hydrodynamic radius of the aggregates. Equation 6.8 implies that the scattered intensity has reached an asymptotic timeindependent behavior once the fractal structure of the clusters is fully developed. The cluster fractal dimension can then be easily determined from the slope of these asymptotic curves in a logarithmic plot of S(q). The fractal dimension of the permanent chains made up of the previous magnetic polystyrene particles was measured at 20 mM KBr by means of SLS. Linear aggregates were achieved by applying a homogeneous magnetic field of about 400 mT. The electrolyte addition improves chain formation in a sense that it screens the electrostatic repulsion between the particles, allowing a greater fraction of particles to overcome the energy barrier and to participate in the formation of stable filaments. It should be emphasized that the scattered intensity curves were always measured once the field was turned off. In this case, some minor spatial reorganization of the particles contained within the chain-like aggregates may be possible. On the other hand, field absence avoids optical anisotropy that could bias the SLS measurements when the polarized light beam passes through a dispersion of aggregates aligned along the field direction.2 In Figure 6.6, we plot the aggregate fractal dimension as a function of the exposure time to the magnetic field. At short exposure times, a very low fractal dimension close to unity is found. This means that the short filaments are relatively stiff since they remain linear even when the field is removed. For longer exposure times and long filaments, however, the fractal dimension increases monotonously and reaches a final value of df ⬇ 1.2. The slightly higher fractal dimension observed for larger clusters indicates that these aggregates have lost part of their linear character after having been taken out of the magnet. Hence, we interpret the experimentally obtained fractal dimensions close to 1.2 as a result of linear but somewhat bended structures present in the sample.22 According to the results reported in the literature, they should be more flexible and more susceptible to bending caused by the surrounding fluid.24 It should be emphasized that similar fractal dimensions were reported by other authors for magnetic liposomes52 and simulations.53,54 The light-scattering results are corroborated by videomicroscopy images (see Figure 6.4). As can be seen, chain-like aggregates are observed in all the micrographs. As expected, the chains become the larger, the longer the samples remain under the influence of the magnetic field. The smaller

2.0 1.8

df

1.6 1.4 1.2 1.0 0.8 0

100

200 300 t (min)

400

500

FIGURE 6.6 (ⵧ) Fractal dimension of the filaments as a function of the exposure time to the magnetic field. (䊉) Fractal dimension of the clusters formed by the filaments at 1 M KBr as a function of the time spent once the magnetic field is switched off. In the latter case, the magnetic particles were exposed to the external field during the first 30 min.

128

Structure and Functional Properties of Colloidal Systems

aggregates have a relatively straight and linear form, whereas the larger aggregates seem to be more flexible since they look more bended and twisted.

6.5

STABILITY OF MAGNETIC FILAMENTS

In this section, the stability of the permanent filaments is studied, focusing our attention on the role of the electrolyte concentration. Initially, stable samples of monodisperse superparamagnetic polystyrene particles were aggregated in the presence of an external magnetic field and different amounts of electrolyte. The strength of the magnetic field was measured to be H = 23.9 kA/m throughout the scattering volume. The particles are dispersed in water. The formation process of the magnetic filaments has been monitored using DLS. Filament stability is expected to depend on the electrolyte concentration. Indeed, our experiments show that the degree of filament breaking is less pronounced at high electrolyte concentration. Figure 6.7 shows the average length of the linear aggregates as a function of time for different electrolyte concentrations. The final electrolyte concentrations used for reducing the repulsive electrostatic energy barrier were 0.25 and 25 mM. The electrolyte was always added to the colloidal dispersion before exposing the samples to the magnetic field. During the first 30 min of the experiments, the magnetic field was applied. Thereafter, the field was turned off but the DLS measurements were still performed for some additional time. When the magnetic polystyrene particles were exposed to the magnetic field, the average filament size increased with the exposure time to the magnetic field. As can be observed in Figure 6.7, the growth rate also rose for increasing electrolyte concentrations.28 When the magnetic field was turned off, the length of the magnetic filaments was determined from the measured average diffusion coefficients applying the previously described Maeda–Fujime model. This adequate theoretical approach allows the mean particle chain length L to be determined from the experimentally accessible average diffusion coefficient Deff. At low electrolyte concentrations, the linear aggregates formed are not able to survive the absence of the magnetic field. This means that the repulsive electrostatic interactions are strong enough to avoid the formation of permanently bonded filaments. Aggregation is observed but only in a shallow energy minimum that disappears once the magnetic field is turned off. At higher electrolyte concentrations, however, stable filaments remain in the sample. The average chain length decreases when the magnetic field is turned off. This implies that the chains formed disassemble only partially, so stable bonds in a deep primary energy minimum, where the particles are in contact with each other,

15

N

10

5

0 0

10

20 t (min)

30

40

50

FIGURE 6.7 Time evolution of the average number of constituent particles per aggregate N monitored by means of DLS. The electrolyte concentrations were (䉭) 0.25 mM and (䉫) 25 mM.

129

Structure and Stability of Filaments Made up of Microsized Magnetic Particles

and instable bonds in shallower energy minimum, where the neighboring particles within the linear aggregates are a short distance apart from each other, must coexist. The coexistence of secondary and primary bonds can be directly observed by means of the machine force technique. As soon as the magnetic field is applied, the mean distances between the aligned particles can be assessed by analyzing the Bragg scattering patterns. For perfectly aligned particles with a separation d, illuminated by a white light source, the first-order Bragg condition becomes l d = __________ , n(1 - cos q)

(6.9)

(a) 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

(b) 2.4 2.2 2.0 1.8 1.6 1.4 1.2 400

500

600 700 l (nm)

800

400

500

Scattered light

600 700 l (nm)

Reflectance (a.u.)

Reflectance (a.u.)

where n is the refractive index of the suspension medium, l is the wavelength of the light, and q is the angle between the incident and the scattered light beams (Figure 6.8). The field-induced aggregates formed by magnetic particles give rise to a strong Bragg diffraction pattern that allows the interparticle spacing to be measured. Since the repulsive interactions between the colloidal particles are balanced by attractive forces (magnetic dipole–dipole and van der Waals interactions), the interparticle force profile between colloidal particles may be obtained as a function of the interparticle spacing d. For further details, please see the excellent work by Leal-Calderon et al.7 Figure 6.8 shows the spectral distribution of the scattered light at 0 and 20 mM KBr. The measurements were performed using magnetic silica particles of 180 nm size, synthesized by Dr. El-Harrak. The field-induced aggregates were illuminated by a white light source. For 0 mM KBr, the observed spectral distribution only depicts an outstanding maximum at l ⬇ 500 nm (Figure 6.8a). This means that there is a preferential distance d between neighboring particles aligned along the field direction, which, according to Equation 6.9, is approximately d ⬇ 195 nm. At 20 mM, however, two smaller maxima are clearly observed in the spectral distribution (Figure 6.8b): one of them corresponding to d ⬇ 195 nm, and the other corresponding to d¢ ⬇ 180 nm. Since there are two different maxima, there should be two preferential distances. Therefore, at 20 mM, the field-induced aggregation may occur in a primary or a secondary energy minimum where the neighboring particles are either in close contact or at a short distance, respectively. The filaments formed are, of course, still quite polydisperse in length. Figure 6.9 shows the time evolution of the polydispersity index (p.i.). The p.i. is a dimensionless parameter calculated from a cumulant’s analysis of the measured intensity autocorrelation function. It is used to characterize the relative shape of the cluster-size distributions. For the electrolyte-free sample (0 mM), aggregation

800

Scattered light

l

l

q

q

Incident light d

d

d¢

H

FIGURE 6.8 Spectral distribution for 0 mM (a) and 20 mM KBr (b). The data show that field-induced aggregation may occur in a primary or in a secondary energy minimum, depending on the electrolyte concentration.

130

Structure and Functional Properties of Colloidal Systems 1.0 0.9 0.8 0.7

p.i.

0.6 0.5 0.4 0.3 0.2 0.1 0.0 –0.1 0

10

20

30 40 t (min)

50

60

70

FIGURE 6.9 Time evolution of the p.i. at electrolyte concentrations of (䉱) 0.0 mM, (ⵧ) 0.50 mM, (Δ) 10 mM, and (䊊) 50 mM.

is not observed and the p.i. remains constant with an average value of 0.08, close to the value measured before for pure monomerical samples. At low electrolyte concentrations (0.50 mM), fieldinduced aggregation is reversible and the mean value of the p.i. return to its initial value once the magnetic field was turned off. Roughly speaking, the p.i. becomes the larger, the larger the mean length of the linear aggregates becomes. The degree of monodispersity around a desired average size might be improved if an accurate separation method could be found. This could open an experimental way for assembling magnetic filaments of a well-determined length by simply adjusting the strength of the magnetic field and the amount of added electrolyte. Further work on this subject is needed before such a goal could be achieved.6,29 A further purpose of the present chapter is to calculate the probability that a particle that is caught in a secondary bond transforms into a primary bond and to determine its dependency on the height of the energy barrier between the primary and secondary minima. As was explained above, particle aggregation may occur in the primary minimum, where the particles are in contact with each other, or in the secondary minimum, where the neighboring particles within the linear aggregates are a short distance apart from each other. In summary, the magnetic particles can be found in three different configurations during the field-induced aggregation process: 1. Aggregated in the primary minimum (close contact) 2. Aggregated in the secondary minimum (short distance) 3. Unlinked (large distance) If np, ns, and nf denote the number of primary bonds, secondary bonds, and open bonds (not yet established bonds), respectively, n = ns + np becomes the total number of established links. Neighboring particles within the linear aggregates must overcome the energy barrier in order to go from a metastable secondary bond to a stable primary bond. The number of secondary bonds increases in time when two particles or aggregates establish a new bond. It diminishes when two neighboring particles contained within a linear aggregate overcome the energy barrier and go from a metastable secondary bond to a stable primary bond. Hence, a suitable system of differential equations for calculating the reaction probability is55 dns(t) dnf(t) _____ = - _____ - kspns(t), dt

dt

(6.10a)

Structure and Stability of Filaments Made up of Microsized Magnetic Particles

dn (t) dt

p _____ = - kspns(t),

131

(6.10b)

Secondary bonds may turn into primary bonds when the corresponding particles overcome the potential barrier. The rate constant ksp parameterizes the probability per unit time that a secondary bond turns into a primary bond. The rate coefficient includes all the factors that affect the reaction rate, except for the number of secondary bonds, which is explicitly accounted for. The primary bonds, formed due to short-range van der Waals interactions, are stable enough so that the backrate constant kps may be neglected completely. The proposed equations also assume that bond formation between particles can be considered as independent events. Moreover, all the particles initially heading from the free unbounded state toward the primary minimum will indeed pass through a secondary bond for at least a short time. This assumption is only correct when the energy barrier between the primary and the secondary minimums is high enough, that is, at not too high electrolyte concentration. Since the rupture of the linear aggregates is forbidden in these equations, it adequately describes the experimental observations only as long as the magnetic field is applied. The primary bonds formed are found to be thermally activated, and so the corresponding rate constant should be given by an Arrhenius law56 1 -E /k T ksp = __ t e a B. 0

(6.11)

The Arrhenius equation determines the dependence of the rate constant ksp on the temperature T and the activation energy Ea, that is, the height of the energy barrier. The rate constant ksp depends on temperature and also on the ionic strength.

6.6

AGGREGATES OF FILAMENTS

Bottom-up techniques, where filaments are employed in a controlled manner to obtain more complex colloidal architectures, are investigated.24 At high electrolyte concentrations, the chains withstand the absence of the magnetic field. The measured filament size does not change once samples aggregated below 50 mM were removed from the magnet. This means that the chains remaining in the sample at not too high electrolyte concentrations are stable and neither break nor continue to grow. Electrolyte concentrations lower than 50 mM do not affect the stability of the samples when the magnetic field is absent. In other words, removing the samples from the magnet allows samples with a given or frozen composition and average chain size to be obtained. At high electrolyte concentrations, the electrostatic repulsion between the particles is completely overcome by the magnetic interaction when the magnetic field is present. Hence, the field-induced aggregation process is totally controlled by the magnetic dipole–dipole interaction among the particles. Therefore, the growth behavior follows the same tendency already observed for not too high electrolyte concentrations. However, at electrolyte concentrations of 50 mM or above, the magnetic filaments continue to grow even when the field is turned off. Due to the superparamagnetic character of the magnetic colloidal particles, the dipole–dipole interaction between the particles disappears in the absence of the field. This means that the electrostatic energy barrier that prevented the particles from aggregation in the primary minimum is now almost completely suppressed and all the field-induced bonds are stable. Hence, the already formed linear chains are expected to aggregate further following a reaction–diffusion aggregation scheme. In this case, fractal aggregates made of chain-like aggregates are in fact observed. Figure 6.6 shows the fractal dimension of the clusters as a function of the time spent once the magnetic field is switched off. As before, the fractal dimensions were measured by SLS as a function of the time. The electrolyte concentration employed was 1.0 M. At long times, the

132

Structure and Functional Properties of Colloidal Systems

fractal dimension increases monotonously and reaches a final value of df ⬇ 1.6. Hence, the linear clusters continue to aggregate forming more and more branched structures.

6.7

CONCLUSIONS

Usually, field-induced aggregation is reversible and microsized superparamagnetic magnetic particles return to their original freely monomeric state once the external magnetic field is removed. If the bonds between the magnetic particles are, however, reinforced by additional linking mechanisms, then linear structures able to withstand the field absence are formed. The morphology and the stability of such permanent filaments were studied. At high enough electrolyte concentrations, stable filaments made up of EDLMP remain in the sample once the magnetic field is turned off. The final filament size may be reliably monitored by light-scattering techniques, when the filaments are either aligned due to the action of the magnetic field, or when they are freely diffusing. Rate equations are proposed, which allow the time evolution of the average number of primary and secondary bonds per chain to be modeled as a function of the exposure time. The morphology of the permanent field-induced aggregates is characterized by a fractal dimension close to unity. For long linear aggregates, however, the cluster fractal dimension increases to ~1.2. This increase is mainly a consequence of the higher degree of internal flexibility of the larger aggregates. The results are corroborated qualitatively via transmission electron microscopy. At high electrolyte concentrations, linear aggregates continue to grow when the magnetic field is turned off. More and more branched clusters are formed now.

ACKNOWLEDGMENTS Financial support from the Spanish Ministerio de Ciencia e Innovación [Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I + D + i), projects MAT200612918-C05-01 and MAT2006-13646-C03-03], the European Regional Development Fund (ERDF), the Junta de Andalucía (Excellency Project P07-FQM-2496), and the Acción Integrada (Project HF2007-0007) is gratefully acknowledged. We also express gratitude to Dr. Jose Manuel LopezLopez for fruitful discussions.

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7

Glasses in Colloidal Systems: Attractive Interactions and Gelation Antonio M. Puertas and Matthias Fuchs

CONTENTS 7.1 7.2 7.3 7.4

Introduction ...................................................................................................................... Theory ............................................................................................................................... Model and Simulation Details .......................................................................................... Results ............................................................................................................................... 7.4.1 Theoretical Predictions for Repulsion- and Attraction-Driven Glasses ............... 7.4.1.1 The Input to MCT: The Equilibrium Structure Factor ......................... 7.4.1.2 MCT of Repulsion- and Attraction-Driven Vitrification ....................... 7.4.1.3 MCT of Bond Formation ....................................................................... 7.4.2 Simulations of Attractive Glasses and Gels .......................................................... 7.4.2.1 Competition of Liquid–Gas Separation with Attraction-Driven Vitrification .............................................................. 7.4.2.2 Simulations of Attractive Glasses .......................................................... 7.4.2.3 Dynamical Heterogeneities .................................................................... 7.4.2.4 High-Order Singularities ....................................................................... 7.4.2.5 Attractive Glasses at Low Density: Gelation ......................................... 7.4.2.6 Discussion: Gels and Attractive Glasses ................................................ 7.5 Conclusions ....................................................................................................................... Acknowledgments ...................................................................................................................... References ..................................................................................................................................

7.1

135 138 140 141 141 142 144 147 148 149 151 155 156 158 159 160 161 161

INTRODUCTION

When an atomic system is cooled below its glass temperature, it vitrifies, that is, it forms an amorphous solid [1]. Upon decreasing the temperature, the viscosity of the fluid increases dramatically, as well as the time scale for structural relaxation, until the solid forms; concomitantly, the diffusion coefficient vanishes. This process is observed in atomic or molecular systems and is widely used in material processing. Several theories have been developed to rationalize this behavior, in particular, the mode coupling theory (MCT) that describes the fluid-to-glass transition kinetically, as the arrest of the local dynamics of particles. This becomes manifest in (metastable) nondecaying amplitudes in the correlation functions of density fluctuations, which are due to a feedback mechanism that has been called “cage effect” [2]. In colloids, glasses induced by the steric hindrance are also found at large concentration of particles. Interestingly, the hard spheres (HS) system, widely studied theoretically, can be realized 135

136

Structure and Functional Properties of Colloidal Systems

FIGURE 7.1 Experimental intermediate scattering function (left) and mean-squared displacement (MSD) (right) for hard colloids approaching the glass transition. From left to right volume fraction increases from 0.466 to 0.583. The bold line represents the single particle dynamics. (From van Megen W. et al. 1998. Phys. Rev. E 58: 6073–6085. With permission.)

using hard neutral colloids. The seminal work by Pusey and van Megen [3] showed that the dynamics of a (polydisperse) HS system arrests, inhibiting the crystallization of the system, which is thermodynamically favored at those high densities. This glass transition has been widely analyzed in terms of the predictions from MCT, finding excellent agreement (see Figure 7.1, and references [4,5]). The typical two-step decay of the intermediate scattering function, or density correlation function, implies a significant decoupling between short-time dynamics and structural relaxation. This differentiation of time scales increases as the glass transition is approached, until the structural relaxation (second decay) does not take place (ideally) beyond the critical density. In this case, the system is kinetically arrested, therefore nonergodic, and behaves like a solid (the elastic modulus grows and overtakes the viscous one). However, in addition to these repulsion-driven glasses, equivalent to atomic or molecular glasses, colloidal systems also present low-density amorphous solids, known as gels, when attractive interactions are present [6]. Gelation is the formation of a percolating network (typically fractal) of dense and more dilute regions of particles with voids, which coarsen up to a certain size and freeze when the gel is formed. It is found in systems with strong short-range attractions, and is a universal phenomenon observed experimentally in many different systems, ranging from colloid–polymer mixtures to protein systems and clays. It is also connected with colloidal flocculation. Because short-range attractions (of comparable short range) are not present in atomic systems, gelation is found only in macromolecular systems. In addition to the desirable complete fundamental comprehension of the process, gels have important applications in food industry, stability of commercial colloids, or paints. Many detailed experiments have been performed using mixtures of colloids with nonadsorbing polymers, where the attraction strength is set by the polymer concentration, and the range by the polymer size. An effective attraction between the colloids appears due to the depletion of polymers between two particles, when they approach each other beyond the polymer diameter. This effective depletion attraction has been studied theoretically since the 1950s [7]. In fluids with short-range attraction, MCT predicts the existence of two glasses with different driving mechanisms [8,9]. At high density and high temperature, a glass transition induced by core repulsion is found, which is identical to the glass transition found in HS or atomic systems. On the other hand, at low temperatures and all densities, vitrification is driven by the formation of longlived physical bonds between the particles, which hinder their motion. Although this attractiondriven glass is exactly in the region where gelation is found experimentally, identification of gelation with it is, however, not straightforward. At low temperatures, a fluid–fluid transition, equivalent to the liquid–gas transition in atomic systems, is also present—typically, this fluid–fluid transition is metastable inside crystal–fluid separation for small attraction ranges. The interplay of the attractive

137

Glasses in Colloidal Systems: Attractive Interactions and Gelation 0.3

12

Single liquid phase Gel boundary Gel phase

0.2 Cp/C*p

0.2

8 6

Rg/R 0.025D

0.1 Rg/R = 0.061

Cp/C*p

Cp/mg cm–3

10

0

0

0.1

0.061D

0.2 0.3 0.4 fc Gel phase

0.090D

0.1

4

0.026T

2 Homogeneous ﬂuid 0

0 0.00

0.10

0.20

0.30 j

0.40

0.50

0

0.1

0.2

0.3

0.4

0.5

fc

FIGURE 7.2 Experimental phase diagrams of colloid–polymer mixtures. The left panel shows fluid states (circles), fluid–crystal coexistence (squares), nonequilibrium nonpercolating clustering (asterisks), and gels (triangles). (From Poon et al. 1999. Faraday Discuss. 112: 143–154. With permission.) The right panel shows phase diagrams with polymers of different sizes. In the inset, only fluid and gel states are observed. (From Shah S.A. et al. 2003. J. Chem. Phys. 119: 8747–8760. With permission.)

glass transition with fluid–fluid separation poses a major problem for the interpretation of experimental results of gelation. Figure 7.2 shows the phase diagram of colloid–polymer mixtures [10,11]. Upon increasing the attraction strength (polymer concentration), the following phases are observed (left panel of Figure 7.2): fluid states, phase separation (crystal–fluid or fluid–fluid), nonequilibrium clustering, and finally, gelation (nonequilibrium clustering is distinguished from gelation since the former leads to independent nonpercolating long-lived clusters that sediment). In some cases, however, the transition from the fluid state to the gel occurs directly, without other phase transitions taking place between them (right panel). In all cases, for strong attractions, phase separations are arrested. The interplay between gelation and equilibrium phase transitions is therefore fundamental to explain this phenomenology, but is as yet an open question. Recent reviews are available in the literature, giving the current state of the art where attractive glasses, gels, and phase separation meet [12–14]. In this chapter, we will present results for gels and attractive glasses mainly from computer simulations, where the effects from phase separation or vitrification can be separated. Specific systems have been devised where the fluid–fluid transition is avoided, or shifted to lower densities, allowing the study of the transition from homogeneous fluids to the attractive glass. At high density, the results will be analyzed using MCT, finding good agreement with the predictions for attraction-driven glasses. It will be also shown, however, that the systems are dynamically nonhomogeneous, induced by the structural heterogeneity at the particle level, which have been observed by simulations and experimentally. At low density, the decay of the correlation functions at wavevectors around the neighbor peak in the structure factor occurs via joint motions in clusters or flapping of branches, which, however, do not cause structural relaxation at long distances. This chapter is organized as follows: the next section will introduce MCT, giving the basic equations and ideas of the theory. Next, several systems that have been used to study the attractive glass transition and gelation in the literature will be presented, giving some details. Section 7.4 will present the main results of this chapter. First, the MCT predictions for the attractive glass will be reviewed; next, the competition of fluid–fluid separation with the attractive glass will be shown. Then, the MCT analysis of the transition found in simulations in systems without fluid–fluid separation, with the study of the dynamical heterogeneities, will be presented. Finally, the results at low density and the connection with experimental gels will be discussed. Section 7.5 will present the main conclusions of this chapter.

138

7.2

Structure and Functional Properties of Colloidal Systems

THEORY

In HS colloidal dispersions, the liquid–glass transition has been studied by dynamic light scattering [15]. Autocorrelation functions for density fluctuations have been measured over about four decades. It was found that these correlations decay quickly to zero, as expected for a liquid, only for densities below a critical value. Close to, and especially above, this value, there opens an extended intermediate time window, where the correlators stay (almost) constant at a finite amplitude (see Figure 7.1, right panel). This observation is equivalent to the arrest of the dispersion into an amorphous solid, namely, a glass. The amplitude is the Debye–Waller factor of the amorphous solid, also called glass form factor, and generalizes the order parameter introduced by Edwards and Anderson in the theory of spin glasses. The evolution of the glassy dynamics for the HS system was studied comprehensively by Pusey and van Megen and coworkers [4,5]. The data suggest that it is the well-known cage effect that causes the glassy dynamics and the arrest of density fluctuations in this HS dispersion. The cage effect is the essential physical concept underlying the MCT for the evolution of glassy dynamics in simple supercooled liquids [16,17]. This theory allows the calculation of the structural dynamics from the equilibrium structure factor [2]. In reference [4], detailed quantitative comparisons of the data for HS colloids with the MCT predictions were presented. It is shown that the theory accounts for the experimental facts within a 15% accuracy level. The evolution of glassy dynamics was also studied for polymeric micronetwork colloids that behave quite HS-like [18], and similar conclusions on the validity of MCT were reached. Recently, also the linear and nonlinear rheology of HS-like dispersions was compared quantitatively with MCT, supporting the conclusions from light scattering [19]. MCT gives a self-consistent equation of motion of the (normalized) intermediate scattering function. This is the autocorrelation function of coherent density fluctuations with wavevector q, defined by

Fq (t ) =

1 Sq

1 N

N

Âe

- iq ◊ ri ( t ) - rj (0)

(

) .

i, j

The number of particles is denoted by N. The normalization to unity at time t = 0 is provided by the static structure factor

Sq =

N

1 N

Â

e

- iq ◊ ri - rj

( ) ,

i, j

which captures equilibrium density correlations. The equation of motion for the correlators takes the form of a relaxation equation with retardation [the retardation or non-Markovian effects are contained in the memory kernel mq(t)]: t

(t ) + F (t ) + m (t - t ′ )F (t ′ ) dt ′ = 0. GqF q q q q

Ú 0

Without retardation, correlators would show fast diffusive relaxation on the timescale Gq = (Dsq2/Sq)-1 determined by the short-time collective diffusion coefficient Ds; it captures instantaneous particle interactions, including solvent mediated, so-called hydrodynamic ones, which do not require structural particle rearrangements to act. This decay mechanism will not play an important role at the glass transition.

Glasses in Colloidal Systems: Attractive Interactions and Gelation

139

The central quantity capturing slow structural rearrangements close to glassy arrest is the memory function mq(t). It can be regarded as a generalized friction kernel, as can easily be verified after Fourier transformation. In MCT approximation, it is given by

mq (t ) =

1 2

ÂV k,p

qkp

Fk (t ) Fp (t ).

The vertices Vq k p couple density fluctuations of different wavelengths and thereby capture the “cage effect” in dense fluids [2]. It thus enters the theory as a nonlinear feedback mechanism where density fluctuations slow down because of increased friction, and where the friction (precisely the time integral over the memory kernel that dominates the long-time collective friction coefficient) increases because of slow density fluctuations. MCT is a first-principles approach as the vertices are calculated from the microscopic interactions:

= Sq S k S p Vqkp

2 r2 È (q ◊ k )ck + (q ◊ p)c p ˚˘ d(q - k - p). 4 Î Nq

The mode coupling approximation for mq(t) yields a set of equations that needs to be solved selfconsistently. Hereby the only input to the theory is the static equilibrium structure factor Sq that enters the memory kernel directly and via the direct correlation function cq that is given by the Ornstein–Zernicke expression cq = (1 - 1/Sq)/r, with r being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. The MCT equations show bifurcations due to the nonlinear nature of the equations. A bifurcation point is identified with an idealized liquid-to-glass transition. The quantity of special interest is the glass form factor or Edwards–Anderson nonergodicity parameter, fq. It describes the frozen-in structure of the glass and obeys fq 1 = 1 - fq 2

ÂV k,p

qk p

fk f p .

In the fluid regime, density fluctuations at different times decorrelate, so that the long-time limit vanishes. On approaching a critical packing fraction or a critical temperature, MCT finds that strongly coordinated movements are necessary for structural rearrangements to relax to equilibrium. MCT identifies two slow structural processes, b and a process, when the glassy structure becomes metastable and takes a long time to relax. In the idealized picture of MCT, the glass transition takes place when the particles are hindered to escape from their neighboring environments. The nonergodicity parameter jumps from zero in the liquid to a finite value. This also is accompanied by diverging relaxation times. Although experiments on molecular glass formers have revealed that the dynamics very close to the transition point is dominated by thermally activated hopping processes, which the described (idealized) MCT cannot account for, MCT has been quite successful in describing the approach to glassy arrest in colloidal dispersions, as mentioned above.

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Structure and Functional Properties of Colloidal Systems

For liquid states close to the glass point and long times the correlator approaches the a-scaling law, where the shape of the scaling or master functions are independent of density or temperature. The a time scale t diverges in MCT with a power law, t ~ e -g , with g =

1 1 + ; 2 a 2b

thus, t depends only on the separation e = (j - j c)/j c from the critical point. The anomalous exponents a and b follow from the structure factor at the transition, and take values around a 0.3 and b 0.6 for HS. In the vicinity of the critical point, von Schweidler’s power law describes the initial a-relaxation from the nonergodicity plateau to zero:

(

)

Fq (t ) = fq - hq t b 1 + kq t b + O(t 3b ), where ~t is a rescaled time, ~ t = t/t, with t following the power law shown above. The coefficients hq and kq are called critical and correction amplitudes, respectively [20]. Von Schweidler’s law is the origin of stretching (viz., nonexponentiality) in the a process of MCT. More details about MCT, the asymptotic expansions, and the scaling laws can be found in references [2,20,21].

7.3

MODEL AND SIMULATION DETAILS

In order to model experimental systems with short-range attractions, different models have been used. Spherically symmetric interaction potentials, such as the simple square well (SW) [22], or the Asakura Oosawa (AO) depletion potential [23], model the colloid–polymer mixture considering that the polymers are ideal [24]. However, due to the short range of these simple potentials, crystallization and fluid–fluid phase separations occur in the same region where gelation is expected, what makes more difficult the interpretation of the data. Therefore, strategies to avoid this equilibrium phase separation have been devised. Keeping the spherical symmetry of the interaction potential a long-range repulsive barrier can be added to the short-range attraction, which destabilizes the fluid–fluid separation, and allows the study of fluid states close to gelation even at low densities [25,26]. Polydispersity is used to avoid crystallization, either a continuous distribution or binary mixtures of particles. Figure 7.3 presents an interaction potential based on the AO potential, with a repulsive barrier and a polydisperse system. Alternatively, nonspherical potentials have also been used, where active spots are located on the particle surface [27], or many body interactions, such as a maximum number of neighbors attracted [28], or combination of both [29]. Using these structural or energetic constraints, the formation of a dense phase is hindered, thus impeding crystallization and liquid–liquid crystallization. Here, we will concentrate on spherically symmetric potentials, which are closer to both the experiments and the theoretical developments. For the AO potential, the potential at contact is Vmin = -k BTj p (3/2x + 1), where j p is the polymer volume fraction and x is the ratio of polymer to particle size (which sets the interaction range). The density of colloidal particles is the other key parameter for the phase diagram of the system, which we will report as volume fraction: j = Âvi ri, where the sum runs over populations of particles, vi is the particle volume, and ri is the number density of population i (remember that polydisperse systems are used). For colloidal systems, the microscopic dynamics for particle j is given by the Langevin equation, mr j =

ÂF

ij

i

- gr j + f j ,

141

Glasses in Colloidal Systems: Attractive Interactions and Gelation 10 2.5

7.5 Vtot

0

5

–2.5 –5

2.5 Vtot

1.75

2

2.25

2.5

r/a 0 –2.5 –5 –7.5

2

2.5

3

3.5

4

r/a

FIGURE 7.3 Interaction potential with a short-range attraction (given by the Asakura–Oosawa depletion interaction with x = 0.1) and a long-range repulsion of height 1 k BT (distance is measured in units of the particle radius, a). The different lines correspond to the interaction of particles with different sizes. More details of the interaction potential can be found in reference [25]. (From Puertas A.M. et al. 2003. Phys. Rev. E 67: 031406. With permission.)

where Fij is the interaction force between particles i and j, g is the solvent friction coefficient, and f j is a random force due to the collisions with the solvent molecules. The friction and viscous forces must obey the fluctuation dissipation theorem: · f j(t) f j(t¢)Ò = 6k BTgd ijd(t – t¢). It was shown by Gleim et al. [30,31] that the structural relaxation of a fluid does not depend on the microscopic dynamics, as predicted by MCT. Thus, the slow microscopic Brownian dynamics is usually replaced in simulations by the much faster microscopic Newtonian dynamics (in the equation above the friction and Brownian forces are zero), without affecting the structural relaxation. The dynamical quantities used to study the dynamics of glass transitions are generally the MSD, the intermediate scattering function, or the density autocorrelation function, presented above. The MSD is given by

dr 2 ( t ) =

1 N

i

i

2

Â (r (t ) - r (0))

,

i

where N is the number of particles, and the summation runs over all particles in the system. ·ºÒ indicates averaging over time origins. Its Fourier transform is the self part of the density autocorrelation function, Fqs(t), which is calculated restricting the summation to i = j in the definition of Fq(t) above.

7.4 7.4.1

RESULTS THEORETICAL PREDICTIONS FOR REPULSION- AND ATTRACTION-DRIVEN GLASSES

The theoretical results to be reviewed deal with dispersions of particles whose pair-interaction potential includes a short-ranged attraction. The quintessential system, also studied intensely experimentally, is characterized by a hard-core repulsion, and a short-ranged attraction potential

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Structure and Functional Properties of Colloidal Systems

induced by the depleted polymers. The theory used, namely MCT, was developed in the high-density regime so that the cage effect is essential for the dynamics. Its extrapolation to lower densities is thus quite speculative and requires that solidification is dominated by considerations of local interaction effects such as caging and the formation of physical bonds. The relative attraction well width is assumed to be small, say less than 10% of the average particle separation. This criterion is based on the Lindemann ratio typically observed at solidification in atomic/molecular systems. There the considerations of packing (viz., the repulsive interactions) dominate solidification and the (possibly present) attraction just affects the adhesion energy and the overall compressibility. Solidification by repulsion commonly allows each particle to explore a region around its site of the size of 10% of the average particle separation. If the attraction range is shorter and the strength of the attraction is sufficiently strong, which is possible in the mentioned colloidal solutions, then crystalline solids with much tighter packing (corresponding to higher density) and amorphous solids dominated by physical bond formation can be observed. Then the particle localization length measured in the MSD is (far) smaller than that predicted by Lindemann, and is of the order of the attraction range. Reviewing the theoretical predictions of this physical bond formation, and its subsequent observation in experimental and simulations studies, is the topic of this chapter. The main outcome of the theory [8,32,33] is the prediction of two different glassy states, arising because of two different physical mechanisms; on the one hand, caging driven by the excluded volume repulsion, and on the other hand, physical bond formation driven by the short-ranged attraction. The theory predicts a quite complicated diagram of metastable states, glass-to-glass transitions (not yet observed experimentally), and higher-order glass-transition singularities at densities somewhat above the glass transition value of the HS system. The rich results reflect the interplay of two mechanisms for particle localization, that is, for the arrest of density fluctuations. Both mechanisms arise because of qualitative features in the only input to the theory; the structure factor S(q), and thus their origin can be traced back (quantitatively) to the underlying particle interactions. 7.4.1.1 The Input to MCT: The Equilibrium Structure Factor The equilibrium structure factor S(q) is the essential input information required in the MCT equations. While in latter sections we discuss the actual S(q) from simulations (some including a weak long-ranged barrier), in this section we discuss the quintessential effect of a short-ranged attraction on S(q) at high density. The interaction potential consists of a hard-core repulsion and an attraction, where we consider both SW (viz., a constant attraction strength) and the discussed AO depletion attraction. The structure can be specified by three control parameters: the packing fraction j of the hard cores, the ratio -Ua(r = d)/k BT of attraction strength at contact relative to thermal energy, and the relative width d of the attraction shell compared with the particle diameter d. The spinodal lines of the particles with an attraction of constant depth (the SW system) are shown in Figure 7.4 for three representative values of the well width. They specify the divergence points of the compressibility caused by the liquid–vapor transition. The local physics of glass formation according to MCT (caging, bonding) is not affected by the long wavelength fluctuations at the liquid–vapor phase transition, which may only mask the glass lines. MCT can presently address states in the one-component region only, and the complete scenario including phase separation and vitrification is yet unknown. Figure 7.5 exhibits structure factors calculated within the mean spherical approximation for states marked by diamonds in Figure 7.4 (left panel). The S(q) exhibits a principal refraction peak as known from other simple liquids. It is caused by the hard-core-driven excluded volume phenomenon. The high-temperature curves 1 and 2 exhibit peaks, which are only slightly smaller and somewhat broader than the peaks of an HS system at the same densities. The attraction modifies the pair correlations and thus the excluded volume effects, as can be inferred by comparing the curves 1 and 3. Lowering the temperature, the short-ranged attraction causes the particles to move closer;

Glasses in Colloidal Systems: Attractive Interactions and Gelation

0.5

143

2

1

0.4

q

0.3

0.2 4

3

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

j

FIGURE 7.4 Dimensionless temperature J = 1/K = -k BT/Ua(r/d) versus packing fraction j for the SW system. The lines show the spinodal calculated within the mean spherical approximation for different relative attraction well widths: d = 0.03 (dashed line), d = 0.05 (continuous line), and d = 0.09 (dash-dotted). Diamonds mark the state parameters for which structure factors are shown in Figure 7.5 (left). (From Dawson K.A. et al. 2000. Phys. Rev. E 63: 011401. With permission.)

therefore, the peak position shifts to higher wavevector upon cooling. The radial pair distribution function develops a more rapidly varying structure at distances that are multiples of the particle diameter (not shown) [8], and this explains the decrease of the peak height and the increase of the peak wings in S(q).

FIGURE 7.5 Structure factors as a function of the wavevector q (in units of the diameter) calculated within the mean spherical approximation. Left panel: SW system with relative well width d = 0.05. The labels 1 to 4 correspond to the states indicated by the diamonds in Figure 7.4. (From Dawson et al. 2000. Phys. Rev. E 63: 011401. With permission.) Right panel: AO model with x = 0.08, at fixed polymer concentration of j p = 0.15, and varying colloid volume fraction, from top to bottom: j = 0.1, 0.2, 0.3, and 0.4. The symbols are canonical Monte Carlo simulation results and the dashed lines show the simple asymptotic result and the solid lines the full mean spherical approximation. (The curves and data have been offset for clarity; they oscillate about unity in all cases.) (From Bergenholtz J. et al. 2003. Langmuir 19: 4493–4503. With permission.)

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Structure and Functional Properties of Colloidal Systems

The large wavevector tail of S(q) will be of importance for physical bonding in the following. In the right panel of Figure 7.5, it is shown that a short-ranged attraction causes large and slowly decaying oscillations extending to very high wavevectors; here, theoretical and simulations results for the AO potential are shown (colloidal density increases from top to bottom at constant attraction strength) [34]. MCT finds that the strength of the primary peak in S(q) determines repulsion-driven glass formation, while the strength of the large-q tail in S(q) present for short-ranged attractions determines attraction-driven glass transitions [35]. Long-ranged attractions, which affect the equilibrium phase diagram and the small-q region in S(q) are unimportant for glass formation in MCT [36]. In the presence of a short-ranged attraction, the large-q tail of S(q) generically takes the approximate form

S (q ) - 1 ª nc(q ) ª

jK Ê sin q + q / b cos q ˆ , bq ÁË (q / b)2 + 1 ˜¯

where K measures the strength of the attraction and b ~ 1/d is inversely proportional to the range of the attraction. This form captures the large-q oscillations, as shown in Figure 7.5 (right panel) by the dashed lines. Equivalent expressions have been found for different interaction potentials, using different liquid state approximations, and when working at low and high densities [8,33,34,37]. For an attraction of shorter and shorter range, corresponding to b Æ •, a tail ~sin(q)/q appears in S(q), which extends to high wavevectors and dominates the glass transition in MCT [35]. It is the origin of bond formation in MCT. 7.4.1.2 MCT of Repulsion- and Attraction-Driven Vitrification The phase diagram for the SW system is shown in Figure 7.6 based on the mean spherical approximation. The two states 1 and 2 from Figures 7.4 and 7.5 are included for d = 0.06. State 1 refers to the liquid phase, increasing j to state 2 increases the height of the first sharp diffraction peak of S(q), and leads to arrest in a glass state, as known from HS. If one cools state 1 at fixed j = 0.50 down to state 3, the primary peak in S(q) decreases, yet the large-q tail increases. As a result of this compressibility increase on the wings of the structure factor peak, the liquid freezes to a glass upon cooling. For large temperatures, S(q) depends only weakly on T; this explains why the transition lines are almost vertical in Figure 7.6. On the other hand, the large-q tails in S(q) are not very sensitive to density changes. This explains why the transition lines in Figure 7.6 are rather flat as a function of temperature. Repulsion-driven glass melts due to cooling, if the decrease of the primary peak in S(q) is not overcompensated by the increase of the structure-factor-peak wings. This leads to the slant of the repulsive glass transition line to higher densities upon lowering the temperature, or equivalently increasing the attraction strength. The attraction causes bonding, in the sense that the average separation of two particles is smaller than that expected for HS. Therefore, the average size of the holes increases and this favors the long-distance motion characteristic for a liquid. This melting of a glass upon cooling occurs only for short-ranged attractions, because they distort the local cage. The system remains in the liquid upon further cooling until it re-enters the glass when physical bonds become (infinitely) long lived. The described re-entry phenomenon (viz., the path glass–fluid–glass possible, e.g., at j = 0.525 when changing the temperature in Figure 7.6) is a manifestation of the two local mechanisms for localization. Figure 7.7 shows a quantitative calculation for the AO system compared with experimental data from the Edinburgh group [38,39]. The theoretical curves for varying attraction range (proportional to the relative polymer size x = 2Rg/d), exhibit the melting of the repulsion-driven glass upon increasing the strength of the depletion attraction, the almost horizontal line of bond formation, and the

Glasses in Colloidal Systems: Attractive Interactions and Gelation

145

FIGURE 7.6 The glass transition lines of the SW system using S(q) from the mean spherical approximation showing cuts through the control parameter space for fixed relative attraction well width d; the values are d = 0.09, 0.06, 0.0465, 0.035, and 0.03 (from left to right). Endpoints of higher-order MCT singularities are marked by open circles (A3) and by an asterisk (A4); for information see reference [8]. The vertical dashed line marks the transition line for the HS system, and states 1 and 2 from Figure 7.4 are included as diamonds. (From Dawson K.A. et al. 2000. Phys. Rev. E 63: 011401. With permission.)

short piece of glass-to-glass transitions ending in a higher-order MCT singularity; it appears only for short enough attraction ranges and is discussed in detail in references [37,40]. The two glass states differ in their local packing and consequently stiffness. While repulsion localizes particles according to Lindemann criterion, bond formation allows particles less local free volume. Figure 7.8 shows the localization lengths (left panel), calculated from the MSD at infinite time, along the transition lines in Figure 7.7. The rapid decrease of the localization length at somewhat higher polymer concentrations is caused by the effect that the dominant mechanism of structural arrest changes from caging to bonding. For sufficiently small values of x, the change is discontinuous, and the localization length jumps from higher values in the glass to lower values in the gel. This jump occurs at the crossing of the two glass lines in Figure 7.7 (best seen for x = 0.07). For somewhat larger values of x, there is a continuous, albeit abrupt, changeover. The typical localization length in the gel state is some fraction of the attraction range, indicating that bonds with lengths set by the attraction are formed among particles. On crossing a glass transition line the viscous fluid changes to a solid that deforms in an elastic manner. The frequency-dependent storage or elastic shear modulus G¢(w), which vanishes in the low-frequency limit when the dispersion is in a fluid state, acquires a plateau at low frequencies G¢(w Æ 0) = G•. The shear modulus G •, namely, the elastic constant of the glass, is another quantity that picks up the difference between the glass formation mechanisms. At low polymer concentration the shear modulus is close to the HS value. For the shorter-range attractions, the shear modulus jumps to much larger values. The discontinuity lies again at the crossing of both nonergodicity lines. For somewhat longer-range attractions, there is a steep but continuous increase of the shear modulus as the polymer concentration is increased. The maximum in the modulus arises from the competition between particle packing and bond strength. At low polymer concentrations the high particle concentration imparts elasticity; increasing polymer concentration

146

Structure and Functional Properties of Colloidal Systems

3

cp (10–3 g cm–3)

2.4 x = 0.07 1.8 x = 0.08 1.2 x = 0.10 0.6

0 0.9

0.95

1 j/jg

1.05

1.1

FIGURE 7.7 Comparison of MCT predictions (continuous lines) in the colloid-rich part of the phase diagram with experimental data from Pham et al. [38,39]. The lines separate equilibrium states from nonergodic glass and gel states (repulsive and attractive glasses), respectively, in terms of the mass concentration of polymer cp and the colloid volume fraction j relative to the glass transition jg for three values of the polymer– colloid size ratio x. The symbols are experimental data: fluid-crystal coexistence (diamond), fully crystalline dispersions (triangle down), repulsion-driven glass states (filled circles), and attraction-driven glass states (filled squares). The gray regions highlight where nonergodic states were observed experimentally. (From Bergenholtz J. et al. 2003. Langmuir 19: 4493–4503. With permission.)

FIGURE 7.8 Left panel: localization length or root-mean-square displacement in the glass/gel state as a function of polymer concentration for three values of x along the transition lines in Figure 7.7. With polymer concentration, the attraction strength increases. The rapid change of rs is apparent between the Lindemann value characteristic for HS glasses at low attraction strength and the attraction range at large strength where physical bonds are formed. Right panel: zero-frequency shear modulus as a function of polymer concentration for three values of polymer to colloid size x along the glass transitions lines, which are shown in Figure 7.7. Note that the particle concentration changes along the curves shown. The dashed lines correspond to asymptotic results for the small density and attraction range limit. (From Bergenholtz J. et al. 2003. Langmuir 19: 4493–4503. With permission.)

Glasses in Colloidal Systems: Attractive Interactions and Gelation

147

leads to an increased elasticity because of the increasing bond strength, whereas at sufficiently high polymer loadings in the gel state, the particle concentration is so low that the shear modulus eventually begins to decrease. 7.4.1.3 MCT of Bond Formation Formation of physical bonds within the framework of MCT hinges on short range, local correlations, while the long-range or long-wavelength structure is considerably less important. The content of bond formation according to MCT can be compared with the familiar Smoluchowski theory of coagulation of charge-stabilized colloids [41]. For the sake of simplicity, we restrict the discussion to low concentrations where only isolated pairs of particles need be considered. Two charged colloidal spheres interact via the Derjaguin–Landau–Verwey–Overbeek potential, which exhibits a deep and narrow primary minimum near contact as well as a repulsive barrier at somewhat larger separations, caused by the interplay between dispersion and screened Coulomb interactions. The aggregation is described as an activation problem, that is, whether the particle pair can overcome the repulsive barrier and enter the primary minimum near contact. The attraction strength here is large compared to thermal energy. In the Smoluchowski theory of aggregation (doublet formation in the present context), a nonequilibrium situation is postulated at the outset by the choice of the value of the radial distribution function at contact, which differs from the value obtained from the interaction potential via equilibrium statistical mechanical averaging. While equilibrium theory predicts a finite, actually a rather high, contact value for aggregation, its vanishing is enforced because two particles in contact are interpreted as a doublet, and thus drop out of the description provided by g(r). A nonequilibrium flux of particles forming doublets upon overcoming the barrier and reaching contact is established, while kinetic stabilization is regulated by the Coulombic barrier height. This familiar Smoluchowski picture should be contrasted with the situation addressed by MCT: bond formation for dilute dispersions of particles interacting via the depletion interaction (or another short-range attraction) of only moderate strength. Since the attraction strength is of the order of a few k BT, the loss of equilibrium cannot be assumed a priori; rather the theory itself must deliver a criterion for when the attraction strength suffices for the formation of long-lived particle aggregates. Thus, the challenge is to find within an equilibrium statistical mechanics approach the threshold interaction beyond which the system falls out of equilibrium and where nonequilibrium type of approaches, such as diffusion-limited cluster aggregation (DLCA), start to become meaningful. The discussion of the large-q tail in S(q) in Section 7.4.1.1, which is characteristic for a shortranged attraction, enables one to formulate a simplified theory of bond formation within MCT with the result that the long-time limit of the dynamic structure factor is controlled by a single interaction parameter, G = K2j/b. Bond formation occurs at Gc = 3.02. . . [34]. For small values of G, the dynamic structure factor decays to zero for all wavevectors. Physically, this means that concentration fluctuations decay into equilibrium at long times, just as expected for a colloidal fluid. However, for G ≥ Gc, the solutions yield a nonzero glass form factor, namely, the system arrests in a metastable state. This simple result requires the approximate expression for S(q) given above and needs to be replaced by a full numerical solution whenever this approximation fails. Figure 7.9 shows that sufficient polymer concentration, namely, attraction strength, suppresses crystallization at a nonequilibrium boundary, beyond which different types of aggregation behavior are observed. Crystallization and fluid crystal coexistence would be the behavior expected from the equilibrium phase diagram, with fluid–fluid separation metastable, as shown by the critical point (open squared) calculated with the PRISM theory [42]. Immediately across the nonequilibrium aggregation boundary, disordered clusters of colloids are formed in the Edinburgh system, which rapidly settle under the influence of gravity to give amorphous sediments. Deeper inside this nonequilibrium region, a rigid gel forms, which settles suddenly after an induction time. Samples in this “transient gel” region not only exhibit a transient rigidity, but also nonergodic dynamics. Yet,

148

Structure and Functional Properties of Colloidal Systems 12 Poon et al. (1999): Fluid Fluid−crystal coexistence Aggregation Transient gelation

10

Cp (10–3 g cm–3)

8

6 x = 0.057 4

2 x = 0.06 0 0.00

0.10

0.20

0.30 f

0.40

0.50

0.60

FIGURE 7.9 Comparison of MCT predictions with the experimental phase diagram of reference [10] (see Figure 7.2) in terms of the mass concentration of polymer cp and the colloid volume fraction for x 0.06. The analytical MCT predictions are shown as broken lines, while a numerical MCT solution is shown as solid line. Also shown is the PRISM prediction for the metastable fluid–fluid spinodal with an open square marking the location of the critical point [42]. (From Bergenholtz J. et al. 2003. Langmuir 19: 4493–4503. With permission.)

MCT is in nearly quantitative accord with the nonequilibrium aggregation boundary, along which crystallization stops, instead of the transient gelation boundary, which is the experimental nonergodicity line. The theoretical prediction is sensitive to the value of attraction range but even on treating x as a freely adjustable parameter, the theoretical line cannot be brought in agreement with the transient gelation boundary. This observation is one of the indications that at low concentrations the role of the MCT bond formation transition with respect to colloidal gelation is yet unclear. The mesoscopic behavior, formation of clusters and/or ramified solids appears to require additional theoretical considerations. Nonergodicity of the local dynamics as described by MCT, however, appears still a necessary ingredient for nonequilibrium phenomena, as may be concluded from the accord with the nonequilibrium aggregation line in experiment. More information can be gained from simulations as will be discussed in the remainder of this review.

7.4.2

SIMULATIONS OF ATTRACTIVE GLASSES AND GELS

Attraction-driven glasses as predicted by MCT should extend into the same region as fluid–fluid (liquid–gas) separation, and therefore a competition between both processes can be found. In fact, the current understanding of most experimental gels is based on the arrest of the spinodal decomposition, which complicates the complete rationalization of the vitrification process induced by attractive interactions. In the following, we will show how this arrest of the demixing takes place, and several models where the fluid–fluid separation is inhibited thermodynamically will serve as benchmarks for the theoretical predictions. These models will show that the glass transition is found in the region predicted by MCT, with the correct properties at high density, and that the high-order singularity indeed exists. The extension of the glass transition to low density, however, presents new features that are currently absent in the theory. Two snapshots of gels at high and low density are presented in Figure 7.10.

149

Glasses in Colloidal Systems: Attractive Interactions and Gelation

FIGURE 7.10 Snapshots of the AO system with arrested phase separation, at volume fraction j = 0.10 (left) and j = 0.40 (right), with attraction strength equal to 16 k BT.

7.4.2.1 Competition of Liquid–Gas Separation with Attraction-Driven Vitrification Fluid–fluid phase separation can be monitored using the structure factor at low wavevectors, or by some specific parameter that measures the macroscopic density fluctuations at long times, such as the standard deviation of the density in subsystems. Figure 7.11 shows the evolution of the structure of the AO system (x = 0.10) for different attraction strengths (the critical point in this system is at 0.4 fp = 0.35

S (q)

3

0.3 2

y4

1

0.2

0

S (q)

tw = 8 tw = 32 tw = 128 tw = 512 tw = 2048 tw = 8192

fp = 0.50

3 2 1

0.1

0.0 0.0

0.2

0.4 fp

0.6

0.8

0 fp = 0.80

S (q)

3 2 1 0

0

5

10

15

qa

FIGURE 7.11 Arrest of phase separation for strong attractions at volume fraction f = 0.40 in the AO system. Evolution of the structure factor for different times after the quench (left; peaks increasing with tw) and standard deviation of the density in subboxes of size L/4 as a function of the attraction strength (right). The filled circles mark three states in the left panel. (From Puertas A.M. et al. 2007. J. Phys.: Condens. Matter 19: 205140. With permission.)

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Structure and Functional Properties of Colloidal Systems

j p 0.29), with the time elapsed since the quench, waiting time tw. It is expected that upon increasing the attraction, the fluid–fluid separation boundary is reached and the system separates in two phases with more and more different densities. For large attractions, however, the separation is prevented, as noticed by the arrest of the low wavevector peak in the structure factor, or by the decrease of the inhomogeneity parameter. The structure of the system is nevertheless not homogeneous, but resembles a network of particles with branches and voids, as can be observed in the snapshots presented in Figure 7.10 for two different volume fractions [23]. The arrest of the phase separation implies obviously an arrest of the dynamics of the system, as can be observed using the self part of the intermediate correlation function, Fqs(t). In Figure 7.12, Fqs(t) is presented for the AO system for different waiting times after the quench, for the three states at j = 0.40 presented in Figure 7.11. The system that undergoes equilibrium phase separation at j p = 0.35 reaches a stage where the dynamics is independent of the waiting time, and the structural relaxation occurs via diffusion of dense fluid clusters and/or by the exchange of particles between the liquid and gas phases. On the other hand, the systems quenched at states with stronger attractions slow down continuously, without saturating. The evolution of Fqs (t) with waiting time does not change the height of the plateau, since the structural evolution from the homogeneous fluid to these

1.0

qa = 9.9 fq

F sq (t)

0.8 fp = 0.35

0.6

1.0 0.8

0.4

0.6

0.2

0.4

0.0 1.0

fp = 0.50

fp = 0.35 fp = 0.50 fp = 0.80

0.2

0.8 F sq (t)

0.0 0.6

0

10

20

t

30 qa

40

50

0.4 104 0.2 103 0.0 1.0

fp = 0.80

F sq (t)

0.8 0.6 0.4 0.2

102 101

tw = 8 tw = 32 tw = 128 tw = 512 tw = 2048 tw = 8192 tw = 16384

fp = 0.35 fp = 0.50 fp = 0.80

100 10–1

101

102

103

104

105

tw

0.0 10–2 10–1 100 101 102 103 104 105 t¢

FIGURE 7.12 Self part of the density correlation functions for the three circles in Figure 7.11, for different waiting times as labeled (left; correlators slowing down with tw). The panels in the right show the nonergodicity parameter (upper panel), where the dashed line is a Gaussian fit giving the localization length for j = 0.50, and the relaxation time (the lower one) for the three states, with power-law fittings for the two states with stronger attractions. (From Puertas A.M. et al. 2007. J. Phys.: Condens. Matter 19: 205140. With permission.)

Glasses in Colloidal Systems: Attractive Interactions and Gelation

151

FIGURE 7.13 Evolution of the structure factor for an SW binary mixture (50/50 with a size ratio of 1.2) quenched at temperature T = 0.05 (the energy scale is set by the well depth) for volume fraction j = 0.10 (left panel) and j = 0.25 (right panel). The insets show the evolution of the fi rst maximum height S max and its position ql with waiting time, and the closed symbols indicate the times where the S(q) in the main panel is shown. (From Foffi G. et al. 2005. J. Chem. Phys. 122: 224903; Sear R.P. and Gelbart W.M. 1999. J. Chem. Phys. 110: 4582–4588. With permission.)

high-density gels only requires local rearrangements (see the structure factor), but only increases the relaxation time (following a power law with the waiting time, as shown in the lower panel). The correlation functions can be time-rescaled to collapse onto a master decay, what allows an unambiguous determination of the relaxation time [23,43]. For stronger attractions, the localization length decreases, as deduced from the increase of the nonergodicity parameter (the upper panel in the right), in agreement with states deeper in the glass phase. These results indicate that the dynamics of the system is similar to glass aging. At lower density, on the other hand, the structural evolution of the system is more dramatic. In Figure 7.13, the evolution of the structure factor is presented for an SW mixture at low temperatures [44]. The change in S(q), until it finally stops evolving, is more pronounced for lower density. This implies that the structure is much more heterogeneous at lower density than at higher ones, as shown in the snapshots in Figure 7.10. In other words, the fluid–fluid separation has been arrested at a later stage, and it affects more strongly the dynamics of the system. In order to avoid the difficulties associated with the structural heterogeneities at low density, the attractive glass transition has been investigated at high density in systems or states where the fluid– fluid transition is not present. In Section 7.4.2.5, we will go back to the problem at low density. 7.4.2.2 Simulations of Attractive Glasses Different strategies have been used to reach the attractive glass transition from fluid states avoiding the possible effects from the fluid–fluid demixing (crystallization is always avoided using a binary or polydisperse mixture of particles). One possibility is to study the attractive glass in systems with very high density, larger than the denser fluid in the phase separation. On the other hand, interaction potentials that hinder phase separation, keeping the short-range attraction, have been used. The left panel of Figure 7.14 shows the isodiffusitivity lines—lines joining states where the diffusion coefficient is constant—for the SW mixture of Figure 7.13 in the high-density region up to the liquid branch of the spinodal [22]. The overall shape of the lines, and its extrapolation to zero diffusion coefficient, follows the theoretical predictions by MCT (inset), particularly in finding the fluid pocket between the attraction-driven (low T) and repulsion-driven (high T) glasses. The MSD along an isodiffusivity line shows that the trapping distance is different at low or high temperatures, signaling the different origin of both glasses (steric hindrance for the repulsive glass and bonding for the attractive one).

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FIGURE 7.14 Attractive glass in the SW system. Isodiffusivity lines (left) finishing in the liquid–gas spinodal (the inset shows the MCT prediction for the same system), and the MSD for different states along the isodiffusivity line with lowest D (right). (From Zaccarelli E. et al. 2002. Phys. Rev. E 66: 041402. With permission.)

These results, however, are restricted to high densities, beyond the liquid branch of the fluid–fluid separation. In order to study the transition at lower density, it is necessary to prevent the phase separation. Figure 7.15 presents the MSD for different states along the isochore j = 0.50 for the AO system with the repulsive barrier presented in Figure 7.3. Similar to the previous figure, this one shows the re-entrant glass (note the disappearance and reappearance of dynamical slowing down as the attraction strengthens), with different driving mechanisms for weak and strong attractions [38]. The density is lower in this case than in Figure 7.14, since the repulsive barrier inhibits the fluid– fluid separation. This shows that the rich scenario predicted by MCT does not change qualitatively due to the long-range repulsion added to the system, and is unaffected by the details of the attraction. This system, therefore, can be used to analyze the properties of the glass transition reached from the fluid side, without effects from the phase transition at moderate density. Before studying the dynamics of the system, it is necessary to analyze its structure to answer the question: How does the system avoid the phase separation? The structure factor is presented in Figure 7.16 for different

(a)

100

(b) bUmin= 0 bUmin= –0.25 bUmin= –0.81 bUmin= –1.98

6

10–1 Ds/DsHS

/R2

101

10–2

10–3

0

1

–bUmin 3 2

4

5

bUmin= –2.88 bUmin= –3.39 bUmin= –4.11 bUmin= –4.48 bUmin= –4.82

4 2 0

10–1

100

101

t(arb. units)

102

10–1 100

101

102

103

104

t(arb. units)

FIGURE 7.15 MSD for the AO system with a repulsive barrier for different states with volume fraction j = 0.50 and the attractions strength as labeled. The inset shows the diffusion coefficient from the long-time slope of the MSD. (From Pham K.N. et al. 2002. Science 296: 104–106. With permission.)

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Glasses in Colloidal Systems: Attractive Interactions and Gelation 3

1 fp = 0.42 fp = 0.35 fp = 0.20 fp = 0

2.5

Increasing fp Fq (t)

S(q)

2

0.8

1.5 1

0.4 0.2

0.5 0 0

0.6

2.5

5

7.5 qa

10

12.5

0 10–2

10–1

100

101 t

102

103

104

FIGURE 7.16 Structure factor of the AO system with a repulsive barrier for different states with j = 0.40, as labeled (small-q peak increasing with j p). The panel in the right presents the density correlation functions for states in the same isochore for the third peak in S(q)–qa = 9.9 (a is the particle radius) from j p = 0 up to j p = 0.42 (j p = 0, 0.10, 0.20, 0.30, 0.35, 0.375, 0.40, 0.405, 0.41, 0.415, 0.42).

states along the isochore j = 0.40. A clear (pre)peak appears at low wavevectors, indicating the formation of holes in the system, and denser regions elsewhere. This peak shows the tendency of the system to undergo microphase separation [45,46], although for the states in the figure, the peak is too small to claim that this separation has occurred. Additionally, the increasing amplitude of the oscillations at large wavevectors signals the local reordering due to the short-ranged attraction (compare with Figure 7.5) [25]. The density correlation functions of this system for different states increasing the attraction up to the glass transition is presented in the right panel of Figure 7.16, for the wavevector qa = 9.9, which corresponds to the third peak in S(q). A two-step decay, typical of glass-forming systems, is observed for high attractions, with an important stretching of the second decay. The height of the plateau, the nonergodicity parameter, is much larger than that for the repulsion-driven glass (comparison shown in Figure 7.18), and is in agreement with the shorter localization lengths of attractive glasses with respect to repulsive ones (see Figure 7.8). The analysis of fluid states close to the glass transition can be performed using the predictions from MCT in Section 7.2. Figure 7.17 presents the von Schweidler fittings to the correlation functions (left panel) and the increase of the time scale for structural relaxation, viscosity, and diffusion coefficient, fitted according to power-law divergences (right panel). The von Schweidler exponent, which measures the stretching of the structural decay, is lower than that for HS, as predicted by MCT: b = 0.37 as compared to b = 0.53 for HS [25,47]. With this value of b, the exponent for the power-law divergence is, within MCT, g = 3.44, which agrees with the value found from free fitting for the time scale, g = 3.23, but disagrees with the value for the diffusion coefficient, g = 1.33; a disagreement also found in other (repulsive) glasses [48]. The increase of the time scale for structural relaxation is also noticeable in the stress correlation function, that is, the time scale for relaxation of local stresses also increases. As stated by the Green–Kubo relation, the viscosity can be calculated as the time integral of this stress correlation function [49], and is shown in Figure 7.17 (right panel) as the transition is approached. A power-law divergence is found, with the same exponent as for the density correlator time scale, g = 3.14, again in agreement with the MCT prediction [50,51]. The Fourier transform of the stress correlation (multiplied by the frequency and imaginary unit) gives the elastic and shear moduli. The elastic modulus shows a plateau appearing at low frequencies, and the viscous one develops a maximum followed by a minimum also at low frequencies [50,51], in agreement with the predictions from MCT but also with experimental results of attractive glasses [52]; they can be compared with corresponding measurements in the HS system [19].

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Structure and Functional Properties of Colloidal Systems 1

104

fpG = 0.4265

0.8

Fq (t)

103 0.6

102 0.4

101 0.2

100 0

10–2

10–1

100

101 t

102

103

104

10–2

10–1 fpG

– fp

FIGURE 7.17 MCT analysis of fluid states close to the attractive glass. The panel in the left shows the von Schweidler fits at different wavevectors (q increases from top to bottom) for the state j = 0.40 and j p = 0.42. (From Puertas et al. 2005. J. Phys. Chem. B 109: 6666–6675. With permission.) The panel in the right shows the structural relaxation time scale (open circles), the viscosity (filled circles), and the self-diffusion coefficient (crosses) as a function of the distance to the transition. (From Puertas, A.M. 2005. J. Phys.: Condens. Matter 17: L271–L277. With permission.)

A more quantitative comparison can be performed taking the structure factor from the simulations and using it as input of MCT—the system has to be made monodisperse to provide the correct input for the conditions of the theory, but the repulsive barrier is present. In Figure 7.18, the nonergodicity parameter and time scale for structural relaxation calculated with MCT are compared with direct measurements in the simulations, and the effects of polydispersity and the repulsive barrier tested correspond (A) to results for a monodisperse system without a barrier, (B) to a monodisperse one with a barrier, and (C) to the polydisperse one with a barrier. Semiquantitative agreement is found in these properties with no adjustable parameters when a monodisperse system is considered, showing that MCT correctly captures the properties of the transition and that the long-range repulsion does not affect it. For comparison, the nonergodicity parameter from simulations of HS is also

FIGURE 7.18 MCT analysis of fluid states close to the attractive glass. With the structure factor from simulations of the system, the nonergodicity parameter (left panel) and time scale (right panel) from the von Schweidler analysis are compared with the theoretical calculations (continuous lines). The structure factors are included in the left panel, to be read in the right scale. (A) Monodisperse system without the repulsive barrier, (B) with the barrier, and (C) polydisperse. The nonergodicity parameter for HS is also included for comparison. (From Henrich O. et al. 2007. Phys. Rev. E 76: 031404. With permission.)

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Glasses in Colloidal Systems: Attractive Interactions and Gelation 1

0 (b) nmax= 4

s(0)=1.3 s(0)=2.7 spinodal percolation DT –1/2=0.1 DT –1/2=0.01 DT –1/2=0.001 DT –1/2=0.0001 fc

T

0.6 0.4

(b) nmax= 4

–1 –2 log10D

0.8

–3

f = 0.20 f = 0.25 f = 0.30 f = 0.35 f = 0.40 f = 0.45 f = 0.50 f = 0.55

–4 –5

0.2 0

–6

gel 0

0.1

0.2

0.3

f

0.4

0.5

–7

0

2

4

6 1/T

8

10

12

FIGURE 7.19 Analysis of the limited valance model with nmax = 4. Isodiffusivity lines with the spinodal (left panel) and dependence of the diffusion coefficient along the isochores indicated as a function of the inverse temperature (right panel). The thick line at the bottom of the left panel shows the expected glass line from the Arrhenius plots. (From Zaccarelli E. et al. 2006. J. Chem. Phys. 124: 124908. With permission.)

included (black points), showing the big difference between fq in the attractive and repulsive glasses. However, the transition point is overestimated by a factor close to two (j pc = 0.426 in the simulations and j pc = 0.246 in MCT), following the typical trend of MCT of overestimating the driving force for the transition [48]. The von Schweidler exponent is also incorrectly calculated in the theory, probably due to different distance from the transition point to the high-order singularity (where the exponent g diverges, and thus b goes to zero) [53]. The time scale for structural relaxation (right panel) shows that the slowest modes to relax are those connected with the prepeak in S(q), although the transition is not driven by them. Due to the wrong determination of the exponents within the MCT calculation, the large-q behavior of tq is incorrectly reproduced (see the inset). Alternatively, the fluid–fluid separation has been avoided using models with a maximum number of interacting neighbors, nmax [54]. Figure 7.19 shows the phase diagram of this system for nmax = 4, with several isodiffusivity lines. Note that the liquid–gas separation is restricted to low density, and the liquid branch of the spinodal has a volume fraction below j = 0.30 for this value of nmax that allows the fluid phase to extend to low temperatures. In this system, however, the diffusion coefficient decreases with temperature, according to the Arrhenius law (right panel), showing that the glass transition driven by the attractive interactions is found here only at zero temperature [54]. This proves that a large number of bonds per particle must be formed in order to completely arrest the local dynamics of the particle at a finite temperature; this is achieved in systems where a larger number of neighbors are allowed, or where a dense liquid phase is formed. 7.4.2.3 Dynamical Heterogeneities Although the results presented in the previous section show that MCT provides a good description of the attraction-driven glass, and even anticipated its main properties, MCT considers the system dynamically homogeneous and does not describe heterogeneities at the particle level. Dynamical heterogeneities of increasing importance, as the transition point is reached, have been described in simulations of repulsion-driven glasses (particles of increased mobility group together forming movable regions [55]). In attraction-driven glasses, dynamical heterogeneities are also present, as can be seen in Figure 7.20, where the distribution of squared displacements is shown for different times for four states close to the glass transition in the AO system with a repulsive barrier. Note that upon increasing the attraction strength, the system becomes very heterogeneous from the dynamical point of view; the structural relaxation occurs in widely different time scales for different particles, and two types of particles can be recognized for the closest state to the transition: fast and slow ones, according to the

156

Structure and Functional Properties of Colloidal Systems

Distribution

0.15

fp = 0.39

fp = 0.40

fp = 0.41

fp = 0.42

0.1 0.05 0

Distribution

0.15 0.1 0.05 0

10–2 10–1 100 dr 2

101

102

103 10–2 10–1 100

101

102

103

dr 2

FIGURE 7.20 Analysis of the squared displacements of a system close to the attractive glass in the AO system with barrier. The distributions for different states with volume fraction j = 0.40 as labeled, different curves correspond to equispaced times such that the MSD is a2 for the shortest 50a2 for the largest, increasing from left to right (the attractive glass for this system is at j p = 0.4265). (From Puertas A.M. et al. 2005. J. Phys. Chem. B 109: 6666–6675. With permission.)

two maxima in the distribution of displacements in the figure [56,57]. This binary distribution of relaxation times has also been observed in other gel-forming systems [58,59]. Detailed analysis of the systems shows that the particles that take longer to diffuse are in the inner parts of the gel, forming the skeleton of the network, whereas the fast ones are in the outer parts, forming the skin. The structural heterogeneity of the system is therefore responsible for the dynamical one, in contrast to repulsive glasses, where the structure is homogeneous. Experimental confirmation of the dynamical heterogeneities in colloidal repulsive glasses was given by Weeks et al. [60] using confocal microscopy in a colloidal system, which allows access to the position of all particles. In attractive colloids, similar experiments have also shown the existence of dynamical heterogeneities [61,62], revealed by studying the distribution of displacements (in Figure 7.21 for one dimension), which do not obey a Gaussian distribution. Different populations of particles with different mobilities can be identified, associated with the structural heterogeneities [63]. 7.4.2.4 High-Order Singularities As described in the theoretical section, MCT predicts a high-order singularity when two glass transitions with different driving mechanisms coexist, in the vicinity of the merging point [8]. In attractive colloids, the attraction- and repulsion-driven glasses meet in the region of high density and strong interactions, and the particular type of singularity depends on the details of the interactions [37]. Perhaps the most impressive consequence of the presence of this high-order singularity is noted in the density correlation function, where a logarithmic decay from the short-time relaxation to the structural one is predicted. This signature has been thoroughly looked for and found in the predicted region. In Figure 7.22, evidence from simulations of the binary SW system is shown [64], where the logarithmic decay is clearly visible for the marked wavevectors. The curvature of the correlation function changes from concave to convex at a particular wavevector q*, where the logarithmic trace is most clear. All these features are correctly predicted by MCT and found in different simulations irrespective of the details of the attraction (AO or SW) [64,65].

157

Glasses in Colloidal Systems: Attractive Interactions and Gelation

10–1

(a)

10–3

10–5

Gs(x, t)

10–1

(b)

10–3

10–5 (c)

s(t)/a

10–1

0.6 0.3 0

10–3

10–5 –4

–2

0

2

0

4 2 t [s] ¥ 104

4

6

x/a

FIGURE 7.21 One-dimensional distribution of particle displacements at different times increasing from (a) to (c). Slow and fast populations can be recognized from the Gaussian fits to the data (spiky for the slow and wide for the fast ones). The inset in the lower panel shows the width of both Gaussians as a function of time. (From Gao Y. and Kilfoil M.L. 2007. Phys. Rev. Lett. 99: 078301. With permission.)

1

Fq(t)

0.5

q1*

0 1

0.5

q2*

0 10–2

10–1

100

101

102

103

104

105

106

t

FIGURE 7.22 Density correlation function for two SW systems: 3.1% width (upper panel) and 4.3% width (lower panel), both with the same volume fraction and temperature: j = 0.6075 and T = 0.4. The former has an A3-type singularity and the latter an A4-type one. Different lines correspond to different wavevectors and dashed lines are fits to a second-order expansion in log(t) in the range marked by the vertical lines. (From Sciortino F. et al. 2003. Phys. Rev. Lett. 91: 268301. With permission.)

158

Structure and Functional Properties of Colloidal Systems 1.0

1.000

F(k,t)

f (q, t)

0.9 0.8 0.999

0.7 0.998 0.997 0

0.2

0.4

0.6

0.8 ¥10–3

0.6 10–6 10–5 10–4 10–3 10–2 10–1 100 101 102 103 104 (qR)2 t/h

r (s)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 295 K 0.3 298 K 300 K 0.2 321 K 0.1 0.0 –1 100 101 10

A

f = 0.535

102

103 104 t (ms)

105

106

107

FIGURE 7.23 Experimental evidences of the logarithmic decay of the density correlation functions. In the left panel, a colloid–polymer mixture with constant colloid volume fraction j 0.64, the lower curve is a repulsive glass, the dark one is a fluid state between the two glasses, and the upper one is an attractive glass [39]. (From Pham et al. 2004. Phys. Rev. E 69: 011503. With permission.) In the right panel, micellar systems where the attraction is induced by increasing temperature are studied [67]. (From Chen S.-H. et al. 2003. Science 300: 619–622. With permission.)

Experiments have been performed on different systems where attractive interactions could be induced, such as colloid–polymer mixtures [39,66] or micellar systems [67]. Figure 7.23 shows the correlation functions of these systems obtained from dynamic light scattering, confirming the existence of the high-order singularity in the region where MCT predicts it. 7.4.2.5 Attractive Glasses at Low Density: Gelation In the previous sections, we have shown the attraction-driven glass transition that is present at a high density, where MCT correctly predicts many of its properties, even at a quantitative level. In this section, we discuss how this transition extends to lower density and eventually connects to the experiments of colloidal gelation and with the well-known aggregation regime DLCA [41]. As the density is lowered, fewer particles are available to form the percolating structure, resulting in thinner or less branched gels. Therefore, stronger attractions are necessary to provoke dynamic arrest, that is, the glass line moves to higher attraction strengths, or lower temperatures. Accordingly, for these transitions MCT predicts that the localization length decreases with decreasing density. However, because the density of the gel is lower, it is more flexible and flapping motions of branches or arms cause decorrelation at larger distances for lower density, even though the whole structure may not relax. This is indeed observed in simulations of attractive systems, as shown in Figure 7.24 for the ideal models presented above, the maximum valence, nmax (left panel), and the AO system with long-range repulsion (right panel). The results for the nmax model show that the plateau in the MSD increases with decreasing density (at constant temperature T = 0.125; note that the transition in this model is at T = 0, as shown in Figure 7.19), signaling less tight structures (the localization length in the inset shows this trend). Accordingly, the nonergodicity parameter decreases at constant wavevector, as shown in reference [54] for the nmax model, and in the right panel for the AO system with a repulsive barrier. It is interesting to note that in the AO system, upon decreasing the density from j = 0.55, the nonergodicity parameter first increases (following the trend in MCT), and then decreases, whereas in the nmax model it decreases continuously, in agreement with the different mechanisms in both systems, which ultimately causes the transition line at zero temperature in the latter. With competing interactions, it has been shown that the fluid– fluid transition can be avoided also at low density, but ordered compact local structures form, namely, the Bernal spiral, which are capable of provoking dynamic arrest, even with a small number of neighbors [28,68].

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Glasses in Colloidal Systems: Attractive Interactions and Gelation

/s2

100 10–1 10–2

1

f=0.30 f=0.35 f=0.40 f=0.45 f=0.50 f=0.51 l02 f=0.52 f=0.53 f=0.54 f=0.55

1

0.4

l 0.5 0 0.2

10–2 10–1

0.6

Increasing f

10–3 10–4

fc=0.55 fc=0.40 fc=0.25

0.8

fq

101

100

101

102 t

0.2 0.3

0.4 f 0.5

103

104

105

0

0

10

20 qa

30

40

FIGURE 7.24 Extension of the attractive glass to low densities. MSD of fluid states close to the attractiondriven glass (at constant temperature T = 0.125) for the limited valence model (nmax = 4): at different densities— left panel. The inset in this panel shows the evolution of the localization length with packing fraction. Critical nonergodicity parameters for the AO system with barrier at the densities indicated (right panel). (Reproduced from Zaccarelli E. et al. 2006. J. Chem. Phys. 124: 124908. With permission.)

Calculations with MCT have been attempted by Wu et al. [69] using a model with a short-range Yukawa attraction supplemented by a long-range Yukawa repulsion. The structure factors and nonergodicity parameters reproduce the low wavevector peak, or cluster peak, which grows and exceeds the neighbor peak, that is, compatible with arrested phase separation. However, although these calculations were extended to very low volume fractions, the trend observed in the figure above for the localization length was not reproduced, that is, the flapping modes are missing in the theory. It is therefore necessary to incorporate them in MCT “by hand” or, alternatively, develop a scheme where these modes appear naturally. These results introduce an important difference between attraction-driven glasses and gels with regard to the mechanisms for structural relaxation. This difference is also apparent in the formation of the gel. When a fluid is “quenched” beyond its glass point, the dynamics arrests progressively; the structural relaxation takes longer and longer, but the nonergodicity parameter does not grow, that is, structural changes are not occurring. This phenomenology has been described with computer simulations in atomic glasses or repulsion-driven colloidal glasses [70], and recently in attraction-driven glasses [43]. For the latter, the typical behavior of the density correlation function is shown in Figure 7.25 for the AO system with a repulsive barrier (left panel). In contrast, in colloidal gels, the formation of the structure competes with structural relaxations, and a first process where the nonergodicity parameter increases with the waiting time since the quench is found, followed by the typical aging of glasses (increase of structural relaxation time with the age of the sample). The right panel in the figure shows the formation of bonds in an SW mixture [44], but not the increase of the structural relaxation time. Evidence from experiment and simulations for this type of aging of colloidal gels has been provided by several groups [71,72]; but also low-density gels with the typical aging of dense glasses are observed [73]. 7.4.2.6 Discussion: Gels and Attractive Glasses The results presented above show the existence of a glass transition that is driven by the formation of quasi-permanent bonds between particles due to a short-range attraction between particles. This transition can occur in the region where fluid–fluid separation is favored thermodynamically, and therefore a competition between both processes can take place. Several models have been designed to avoid this competition and to isolate the effects of the glass transition. In experiments, however, one cannot prevent phase separation, and the final stage of the system is controlled by the competition. It is now generally accepted that most of the gels found experimentally

160

Structure and Functional Properties of Colloidal Systems 1.0

1

0.8

0.75

tw=610

fq(t)

Fqs (t)

tw=164 0.6

0.5

0.4

tw=97 0.25

0.2 0.0 –2 10–1 10

100

101 t

102

103

104

f=0.25 qsB=20

0 10–2

10–1

tw=15 100

101

102

103

104

105

t–tw

FIGURE 7.25 Density correlation function for the AO system with repulsive barrier at j = 0.40 and j p = 0.80, for different waiting times tw = 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16,384, from left to right (left panel) and for the SW system at low density (liquid–gas separation stops due to the dynamic arrest) (right panel). Note that the plateau increases with waiting time after the quench, similar to experimental gels, due to the formation of denser and more arrested structures. (Right panel reproduced from Foffi G. et al. 2005. J. Chem. Phys. 122: 224903. With permission.)

are the result of arrested demixing due to the crossing of the attractive glass line with the liquid branch of the spinodal [74,75]. In this picture, liquid–gas separation would be obtained in the temperature range between the critical point and the crossing point, and gels for lower temperatures (stronger attractions). Only when the attractive glass line is crossed, phase separation is arrested even for short times, and more homogeneous systems are obtained [76]. It is also argued that for short-range interactions, the fluid–fluid binodal is very flat, and the glass line follows it; therefore, any quench inside the phase separation region results in a gel, according to the absence of liquid–gas separation in many experimental systems [77]. In any case, it is now evident that a process that produces density fluctuations and stabilizes dense regions in the system is necessary in order to have an attractive glass and/or gel transitions, either in fluid–fluid separation or in short-range attractions with longer-range repulsive interactions.

7.5

CONCLUSIONS

In this chapter, we have presented the analysis of gelation in systems with short-range attractions, mainly from the point of view of computer simulations and theory. The analysis presented here indicates that gelation at high density should be viewed as an attraction-driven glass transition in agreement with the theoretical predictions from MCT. The fluid–fluid separation, which may take place in the same region of the phase diagram, is completely arrested, and the dynamics of the system is controlled by the glassy dynamics. At low density, new relaxation mechanisms play a dominant role, which are absent at higher density, needing further theoretical developments. For the rationalization of experiments on gels, the competition between vitrification and fluid–fluid separation must be considered, especially at low density. Within MCT, the fluid structure factor controls the equations of motion of the structural relaxation, and glass transitions are identified as bifurcation points. For fluids with short-range attractions, MCT predicts an attraction-driven glass transition induced by the formation of quasi-permanent bonds between particles, in addition to the repulsion-driven one, caused by the caging of hard particles. In the region where these two glass transitions merge, a high-order singularity is predicted, with peculiar properties of the relaxation mechanism. Experiments and simulations have shown that these predictions are fulfilled, even nearly at a quantitative level when the structure factor from simulations is taken as input for MCT. Yet, some quantitative discrepancies remain, as usually found in MCT analysis of the glass transition: the transition point is incorrectly located and different exponents are found for the divergence of the time scale and diffusion coefficient.

Glasses in Colloidal Systems: Attractive Interactions and Gelation

161

At the particle level, the system is heterogeneous, both structurally and dynamically. Structural heterogeneities are caused by the local clustering induced by the bonds that open up holes in the elsewhere system—different mobilities are thus observed for particles in different environments. These dynamical heterogeneities are more dramatic in attraction-driven glasses than in repulsiondriven ones, but are absent in MCT, what poses an interesting dichotomy. Interestingly, for low densities, a new mechanism for the relaxation of mesostructures (the size of a few particles) appears, that is, cluster motion or flapping of branches. This provokes an increase in the localization length and a concomitant decrease in the nonergodicity parameter. It must also be noticed that theoretical calculations using MCT miss these relaxation modes. This relaxation mechanism can serve to distinguish gels from attractive glasses, as observed in dilute systems in contrast to the dense ones, with implications for the elasticity of the final structure. Finally, several routes to gelation have been presented: systems with arrested spinodal decomposition and without states of fluid–fluid coexistence, systems where the fluid–fluid separation is arrested only at very low temperatures (but can be observed at higher ones), and systems where the fluid–fluid transition is suppressed. Whereas the former probably applies to most experiments with short-range attractions (colloid–polymer mixtures), the second mechanism is found when longerranged attractions are present (particularly in the case of proteins). The latter has been extensively studied by computer simulations since it allows access to the attractive glass transition from homogeneous fluid states and is the main objective of the present work.

ACKNOWLEDGMENTS Financial support from the DAAD and the Spanish Ministerio de Educación y Ciencia, project HA2004-0022, the IFPRI (MF), and the MEC—project MAT2006-13646-C03-02- (AMP) is acknowledged. We thank a long list of colleagues for scientific collaboration and for very pleasant discussions, particularly Mike Cates, Johan Bergenholtz, Francesco Sciortino, and Wilson Poon.

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8

Phase Behavior and Structure of Colloidal Suspensions in Bulk, Conﬁnement, and External Fields A.-P. Hynninen, A. Fortini, and M. Dijkstra

CONTENTS 8.1 8.2

Introduction ...................................................................................................................... Steric and Charge-Stabilized Colloids ............................................................................. 8.2.1 Binary Mixtures of Large and Small Hard Spheres ............................................. 8.2.2 Primitive Model .................................................................................................... 8.2.3 DLVO Theory ....................................................................................................... 8.2.4 Phase Behavior of Hard-Core Yukawa Particles .................................................. 8.2.5 Crystal Structures of Oppositely Charged Colloids with Size Ratio 0.31 ............ 8.2.6 Oppositely Charged Colloids with Size Ratio 1 ................................................... 8.3 Colloids in External Fields ............................................................................................... 8.3.1 Manipulation of Colloidal Crystal Structures by an Electric Field ...................... 8.3.2 Manipulation of Colloidal Crystal Structures by Confinement ............................ References ..................................................................................................................................

8.1

165 166 167 168 168 170 175 180 184 184 190 196

INTRODUCTION

Colloids are particles with a size in the nano- to micrometer range that are dispersed in a solvent, and that due to collisions from the solvent molecules undergo Brownian motion [1]. As colloids are much larger than molecules and much smaller than macroscopic objects, they possess peculiar thermodynamic, rheological, and optical properties. Examples of colloids can be found in almost every aspect of life. In biology, red blood cells, bacteria, and viruses are all in the colloidal regime. Colloids are used in paint, food (e.g., mayonnaise, butter, and milk), and makeup products as stabilizers that keep the structure of the product unchanged. Magnetic colloids can be found in stateof-the-art bridges where they act as cable dampers. The most familiar of the optical properties is the scattering of light. Colloids arranged on a regular crystal lattice act as a three-dimensional diffraction grating that splits white light into its components. In nature, elegant examples of this are the colors of peacock feathers, butterfly wings, and gem opals. Optical properties of colloids also have industrial applications as photonic crystals, which can be built from colloids by utilizing their ability to self-organize. The tunability of the interactions, size, shape, and composition have made colloids an important model system for atoms and molecules as they, like atoms and 165

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Structure and Functional Properties of Colloidal Systems

molecules, have a well-defined thermodynamic temperature and manifest many of the same phenomena, but are easier to study due to their greater size and longer time scales. For example, laser confocal microscopy enables real-time and real-space studies of colloids in three dimensions on a single particle level. This makes it possible to perform detailed studies of, for example, crystal nucleation and growth [9], glass transition [2], and solid–solid transitions [3]. The tunability of the interactions can be achieved in many ways. For example, in charged colloidal suspensions, the colloid pair potential can be tuned from soft long-range repulsive to essentially hard-sphere-like [3–9] by adding salt. In a binary mixture of two species of colloids, added salt can even reverse the charge on one of the species, resulting in oppositely charged colloids [10]. On the other hand, in uncharged colloidal suspensions, addition of polymer gives rise to a short-range depletion attraction [11]. Also, nonspherically symmetric interactions can be realized. A dipole–dipole interaction between colloids can be achieved either by using specially prepared ferromagnetic particles [12] or by exposing the colloids to an external electric [9] or magnetic field [13]. Colloids with a well-characterized size, shape, and interactions are a perfect subject for computer simulations. Computer simulations, which are typically quicker, easier, and cheaper to perform than experiments, have become indispensable for predicting the phase behavior of colloids and explaining the experimental results. The rest of this chapter is organized as follows. In Section 8.2, we give an introduction to sterically and charge-stabilized colloids and present theoretical models that are used to describe them. In Section 8.3, we present examples of external fields that can be used to manipulate colloids.

8.2

STERIC AND CHARGE-STABILIZED COLLOIDS

In order to stabilize a colloidal suspension against irreversible aggregation due to strong van der Waals attractions between the colloids, two mechanisms are common: charge and steric stabilization. In the case of steric stabilization, the colloidal particles are coated with a polymer layer, which leads to a steep repulsive interaction between the colloids when they approach each other; the colloidal pair potential can therefore be regarded as hard-sphere-like. The hard-sphere system has been studied in great detail. In 1957, Wood and Jacobson [14] and Alder and Wainwright [15] showed by computer simulations that a system of purely repulsive hard spheres has a well-defined freezing transition. The origin of this freezing transition is purely entropic and occurs because the entropy of the crystalline phase is higher than that of the fluid phase at sufficiently high densities. The location of the hard-sphere freezing transition was determined from simulations by Hoover and Ree, who found that the packing fractions of the coexisting fluid and face-centered-cubic (fcc) solid phase are given by h = ps3N/6V = 0.494 and 0.545 with N the number of particles and V the volume, which corresponds to a pressure Ps3/k BT = 11.69 with s the diameter of the hard spheres [16]. On the other hand, colloidal particles can also be stabilized by a net surface charge that they acquire when the particles are dispersed in a solvent. For simplicity, we assume that the surface charge is homogeneous, although in reality the charge is most likely concentrated on charge centers. The ions that are released from the colloidal particles due to dissociation of chemical groups at the surface are called counterions as they carry a charge that is opposite to the colloid charge. In addition to the counterions, most solvents also contain a certain concentration of dissociated coions and counterions. For example, water in room temperature contains 10-7 M of H+ and OH- ions. The total excess amount of co- and counterions is called “added salt” or just simply “salt.” The salt concentration can be lowered by deionizing the solvent and increased by adding a salt solution. Because the counterions undergo thermal motion but are still attracted to the colloids electrostatically, they build up a layer of opposite charge around each colloid. Thus, we obtain what is called an electric double layer where the first layer is the colloid surface charge and the second is the cloud of counterions surrounding the colloid surface. This section is divided into four subsections. In Section 8.2.1, we discuss the phase behavior of a binary mixture of large and small steric-stabilized colloidal hard spheres. In Section 8.2.2, we discuss the so-called primitive model, where the co- and counterions

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are taken into account explicitly, and we discuss the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory that is standardly used to describe charged colloids and where the co- and counterions are integrated out. We then present results on the phase behavior of charged colloidal particles and of oppositely charged colloids.

8.2.1

BINARY MIXTURES OF LARGE AND SMALL HARD SPHERES

We already discussed that a system of pure sterical stabilized colloidal hard spheres exhibits a fluid– solid transition with coexisting packing fractions at hfluid = 0.494 and hsolid = 0.545. We now determine the phase diagram of a binary mixture of sterical stabilized colloidal hard spheres with a size ratio of 0.82. To this end, we determine the Gibbs free energy G = F + PV, where V is the volume, P is the pressure, and F is the Helmholtz free energy. We calculated the Helmholtz free energy and the equation of state numerically for the so-called Laves phases, for example, MgCu2, MgNi2, and MgZn2 structure, using thermodynamic integration. The free energies of the fluid and fcc phases are taken from analytical results [17,18]. Subsequently, the phase diagram was constructed from the Gibbs free energy data using the common tangent construction in the Gibbs free energy G, composition x = NS/(NL + NS)-plane. Figure 8.1 shows the phase diagram of binary hard spheres with a size ratio of 0.82 in the reduced pressure p = Ps 3L/k BT, composition x presentation, where k B is Boltzmann’s constant and T is the absolute temperature. The phase diagram consists mainly of coexistence regions where the tie lines are horizontal. For example, in the region marked “fccL + Laves,” the stable system has an fcc crystal of large spheres at x = 0 in coexistence with Laves structures at x = 2/3 = 0.667. The pure one-phase regions are at x = 0, x = 2/3, and x = 1 and in the fluid phase. The phase diagram can also be drawn, for example, in the packing fraction h, composition x plane, where the pressure region p = 22.8–59.6 of stable Laves structures corresponds to packing fractions h = 0.59–0.66. We found the Laves structures to be stable at size ratios in the 0.74–0.84 range. Thus, the main features of the phase diagram in Figure 8.1 are expected to hold in this size ratio range. Recently, the Laves structures MgNi2 and MgZn2 have also been observed in experiments with binary nanoparticle suspensions [19]. The relative stability of the three different Laves structures was studied by calculating their Helmholtz free energies [21] with extrapolation to the infinite system limit [22]. As all of the Laves structures pack with the same volume fraction (h), it is not

50

p

fccL + fccS

+

60

Laves + fccS

fccL + Laves

+

40 + 30

Laves + fluid

20

Fluid + fccS

fccL + fluid Fluid

10

0

0.2

0.4

x

0.6

0.8

1

FIGURE 8.1 Phase diagram of binary hard spheres with a small-to-large size ratio of 0.82. The phase diagram is shown in the composition x, reduced pressure p, representation, where x = NS/(NS + NL) is the number fraction of small spheres. The labels “fccL” and “fccS” denote the fcc crystals of large and small particles, respectively. (From Hynninen AP et al. 2007. Nature Materials 6: 202–205. With permission.)

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Structure and Functional Properties of Colloidal Systems

surprising that the free energy difference between them turned out to be very small, of the order of 10-3 k BT per particle (at h = 0.6 and a size ratio of 0.82). Owing to the small free energy differences between the Laves structures, we expect to observe a mix of all three structures in experiments, analogously to single-sized hard spheres where, instead of a pure fcc or hexagonal-close-packed (hcp), a random hexagonal-close-packed (rhcp) crystal is observed [23]. The stability of the MgCu2 is particularly interesting as the small spheres in this structure form a pyrochlore structure and the large spheres a diamond structure. Both the diamond and the pyrochlore structure could exhibit a photonic bandgap in the visible region. When the large and small spheres consist of different materials, one can remove selectively one of the species by burning or dissolution. In summary, we have proposed here a promising route to fabricate photonic crystals with a diamond or pyrochlore structure through self-assembly of the MgCu2 structure using a binary mixture of colloidal spheres and subsequently, the removal of one of the species.

8.2.2

PRIMITIVE MODEL

Colloidal suspensions can also be charge-stabilized, when the particles carry a surface charge. A detailed theoretical description of charged colloidal suspensions is given by the so-called primitive model. In this model, the colloids are charged spheres with diameter s and charge -Ze; the co- and counterions are charged spheres with diameter sI and charges -e and e, respectively. The solvent is treated as a uniform continuum with dielectric constant e s and temperature T. The interactions between all the species are taken into account via the Coulomb potential given by Ï Zi Z j l B u(r ) Ô = r kBT ÌÔ • Ó

for r ≥ s ij ,

(8.1)

for r < s ij ,

where lB = e2/e sk BT is the Bjerrum length, Zi and Zj are the charge numbers (-Z or ±1) of particles i and j, and sij is the hard-core distance between them: For colloid–colloid interaction sij = s, for colloid–microion interaction sij = (s + sI)/2, and for microion–microion interaction sij = sI. Although the primitive model is typically considered to capture most of the features of real charged colloidal suspensions, it is clearly a simplification. For example, systems that include hydrodynamic effects cannot be studied with the primitive model due to the simple treatment of the solvent. The main problem in using the primitive model is the difficulty of obtaining accurate predictions. In computer simulations of charged colloids, for example, simulations at the primitive model level typically become too demanding at Z > 100 due to the great number of counterions that have to be included to neutralize the colloid charge. Therefore, much of our understanding of charged colloids have come from studying simpler models where the co- and counterions are coarsegrained out. The most famous of these models is the DLVO theory, which is described in the following section.

8.2.3

DLVO THEORY

As mentioned above, the presence of counterions builds up an electric double layer around each colloid. The most important feature of the double layer is that it provides a repulsive interaction between two charged colloids. Without such repulsive interaction, colloids would aggregate due to London–van der Waals attractions. This charge stabilization mechanism was first introduced in the 1940s by Derjaguin and Landau [25], and independently by Verwey and Overbeek [24]. The DLVO theory assumes a single spherical colloid with charge -Ze and diameter s suspended in a continuum solvent with dielectric constant e s and temperature T. The solvent contains point-like

169

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(sI = 0) co- and counterions, whose density far from the colloid is 2rs. In the DLVO theory, the co- and counterion densities are given by the Boltzmann distribution and written as coion, r - (r) = rs exp[F(r)] r + (r) = rs exp[-F(r)] counterion,

(8.2)

where exp[F(r)/k BT ] is the dimensionless electric potential at distance r from the colloid center. Next, the Boltzmann distributions (Equation 8.2) are combined with the Poisson equation to give the Poisson–Boltzmann (PB) equation —2F(r) = -k2 sinh[-F(r)],

(8.3)

which is subject to boundary conditions F(r ) = 0 4l B Z n ◊ —F(r ) = s2

for r Æ •, for r =

s , 2

(8.4)

where k = (8plBrs)-1/2 is the inverse Debye screening length and n is a unit vector normal to the particle surface. For spherical particles, the PB Equation 8.3 has to be linearized in order to solve it analytically. The result of the linearization is — 2 F(r) = k2F(r).

(8.5)

The solution of the linearized PB Equation 8.5 with the boundary conditions in Equation 8.4 reads as F(r ) = -

Z l B exp(ks / 2) exp(-kr ) . s 1 + ks / 2 r /s

(8.6)

One readily checks from Equation 8.6 that the linearization is acceptable provided that max|F(r)| = |F(s/2)| Ⰶ 1 or if the constant proportional to ks is ignored, F(s/2) μ ZlB/s Ⰶ 1. That is, the linearization is permissible for low Z and/or for low lB/s. Equation 8.6 shows that the electric potential surrounding a colloid is given by a Coulomb potential which is screened by a factor exp(-kr). The pair potential between two charged colloids can be derived from the electric potential in Equation 8.6 by assuming that the two double layers do not disturb each other. The result is a screened Coulomb (or repulsive Yukawa) potential given by Ï Z 2l B exp[ -k (r - s )] for r ≥ s u(r ) Ô r = Ì (1 + ks / 2)2 kBT Ô for r < s Ó•

(8.7)

where we have included a hard-core repulsion that makes sure that the colloids do not overlap, and neglected the London–van der Waals attractions. The London–van der Waals attractions can be neglected if the refractive indices of the colloids and the solvent are matched or if Z is so high that the colloid–colloid pair interaction is dominated by the repulsive term, and therefore the colloids never come close enough to feel the attractive forces. In this chapter, we always assume that at least one of the above conditions is fulfilled and neglect the London–van der Waals forces. As can be seen from

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Structure and Functional Properties of Colloidal Systems

Equation 8.7, the screening length k determines the range of the pair interaction analogously to the thickness of the double layer: At high salt concentration, k is large and the repulsion is short-ranged, while at low salt concentration, k is small and the repulsion is long-ranged. Therefore, the interactions between charged colloids can be tuned by the amount of added salt. Due to its simplicity and accuracy, the DLVO potential (Equation 8.7) has become the most used model for spherical charged colloids. It correctly predicts the experimental phase diagram of charged colloids with stable fluid, body-centered-cubic (bcc), and fcc phases [5,8,26–28]. The DLVO theory is routinely used outside its original range of applicability. For example, in the case of highly charged colloids, where the linearization of the PB equation cannot be justified close to the colloid surface, it is customary to use a renormalized charge Z eff < Z that takes into account the nonlinear effects arising close to the surface [29]. Another example is the use of the DLVO theory at high colloid densities or low salt concentrations, where the nearby double layers overlap and the assumption of pairwise additivity becomes suspect. Although the phase diagram of point Yukawa particles is known from earlier studies [26,27], much less is known about the phase behavior of hard-core Yukawa particles. Intuitively, two limiting cases for the effect of the hard core can be considered: (i) In the limit of highly charged colloids or low density of colloids (or both), the particles hardly ever come sufficiently close to each other and therefore the effect of the hard-core diameter is minimal. (ii) In the other extreme of low charge or high density (or both), the hard-core interaction should play a large role. However, in order to make a more precise analysis, we decide to study the phase behavior of hard-core Yukawa particles systematically.

8.2.4 PHASE BEHAVIOR OF HARD-CORE YUKAWA PARTICLES In our model, the colloids interact via a hard-core repulsive Yukawa potential, given by Ï exp[ -ks(r / s - 1)] u(r ) Ôbe r/s =Ì kBT ÔÓ•

for r ≥ s

,

(8.8)

for r < s

where be is the value of the pair potential at contact per k BT, k is the inverse Debye screening length, and s is the hard-core diameter. The contact value is given by be =

Z 2lB . (1 + ks / 2)2

(8.9)

The total potential energy of N particles is given by the sum of all pairs as N

U (r N ) =

Â u(r ). ij

(8.10)

i< j

Subsequently, we take be, ks, and h as independent variables, and calculate the phase behavior in the three-dimensional space spanned by them. This choice is made for the sake of simplicity. In experimental charge-stabilized colloidal suspensions, all these variables depend on each other. Experimental parameters can be mapped on our phase diagrams by estimating the effective screening length ks and contact value be. Note that, in our phase diagrams, two phases in coexistence have equal pressure, chemical potential, ks, and be, but different h. We perform Monte Carlo (MC) simulations in a cubic box (with few exceptions) and with periodic boundary conditions. The cut-off radius of the potential was always chosen to be half of the box

171

Phase Behavior and Structure of Colloidal Suspensions

length and a continuous distribution of particles beyond the cut-off was assumed [21]. We limit ourselves to ks ≥ 2, where the effects of the finite cut-off remain small or can be eliminated by moderate increase of the system size. Our purpose is to use a combination of Helmholtz free energy calculations and the so-called Gibbs–Duhem integration method to trace out the phase diagram of the hard-core Yukawa particles. The phase diagram consists of stable regions of fluid, bcc, and fcc phases that are bounded by coexistence regions between any two phases. Therefore, the determination of the phase diagram reduces to the calculation of the coexistence lines. Points on the coexistence line can be determined by calculating, for each phase, the Helmholtz free energy per volume as a function of density and using the common tangent construction to obtain the densities of the coexisting phases. In principle, this could be repeated for every point to obtain a smooth coexistence line. However, this would be computationally very demanding and, as it turns out, not even necessary. This is because, once one point on the coexistence line is known, the rest of the line can be calculated by using the Gibbs–Duhem integration, without performing additional free energy calculations. The Helmholtz free energy of the solid phases is calculated using the Frenkel–Ladd method [21,30] using the Einstein crystal as the reference system, whereas the Helmholtz free energy of the fluid phase employs the hard-sphere fluid as a reference. Subsequently, we employ the Gibbs–Duhem method as first proposed by Kofke [31,32] to determine the phase coexistence lines in the (h, ks) plane for a fixed be. Using these methods, we study the phase behavior of hard-core Yukawa particles, whose interactions are described by the pair potential given by Equation 8.8. The phase diagrams are calculated for fixed contact values be and they are given in the (h, 1/ks) representation. We calculate the phase diagram for four contact

0.5

0.4

1/ ks

0.3

0.2

Fluid

bcc

fcc 0.1

0 0.35

0.45 h

0.55

FIGURE 8.2 Phase diagram of hard-core Yukawa particles with be = 8 presented in the (packing fraction h, Debye screening length 1/ks) plane. The lower part of the diagram (1/ks = 0) is high salt regime and the upper part (1/ks = 0.5) is low salt regime. The solid lines are coexistence lines obtained from Gibbs–Duhem integration and the gray areas denote the coexistence regions, where tie lines are horizontal. We find a stable fluid phase at low h, a stable fcc solid at high h, and in between a stable bcc solid. The dashed lines are the phase boundaries of point Yukawa particles by Hamaguchi et al. [26]. The squares (䊐) mark the starting points for Gibbs–Duhem integration and the circles (䊊) are control points from free energy calculations. (From Hynninen AP and Dijkstra M. 2003. Physical Review E 68: 021407. With permission.)

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Structure and Functional Properties of Colloidal Systems

values be = 8, 20, 39, and 81, and the results are given in Figures 8.2 through 8.5, respectively. In all four phase diagrams, the gray areas bounded by the solid lines give the coexistence regions (tie lines are horizontal), whereas the dashed lines give the point Yukawa phase boundaries of Hamaguchi et al. [26]. We like to refer the reader for more details on the mapping between the point Yukawa model and the hard-core Yukawa model to reference [33]. Let us first describe the structure of the phase diagrams in Figures 8.2 through 8.5. The phase diagrams start from the hard-sphere limit at 1/ks = 0 with coexisting fluid and fcc phases at packing fractions h = 0.491 and h = 0.543, respectively. As the softness and the range of the interactions increase with increasing screening length 1/ks, the fluid–fcc coexistence becomes thinner and moves to lower packing fractions in agreement with reference [34]. A further increase of 1/ks takes us to a fluid–bcc–fcc triple point. Here the softness and the range of the interactions are so pronounced that at lower packing fractions it is more favorable to form a bcc crystal than an fcc crystal. Increasing 1/ks from the triple point, two phase coexistence lines originate, namely, the fluid–bcc and the bcc–fcc. While the fluid–bcc coexistence line is relatively slowly varying, the bcc–fcc line moves quickly to higher packing fraction, producing a broad region of stable bcc phase. This is especially true for be = 20, 39, and 81 (Figures 8.3 through 8.5), and we can see that the steepness of this “shoot-up” behavior becomes more pronounced with large be. After the shoot-up, at higher values of 1/ks, the bcc–fcc coexistence line turns and behaves more or less as a straight vertical line at h ⬇ 0.5. At high 1/ks, the fluid–bcc coexistence line turns to higher packing fractions, that is, here the fluid phase becomes more favorable with respect to the bcc phase. This is because at high 1/ks, where the range of interactions becomes longer than the average interparticle spacing in the crystal, both the fluid and the bcc phases have a similar energetic contribution to the free energy, and thus the fluid phase wins since it has larger entropy. It is worthwhile to note that in all the phase diagrams in Figures 8.2 through 8.5, both the fluid–bcc and the bcc–fcc coexistence regions are very narrow, or in other words, the density difference between the two coexisting phases is small. In particular, the bcc–fcc coexistence region is extremely narrow: The largest density jump between the bcc and the fcc

0.5

0.4

Fluid

bcc

1/ ks

0.3

0.2

fcc

0.1

0 0.2

0.3

0.4 h

0.5

0.6

FIGURE 8.3 Phase diagram of hard-core Yukawa particles with be = 20 presented in the (h, 1/ks) plane. The symbols and lines are the same as in Figure 8.2. Note the difference in the h scale compared to Figures 8.2 through 8.5. (From Hynninen AP and Dijkstra M. 2003. Physical Review E 68: 021407. With permission.)

173

Phase Behavior and Structure of Colloidal Suspensions 0.5

0.4

bcc Fluid

1/ ks

0.3

fcc

0.2

0.1

0 0.1

0.2

0.3

0.4

0.5

0.6

h

FIGURE 8.4 Phase diagram of hard-core Yukawa particles with be = 39 presented in the (h, 1/ks) plane. The symbols and lines are the same as in Figure 8.2. Note the difference in the h scale compared to Figures 8.2 through 8.5. (From Hynninen AP and Dijkstra M. 2003. Physical Review E 68: 021407. With permission.)

0.5 bcc Fluid

0.4

fcc

1/ ks

0.3

0.2

0.1

0

0

0.1

0.2

0.3 h

0.4

0.5

0.6

FIGURE 8.5 Phase diagram of hard-core Yukawa particles with be = 81 presented in the (h, 1/ks) plane. The symbols and lines are the same as in Figure 8.2. Note the difference in the h scale compared to Figures 8.2 through 8.4. (From Hynninen AP and Dijkstra M. 2003. Physical Review E 68: 021407. With permission.)

174

Structure and Functional Properties of Colloidal Systems

phases is at be = 8, where it is less than 0.3%. In the case of the lowest contact value be = 8 (Figure 8.2), the bcc region ends at another triple point around 1/ks = 0.28. The presence of a second triple point for hard-core Yukawa particles was already found in reference [35], where the phase diagram for be = 8 was presented. The tendency of the bcc region to close up can also be seen in the phase diagram for be = 20 (Figure 8.3), where the fluid–bcc and the bcc–fcc coexistence lines turn toward each other at around 1/ks = 0.5. Note also that this tendency moves to higher 1/ks with increasing contact value be. Based on our results, we expect another triple point for all be at high values of 1/ks, although our calculations could only reach it at be = 8. We also predict that with increasing be, this other triple point escapes very quickly to high values of 1/ks, where numerical calculations are difficult to carry out. Figure 8.6 summarizes the results from Figures 8.2 through 8.5 by plotting all the phase diagrams in one figure. As can be seen from Figure 8.6, the low 1/ks triple point moves to lower h and higher 1/ks with increasing contact value be. Another observation is that the region of stable bcc phase broadens mainly because the fluid–bcc coexistence line moves to lower packing fractions, whereas the bcc–fcc coexistence line moves only slightly to higher h and seems to saturate around h ⬇ 0.5. The dashed line connecting the triple points in Figure 8.6, is discussed later. Now, we turn our attention to the comparison of our results on hard-core Yukawa particles with those obtained for point Yukawa particles by Hamaguchi et al. [26]. Note that the calculations of Hamaguchi et al. did not include the determination of phase coexistence regions. Therefore, in the case of point Yukawa particles only phase boundaries are considered. In Figures 8.2 through 8.5, these phase boundaries are plotted with dashed lines. Figures 8.2 through 8.5 show that the phase boundaries of hard-core Yukawa particles approach those of the point Yukawa particles with increasing be. This is because at high values of be, the particles hardly ever get close enough to feel the hard-core interaction. High be corresponds to highly charged colloids (see Equation 8.9). The deviation between the point and hard-core Yukawa results is particularly pronounced for be = 8 (see Figure 8.2). For the phase diagrams with higher values of be, the description with point Yukawa particles improves. Especially, the fluid–fcc line at high 1/ks, the fluid–bcc line, and the beginning of the bcc–fcc line are well predicted by the point Yukawa picture. However, the vertical rise of the bcc–fcc

0.5

0.4

1/ ks

0.3 be=81

0.2

be=39

be=20 be=8

0.1

0

0

0.1

0.2

0.3 h

0.4

0.5

0.6

FIGURE 8.6 Phase diagrams of Figures 8.2 through 8.5 plotted in one figure. The dashed line gives the line of triple points for point Yukawa particles [26], the diamonds (‡) highlight the triple points at be = 8, 20, 39, and 81, and the squares (䊐) mark the position of the triple points used in our calculations. (From Hynninen AP and Dijkstra M. 2003. Physical Review E 68: 021407. With permission.)

Phase Behavior and Structure of Colloidal Suspensions

175

line at high 1/ks is completely missing in the point Yukawa phase diagram. Instead, the bcc region is predicted to become indefinitely broad in the point Yukawa system, and hence the second triple point is absent. Thus, the closing of the bcc region by a second triple point is caused solely by the presence of the hard core. Next, we make a small excursion to study the position of the low 1/ks triple point. The position of the triple point for point Yukawa particles can be mapped to any hard-core Yukawa system for a fixed be. The resulting line of triple points is denoted by the dashed line in Figure 8.6, where the diamonds highlight the triple points at be = 8, 20, 39, and 81, and the squares give the triple points used in our calculations. We see that the agreement between the two results (the squares and the diamonds) does not depend much on the value of be and therefore we can conclude that the position of the lower triple point is given quite precisely by the point Yukawa results. In conclusion, we have shown that the phase diagram of charged colloids, where the interactions are given within the DLVO theory by hard-core repulsive Yukawa pair potential, can be obtained for any sufficiently high contact value be by mapping the well-known phase boundaries of the point Yukawa particles onto those of the hard-core repulsive Yukawa system and bearing in mind that the stable bcc region is bounded by a bcc–fcc coexistence at h ⬇ 0.5.

8.2.5

CRYSTAL STRUCTURES OF OPPOSITELY CHARGED COLLOIDS WITH SIZE RATIO 0.31

Recently, a new model system was presented that consists of oppositely charged colloids that form equilibrium crystals [10,36]. A profound difference with atomic systems is that ionic colloidal crystal structures are not dictated by charge neutrality as the charge balance is covered by the presence of counterions. This severs the link between charge ratio and stoichiometry and enlarges the number of possible crystal structures. Predicting these is a computational challenge, not only because of the overwhelming number of possible structures and system parameters (charge, size, solvent, salinity, composition, etc.) but also because of the intricate interplay between attractive and repulsive interactions, entropy, and packing effects. In this section, we develop an interactive simulation method to predict binary crystal structures of oppositely charged colloids, based on simulated annealing [37]. Employing this method, we are able to predict a whole variety of new binary crystal structures with different stoichiometries, which we used in Madelung energy calculations to map out the ground-state phase diagram. Results are presented for binary mixtures of small and large oppositely charged colloids with size ratio 0.31, corresponding to one of the experimental systems that triggered our theoretical interest [10]. The calculated phase diagram exhibits a plethora of different crystal structures, some of which have atomic and molecular analogs, while others do not. We are able to confirm the stability of three of the crystal structures experimentally. We model this system as NL large colloids in a volume V with a radius aL carrying a negative charge Z L e (where Z L < 0 and e is the proton charge) and NS small colloids with a radius aS carrying a positive charge Z Se (Z S < |Z L|). We assume that the pair potential uij(r), between colloid i and j at interparticle separation r, is given by the linear superposition approximation (LSA) [38] of the DLVO theory [24,25] as Ï Z i Z j l B exp[ -k (ai + a j )] exp[ -kr ] for r ≥ ai + aj uij (r ) Ô r = Ì (1 + kai )(1 + kaj ) , kBT Ô• for r < ai + aj Ó

(8.11)

where Zi (Zj) and ai (aj) are the charge number and radius of colloid i (j), lB = e2/eskBT is the Bjerrum length, k = (8plBrs)-1/2 is the inverse Debye screening length, es is the dielectric constant of the solvent, and rs is the salt concentration. We denote the composition by x = NS/(NL + NS), the packing fraction of large colloids by hL = 4paL3NL/3V, the dimensionless large–small colloid contact potential by G = |uLS(aL + aS)|/kBT, and the charge ratio by Q = |ZL/ZS|. We fix the size ratio to the experimental

176

Structure and Functional Properties of Colloidal Systems

value aS/aL = 0.31 and determine the phase diagram of this system for varying Q and k as these parameters turned out to be hard to measure under crystallization conditions and are likely to vary for different experiments. We use a simulated annealing approach to predict binary crystal structure candidates. The simulations are performed using periodic boundary conditions. The simulations were started in the NVT (fixed number of particles N, volume V, and temperature T) ensemble with the large colloids in an fcc, bcc, hcp, simple-cubic (sc), simple hexagonal (sh), or base-centered (bc) crystal, and the small colloids at random (nonoverlapping) positions. Next, we used simulated annealing to increase G slowly from 0 (high temperatures) to G ⬇ 10–20, which gives us the ground-state structure. The search for new stable binary structures was performed by trying all the above-mentioned large colloid crystals as starting configurations, at varying packing fractions and at various initial shapes of the simulation box. This was repeated for small–large stoichiometries n = 1–8 (denoted by LSn), where n is related to the composition by x = n/(n + 1). The simulations were performed in a box with 4–16 large colloids. Such a small simulation box with periodic boundaries facilitates finding structures with crystalline order, as it reduces the probability of stacking faults. Using this approach we found a variety of crystal structures for each stoichiometry and large-colloid starting configuration. Structures, which have no atomic or molecular analog, are named LSlat n , where “lat” is the lattice symmetry of the large colloids. In the colloidal analogs of the fullerene structures, A4C60 [39], and the bcc, A6C60 [40], the large colloids correspond to C60 and the small ones to A (= K, Rb, or Cs). We note that the size ratio of the doped fullerene systems is very similar to that of our system, for example, aRb/aC60 = 0.3. We have not considered primitive unit cells with more than two large colloids, and have hence excluded n = 5 and 7. Predicting the phase diagram of a binary mixture often involves a calculation of the Gibbs free energy per particle, since its x-dependence at fixed P and T allows for common tangent constructions. The Gibbs free energy reduces, however, to the enthalpy for the ground-state properties of interest here, where entropic effects are ignored. Moreover, since many of our experiments show crystals that are self-supported by their cohesive energy, we restrict attention to the zero-pressure limit. The present phase diagram follows therefore from the internal energy, which for a crystal with all particles at their ideal lattice positions is the composition-dependent Madelung energy U(x) [10]. In the zero-pressure ground-state approximation, phase stability depends only on the charge ratio Q and the screening parameter k and not on the absolute values of ZL, ZS, and lB separately. We calculated U(x) for all the structures that we found using simulated annealing and for many known structures: NaCl, CsCl, NiAs, CuAu, AlB2, Cu3Au, Al3Ti, CaCu5, and CaB6 [41]. In the calculations, the simulation box was repeated periodically such that the relative error in U(x) was smaller than 10-7. The lowest U(x) per colloidal particle was found for given Q and k by optimizing each possible structure with respect to hL and the shape of the unit cell. We then performed, for fixed Q and k, the common tangent construction. Figure 8.7 shows the common tangent construction for Q = 1.8 and kaL = 2.5, where the circles denote the Madelung energies U(x) for the Wurtzite, CaF2, Cr3Si, and A4C60 structures, and the dashed line gives the actual common tangent construction. Note that at the two extremes where x = 0 or x = 1, the Madelung energy is zero (U(0) = U(1) = 0) because of the repulsive character of like species. One reads Figure 8.7 as follows. At x 僆 [0, 0.67], we have an infinitely dilute gas of large colloids in coexistence with the CaF2 phase, at x 僆 [0.67, 0.75] a CaF2–Cr3Si coexistence, at x 僆 [0.75, 0.8] a Cr3Si– A4C60 coexistence, and finally, at x 僆 [0.8, 1] a A4C60 phase in coexistence with an infinitely dilute gas of small colloids. Figure 8.7 also tells us that the phase diagram is three dimensional because all three parameters, x, k, and Q, affect the phase behavior. We repeated the common tangent construction for each Q 僆 [1, 8] and kaL 僆 [0.5, 5] to obtain the full phase diagram. Figure 8.8 shows a x–Q slice of the phase diagram at kaL = 2.5. One readily checks from Figure 8.8 that at Q = 1.8, the phase sequence corresponds to the common tangent construction in Figure 8.7. Another way of studying the three-dimensional phase diagram is to look at the two extremes cases where the crystal coexists with an infinitely dilute gas of pure large colloids (x = 0) and pure small colloids (x = 1). Figures 8.9 and 8.10 show k–Q slices of the phase diagram at x = 0 and x = 1, respectively. The dashed line in Figures 8.9 and 8.10 at kaL = 2.5 depicts the location

177

Phase Behavior and Structure of Colloidal Suspensions

0

0

0.1

0.2

0.3

0.4

x 0.5

0.6

0.7

0.8

0.9

1

U(x)/NkBT

–0.2

Wurtzite

–0.4

–0.6

CaF2

–0.8

A4C60 Cr3Si

FIGURE 8.7 Common tangent construction for Q = 1.8 and kaL = 2.5. The circles denote the Madelung energies and the dashed line the common tangent construction.

where the x–Q slice shown in Figure 8.8 is made. Figure 8.8 shows an increasing stoichiometry with increasing Q at low x, and Figures 8.9 and 8.10 show the same at low kaL. In Figure 8.8, one has gas– crystal and crystal–crystal coexistence regions, whereas in Figures 8.9 and 8.10, all crystals are in coexistence with an infinitely dilute gas of large or small colloids. Note that the phase lines in Figures 8.9 and 8.10 are triple-point lines where a gas and two crystals coexist. From Figures 8.9 and 8.10, one observes that an excess of small colloids favors structures with high stoichiometry. Moreover, we see that ReO3, Li3N, SiF4, LS6fcc, and LS 6hcp structures are only stable for relatively long-ranged interactions with kaL ~ 0.5. In these structures, shown in Figure 8.11, the small colloids are in between pairs of neighboring large colloids in the sc (ReO3), sh (Li3N), bcc (SiF4), fcc (LS6fcc), or hcp (LS 6hcp) structure. Although such structures are electrostatically favorable, they pack inefficiently and thus destabilize at stronger screening. At Q ⬇ 1, we find the Wurtzite structure, which has dimers of large and small colloids stacked in an hcp lattice (see Figure 8.11f). The CaF2 structure, depicted in 8

gasL + LS8fcc/LS8hcp

7

gasL + LS6hp

6

fcc hcp A6 C6bcc 0 +LS8 /LS8

bcc

gasL + A6C6 0 5

gasL + LS4bct

4

gasL + A4C60

Q

LS4bct+A6C6bcc 0 A4C60+A6C6bcc 0 A4C60+LS6hp

gasL + Cr3Si

1

gasL + CaF2 0

0.1

0.2

0.3

0.4

gasS+A6C6bcc 0 Cr3Si+A4C60

3 2

gasS + LS8fcc / LS8hcp

0.5 x

0.6

gasS+LS6hp

gasS+A4C60 CaF2 + Cr3Si gasS+Cr3Si 0.7 0.8 0.9 1

FIGURE 8.8 x–Q slice of the ground-state phase diagram at kaL = 2.5. “gasL” and “gasS” refer to an infinitely dilute gas of large and small colloids, respectively. (From Hynninen AP, et al. 2006. Physical Review Letters 96: 138308. With permission.)

178

Structure and Functional Properties of Colloidal Systems 8 LS8fcc/LS8hcp 7

LS6hp

LS6hcp 6

bcc

A6C60 5

LS4bct

SiF4

Q 4

A4C60

3

Li3N

2

Cr3Si

Wurtzite

1 0.5

CaF2

1

1.5

2

2.5

3

3.5

4

4.5

5

kaL

FIGURE 8.9 k–Q slice of the ground-state phase diagram at x = 0 where the structures coexist with an infinitely dilute gas of large colloids. (From Hynninen AP, et al. 2006. Physical Review Letters 96: 138308. With permission.)

Figure 8.11g, has the large colloids in an fcc lattice and the small colloids in the tetrahedral holes. The Cr3Si structure has the large colloids in a bcc lattice and the small colloids as shown in Figure 8.11h. LS6hp, shown in Figure 8.11i, consists of hexagonal planes of large colloids, which are slightly compressed (6–10%) with respect to an ideal hcp crystal, and of small colloids, which are in two kagome patterns [43] above and below each large-colloid hexagonal layer. In the A4C60 structure [39], the large colloids are in a body-centered-tetragonal (bct) lattice and the small colloids are between two large colloids such that they form a square parallel to the square plane of the tetragonal unit cell (see Figure 8.11j). The LSbct 4 structure, shown in Figure 8.11k, differs from A4C60 in that the small colloids are turned 45° in the square plane. To verify the theoretical phase diagram, we studied the crystal structures and their formation in our experimental system, with total particle volume fraction 0.2 and 8 7

LS8fcc/LS8hcp

6

A

6 C bcc 60

5

LS fcc

Q

6

4 LS4bct LS6hp

3 R eO

A4C60 Li3N Cr Wurtzite 3 Si CaF2 3

2

1 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

kaL

FIGURE 8.10 k–Q slice of the ground-state phase diagram at x = 1 where the structures coexist with an infinitely dilute gas of small colloids. (From Hynninen AP, et al. 2006. Physical Review Letters 96: 138308. With permission.)

179

Phase Behavior and Structure of Colloidal Suspensions (a)

(b)

(e)

(h)

(c)

(f)

(i)

(d)

(g)

(j)

(k)

FIGURE 8.11 Theoretically predicted stable binary crystal structures: (a) ReO3, (b) Li3N, (c) SiF4, (d) LS6fcc, (e) LS6hcp, (f) Wurtzite, (g) CaF2, (h) Cr3Si, (i) LS6hp, (j) A4C60, and (k) LS4bct. (From Hynninen AP, et al. 2006. Physical Review Letters 96: 138308. With permission.)

added salt (tetrabutylammonium bromide) concentrations in the range 15–35 mM. Interestingly, we bcc, LSfcc, and LShcp, experimentally found three of the predicted crystal structures, namely, A6C 60 8 8 which are indeed rather dominant in the phase diagram of Figures 8.9 and 8.10 at large Q. In the LS8hcp structure the large colloids form an hcp lattice. In the hexagonal layers, each large colloid is surrounded by a ring of six small ones occupying the trigonal interstices. Above and below each layer, there are two planes of small colloids ordered in a kagome pattern (Figure 8.12). The position of the small colloids in the kagome pattern is the same as in the LS6hp structure (see Figure 8.11i). Figure 8.11b shows the LS8hcp structure as a superposition of 5 layers of large colloids and 15 layers of small colloids. We observe two kagome patterns of small colloids superimposed that are formed because the hcp lattice has two kinds of hexagonal layers of large colloids. The hexagonal structure of the large colloids can be seen below the small colloid kagome patterns. In the other binary structure with stoichiometry 8, the LS8fcc crystal, the large colloids are ordered in an fcc lattice and in every octahedral hole there are eight small colloids forming a cube. Figure 8.12 shows the (100) plane of the LS8fcc unit cell obtained from experiments and predicted by our simulations, revealing excellent agreement. Experimentally, the LS8hcp and LS8fcc structures were found to coexist. This is also in agreement with our calculations that predict essentially equal Madelung energies (although LS8fcc is more stable, but bcc structure [40] consists of a bcc lattice of the large colloids with four only by 0.005–0.1%). The A6C60 small colloids in a square situated in the plane between every large next nearest neighbors. Figure bcc structure unit cell together with the corresponding theoreti8.12 shows the (100) plane of the A6C 60 cal predictions. In summary, we used a simulated annealing approach to predict binary crystal structures of oppositely charged colloids with a size ratio of 0.31. Our ground-state phase diagram displays novel structures, but also colloidal analogs of simple salt structures and of doped fullerene C60 structures. bcc, LS hcp, and LS fcc, were also observed experimentally, Three of the predicted structures, A6C60 8 8 thereby providing confidence that the proposed method with the screened Coulomb potential yields reliable predictions, even though the experimentally observed Brownian motion suggests that entropy should not be ignored.

180

Structure and Functional Properties of Colloidal Systems

A6Cb6cc 0

LS8fcc

LShcp 8

FIGURE 8.12 Experimental observations (top) and the corresponding theoretical predictions of binary bcc, LS fcc, and LS hcp. structures (bottom). From left to right: A6C60 8 8

8.2.6

OPPOSITELY CHARGED COLLOIDS WITH SIZE RATIO 1

An important special case of the primitive model is obtained when one considers solely the electrolyte part, that is, co- and counterions without colloids. In this case, the primitive model reduces to what is called “restrictive primitive model” (RPM), because it is restricted in a sense that all particles have the same size and the same magnitude of charge. The RPM is widely used to model electrolytes and ionic liquids, and in the past the phase behavior of the RPM has been studied extensively. By now, it is known that the RPM exhibits a gas–liquid phase separation and the location of the critical point is known to high accuracy [44]. Also the global phase diagram, consisting of fluid, CsCl (or bcc), tetragonal, and fcc disordered phases [45–47], has been constructed. Related to the RPM is a system of oppositely charged screened Coulomb particles. The interest in this system has started only very recently, when it became possible to realize it experimentally using oppositely charged colloids that form stable crystal structures instead of aggregates [10,36]. The phase behavior of this system can be understood on the basis of screened Coulomb potentials, in which the screening is due to the presence of co- and counterions in the solvent. We calculate the phase diagram of screened Coulomb particles and reexamine the phase behavior of the RPM. We find that the two phase diagrams are qualitatively similar, and, more importantly, that both contain a novel crystal structure where the particles are arranged in a CuAu-type crystal. Remarkably, we are able to observe the CuAu structure also experimentally in a system of oppositely charged colloids. We present the phase diagrams of the RPM and screened Coulomb particles and show the experimental observations of the CuAu structure. Our simulations consist of N spheres with a diameter of s = 2a in a volume V, half of which carry a positive charge Ze and the other half a negative charge -Ze. The particles interact via the screened Coulomb potential (Equation 8.11), and we define a reduced temperature T*=

(1 + ka )2 s , Z 2lB

(8.12)

which is the inverse of the contact value of the potential in Equation 8.11. The RPM is achieved by setting k = 0, because then the screened Coulomb potential in Equation 8.11 reduces to the Coulomb potential. The overall (of both the positive and the negative particles) symmetry of a CsCl crystal is bcc (Figure 8.13a). In the CuAu crystal, shown in Figure 8.13c, the overall crystal symmetry is face-centered-tetragonal (fct), and the oppositely charged particles are arranged in alternating layers. In the tetragonal phase, the overall symmetry is fct, and the substitutional order can be described by two fct cells on top of each other (Figure 8.13b). The fcc disordered phase has no substitutional

Phase Behavior and Structure of Colloidal Suspensions (a)

181

(b)

z x (c)

y (d)

b

a

a

FIGURE 8.13 Unit cells of (a) CsCl, (b) tetragonal, and (c) CuAu structures, where the light and dark spheres have opposite charges. In (d), the tetragonal cell of the CuAu structure is highlighted. (From Hynninen AP, et al. 2006. Physical Review Letters 96: 018303. With permission.)

order, and the two particle species sit in an fcc lattice. We used N = 250 for the CsCl phase and N = 256 for the CuAu, fcc disordered, and tetragonal phases. We performed MC simulations using the canonical (NVT) ensemble and periodic boundary conditions. The interactions were truncated at one half of the smallest box side length, L min. In the case of the RPM, we used the Ewald summation method [21] with k-space cut-off at kcut = 10p/L min. Our Ewald summation method was tested to reproduce the Madelung energies in reference [45] for the CsCl, CuAu, and tetragonal crystals. The phase diagrams were determined from the Helmholtz free energies that were calculated for the fluid, CsCl, CuAu, and tetragonal phases. We found that the CsCl–CuAu transition is a weakly firstorder martensitic phase transition and defined an order parameter a/c, which is the ratio of the CuAu unit cell side lengths. In order to understand this martensitic transition, we note that the CuAu to CsCl transformation occurs by a continuous isochoric deformation of the tetragonal cell from a/c = 1 to a/c = ÷2, where the cell has become cubic. Figure 8.13d shows the connection between the bcc and the CuAu cells. To distinguish between the two structures, we define the threshold value to be in between, that is, a/c = (1 + ÷2)/2 ⬵ 1.2. Figures 8.14 and 8.15 show the theoretical phase diagrams of the RPM and the screened Coulomb particles, respectively, in the packing fraction h— reduced temperature T* representation. The phase coexistence regions were determined from the Helmholtz free energies using a common tangent construction. In Figure 8.14, we only calculated the CsCl–CuAu and the CuAu–tetragonal phase lines, as the other parts of the phase diagram are known from earlier work [47,49,50]. In Figure 8.15, the weakly first-order tetragonal/CuAu–fcc disordered phase line [51] was obtained from the jump in the internal energy U, which occurs when crossing the phase boundary [45]. The same method was also used for the CsCl–fcc disordered phase transition, although the order of this phase transition has not yet been well characterized. Qualitatively, the two phase diagrams are quite similar. Both have a fluid phase in the low packing fraction region, a fluid–fcc disordered phase coexistence in the high temperature–high packing fraction region, a broad gas–CsCl phase coexistence in the low temperature region, and in the intermediate temperature region, a sequence of CsCl, CuAu, and tetragonal phases with increasing packing fraction. A comparison between the two phase diagrams shows that screening enlarges the CuAu region. In the

182

Structure and Functional Properties of Colloidal Systems 0.4 fcc disordered

0.35 0.3 0.25 T*

Tetragonal 0.2

Fluid

0.15

CuAu

0.1 CsCl 0.05 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

h

FIGURE 8.14 Phase diagram of the RPM in the packing fraction reduced temperature T* plane. Phase lines for fluid–CsCl (+), fluid–fcc disordered (D), fluid–tetragonal (×), and fcc disordered–tetragonal (䊐) are from reference [47]. Gas–liquid phase envelope (—) is from reference [49] and liquid points (䉰) are from reference [50]. The circles (䊊) and diamonds (‡) mark the CsCl–CuAu and CuAu–tetragonal phase transitions, respectively. The black circles (䊉) mark the position of the triple points and the gray areas denote the coexistence regions where tie lines are horizontal. The solid lines are guide to the eye. (From Hynninen AP, et al. 2006. Physical Review Letters 96: 018303. With permission.)

RPM phase diagram, the CuAu phase forms a narrow pocket that has a low-T* triple point and seems to close up before reaching the fcc disordered region. In the screened Coulomb phase diagram, however, the CuAu pocket is broad and extends from T* = 0 all the way to the fcc disordered region at T* = 1.2. This makes it possible to observe CuAu–fcc disordered phase coexistence, which may explain the experimental observations (see below). We like to point out that the RPM and the screened Coulomb phase diagrams display a slightly different behavior in the T* Æ 0 limit: Although both exhibit a broad gas–CsCl coexistence at h < 0.68, at higher h the RPM has a stable tetragonal phase, whereas the screened Coulomb system has a stable CuAu phase, only to be followed by the tetragonal phase at h ≥ 0.73. Screening also affects the stability of the gas–liquid critical point. The RPM has a stable gas–liquid critical point at T* ⬵ 0.05, but in the screened Coulomb phase diagram, the critical point (at h = 0.137 and T* = 0.176) is metastable with respect to the gas– CsCl phase coexistence (see Figure 8.15). According to our preliminary results, the gas–liquid critical point of the screened Coulomb system is stable at ks £ 4. In the following, we compare our theoretical findings with the experimental observations carried out in the system of oppositely charged colloids. Figure 8.16 shows a selection of representative confocal images, where the packing fraction is h = 0.58 ± 0.04. Figure 8.16a and b show the CuAu phase in the presence of a rhcp disordered phase. The CuAu structure is easily recognized as it has alternating stripes of dark and bright particles that make up the (111) hexagonal plane. Remarkably, whereas the CuAu crystallites were strictly fcc, the substitutionally disordered structures were rhcp. Note that on the basis of hard spheres, one expects to see a rhcp because of the small free energy difference between fcc and hcp crystals [52]. The simultaneous observation of the CuAu and fcc disordered phases agrees with the screened Coulomb phase diagram in Figure 8.15, in which we find the CuAu and fcc disordered phases connected by a weakly first-order phase line. Unfortunately, a more quantitative comparison between the simulations and the experiments is difficult as it is not known whether the experimentally observed structures are in equilibrium, and the experimental values of ks and T* are not known accurately. However, the relatively low packing fraction (h ⬇ 0.58) of the observed CuAu phase and the presence of the rhcp disordered phase point toward the intermediate temperature region of the phase diagram in Figure 8.15 where T* ⬇ 1. Thus, the experiments fall in between the

183

Phase Behavior and Structure of Colloidal Suspensions 2 1.8

fcc disordered

1.6 1.4

1

CuAu

Fluid

0.8

Tetragonal

1.2 T*

CsCl

0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

h

FIGURE 8.15 Phase diagram of screened Coulomb particles with an inverse Debye screening length of ks = 6 in the packing fraction h–reduced temperature T* plane. The dashed lines and the cross (×) show the metastable gas–liquid phase envelope and the critical point, respectively. The rest of the symbols denote the phase lines for fluid–CsCl (䊊), fluid–fcc disordered (D), Fluid–tetragonal (×), fcc disordered–tetragonal (䊐), CsCl–CuAu (|), and CuAu–tetragonal (‡). The black circles (䊉) mark the position of the triple points and the gray areas denote the coexistence regions where tie lines are horizontal. The solid lines are guide to the eye. (From Hynninen AP, et al. 2006. Physical Review Letters 96: 018303. With permission.)

FIGURE 8.16 Confocal microscope images of CuAu-type crystallites: (a), (b) in the presence of an rhcp disordered crystal; (c), (d) a (111) plane; (f) a (100) plane; and (i) a (001) plane. In (c), one can observe a line defect. Illustrations of CuAu-type crystal planes: (e) the (111) plane, (g) the (100) plane, and (h) the (001) plane. The scale bars are 25 mm in (a), 10 mm in (b), 5 mm in (c), and 2.5 mm in (d), (f), and (i). (From Hynninen AP, et al. 2006. Physical Review Letters 96: 018303. With permission.)

184

Structure and Functional Properties of Colloidal Systems

energy- (T* Ⰶ 1) and entropy (T* Ⰷ 1)-dominated limits. Figure 8.16c and d show closeup images of the (111) plane, now with the stripes running vertically. Figure 8.16e illustrates the (111) plane in an ideal crystal. Although most of the CuAu-type crystallites were oriented with their (111) plane parallel to the bottom wall of the sample cell, we were able to observe (100) and (001) planes as well. Figure 8.16f shows the (100) plane that consists of alternating layers of oppositely charged particles, and Figure 8.16g shows the corresponding plane in an ideal crystal. The (001) plane is shown in Figure 8.16h and i. In summary, we studied the phase behavior of oppositely charged particles described by the RPM and screened Coulomb potentials with equal size and magnitude of charge and made comparisons with experimental observations in a system of oppositely charged colloids. We showed that the two phase diagrams are qualitatively similar, and more importantly, that both contain a novel solid phase, which is analogous to the CuAu structure. We also observed the CuAu structure in our experiments with oppositely charged colloids, which can be seen as an experimental realization of screened Coulomb particles.

8.3

COLLOIDS IN EXTERNAL FIELDS

An intriguing feature of colloids is the possibility to use external fields to manipulate their interactions, structure, and dynamics. Examples of external fields include confinement, gravity, temperature, shear flow, optical tweezers, and external electric or magnetic fields. A full description of all these external fields would be outside the scope of this chapter, and therefore, in the following, we will restrict our discussion to external electric or magnetic fields and confinement. In an external electric or magnetic field, colloids whose dielectric constant or magnetic susceptibility is different from that of the solvent acquire a (electric or magnetic) dipole moment parallel to the field. The colloid pair potential is governed by the dipole–dipole interaction, whose strength can be tuned by the magnitude of the field. Such suspensions are called electrorheological (ER) and magnetorheological (MR) fluids and are used in industrial applications as dampers, hydraulic valves, clutches, brakes, and displays. An important feature of ER/MR fluids is their ability to crystallize above a certain critical field strength [53]. The high-field crystal structure is bct [54], but at lower field strengths, other structures are also possible [9]. The tunability of the crystal structure via an external field makes ER/MR suspensions appealing for photonic applications [9,55,56]. One can also manipulate the crystal structure by just confinement. Recent experiments on colloidal particles in a wedge geometry show an enormous rich phase behavior as a function of the position in the wedge [57].

8.3.1

MANIPULATION OF COLLOIDAL CRYSTAL STRUCTURES BY AN ELECTRIC FIELD

Colloidal particles in an external electric or magnetic field whose dielectric constant or magnetic susceptibility is different from that of the solvent acquire a (electric or magnetic) dipole moment parallel to the field. The behavior of the colloids is governed by the dipole–dipole interaction, whose strength can be tuned by the magnitude of the field. Due to their unique shearing properties, such suspensions are called ER and MR fluids. ER/MR fluids have potential use in industrial applications as hydraulic valves, clutches, brakes, and displays. Moreover, the possibility to tune the crystal structure of these suspensions by an external field makes these suspensions appealing for photonic applications [9,55,56]. The equilibrium structure of these fluids has been the subject of many experimental [3,9,56,58,59], theoretical [53,54,60–64], and simulation studies [65–69]. In an early theoretical study, Tao et al. proposed that above a certain critical field strength the system experiences a phase transition to a solid structure [53]. In reference [54], Tao and Sun studied the crystal structure of the solid phase and found that, of the structures that they considered, a bct structure was the one with the lowest energy. The bct structure predicted by Tao and Sun has been observed by both computer simulations [66,68,69] and experiments [9,56,58,59]. We determine the phase diagram of two dipolar systems: (i) colloids without charge (or dipolar hard spheres) and (ii) colloids with a charge

Phase Behavior and Structure of Colloidal Suspensions

185

of Ze = 300e and an inverse Debye screening length (in units of the colloid diameter s) of ks = 10 (dipolar soft spheres). The phase diagram of dipolar hard spheres shows fluid, fcc, hcp, and bct phases. The phase diagram of dipolar soft spheres shows, in addition to the above-mentioned phases, a body-centered-orthorhombic (bco) phase. As we will show, the phase diagram of dipolar soft spheres is in good agreement with the experimental phase diagram in reference [9]. We are able to explain the appearance of a bco phase based on simple arguments. Furthermore, our calculations propose hcp crystal as the high-density stable phase in systems with dipolar interactions. We use a dipole approximation to describe the pair potential between two dielectric and magnetic particles. In the case of an electric field, the particles and the solvent have a dielectric constant of ep and es, respectively, whereas in the case of a magnetic field, the particles and the solvent have a magnetic susceptibility of mp and m s, respectively. The external field induces a (electric or magnetic) dipole moment on the particles, which is parallel to the field direction (the z axis). The dipole–dipole interaction is given by 3 udip ( r ) g Ê s ˆ = Á ˜ (1 - 3cos2 q), kBT 2Ë r¯

(8.13)

where k B is the Boltzmann constant and T is the temperature, r is the vector separating the two particles, and q is the angle that r forms with the z axis. In the case of electric dipoles, the prefactor g is given by

g=

ps 3a 2 e s | Eloc |2 , 8kBT

(8.14)

where the local electric field is given by Eloc = E + Edip, which is the sum of the external field E, and the field induced by the other dipoles Edip [54,67,70]. In Equation 8.14, a = (ep - es)/(ep + 2e s) is the polarizability of the particles. In the case of magnetic dipoles, g is given entirely symmetrically and is written as

g=

ps 3a 2 m s | Hloc |2 , 8kBT

(8.15)

where a = (mp - m s)/(mp + 2μs) and Hloc = H + Hdip is the local magnetic field [71]. For charged colloids, we supplement the dipole–dipole interaction in Equation 8.13, with a soft repulsion caused by like-charge repulsion. According to the DLVO theory [24,25], the pair interaction between two charged colloids is given by a repulsive Yukawa (or screened Coulombic) plus the hard-core potential (Equation 8.8). Since we are interested in modeling systems where the van der Waals attraction is very small due to refractive index matching, we have neglected it in Equation 8.8. Using the pair potentials in Equations 8.8 and 8.13, we perform MC simulations in the canonical ensemble (NVT), where we fix the number of particles N, the volume V, and the temperature T [21]. The simulation box is periodic in all three directions. Typical number of particles in our simulations is N = 144– 288, and cubic, or nearly cubic, simulation boxes are used. Because of the long-range nature of the dipolar interactions, we use the Ewald summation method to evaluate the potential in Equation 8.13 [21]. Both the Yukawa and the dipolar potential are truncated at half of the shortest box side length. The width of the Gaussian distribution, the tunable parameter in the Ewald sum, is optimized according to the analytical estimates given in reference [72]. In our simulations, we consider the fluid, bcc, bct, bco, fcc, and hcp phases. The body-centered (bcc, bct, and bco) structures are aligned such that the particles form strings parallel to the z axis (the field direction) (Figure 8.17a). The

186

Structure and Functional Properties of Colloidal Systems

(a)

(b)

a

bco

z

a

z hcp

z

x

a

x fcc

(d)

A

A

B

B

A

C

B

y

A

z x

z

a=b

b

a

x

(c)

bct

b

y

b

y

aπb

b

c

x

y x

z

FIGURE 8.17 (a) Body-centered structure in three dimensions, whose conventional unit cell is a × b × c. The field is along the z axis. The white arrows show the direction of the field-induced dipole moments. The bct structure corresponds to a = b π c and the bco to a π b, c π a, and c π b. (b) Top-view of the body-centered structure that can be constructed by placing strings of particles shifted by c/2 into two interpenetrating rectangular lattices. (c), (d) The hcp and fcc structures shown in side and top views. The hcp structure has AB stacking of the hexagonal planes; the fcc ABC. (From Hynninen AP and Dijkstra M. 2005. Physical Review Letters 94: 138303. With permission.)

body-centered box side lengths are given in the x, y, and z axis directions by a, b, and c, respectively. Shown in Figure 8.17b are the bco and bct structures viewed along the z axis. As can be seen from Figure 8.17b, the bco and bct structures can be constructed by placing strings of particles into two interpenetrating rectangular (a × b) lattices. The particles in the strings are displaced by c/2 in the z direction. The bco structure can be thought of as an asymmetric version of the bct structure. The maximum packing of the bct structure is obtained when a = b = (÷6/2)s and c = s, corresponding to a packing fraction h = ps3N/6V = 2p/9 ⬵ 0.698. Note that, the bcc phase has a = b = c. The fcc and hcp structures are depicted in Figure 8.17c and d, respectively. Both fcc and hcp are oriented with the (111) plane perpendicular to the z axis. The phase behavior was studied by MC simulation runs and Helmholtz free energy calculations. We used the MC simulation runs to obtain a rough estimate of the phase behavior, after which the more accurate free energy calculations were performed to check the result and to determine the phase boundaries more exactly. Phase coexistence regions were determined by a common tangent construction from Helmholtz free energies that were calculated using thermodynamic integration methods. We used the l-integration method for the fluid phase and the Frenkel–Ladd method for the solid phase [21,30]. As a reference state, we used the hard-sphere fluid for the fluid phase and the noninteracting Einstein crystal for the solid phase. Due to the dipole–dipole interaction, compression along the z axis lowers the energy of all our crystal phases. Therefore, in order to get reliable results, we need to optimize the z axis side length. For the bct, fcc, and hcp phases, we calculated the free energies for various z axis side lengths and used the minimum value to determine the phase boundaries. In the case of soft repulsions, the bco unit box symmetry, given by c and the ratio a/b, was determined by varying a/b and c to find the minimum of the Madelung energy (energy of an ideal crystal per particle). If the minimum state had strings in touching configurations (c = s), we set c = 1.01s to ensure that efficient MC sampling of the system is still possible. For some systems, also other choices like c = 1.04s and c = 1.005s were

187

Phase Behavior and Structure of Colloidal Suspensions

0.7

fcc

tried, but in general, the results did not depend strongly on the choice of c. Our model presented can be described by four independent dimensionless parameters: the packing fraction h, the strength of the dipolar interaction g, the colloidal charge Z, and the inverse screening length ks. We fix the colloidal charge Z and study the phase behavior in a constant ks plane, that is, our phase diagrams are plotted in the (g, h) representation. In Figure 8.18, we show the phase diagram of the dipolar hard spheres (i.e., be = 0) in the (g, h) representation. At zero dipole moment strength (g = 0), the well-known hard-sphere fluid–fcc coexistence with the coexisting phases at hfluid = 0.494 and hfcc = 0.545 is recovered. At g > 0, the fluid– fcc coexistence switches to fluid–hcp coexistence. Increasing the dipole moment strength from g = 0 to g = 6.5 does not change the fluid–hcp coexistence much. At g > 6.5, the bct phase is the stable crystalline phase at low densities, while the hcp phase is, due to more efficient packing, still the stable phase at packing fractions h ≥ 0.57. At the dipole moment strength g ⬇ 8.0, the system phase separates into a (string) fluid phase and a bct phase. The fluid–bct phase coexistence region broadens with increasing dipole moment strength and at g = 13.1; the fluid phase has turned into a very low-density gas phase (or void). At g > 13.1, the gas–bct coexistence broadens further and at g ⬇ 38, the bct phase has packing fraction hbct = 0.66 close to the maximum packing. We expect at higher dipole moment strengths (i.e., g > 38), a very broad gas–bct coexistence between a void (lowdensity gas phase) with packing fraction h ⬇ 0.0 and a bct phase at the maximum packing h = 0.698. As can be seen from Figure 8.18, the stability of the hcp phase is reduced when the dipole moment strength is increased, and beyond g ⬇ 25 the hcp phase is stable only above the maximum bct packing. We want to point out that the bct–hcp and the hcp–fcc coexistence regions exist but are too narrow to discern in the graph. The ground-state (g = •) phase behavior of dipolar hard spheres has been studied in reference [65], where the authors found a gas–bct coexistence at h 僆 [0, 0.698], a stable high-density bco phase at h 僆 [0.698, 0.724], a bco–hcp coexistence at h 僆 [0.698, 0.740], and a stable hcp phase at h = 0.740. This sequence of phases and in particular the presence of the bco phase should also be present in our dipolar hard-sphere phase diagram at g >> 40, but we did not extend our simulations to high enough g to observe it. The reason for this is that at very high

hcp

0.6

bct

0.5 Fluid-hcp

(c) (String) fluid-bct

h

0.4 (b)

(String) Fluid

0.3

(d) 0.2 0.1 (a) 0

0

5

10

15

20 g

25

30

35

40

FIGURE 8.18 Phase diagram of dipolar hard-sphere particles in the (dipole moment strength g, packing fraction h) representation. The circles denote points where the phase boundary was determined and the gray areas denote coexistence regions (where tie lines are vertical). (From Hynninen AP and Dijkstra M. 2005. Physical Review Letters 94: 138303. With permission.)

188

Structure and Functional Properties of Colloidal Systems

values of g, the simulations are hampered by sampling problems as the displacement moves become difficult due to strong dipole–dipole interactions. Note that for dipolar soft spheres (which are discussed next) it is possible to reach higher g because the soft repulsion compensates partly for the dipolar interaction. Next, we consider the case where the particles, in addition to the dipole–dipole interaction, interact via the Yukawa repulsion in Equation 8.11 with parameters ks = 10 and Z = 300. For a solvent with Bjerrum length lB/s = 0.005, these parameters correspond to a contact value be = 12.54. The phase diagram for this dipolar soft-sphere system is shown in Figure 8.19. At zero electric field, the phase diagram in Figure 8.19 shows a fluid–fcc phase coexistence with the two phases at packing fractions hfluid = 0.31 and at hfcc = 0.32. At g > 0, the fcc phase is replaced by the hcp phase. As can be seen from Figure 8.19, even a small amount of electric field (g > 4) is sufficient to suppress the stability region of the hcp phase considerably and to replace it by a bco phase, which is stable at low densities for 4 < g < 10. In the bco phase, the ratio a/b varies from slightly greater than 1 up to 1.30. In the phase diagram in Figure 8.19, at g ⬇ 17, the bct phase emerges as the stable low-density crystal. Further increase in the dipole moment strength from g ⬇ 17 reduces the significance of the soft repulsion relative to the dipolar attraction, thereby increasing the region of the stable bct phase. Finally, at g ⬇ 67, the bco phase vanishes completely. Increasing the dipole moment strength reduces the stability region of the hcp phase and at g > 100, the hcp phase is only stable at packing fractions higher than the maximum body-centered packing. The phase diagram in Figure 8.19 shows that the fluid–bct coexistence region, starting at g ⬇ 38, broadens quickly with increasing dipole moment strengthg. At g ⬇ 67, the fluid phase in coexistence with the bct phase consists of strings of particles (string fluid phase), while the fluid phase is extremely dilute, that is, h ⬇ 0, for

Erms(V/mm) 0

0.16

0.23

0.29

0.33

0.37

0.40

0.44

0.47

0.49

0.7 fcc hcp 0.6 bco

0.5

bct

h

0.4 0.3 (String) fluid-bct

0.2 (String) fluid

0.1 0

0

20

40

60

80

100

120

140

160

180

g

FIGURE 8.19 Phase diagram of dipolar soft-sphere particles with Yukawa parameters ks = 10.0 and Z = 300 (that with lB/s = 0.005 correspond to a contact value be = 12.54) in the (dipole moment strength g, packing fraction h) representation. The circles denote points where the phase boundary was determined and the gray area denotes the coexistence region (where tie lines are vertical). The upper horizontal axis gives an estimate of the root-mean-square external electric Field (see text for details). (From Hynninen AP and Dijkstra M. 2005. Physical Review Letters 94: 138303. With permission.)

189

Phase Behavior and Structure of Colloidal Suspensions

g > 100. At g > 180, we expect that the coexisting bct phase reaches the maximum packing h ⬇ 0.698. Similar to Figure 8.18, the hcp–fcc, bco–hcp, and the bct–hcp phase coexistence regions are too narrow to show up in the graph. We now compare our results with the experimental phase diagram of reference [9], which is reproduced in Figure 8.20. We have renamed some of the phases in the experimental phase diagram. Firstly, we call the space-filling tetragonal (sft) phase of reference [9] bct. Secondly, the nonspace-filling bct phase of reference [9], which consists of small bct crystallites with voids, corresponds to our gas–bct coexistence. Experimentally, one observes a microscopic phase separation instead of a macroscopic phase separation, which depends strongly on the kinetics and dynamics of the phase separation and on how fast the electric field is switched on. For comparison with the experimental phase diagram of reference [9], the upper horizontal axis in Figure 8.19 gives the (root-mean-square) electric field strength Erms. The g to Erms conversion is carried out using

Erms = 2

|1 - a(p / 6) | |a |

kBT g , es s3 p

(8.16)

where we used parameter values that correspond to the experimental system of reference [9]: a = -0.105, T = 300 K, es = 5.6, and s = 2 mm. For the derivation of Equation 8.16, see for example references [54,67,70]. We have assumed a cubic lattice, for which the local field is given by Eloc = E/(1 - ap/6) [54]. Our phase diagram for dipolar soft spheres in Figure 8.19 shows a remarkable structural agreement with the experimental phase diagram in Figure 8.20. Both phase diagrams show, at low electric field strength, the same sequence of fluid, bct, and bco phases upon increasing h, and at high electric fields, phase separation between a gas (void) and a bct phase. The main difference between the two phase diagrams is the bcc phase, which is seen experimentally at zero electric field but which is not present in the theoretical phase diagram. This dissimilarity is due to the different Z and ks in the experiments and the simulations. In summary, we determined the phase diagrams of dipolar hard and soft spheres using MC simulations. Two systems were considered: (i) colloids without charge (or dipolar hard spheres) and (ii) colloids with a charge of Ze = 300e and an inverse Debye screening length of ks = 10 (dipolar soft spheres). The simulations correspond to dielectric (or magnetic) particles in an external electric (or magnetic) field. The phase diagrams were plotted as a function of the dipole moment strength g

0.4

0.3

h

bco

fcc

0.2 bct 0.1

bcc

gas–bct String ﬂuid

0

Fluid 0

0.2

0.4

0.6

0.8

Erms(V/mm)

FIGURE 8.20 Reproduction of the experimental phase diagram. (From Yethiraj A and van Blaaderen A. 2003. Nature 421: 513–517. With permission.)

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Structure and Functional Properties of Colloidal Systems

and the packing fraction h. In the phase diagram of dipolar hard spheres, we found stable regions of (string-) fluid, fcc, hcp, and bct phases, and regions of fluid–hcp and fluid–bct coexistence. In the phase diagram of soft spheres, we found all the above phases and also a stable region of the asymmetric bco phase. We found the hcp phase as the new stable phase in the high packing fraction region. Our results show that bulk hcp, bct, and bco crystals can be stabilized and therefore realized experimentally by applying an external electric or magnetic field. It is important to remember that these crystal phases are unstable in the absence of a field.

8.3.2

MANIPULATION OF COLLOIDAL CRYSTAL STRUCTURES BY CONFINEMENT

The physics of confined systems is important in different fields of modern technology, such as lubrication, adhesion, and nanotechnology. The study of simple models is instrumental in understanding the behavior of complex systems. As such the hard-sphere system plays an important role in statistical physics; it serves as a reference system for determining the structure and phase behavior of complex fluids, both in theory and in simulations. The bulk-phase behavior of hard spheres is now well understood. At sufficiently high densities, the spheres can maximize their entropy by forming an ordered crystal phase [6,16]. The insertion of a hard wall in such a fluid decreases the number of hard-sphere configurations. The system can increase its entropy by the spontaneous formation of crystalline layers with triangular symmetry, the (111) plane, at the wall, while the bulk is still a fluid. This effect is known as prefreezing, and is analogous to complete wetting by fluids at solid substrates [75]. It is induced by the presence of a single wall and should not be confused with capillary freezing. Capillary freezing denotes the phenomenon of confinement-induced freezing of the whole fluid in the pore at thermodynamic state points where the bulk is still a fluid. This transition depends strongly on the plate separation. The opposite phenomenon, called capillary melting, can also occur. The capillary induces melting for thermodynamic state points that correspond to a crystal in the bulk. Confinement can also change dramatically the equilibrium crystal structure. In 1983, Pieranski reported a sequence of layered solid structures with triangular and square symmetry for colloidal hard spheres confined in a wedge [76]. The sequence of high-density structures is determined more accurately in recent experiments [57,77], reporting the observation of prism phases with both square and triangular symmetry. Recently, Cohen studied configurations of confined hard spheres under shear, demonstrating the importance of the equilibrium configurations in the rheological properties [78]. Despite the great number of theoretical and simulation studies on confined hard spheres [79–81], the full equilibrium phase behavior is yet unknown. In fact, many of the previous studies were based on an order parameter analysis, which fails dramatically in discriminating the different structures at high densities and large plate separations. More importantly, free energy calculations of confined hard spheres are prohibited so far due to the lack of an efficient thermodynamic integration path that relates the free energy of interest to that of a reference system, while a further complication arises from the enormous number of possible solid phases that has to be considered. Hence, it is unresolved whether the experimentally observed phases are stabilized kinetically or are thermodynamically stable. In this work, we present a novel efficient thermodynamic integration path that enables us to calculate the free energy of densely packed and confined hard spheres, with high accuracy close to the fluid–solid transition. This method allows us to determine for the first time the stability of the structures found in experiments. To this end, we perform explicit free energy calculations to map out the full phase diagram for plate separations from 1 to 5 hard-sphere diameters. We report a dazzling number of thermodynamically stable crystal structures (26!) including triangular, square, buckling, rhombic, and prism phases, and a cascade of corresponding solid–solid transformations. In addition, the free energy calculations allow us to determine the chemical potential at coexistence that was not accessible in previous simulations. From the analysis of the chemical potential, we find an intriguing sequence of capillary freezing and melting transitions coupled to a structural phase transition of the confined crystal. We note that our new method and results are also relevant for confined simple

191

Phase Behavior and Structure of Colloidal Suspensions

fluids [82–84] and self-assembled biological systems [85]. In addition, the structure of dense packings of spheres explains the shape of, for instance, snowflakes, bee honeycombs, and foams, and it is of great importance for fundamental research, for example, solid-state physics and crystallography, and for applications such as communication science or powder technology [86,87]. Our model system consists of N hard spheres with diameter s, confined between two parallel hard plates of area A = L xLy (Figure 8.21). In each layer, we used approximately 200 particles. We use the packing fraction h = ps3N/(6AH) as a dimensionless density, where H is the distance between two plates. We determine the equilibrium phase diagram by performing MC simulations in a box, which is allowed to change its shape to accommodate different types of crystals; the ratio L x/Ly may vary while H and A are fixed. Trial solid structures are obtained from crystals with triangular or square symmetry relaxed with MC moves while slowly increasing the density by expanding the spheres. The free energy F for the resulting equilibrated structures is calculated as a function of h and H. We use the standard thermodynamic integration technique [21], but with a new and efficient path based on penetrable potentials that enable us to change gradually from a noninteracting system to the confined hard-core system of interest. The sphere–sphere potential reads as ÏÔe exp(−ARij ) if Rij < s vij ( Rij ) = Ì otherwise ÔÓ0

(8.17)

Ïe exp(−Bzi ) if zi < s / 2 vwi ( zi ) = Ì , otherwise Ó0

(8.18)

and the wall–fluid potential

where Rij is the distance between spheres i and j, zi is the distance of sphere i to the nearest wall, A and B are adjustable parameters that are kept fixed during the simulations, and e is the integration parameter. The limit e Æ • yields the hard-core interaction, but convergence of the thermodynamic integration is already obtained for e ⬇ 70 k BT. The reference states (e = 0 k BT) are the ideal gas and the Einstein crystal for the fluid and solid phase, respectively. We use a 21-point Gaussian quadrature for the numerical integrations and the ensemble averages are calculated from runs with 40.000 MC cycles (attempts to displace each particle once), after first equilibrating the system during 20.000 MC cycles. We determine phase coexistence by equating the grand potentials W = F - mN [89]. To validate this approach, we perform simulations of a bulk system of hard spheres and find that the packing fractions of the coexisting fluid and fcc solid phase are given by hf = 0.4915 ± 0.0005

Ly Lx σ

H

FIGURE 8.21 Illustration of hard spheres with diameter s, confined between parallel hard plates of area A = L xLy and a separation distance H. (From Fortini A and Dijkstra M. 2006. Journal of Physics: Condensed Matter 18: L371–L378. With permission.)

192

Structure and Functional Properties of Colloidal Systems

and hs = 0.5428 ± 0.0005, respectively. The pressure and the chemical potential at coexistence are bPs3 = 11.57 ± 0.10 and bm = 16.08 ± 0.10. These results are in good agreement with earlier results [16,90]. Furthermore, to validate the approach for confined systems, we determine at bulk coexistence the wall–fluid interfacial tension bgwf s2 = 1.990 ± 0.007, and the wall–solid interfacial tension 2 100 2 for the (111) and (100) planes of the fcc phase, bg 111 ws s = 1.457 ± 0.018 and bg ws s = 2.106 ± 0.021. Our results are in agreement with previous simulations [91], but the statistical error is one order of magnitude smaller due to our new thermodynamic integration path. Employing this approach, we determine the phase behavior of confined hard spheres for plate separations 1 < H/s £ 5. Figure 8.22 displays the full phase diagram based on free energy calculations in the H–h representation. The white regions of the phase diagram denote the stable one-phase regions. The shaded regions indicate coexistence between fluid and solid or two solid phases, and the dotted region is forbidden as it exceeds the maximum packing fraction of confined hard spheres. At low densities, we observe a stable fluid phase followed by a fluid–solid transition upon increasing the density. The oscillations in the freezing and melting lines reflect the (in)commensurability of the crystal structures with the available space between the walls. For the crystal phases, we follow the convention introduced by Pieranski [76], where nD denotes a stack of n triangular layers, and n䊐 denotes a stack of n square layers. For H/s Æ 1, the stable crystal phase consists of a single triangular layer 1D, which packs more efficiently than the square layer. As the gap between the plates increases, crystal slabs with triangular (Figure 8.23a) and square packings (Figure 8.23b) are alternately stable. We find the characteristic sequence nD Æ (n + 1)䊐 Æ (n + 1)D that consists of an nD Æ (n + 1)䊐 transformation where both the number of layers and the symmetry change followed by an (n + 1)䊐 Æ (n + 1)D transformation where only the symmetry changes. This sequence is driven by a competition of a smaller height of n square layers compared with n triangular layers and a more efficient packing of triangular layers with respect to square layers. When the available gap is larger than that required for the nD structure, but smaller than that required for (n + 1)䊐, intermediate structures may become stable. Similar arguments can be used for the intervention of intermediate structures in the

5

6 h error bar

I10

5

I9 I8

5

H/s

4

I7

4

Fluid 3

I6

3

I5

3 2

2

I4 I3

2 1 1 0.3

0.4

I2 I1 0.5 h

0.6

Forbidden

4

0.7

FIGURE 8.22 Equilibrium phase diagram of hard spheres with diameter s confined between two parallel hard walls in the plate separation H–packing fraction h representation. The white, shaded, and dotted regions indicate the stable one-phase region, the two-phase coexistence region, and the forbidden region, respectively. (From Fortini A and Dijkstra M. 2006. Journal of Physics: Condensed Matter 18: L371–L378. With permission.)

Phase Behavior and Structure of Colloidal Suspensions (a)

193

(b)

(c) (d)

(e)

(f)

(g)

(h)

FIGURE 8.23 Stable solid structures of confined hard spheres: (a) the triangular phase 2D; (b) the square phase 2䊐; (c) the buckling phase 2B; (d) the rhombic phase 2R; (e), (g) the prism phase with square symmetry 3P䊐; and (f), (h) the prism phase with triangular symmetry 3PD. In (a)–(f), the point of view is at an angle of 30° to the z direction. In (g), (h), the point of view is at an angle of 90°. Different shades indicate particles in different planes ((a)–(d)) or particles belonging to different prism structures ((e)–(h)). (From Fortini A and Dijkstra M. 2006. Journal of Physics: Condensed Matter 18: L371–L378. With permission.)

(n + 1)䊐 Æ (n + 1)D transformation. Especially at high packing fractions, the spheres can increase their packing by adopting interpolating structures. In Figure 8.22, we report the boundaries of the interpolating regions In. Each region represents one or more interpolating structures that are listed in Table 8.1 according to the standard notation. Within the resolution of our simulations, it is difficult to draw the phase boundaries of all the intermediate structures in In, but in Table 8.1 the thermodynamically stable structures are listed in the order they appear upon increasing H and h. We also compare our sequence of structures with the experimental one [57]. The experiments considered

194

Structure and Functional Properties of Colloidal Systems

TABLE 8.1 List of Intermediate Structures In as Found in Our Simulations and in the Experiments Phase

Transition

I1

Simulation

Experiment

1D Æ 2䊐

2B

2B

I2

2䊐 Æ 2D

2R

2R

I3

2D Æ 3䊐

2PD + 2P䊐

2PD + 2P䊐

I4

3䊐 Æ 3D

3R + 3P䊐 + 3B

I5

4D Æ 4䊐

3PD + 4B

3R + 3P䊐 + 3PD 3P䊐

I6

4䊐 Æ 4D

4P䊐 + 4R + 4PD

4P䊐 + 4PD

I7

4D Æ 5䊐

4PD

I8

5䊐 Æ 5D

5P䊐 + 4PD + 5PD + 5R

I9

5D Æ 6䊐

5PD

4P䊐 + 4PD + Ha 5P䊐b No data

I10

6䊐 Æ 6D

5PD

No data

Source: Fontecha AB, et al. 2005. Journal of Physics: Condensed Matter 17: S2779–S2786. With permission. a Not fully characterized structure with hexagonal symmetry. b Not fully characterized prismatic structure.

charged particles, but we do not expect that the soft repulsion has a strong effect on the observed structures at high densities. The agreement is excellent at small plate separations. The buckling phase 2B (Figure 8.23c) interpolates between the 1D and 2䊐 phases. In the 2B phase, the 1D structure is split into two sublayers consisting of rows that are displaced in height and that can transform smoothly into 2䊐 structure upon increasing the gap. The rhombic phase 2R (Figure 8.23d) is found between the 2䊐 and 2D phases. The rhombic phase is also stable between the 2䊐 and 2D phases n £ 5, but not in the whole region. In addition, we find that at higher n the interpolating structures are mainly prism phases. In agreement with experiments, we find two types of prisms, one with a square base nP䊐 (Figure 8.23e), and the other with a triangular base nPD (Figure 8.23f), where n indicates the number of particles in the prism base. As shown in Figure 8.23g and h, these structures display large gaps as a result of periodically repeated stacking faults in the packing which, nevertheless, allow particles to pack more efficiently than a phase consisting of parallel planes of particles. For n > 3, differences between simulations and experiments emerge. We find that the stability region of interpolating structures between nD and n䊐 decreases for larger H, becoming invisible on the scale of Figure 8.22 for I9 = 5D Æ 6䊐. On the other hand, the region of stability of the interpolating structures between n䊐 and nD increases while increasing the wall separation, becoming stable also at low packing fractions for the transitions I8 = 5D Æ 5䊐, and possibly I10 = 6D Æ 6䊐. We also note that the solid–solid transitions are first-order with a clear density jump at low h, but they get weaker (and maybe even continuous) upon approaching the maximum packing limit. In addition, the rhombic and buckling phases highly degenerate as we find zig-zag and linear buckling or rhombic phases, and a combination of those. We now turn our attention to the fluid–solid transition. In Figure 8.24, we plot the chemical potential bm cap at the freezing transition of the confined system as a function of H. The freezing for crystal slabs with a triangular symmetry are denoted by triangles, while the square symmetry is displayed by squares. We find strong oscillations in the chemical potential reminiscent to the (in) commensurability of the crystal structures with plate separation. The highest values for bm cap are reached at the transition region nD Æ (n + 1)䊐 corresponding to plate separations where both structures are incommensurate and hence unfavorable. In this regime, bm cap can reach values that are higher than the bulk freezing chemical potential bmbulk (the black vertical line in Figure 8.24), corresponding to capillary melting, while the freezing transitions with bm cap lower than the bulk value

195

Phase Behavior and Structure of Colloidal Suspensions 5

H/s

4

3

2

1 10

12

14

16

18

20

bm

FIGURE 8.24 Chemical potential bm at fluid–solid coexistence, for different wall separations H/s. The symbols are the simulation results for the triangular (D) and square structures (䊐). The thin dashed line is a guide to the eye. The thick continuous line indicate the value of the bulk freezing chemical potential bm bulk = 16.08. The thick dashed and dotted curves are the prediction of the Kelvin equation for the (111) and (100) planes parallel to the walls, respectively. (From Fortini A and Dijkstra M. 2006. Journal of Physics: Condensed Matter 18: L371–L378. With permission.)

correspond to capillary freezing. Hence, we find a re-entrant capillary freezing/melting behavior for wall separations 1 < H/s < 3.5. In addition, we compare our results with the predictions of the Kelvin equation [92]: bm cap = bm bulk -

ps 3 ( g wf - g ws ) , 3 H ( hs - hf )

(8.19)

using the parameters determined in our simulations. The thick dashed line in Figure 8.24 is the prediction of the Kelvin equation for the (111) crystal plane (triangular order) at the walls, whereas the dotted line is that for the (100) plane (square order). The Kelvin equation predicts capillary freezing for the triangular structure and capillary melting for the square structures. The Kelvin equation predictions are in reasonable agreement with our simulations for triangular order for wall separation as small as H/s ⬇ 4, but deviates for smaller H, while the prediction for the square structure is in agreement only at very small H. It is surprising to find qualitative agreement at small H since the Kelvin equation is valid in the limit H/s Æ •. In summary, we calculated the equilibrium phase diagram of confined hard spheres using free energy calculations with a novel integration path. The high-density sequence of structures is in good agreement with experimental results. We find that the prism phases are thermodynamically stable also at lower densities, and this work will hopefully stimulate further experimental investigations for a quantitative comparison at intermediate packing fractions. In addition, our results show an intriguing sequence of melting and freezing transitions upon increasing the distance between the walls of a slit that is in contact with a bulk reservoir. The mechanical behavior is therefore very sensitive on the degree of confinement, and the knowledge of the phase diagram can help the understanding and fabrication of new materials.

196

Structure and Functional Properties of Colloidal Systems

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56. Yethiraj A, Thijssen JHJ, Wouterse A, and van Blaaderen A. 2004. Large-area electric-field-induced colloidal single crystals for photonic applications. Advanced Materials 16: 596–600. 57. Fontecha AB, Schope HJ, Konig H, Palberg T, Messina R, and Lowen H. 2005. A comparative study on the phase behaviour of highly charged colloidal spheres in a confining wedge geometry. Journal of Physics: Condensed Matter 17: S2779–S2786. 58. Chen TJ, Zitter RN, and Tao R. 1992. Laser diffraction determination of the crystalline-structure of an electrorheological fluid. Physical Review Letters 68: 2555–2558. 59. Dassanayake U, Fraden S, and van Blaaderen A. 2000. Structure of electrorheological fluids. Journal of Chemical Physics 112: 3851–3858. 60. Friedberg R and Yu YK. 1992. Energy of an electrorheological solid calculated with inclusion of higher multipoles. Physical Review B 46: 6582–6585. 61. Halsey TC and Toor W. 1990. Structure of electrorheological fluids. Physical Review Letters 65: 2820–2823. 62. Khusid B and Acrivos A. 1999. Phase diagrams of electric-field-induced aggregation in conducting colloidal suspensions. Physical Review E 60: 3015–3035. 63. Klapp S and Forstmann F. 1999. Phase behavior of aligned dipolar hard spheres: Integral equations and density functional results. Physical Review E 60: 3183–3198. 64. Toor WR and Halsey TC. 1992. Surface and bulk energies of dipolar lattices. Physical Review A 45: 8617–8623. 65. Groh B and Dietrich S. 2001. Crystal structures and freezing of dipolar fluids. Physical Review E 6302: 021203. 66. Gross M and Wei C. 2000. Ground state of a dipolar crystal. Physical Review E 61: 2099–2102. 67. Martin JE, Anderson RA, and Tigges CP. 1998. Simulation of the athermal coarsening of composites structured by a biaxial field. Journal of Chemical Physics 108: 7887–7900. 68. Martin JE, Anderson RA, and Tigges CP. 1999. Thermal coarsening of uniaxial and biaxial field-structured composites. Journal of Chemical Physics 110: 4854–4866. 69. Tao R and Sun JM. 1991. Ground-state of electrorheological fluids from Monte-Carlo simulations. Physical Review A 44: R6181–R6184. 70. Tao R. 1993. Electric-field-induced phase-transition in electrorheological fluids. Physical Review E 47: 423–426. 71. Tao R. 2001. Super-strong magnetorheological fluids. Journal of Physics: Condensed Matter 13: R979–R999. 72. Wang ZW and Holm C. 2001. Estimate of the cutoff errors in the Ewald summation for dipolar systems. Journal of Chemical Physics 115: 6351–6359. 73. Hynninen AP and Dijkstra M. 2005. Phase diagram of dipolar hard and soft spheres: Manipulation of colloidal crystal structures by an external field. Physical Review Letters 94: 138303. 74. Hynninen AP and Dijkstra M. 2005. Phase behavior of dipolar hard and soft spheres. Physical Review E 72: 051402. 75. Dijkstra M. 2004. Capillary freezing or complete wetting of hard spheres in a planar hard slit? Physical Review Letters 93: 108303. 76. Pieranski P, Strzelecki L, and Pansu B. 1983. Thin colloidal crystals. Physical Review Letters 50: 900–903. 77. Neser S, Bechinger C, Leiderer P, and Palberg T. 1997. Finite-size effects on the closest packing of hard spheres. Physical Review Letters 79: 2348–2351. 78. Cohen I, Mason TG, and Weitz DA. 2004. Shear-induced configurations of confined colloidal suspensions. Physical Review Letters 93: 046001. 79. Messina R and Lowen H. 2003. Reentrant transitions in colloidal or dusty plasma bilayers. Physical Review Letters 91: 146101. 80. Schmidt M and Lowen H. 1996. Freezing between two and three dimensions. Physical Review Letters 76: 4552–4555. 81. Zangi R and Rice SA. 2000. Hexagonal to square lattice conversion in bilayer systems. Physical Review E 61: 671–681. 82. Gao JP, Luedtke WD, and Landman U. 1997. Layering transitions and dynamics of confined liquid films. Physical Review Letters 79: 705–708. 83. Ghatak C and Ayappa KG. 2001. Solid–solid transformations in a confined soft sphere fluid. Physical Review E 64: 051507. 84. Klein J and Kumacheva E. 1995. Confinement-induced phase-transitions in simple liquids. Science 269: 816–819.

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9

Water–Water Interfaces R. Hans Tromp

CONTENTS 9.1 9.2

Introduction ...................................................................................................................... Theoretical Background ................................................................................................... 9.2.1 Interface Tension ................................................................................................... 9.2.2 Interface Width ..................................................................................................... 9.2.3 Interface Excess .................................................................................................... 9.2.4 Permeability of the Interface ................................................................................ 9.3 Polydispersity .................................................................................................................... 9.4 Experiments ...................................................................................................................... 9.4.1 Interface Tension ................................................................................................... 9.4.2 Interface Width ..................................................................................................... 9.4.3 Permeability and Excess Water at the Interface ................................................... 9.4.4 Coexistence of Curvatures .................................................................................... 9.5 Suggestions for Further Work ........................................................................................... References ..................................................................................................................................

201 202 202 205 207 208 210 210 210 212 212 214 215 216

9.1 INTRODUCTION The water–water interface is a special case of the interface that forms on the phase separation of a solution of two solutes, with their common solvent being water. The result of the phase separation is a situation of two coexistent phases, one enriched (relative to the overall concentration) with respect to one solute, and the other enriched with respect to the other solute. In most cases studied, the two solutes are polymers, but solutions of polymer surfactants may also show phase separation [1], as well as solutions of a single solute, which can form ordered structures that coexist with an isotropically dissolved state [2–4]. The present overview focuses on solutions of two polymers. The interface between two aqueous polymer solutions is interesting from both a practical and a scientific point of view. The tendency toward environmentally friendly industrial processes in which water takes the place of organic solvents is increasing their occurrence in practical systems. The demand for low-fat processed food makes more urgent the question of to what extent a water–water interface can be given the same properties as oil–water interfaces [5]. A third area in which the feasibility of the application of water–water interfaces is emerging is in systems for the controlled release of pharmaceuticals in the human body. From a scientific point of view, water–water interfaces, as well as solvent–solvent interfaces in general, are interesting because of their extremely low interface energy (interface tension). Typically, interface tensions are in the μN/m range [6–9], although values in the nN/m range have been found [2,10]. These low values are due to concentration gradients at the interface that are much lower than those in the case of polymer blends. Such weak gradients are expected on theoretical grounds to be sensitive to curvature and therefore difficult or impossible to measure by classical methods, such as 201

202

Structure and Functional Properties of Colloidal Systems 10

Dextran [%w/w]

8

6

4

2

0

0

2

4 6 Gelatine [%w/w]

8

10

FIGURE 9.1 Representative phase diagram of mixing of dextran (Mw 182 kDa) and pork skin gelatin (Mw 150 kDa) at 50°C. The solid line is the cloud line. The dashed lines are tie lines.

the Wilhelmy plate method, of monitoring the force needed for macroscopic deformation. Another interesting aspect of solvent–solvent interfaces is the fact that they are permeable to the solvent [11]. The particular case of an aqueous solvent has the extra dimension of (partial) permeability for salts and the subsequent possibility of Donnan potentials, which, however, have not been observed yet. The permeability makes water–water interfaces comparable to semipermeable dialysis membranes. This permeability is discussed further in the experimental section. In this overview, water–water interfaces are defined as the composition gradient in the latest stage of phase separation in a ternary system of polymer A, polymer B, and aqueous solvent (which may include salt or other low molar mass solutes) in the absence of gel formation or glass formation. “Late stage” refers to the situation in which further phase separation involves only the decrease of curvature of the interface and the interface area. Most experimental examples are taken from work carried out on aqueous mixtures of gelatin and dextran, which are particularly suitable as a model system for structure-building food biopolymers. Advantages of this system are the possibility of full phase separation at temperatures above the temperature of gelation of gelatin (about 40°C for standard gelatin and about 8°C for fish gelatin), leading to transparent phases, and the possibility of studying simultaneous phase separation and gelation at temperatures at which gelatin gels. A representative phase diagram is shown in Figure 9.1. Examples of other widely studied water–water phase separating systems are pectin–caseinate [12–19].

9.2 THEORETICAL BACKGROUND 9.2.1

INTERFACE TENSION

Phase separation in a solution of two polymers, giving rise to two coexistent solutions that are enriched in one of the two polymers, has been described many times from both a theoretical and an experimental point of view, and is reasonably well understood. The latest stage, which is our present interest, is a concentration gradient, identified as an interface, of both polymers and water. The

203

Water–Water Interfaces

width of the gradient is governed by the balance between the entropy of mixing, which tends to weaken the gradient, and the incompatibility of the polymers, which tends to sharpen the gradient. For a system of a certain composition, the temperature determines the exact position of the equilibrium between the two opposing influences on the width of the interface. The reason for the incompatibility of the polymers in the same solvent [20] could be a repulsive direct interaction between two chemically different species, such as is found for oil and water or a difference in solvent quality. The overlap concentration (in % w/w)

c* ª

3N p

M mon 4 pR 10 4 N Av

(9.1)

3 g

for typical cases such as aqueous solutions of gelatin and dextran (radius of gyration Rg 18 nm and degree of polymerization Np 1000 for both polymers, molar mass of monomers Mmon 120 g, and NAv, Avogadro’s number) is in the range 1–3% (w/w). The lowest concentration where phase separation takes place, that is, the critical concentration of mixing (for the gelatin/dextran system found to be 3.5% dextran and 3.5% gelatin, see Figure 9.1) coincides therefore with the onset of entanglements (if no phase separation would take place) and competition for hydration water. In practice, even for molar masses in excess of 1 MDa, the critical concentration of mixing is less than 10 times c*. If competition for hydration water is causing phase separation, it could be accompanied by a change in the compactness of the chain or even chain conformation [15]. As a consequence, the phase separation of the two polymers is then coupled with a change in radius of gyration, rendering the description of the process less easy than for a binary system. At the interface between coexisting polymer solutions a central layer enriched in solvent is expected, because the two incompatible polymers avoid mutual contact. Using mean field theory extended to include excluded volume effects to describe the mixing of two polymers in a common solvent, the structure of the interface can be estimated [21]. In this approach, the polymer solution is considered a “melt” of ideal chains of “blobs.” Blobs are sequences of monomers that are correlated by excluded volume effects. Beyond a blob, no such correlations exist and the chain of blobs is ideal. The interaction between blobs of the two types of polymers is responsible for phase separation. This interaction is dependent on concentration because the blob size is dependent on concentration. The calculation [21] starts from the density of the free energy of mixing, rescaled on the relevant distance scale, that is, the blob size Ê c ˆ x = Rg Á ˜ Ë c *¯

-3/ 4

= 6

-5/8

b

-5/ 4 -3/ 4

c

Ê 3 M mon ˆ Á 4 p 10 4 N ˜ Ë Av ¯

3/ 4

,

(9.2)

where Rg is the radius of gyration of the polymer chain in dilute solution and c is the monomer concentration in % w/w. The density of the free energy of mixing is then expressed by Fmix c u K = [j A ln j A + j B ln j B ] + 3 j A j B + 3 , kT N x x

(9.3)

where jA and jB are the volume fractions of polymers A and B and u is the interaction energy between polymers in units kT and K 0.024 [22,23]. K /x3 is the interaction energy between polymer and solvent (for the sake of simplicity this interaction is taken to be the same for both polymers), which is concentration-dependent because the size of a blob is concentration-dependent. This term in fact represents the contribution to the osmotic pressure of a number of x-3 blobs per unit volume.

204

Structure and Functional Properties of Colloidal Systems 10 9

Concentration [%w/w]

8 7

ccrit = 1.5%

6

ccrit = 3%

5

ccrit = 4%

4 3 2 1 0 –40

–30

–20 –10 0 10 20 Distance from the interface [nm]

30

40

FIGURE 9.2 Profile of an interface between coexistent phases of solutions of polymers A and B, calculated while taking into account excluded volume effects inside the “blobs” of correlated monomers of semidilute solutions [21]. Polymers A and B are identical with respect to degree of polymerization (1000), radius of gyration in dilute solution (18 nm), critical concentration of demixing, and overall concentration of each (5% w/w). Values are based on those experimentally relevant for phase-separated mixtures of gelatin (Mw 150 kDa) and dextran (Mw 182 Da). Gray lines: profiles of polymer concentrations; black lines: profiles of solvent concentrations.

The value of u is imposed by the experimentally available critical concentration of mixing: 0.3

2 Ê c ˆ u= . N blob ÁË ccrit ˜¯

(9.4)

Nblob is the number of blobs per chain: N blob =

N . cx3

(9.5)

In the case of the presence of gradients ranging over several blob sizes (i.e., weak gradients) the additional free energy due to the gradients can be included in the expression for the free energy of mixing in terms of squares of gradients in total polymer concentration and polymer composition: Fmix c u K 1 1 2 = ÈÎj A ln j A + j B ln j B ˘˚ + 3 j A j B + 3 + G comp (Dj ) + G conc c 2 . kT N 2 2 x x

(9.6)

It has been assumed that polymers A and B are identical with respect to size and interaction with the solvent.* c˙ and j˙ are the derivatives in the distance to the interface. Gcomp and Gconc are the energies (in units of kT) associated with gradients in the polymer composition and the polymer concentration and Dj is the difference between the volume fractions of polymers A and B. To find the equilibrium profile of the interface, the integral of the free energy Fmix over the distance to the interface, in other words the interface tension, is minimized. Figure 9.2 gives an example for three

*

The realism of this assumption is questionable, as the reason for phase separation of the polymers may be the difference in their interaction with the solvent. However, for the essence of the outcome, that is, the enrichment in solvent of the interface, this is irrelevant.

205

Water–Water Interfaces

degrees of incompatibility, corresponding to commonly found critical concentrations of mixing, 1.5%, 3%, and 4% (w/w). Parameters are chosen to mimic the gelatin/dextran/water system. The minimized free energy is the interface tension. Its reduction relative to g •, the value for and interface between phases of infinitely long chains g• =

kT u / 6, x2

(9.7)

(symbols with a bar indicate values far from the interface, that is, “bulk” values), consists of two contributions: D comp from the incomplete incompatibility and D conc from the reduced polymer concentration or excess solvent at the interface: g = g • (1 - D comp - D conc ),

(9.8)

with

D comp = 1 +

1 x 8 6u

. ( D j )2 dz . 2 -μ 1 - ( D j ) μ

Ú

(9.9)

and

D conc =

1 8c

6u x

μ

¸ 6n + c - 1 ÔÏ x 2 (Dj )2 È(Dj )2 - (Dj )2 ˘ Ô˝(c - c )dz, Ì 6u 2 Î ˚ 3n - 1 Ô 1 - (Dj ) Ô˛ -μ Ó

Ú

(9.10)

in which n = 3/5 and c 0.22, so 6n + c - 1 ª 3.52. 3n - 1

(9.11)

Dcomp equals unity when the gradient Dj is zero for all z, due to infinite incompatibility. Dconc equals zero when c(z) is equal to cˉ at all z, due to absence of solvent. It is also equal to zero in the case of infinite incompatibility. The interface tensions calculated from the profiles in Figure 9.2 are in Figure 9.3. The values found for the range of practical concentrations of 5–20% are roughly between 1 and 20 mN/m. A higher incompatibility, that is, a lower ccrit, leads to a higher interface tension. In the example given in Figure 9.2, the total polymer concentration in the middle of the interface is about 9.5%, instead of 10% in the bulk. As expected, the theory predicts that this depletion of polymer at the interface increases (the excess decreases) as the total concentration increases (Figure 9.4).

9.2.2

INTERFACE WIDTH

The value of interface width, defined as

D=

x Dc 6u c z =0

(9.12)

206

Structure and Functional Properties of Colloidal Systems 50 ccrit = 1.5%

Interface tension [mN/m]

40

ccrit = 3% 30

ccrit = 4% ccrit = 6%

20 ccrit = 8% 10

0

0

10 20 Total polymer concentration [%w/w]

30

FIGURE 9.3 Calculated interface tension, using Equations 9.8 through 9.10, of the interface between coexisting phases of two incompatible, but otherwise identical polymers, for five degrees of incompatibility. For further details, see the caption of Figure 9.2.

(Dcˉ is the difference between the coexisting concentrations far from the interface, and z is the distance to the center of the interface), lies between 7 and 10 nm (Figure 9.5) and is, except for the highest incompatibility of ccrit = 1.5%, quite independent of the concentration in the relevant concentration range between 5% and 20%. This is probably due to the interplay between the opposing influences of an increasing polymer concentration on the interface width. In the absence of solvent, an increase in incompatibility would cause the interface to become thinner. The presence of solvent, however, enables the total polymer concentration at the interface to decrease and provide a pathway to reduce interface energy.

Excess concentration at interface [%w/w]

0.00

–0.20

ccrit = 1.5% ccrit = 3%

–0.40

ccrit = 4% –0.60

ccrit = 6% ccrit = 8%

–0.80

–1.00

0

10 20 Total polymer concentration [%w/w]

30

FIGURE 9.4 Depletion of total polymer concentration at the center of the interface versus total polymer concentration for five degrees of incompatibility. For further details, see the caption of Figure 9.2.

207

Water–Water Interfaces

Interface width [nm]

10.00

9.50

ccrit = 1.5%

9.00

ccrit = 3% ccrit = 4%

8.50 ccrit = 6% 8.00

ccrit = 8%

7.50

7.00

FIGURE 9.5

9.2.3

0

10 20 Total polymer concentration [%w/w]

30

Calculated interface width. For parameter values, see Figure 9.2.

INTERFACE EXCESS

The amount of polymer depleted from the interface region, enriching the bulk of the phases, can be approximated by integrating the total polymer concentration along the distance from the center of the interface. The result for the example of Figure 9.2 is in Figure 9.6. The degree of desorption is consistent with what is usually found for polymer adsorption or desorption. In the case of a monolayer, the amount of polymer adsorption is usually 1 mg/m2 or less [24]. There turns out to be a minimum in the polymer excess at the interface as a function of concentration, that is, a maximum in accumulation of solvent. This minimum becomes deeper, the higher the incompatibility. This minimum is most pronounced for ccrit = 1.5%. At this concentration, there is also a significant influence of the concentration on the interface width (Figure 9.5). The minimum

Polymer depletion at interface [mg/m2]

0.00

ccrit = 1.5%

–0.05

ccrit = 3% –0.10 ccrit = 4% –0.15

ccrit = 6% ccrit = 8%

–0.20

–0.25

0

10

20

30

Total polymer concentration [%w/w]

FIGURE 9.6 Figure 9.2.

Quantity of polymer depleted from one side of the interface area. For parameter values, see

208

Structure and Functional Properties of Colloidal Systems

may therefore be explained by the opposing effects, with increasing concentration, of a deepening depletion at the center of the interface (see Figure 9.4) and a decreasing interface width. The quantity of polymer expelled from the interface causes the bulk phases to become enriched in polymer. In fully phase-separated systems with a single flat interface this enrichment is extremely small. However, when such a system is stirred, or when full phase separation has not taken place yet and there is still a large amount of interface, the enrichment may become significant. The bulk phase concentration in a system of dispersed phase regions cdisp, far away from the interface (i.e., much farther than D), can be estimated by assuming a situation of phase broken up into droplets of size R and taking a fraction y of the total volume: cdisp = c +

M depl y M depl y ªc + , 4 pR - 3jD 4pR

(9.13)

in which Mdepl is the mass of polymer depleted from the interface per unit area (plotted in Figure 9.6), cˉ is the phase concentration in the absence of interfaces, and D is the interface width. For the typical case of c = 10%, Mdepl = 0.17 mg/m2, and R = 10 μm, the increase in concentration far from the interface is about 6%, that is, the concentration goes up from 10% to 10.6%. This suggests that during the late stage of phase separation the systems is actually demixed further than after maximum reduction of the amount of interface. Figure 9.7 shows the results of the calculation of cdisp as a function of droplet size for some values of the incompatibility.

9.2.4

PERMEABILITY OF THE INTERFACE

In the final stage of the phase separation process gradients, initially present all over the volume, are limited to the interface region. Further time dependence arises from the tendency to reduce interface area (coarsening). The interface is locally curved and, as a consequence, there will be local variation in Laplace pressure. In this stage, it may be expected that the permeability for water at the interface introduces a difference between the coarsening behavior of phase-separated polymer blends and polymer solutions. In the case of polymer blends, relaxation of local differences in Laplace pressure can only take place by polymer diffusion or convection, whereas in the case of polymer solutions, solvent swiftly diffuses through interfaces or toward sharply curved interfaces. The result of this diffusion could be an increase in the width of the interface, which reduces the interface tension and the Laplace pressure. Another effect of the permeability of the interface for solvent, or, in other words, the local osmotic compressibility, could be the compensation of Laplace pressure by a difference in osmotic pressure on the two sides of the interface. In that case, the two sides of the interface may still have the same chemical potential, which, however, has different contributions from the different concentrations of the three components. In fact, the permeability of the interface means that the free energy of mixing of two polymers and solvent depends on the amount of curvature per unit volume. As a consequence, the free energy of mixing depends on time because the amount of curvature per volume decreases with time. A rough estimate of the size of this effect, which is analogous to the increase of vapor pressure above a curved area of matter, may be obtained from equating the osmotic pressure to the Laplace pressure: P in - P out =

2g , R

(9.14)

in which Pin and Pout are the osmotic pressures inside and outside a phase droplet inside a continuous volume of the other phase. For concentrations above the overlap concentration c*, P (c ) =

N Av Ê c ˆ kTc * Á ˜ 4 Ë c *¯ 10 M mon

9/4

.

(9.15)

209

Water–Water Interfaces

Because the interface is permeable to water, the Laplace pressure may be relaxed by sorption of water into the droplet causing it to swell, until a new equilibrium is reached with concentrations inside and outside the droplet of c - d and c + d, respectively. Inserting Equation 9.15 into Equation 9.14 and taking into account the swelling of the droplet, we obtain ÈÊ c + d ˆ 9/4 Ê c - d ˆ 9/4 ˘ N av 2g kTc * ÍÁ , -Á ˙= ˜ ˜ -1/3 c * c * Ë ¯ Ë ¯ 10 M mon ÍÎ ˙˚ R (c - d / c )

(9.16)

4

where R is the initial droplet radius, and g is the interface tension, which is assumed to be independent of curvature and to stay the same when the droplet absorbs water. For the typical case of c = 10%, N = 1000, R = 10 mm, and g = 1 mN/m, the value of d is about 1% (w/w) for a phase ratio of 50%, that is, the concentration inside the droplets decreases from 10% to about 9% by swelling. The solution of Equation 9.16 for a droplet size prior to swelling (i.e., R) of 10 mm is in Figure 9.8. For the calculation of the data in Figures 9.7 and 9.8, only the dependence on the overall concentration of the interface tension is taken into account, not the effect of the difference in concentration on the two sides of the interface. Also, polydispersity effects are ignored. Therefore, only the trends in Figures 9.7 and 9.8 should be considered. However, when comparing Figures 9.8 and 9.7, it can be concluded that there are two opposing influences of similar magnitude on the concentration inside a droplet. For high compatibility (ccrit = 8%), the two effects nearly cancel each other, whereas at low compatibility (ccrit = 1.5%), the net effect is a concentration inside a droplet, which is lower than that in the continuous bulk. This is about the same as the increase in concentration due to depletion of polymer from the interface (Figure 9.7). However, it enables the relaxation of the Laplace pressure difference across the interfaces of the droplets. Full relaxation of the Laplace pressure is not possible because there will not be a single solution for d for an arbitrary distribution of droplet sizes. This situation of a range of droplet pressures communicating through a common matrix is reminiscent of the increase of pressure above foam coarsening inside a closed vessel. The time needed to reach the quasi-equilibrium by relaxation of the Laplace pressure is dependent on the diffusion rate of water. Assuming a diffusion constant of 2 ¥ 10-9 m2/s, the typical distance traveled by a water molecule in 1 s is 100 μm. The typical size of a phase droplet after break 15.00 ccrit = 1.5%

cbulk [%w/w]

14.00

ccrit = 3% 13.00 ccrit = 4% 12.00

ccrit = 6% ccrit = 8%

11.00

10.00

0

2

4 6 8 Droplet radius [mm]

10

12

FIGURE 9.7 Total polymer concentration far away from the interface in a phase-separated system of equal phase volumes, with one phase broken up into equally sized droplets. Overall concentration of each polymer is 5% (w/w).

210

Structure and Functional Properties of Colloidal Systems 10 9

ccrit = 1.5%

8

c bulk [%w/w]

7

ccrit = 3%

6 5

ccrit = 4%

4

ccrit = 6%

3

ccrit = 8%

2 1 0

0

2

4

6

8

10

12

Droplet radius [mm]

FIGURE 9.8 Concentrations inside droplets after osmotic swelling calculated by solving Equation 9.16 for d, using the interface tension from Figure 9.3. The radius of the droplets prior to swelling was 10 mm and the phase volume fraction of the droplet phase was 0.5.

up by stirring is 10 μm and the time needed for significant coarsening by coalescence is in the order of minutes. Therefore, a variation in “bulk” concentrations of phase regions, dependent on their average curvature, may exist for long times.

9.3 POLYDISPERSITY All the considerations above were made for monodisperse polymers. In practical systems polymers are nearly always polydisperse. Polydispersity in phase separation is coupled to concentration, leading to fractionation of the molar distribution between the coexisting phases [25–28]. The size distribution in the enriched phase contains more of the high molar mass fraction, and that in the depleted phase more of the low molar mass fraction. Because the low molar mass fraction is less strongly phase separated, it will also be less depleted at the interface and therefore lower the interface tension [29,30]. When the concentration is sufficiently high, the length beyond which excluded volume interactions are screened (i.e., the blob size) may be small enough to cause even the smallest chains to be divided into several blobs (see Equation 9.4). At such a high concentration the molar mass distribution becomes irrelevant for the composition of the coexistent phases. In the typical case of a concentration of 10% (w/w) of polymers with an Rg of 18 nm (for biopolymers this corresponds typically to about 1000 monomers and a molar mass of 150 kDa), there are about 100 blobs per chain. A low molar mass fraction with a molar mass of <15 kDa would have less than 10 blobs per chain and would be expected to affect the phase diagram (relative to that of a system containing only 150 kDa material). In practice, for systems of phase separating polydisperse biopolymers, the material found in the depleted phase has a considerably lower molar mass than the original material [31]. This suggests that in these practical cases, polydispersity is important for phase and interface properties.

9.4 9.4.1

EXPERIMENTS INTERFACE TENSION

Almost all experimental data that are available for water–water interfaces are interface tension results [2,6–9,32–34]. The interface tension is an integral over the concentration profile at the interface and

211

Water–Water Interfaces Capillary length Dextran-rich

1 mm

Gelatin-rich

gelatin/dextran 6%/4%

g @ 0.05 mN/m 5.4%/3.6%

3.9%/2.6%

g @ 0.01 mN/m 3.6%/2.4%

FIGURE 9.9 Confocal microscope images of the interface between coexistent phases of fish gelatin-rich (bottom phase) and dextran-rich solution (top phase). Top: graphic interpretation of the capillary length. Bottom: a dilution series showing the decrease in capillary length with decreasing concentration. The container is a flat capillary (0.5 × 10 mm).

therefore not very sensitive to changes in the interface profile. Direct experimental information on the structure of the water–water interface is hardly available. Electron microscope images of specimens of water–water phase-separated gelatin–dextran mixtures prepared by the freeze fracturing technique suggest that the fracture prefers to follow the interface between the phase regions. This could be interpreted as the interface being more brittle than the bulk of the phase regions due to a lower polymer concentration. This point, however, has not yet been thoroughly investigated. In most cases, with very few exceptions [6,35], the water–water interface tension was derived from spinning drop experiments [36,37] or from the recovery of the droplet shape after a strain jump [38]. The values of the interface tension obtained are therefore out-of-equilibrium observations. Due to the ability of the water–water interface to absorb or desorb water, it is expected that the time scale of deformation can cross the time scale of adjustment by sorption of the profile of the interface to a changing curvature. This will have a consequence for the interface tension, which is measured [11]. Also, the permeability of the interface for water may affect the way in which a droplet reacts to deformation. Observation of the static, equilibrium shape of the interface [39] does not suffer from this complication. This can be done by careful measurement of the shape of the water–water meniscus at the wall of the container. The contact angle at the wall, the level far from the wall, and the density difference between the phases have to be known, and it has to be certain that there is no optical deformation from lens action of the glass container. If the last condition cannot be verified, the interface tension can be roughly estimated from the capillary rise along the wall. Figure 9.9 shows an example of the capillary rise in a flat capillary for a concentration series of phase-separated aqueous mixtures of nongelling fish gelatin and dextran. The interface tension can be estimated in the cases in which the level far from the wall is recorded as a reference point for the capillary rise. The capillary length

Lcap =

2g ~ (c - ccrit )0.7 g Dr

(9.17)

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Structure and Functional Properties of Colloidal Systems

10000

Norm. intensity

(1) 7% dextran

25 mm

1000

slope-4

(2) 7% dextran with gelatin droplets Only gelatin droplets (2)-(1)

100 10 1 1

10

100

Q [1/micron]

FIGURE 9.10 Example of Porod’s behavior of light scattering by water–water phase droplets. Left: a microscope image of droplets of nongelling gelatin-rich phase in a dextran-rich solution. Concentrations in both phases are about 7% (w/w). Right: the light-scattering intensity versus scattering vector Q for the total system, the dextran solution, and their difference, that is, the scattering from only the droplets.

is the distance from the wall where capillary rise and gravitational pull are balanced (g is the acceleration in gravity, 9.8 m/s2). The decrease with decreasing concentration is clearly observed in Figure 9.9. For 3.6% gelatin/2.4% dextran, the experimentally determined density difference Dr is 0.3 kg/m3. The interface tension is estimated at 0.01 μN/m. In a similar way, a value 0.05 μN/m for 3.9% gelatin/2.6% dextran is found.

9.4.2

INTERFACE WIDTH

An upper limit to the interface width can be obtained from light scattering. If the dependence of the intensity of scattered light on the scattering vector Q obeys Porod’s law, I(Q) ~ Q-4

(9.18)

(in which Q = 4pn sin(q/2)/l, q is the scattering angle, l is the wavelength of the light used, and n is the refractive index), the scattering is caused only by variations in the refractive index over distances well below the wavelength of the light used for the experiment. In other words, when Equation 9.18 is obeyed, the interface width is less than 200 nm. In Figure 9.10, an example of a water-inwater emulsion is given, which showed -4 power law dependence according to Equation 9.18. Agreement with Porod’s behavior is generally found in the late stage of phase separation of nongelling systems, also for systems that have more equal phase volumes than the system in Figure 9.10, which are therefore closer to the critical point. This implies correctness of the order of magnitude of the calculated values of the interface width (Figure 9.5).

9.4.3

PERMEABILITY AND EXCESS WATER AT THE INTERFACE

The permeability of the water–water interface between solutions of polydisperse polymers is shown in Figure 9.11. A labeled low molar mass fraction of one of the phase-separated polymers is seen to diffuse freely through the interface between the phase regions of the coexisting phases. The permeability for salt is demonstrated in Figure 9.12. In mixtures of aqueous gelatin and dextran (and possibly also in other biopolymer mixtures), the density of the two phases can be matched by adding salt. Figure 9.12 shows concentration series for the inert salts NaCl (Mw = 58.5 Da) and CsCl (Mw = 168.5 Da). Around the matching point, the interface is seen to become strongly curved due

213

Water–Water Interfaces Contact line between two sample droplets Phase separated high molar mass dextran (light gray) and gelatin (black)

Low molar mass dextran(dark gray)

FIGURE 9.11 Permeability of the water–water interface for low molar mass components. Macroscopic droplets of a phase-separated mixture of gelatin (150 kDa) and labeled dextran (182 kDa, light grey) (left) and a nonlabeled dextran (182 kDa) solution (right) are brought into contact. The dextran solution on the right-hand side contained labeled, low molar mass dextran (9 kDa, dark grey). These small labeled molecules color the dark gelatin-rich droplets on the left by crossing the interfaces of the droplets.

to the increase of the capillary length and the tendency of the gelatin phase to wet the glass container wall. The matching point is near 20 mM salt, irrespective of the molar mass of the salt (the series for KCl, not shown, was found to have the same matching point). The absence of an influence of the molar mass indicates that the salt is distributed evenly over the two phases. Therefore, the interface must be permeable to salt. Apparently, the introduction of salt alters the distribution of water. Water is expelled from the dextran phase to the gelatin phase. Effectively, adding salt worsens

CsCl

NaCl

0M

0.05 M

0.02 M 0.025 M

FIGURE 9.12 Phase-separated fish gelatin–dextran mixtures with increasing salt concentration showing a density matching point that is independent of the molar mass of the salt used. The salt must therefore be evenly distributed over the phases.

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Structure and Functional Properties of Colloidal Systems

the solvent quality of water for dextran and improves the solvent quality of water for gelatin, in spite of the weak polyelectrolyte character of gelatin. The presence of solvent gives the two phase separating polymers an additional means to lower the interface tension between their phase regions, that is, by the accumulation of solvent at the interface. This solvent, though, is extracted from the bulk of the phases. The area far (>D 10 nm) from the interface will therefore have a higher concentration in the presence of interface, and more so the more interface there is in the system. In Figure 9.7, it is shown that in the case of breakup of equal phase volumes into droplets of <10 μm, this increase in concentration may be significant. On the other hand, permeability of the interface may lead to a significant decrease in concentration due to osmotic swelling. A consequence of this dependence on the presence of an interface of phase concentrations is that during the late stage phase separation, when in the classical picture further phase separation is a matter of reduction of interface area, the “coexisting” phase concentrations depend on time. This effect may be noticeable when following phase coarsening by light scattering. Light-scattering intensity for the late stage phase separation is proportional to the square of the concentration difference on the two sides of the interface: I(Q) ~ (Dc)2 S(Q),

(9.19)

in which S(Q) is the structure factor, determined by the shape of the phase regions. The scattering invariant •

Ú I (Q)Q

2

dQ ~ ( Dc)2

(9.20)

0

is dependent on time when Dc is dependent on time. In the early stage of phase separation, Dc increases with time [40,41] because the coexisting phase concentrations are building up. However, during the late stages in which reduction of interface has become the major process, the disappearance of interface area should lead to the release of excess water at the interface and decrease of osmotic swelling. Both should result in a decrease in Dc. This point needs further experimental verification. This verification will be challenging because phase domains are in practice several micrometers larger than that in the late stage of phase separation, and light scattering at ultralow angles is necessary to measure I(Q) over a sufficient range of Q values in order to calculate the integral in Equation. 9.20.

9.4.4

COEXISTENCE OF CURVATURES

It was argued above that the fact that water is expected to freely diffuse across the phases may give rise to relaxation of the Laplace pressure across curved interfaces, and therefore enable different curvatures to coexist. This may give rise to long-time stability of water–water emulsions, which is indeed observed. Emulsions of the type shown in Figure 9.10 are stable for several days. However, such emulsions are expected to be quite stable anyway, because of the very slow “evaporation” of polymer from larger to weaker curvatures [42], by which process the coarsening would proceed, irrespective of relaxation of the Laplace pressure. A possibly more meaningful observation is the coexistence of curvatures surrounding the same phase region. Such a situation can be created by letting the phase separation and coarsening take place in the confined geometry of a capillary tube [43]. When the inner diameter of the capillary tube is smaller than the capillary length, the coarsening process is insensitive to gravity and dominated by differences in wetting tendency between the phases. Figure 9.13 is representative for the type of final states that are found for a gelatin–dextran mixture in which the gelatin-rich phase fully

Water–Water Interfaces

215

FIGURE 9.13 Coexistence of different curvatures of a water–water interface between fish gelatin-rich and dextran-rich coexisting phases. Top: capillary tube (diameter 0.6 mm) with stable plugs (dark, gelatin-rich; light, dextran-rich). Bottom: a single, dextran-rich plug surrounded by gelatin-rich phase fully wetting the glass wall.

wets the glass wall. The final state is a sequence of phase regions (called “plugs” or “bridges”), which is stable for many months. A microscopic image of a single plug of nonwetting dextran-rich phase shows that the gelatin-rich wetting phase, as expected, surrounds the nonwetting dextran-rich phase. Because the plugs are not spherical, the spherical part of the interface between the plugs in the axial direction has to coexist with a much flatter interface in the perpendicular direction. In Figure 9.13, it is also seen that the stress of this coexistence is relaxed by lens formation in the middle. It is postulated that the coexistence of a distribution of curvatures corresponds to local variation in the interface tension. Such a variation is made possible by local changes in the interface structure, in particular the amount of excess water in the interface.

9.5 SUGGESTIONS FOR FURTHER WORK Future research on water–water interfaces could address both fundamental issues, discussed above, as well as practical aspects. Practical aspects are the permeability and the stability of the water– water interface. From a fundamental point of view, direct experimental information on the interface profile would be desirable. To get direct structural information with respect to the interface profile, neutron reflection would be the ideal technique to use. Neutron reflection has been used extensively to shed light on the structure of air–liquid interfaces [44,45]. A liquid–liquid interface brings along the complication that the neutron beam has to pass an air–liquid interface before it reaches the water–water interface. The air–liquid interface is in general a much stronger transition in reflective index than the water–water interface and will obscure the reflection of the water–water interface, which is beyond it. Therefore, a water–water phase-separated system has to be used with a H2O/D2O ratio with zero neutron-scattering cross-section. The polymer in the top phase should also have an H/D composition rendering the polymer in the top phase invisible for neutrons. In principle, the polymer profile at the water–water interface can in this way be obtained, because the polymer composition at the interface differs from that with the “null” H/D composition in the top phase. The solvent profile can only be indirectly determined from the polymer excess. As explained above, indirect evidence of the effect of the amount of interface area per unit volume on coexisting phase composition could be obtained from monitoring the dynamic scaling during phase separation. As explained above, the scattering invariant should decrease with time during late stage coarsening. Theoretical work might extend initial work [46,47] on the possible existence of a preferential curvature of water–water interfaces between phases containing polymers with different affinities for the solvent water and, as a consequence, different amounts of excluded volume. This will be in general the case in real-life systems. The asymmetry in solvent distribution will give rise to an

216

Structure and Functional Properties of Colloidal Systems

asymmetry in the interface profile. This asymmetry could lead to a preferential curvature. A preferential curvature would stabilize droplets of a certain size, and it would influence, in addition, the asymmetry in viscosity, the commonly observed transition from a bicontinuous to a broken-up morphology during spinodal decomposition. Closely related to the issue of preferential curvature is the relation between curvature and interface width. The osmotic compressibility of polymer blends containing solvent enables the reduction of Laplace pressure by adjusting concentration differences between the two sides of the interface, but also by reduction of the interface tension by absorption of solvent at the interface. Curvature might have to be treated as a separate thermodynamic variable (besides interface area, [48]), which leads to the conclusion that measurements of the interface tension using different shapes of the interfaces or different ways of relaxation of the interface shape should be compared with each other with caution. From a practical point of view, it would be desirable to have ways to stabilize the interface by adding interface-active additives. In this way, the interface can be given functionality. The functionality could be just stabilization, for water-in-water emulsions that behave like oil-in-water emulsions stabilized by protein. Alternatively, the interface could be given a gel-like character, by accumulating a compound that forms a (visco) elastic layer at the interface. This interface would survive dilution, certainly for some time, and cause the delayed release of the content of the droplets. For the molecules of such additives, it would have to be energetically favorable to reside at the interface between two aqueous environments, which are only subtly different. Copolymers of the two separated polymers should be tested in this respect. Recently [49], it was shown that in extremely concentrated aqueous systems of methylcellulose as one of the incompatible polymers, and dextran or maltodextrin as the other, fat particles and silica particles accumulate at the interface between the phases. The water–water emulsion was found to be more stable than the system without the surfaceactive additives.

REFERENCES 1. Spyropoulos, F., Frith, W.J., Norton, I.T., Wolf, B., and Pacek, A.W. 2008. Sheared aqueous two-phase biopolymer–surfactant mixtures. Food Hydrocolloids 22: 121–129. 2. Van der Beek, D., Reich, H., van der Schoot, P., Dijkstra, M., Schilling, T., Vink, R.L.C., Schmidt, M., van Roij, R., and Lekkerkerker, H.N.W. 2006. The isotropic–nematic interface and wetting in suspensions of colloidal platelets. Phys. Rev. Lett. 97: 087801. 3. Poulin, P., Stark, H., Lubensky, T.C., and Weitz, D.A. 1997. Novel colloidal interactions in anisotropic fluids. Science 275: 1770–1773. 4. Simon, K.A., Sejwal, P., Gerecht, R.B., and Yuk, Y.-Y. 2007. Water-in-water emulsions stabilized by nonamphiphilic interactions: Polymer-dispersed lyotropic liquid crystals. Langmuir. 23: 1453–1458. 5. Tolstoguzov, V.B. 1993. Thermodynamic incompatibility of food macromolecules. In: E. Dickinson and P. Walstra (Eds). Food Colloids and Polymers: Stability and Mechanical Properties, Vol. 113, pp. 94–102. Cambridge: Royal Society of Chemistry. 6. Tolstoguzov, V.B., Mzhels’sky, A.I., and Gulov, V.Y. 1974. Deformation of emulsion droplets in flow. Colloid Polym. Sci. 252: 124–132. 7. Ryden, J. and Albertsson P.-A. 1971. Interfacial tension of dextran-polyethylene glycol-water two-phase systems. J. Colloid Interface Sci. 37: 219–222. 8. Scholten, E., Tuinier, R., Tromp, R.H., and Lekkerkerker, H.N.W. 2002. Interfacial tension of a decomposed biopolymer mixture. Langmuir 18: 2234–2238. 9. Antonov, Y.A., Van Puyvelde, P., and Moldenaers P. 2004. Interfacial tension of aqueous biopolymer mixtures close to the critical point. Int. J. Biol. Macromolecules 34: 29–35. 10. Aarts, D.G.A.L., Schmidt, M., and Lekkerkerker, H.N.W. 2004. Direct visual observation of thermal capillary waves. Science 304: 847–850. 11. Scholten, E., Sprakel, J., Sagis, L.M.C., and van der Linden, E. 2006. Effect of interfacial permeability on droplet relaxation in biopolymer-based water-in-water emulsions. Biomacromolecules 7: 339–346.

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12. Rediguieri, C.F., de Freitas, O., Lettinga, M.P., et al. 2007. Thermodynamic incompatibility and complex formation in pectin/caseinate mixtures. Biomacromolecules 8: 3345–3354. 13. De Kruif, C.G. and Tromp, R.H. 2008. Interaction of pectin with casein. Foods Food Ingredients J. Jpn 213: 281–285. 14. Capron, I., Costeux, S., and Djabourov, M. 2001. Water in water emulsion: Phase separation and rheology in biopolymer solutions. Rheol. Acta 40: 441–456. 15. Loren, N., Hermansson, A.-M., Williams, M.A.K., Lundin, L., Foster, T.J., Hubbard, C.D., Clark, A.H., Norton., I.T., Bergstrolm, E.T., and Goodall, D.M. 2001. Phase separation induced by conformational ordering of gelatin in gelatin/maltodextrin mixtures. Macromolecules 34: 289–297. 16. Williams, M.A.K., Fabri, D., Hubbard, C.D., Lundin, L., Foster, T.J., Clark, A.H., Norton, I.T., Loren, N., and Hermansson, A.-M. 2001. Kinetics of droplet growth in gelatin/maltodextrin mixtures following thermal quenching. Langmuir 17: 3412–3418. 17. Stokes, J.R., Wolf, B., and Frith, W.J. 2001. Phase separated biopolymer mixture rheology: Prediction using a viscoelastic emulsion model. J. Rheol. 5: 1173–1191. 18. Forciniti, D., Hall, C.K., and Kula, M.-R. 1991. Influence of polymer molecular weight and temperature on phase compositions in aqueous two phase systems. Fluid Phase Equilib. 61: 243–262. 19. Edelman, M.W., van der Linden, E., and Tromp, R.H. 2003. Phase separation of aqueous mixtures of PEO and dextran. Macromolecules 36: 7783–7790. 20. Bergfeldt, K., Piculell, L., and Tjerneld, F. 1995. Phase separation phenomena and viscosity enhancements in aqueous mixtures of poly(styrenesulfonate) with poly(acrylic acid) at different degrees of neutralization. Macromolecules 28: 3360–3370. 21. Broseta, D., Ould Kaddour, L., Leibler, L., and Strazielle, C. 1987. A theoretical and experimental study of interfacial tension of immiscible polymer blends in solution. J. Chem. Phys. 87: 7248–7256. 22. Broseta, D., Leibler, L., and Lapp, A. 1986. Universal properties of semidilute polymer solutions—a comparison between experiments and theory. Europhys. Lett. 2: 733–737. 23. Broseta, D., Ould Kaddour, L., and Joanny, J.-F. 1987. Critical properties of incompatible polymer blends dissolved in a good solvent. Macromolecules 20: 1935–1974. 24. Fleer, G.J., Cohen Stuart, M.A., Scheutjens, J.M.H.M., Cosgrove, T., and Vincent, B. 1993. Polymers at Interfaces. London: Chapman & Hall. 25. Koningsveld, R., Chermin, H.A.G., and Gordon, M. 1970 Liquid–liquid phase separation in multicomponent polymer solutions. VIII. Stability limits and consolute states in quasi-ternary mixtures. Proc. R. Soc. Lond. A 319: 331–349. 26. Hsu, C.C. and Prausnitz, J.M. 1974. Thermodynamics of polymer compatibility in ternary systems. Macromolecules 7: 320–324. 27. Koningsveld, R., Šolc, K., and MacKnight, W.J. 1993. Thermodynamics of aging of immiscible polymer blends. Macromolecules 26: 6676–6678. 28. Evans, R.M.L., Fairhurst, D.J., and Poon, W.C.K. 1999. Universal law of fractionation for slightly polydisperse systems. Prog. Colloid Polym. Sci. 112: 172–176. 29. Broseta, D. and Leibler, L. 1989. Critical properties of bimodal polymer solutions. J. Chem. Phys. 90: 6652–6655. 30. Broseta, D., Fredrickson, G.H., Helfand, E., and Leibler, L., 1990. Molecular weight and polydispersity effects at polymer–polymer interfaces. Macromolecules 23: 132–139. 31. Edelman, M.W., van der Linden, E., Hoog, E., and Tromp, R.H. 2001. Compatibility of gelatin and dextran in aqueous solution. Biomacromolecules 2: 1148–1154. 32. Spyropoulos, F., Ding, P., Frith, W.J., Norton, I.T., Wolf, B., and Pacek, A.W. 2008b. Interfacial tension in aqueous biopolymer–surfactant mixtures. J. Colloid. Interface Sci. 317: 604–610. 33. Ding, P., Pacek, A.W., Frith, W.J., et al. 2005. The effect of temperature and composition on the interfacial tension and rheology of separated phases in gelatin/pullulan mixtures. Food Hydrocolloids 19: 567–574. 34. Simeone, M., Alfani, A., and Guido, S. 2004. Phase diagram, rheology and interfacial tension of aqueous mixtures of Na-caseinate and Na-alginate. Food Hydrocolloids 18: 463–470. 35. Mitani, S. and Sakai, K. 2002. Measurement of ultralow interfacial tension with a laser interface manipulation technique. Phys. Rev. E 66: 031604. 36. Vonnegut, B. 1942. Rotating bubble method for determination of surface and interfacial tension. Rev. Sci. Instrum. 13: 6–9. 37. Princen, H.M., Zia, I.Y.Z., and Mason, S.G. 1967. Measurement of interfacial tension from the shape of a rotating drop. J. Colloid. Interface Sci. 23: 99–107.

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38. Oldroyd, J.G. 1953. The elastic and viscous properties of emulsions and suspensions. Proc R. Soc. Lond. Ser. A 18: 122–132. 39. Aarts, D.G.A.L. and Lekkerkerker, H.N.W. 2004. Confocal scanning laser microscopy on fluid–fluid demixing colloid–polymer mixtures. J. Phys.: Condens. Matter 16: S4231–S4242. 40. Verhaegh, N.A.M., van Duijneveldt, J.S., Dhont, J.K.G., and Lekkerkerker H.N.W. 1996. Fluid–fluid phase separation in colloid–polymer mixtures studied with small angle light scattering and light microscopy. Physica A 230: 409–436. 41. Tromp, R.H., Rennie, A.R., and Jones, R.A.L. 1995. Kinetics of the simultaneous phase separation and gelation in solutions of dextran and gelatin. Macromolecules 28: 4129–4138. 42. Wolterink, J.K., Barkema, G.T., and Puri, S. 2006. Spinodal decomposition via surface diffusion in polymer mixtures. Phys. Rev. E: 011804. 43. Tromp, R.H. and Lindhoud, S. 2006. Arrested segregative phase separation in capillary tubes. Phys. Rev. E 74: 031604. 44. Geoghegan, M., Nicolai, T., Penfold, J., et al. 1997. Kinetics of surface segregation and the approach to wetting in an isotopic polymer blend. Macromolecules 30: 4220–4227. 45. Penfold, J. 2002. Neutron reflectivity and soft condensed matter. Curr. Opin. Colloid Interface Sci. 7: 139–147. 46. Leermakers, F.A.M., Barneveld, P.A., Sprakel, J., and Besseling, N.A.M. 2006. Symmetric liquid–liquid interface with a nonzero spontaneous curvature. Phys. Rev. Lett. 97: 066103. 47. Van Male, J. and Blokhuis, E.M. 2007. Comment on “symmetric liquid–liquid interface with a nonzero spontaneous curvature”. Phys. Rev. Lett. 98: 039601. 48. Groenewold, J. and Bedeaux, D. 1995. Microscopic integral relations for the curvature dependent surface tension in a two-phase multi-component system. Physica A 214: 356–378. 49. Poortinga, A.T. 2008. Microcapsules from self-assembled colloidal particles using aqueous phaseseparated polymer solutions. Langmuir 24: 1644–1647.

10

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions Julia Maldonado-Valderrama, Antonio Martín-Rodríguez, Miguel A. Cabrerizo-Vílchez, and María Jose Gálvez Ruiz

CONTENTS 10.1 Introduction ...................................................................................................................... 10.1.1 Interfacial Tension, Interfacial Rheology, and Thin Liquid Films ....................... 10.1.2 Stability of Foams and Emulsions ......................................................................... 10.2 Foams and Emulsions Examined with Interfacial Studies ............................................... 10.2.1 Foams Stabilized by Whole Casein and b-Casein ............................................... 10.2.1.1 Foam Stability ........................................................................................ 10.2.1.2 Surface Tension, Surface Rheology, and Thin Liquid Films ................. 10.2.2 Foams Stabilized by Mixtures of Protein and Surfactants ................................... 10.2.2.1 Foam Stability ........................................................................................ 10.2.2.2 b-Casein/Tween 20 System ................................................................... 10.2.2.3 Whole Casein/Tween 20 System ........................................................... 10.2.3 Foams and Emulsions Stabilized by b-Casein ..................................................... 10.2.3.1 Stability of Foams and Emulsions of b-Casein ..................................... 10.2.3.2 Interfacial Rheology of b-Casein .......................................................... 10.3 Conclusions ....................................................................................................................... References ..................................................................................................................................

219 220 221 222 222 223 223 225 225 226 227 229 230 231 232 233

10.1 INTRODUCTION Food emulsions and foams are complex dispersions that constitute a major part of the food products that are consumed daily. The correct elaboration of such colloidal systems determines the functional properties of the final product such as texture or long-term stability. Accordingly, the optimization of the food product depends fundamentally on the comprehension of the structural characteristics of its components. Foams and emulsions are dispersions of air and liquid in another immiscible liquid, respectively. Formation of these dispersions is subject to the presence of amphiphilic molecules, which tend to place themselves at the air–water interface (foams) or the oil–water interface (emulsion). Thus, they constitute molecular barriers that stabilize the dispersion. The composition and structure of these molecular barriers determine ultimately the behavior of foams and emulsions. In order to obtain foams and emulsions, a high amount of energy is required so as to create free air–water interfacial area and oil–water interfacial area, respectively. This energy equals the free energy of the system (gA, where g is the interfacial tension and A is the interfacial area). Since the area 219

220

Structure and Functional Properties of Colloidal Systems

is not minimized in these systems, foams and emulsions constitute thermodynamically unstable dispersions. An effective way of reducing this free energy is having amphiphilic molecules, which tend to adsorb at interfaces with the effect of lowering the surface tension. Proteins are the most important foams and emulsifiers used in food technology since they fulfill both a practical and a nutritional aspect. Often, surfactants and polymers are added to the system to improve its stability. Food dispersions can be very complex, such as solid foams, in which the system is solidified after heating, such as cakes or bread, or after freezing, such as ice creams. Furthermore, ice creams are even more complex since they consist of foam made from an emulsion. However, this chapter is devoted to the study of the structure of foams and emulsions on the basis of their fundamental physicochemical properties. Hence, we will focus on liquid foams and emulsions.

10.1.1

INTERFACIAL TENSION, INTERFACIAL RHEOLOGY, AND THIN LIQUID FILMS

When an amphiphilic molecule is added into water, it tends to place itself at the air–water interface so that the hydrophobic parts of such molecules come in contact with the air and the hydrophilic parts remain in the water. This has an immediate effect of lowering the surface tension (i.e., decreasing the surface free energy), and hence the process of adsorption of this kind of molecules is spontaneous. The decrease in surface tension (g) is identified with the surface pressure (p): p = g 0 - g,

(10.1)

where g0 is the surface tension of pure water (72.7 mJ/m2 at 20°C). The surface tension, or interfacial tension in the case of a liquid–liquid interface, contains information about the composition and structure of the interface. Surfactants adsorb very rapidly and form very compact surface layers giving rise to high surface pressures (Wilde et al., 2004). Adsorption of proteins is more complex since they partially change their conformation upon adsorption exposing hydrophobic groups to the air. Therefore, the structure of proteins determines their adsorption behavior. Flexible proteins such as b-casein adsorb more rapidly and change conformation more easily than globular proteins such as b-lactoglobulin (Martin et al., 2002). All these species adsorb at the air–water and the oil–water interface lowering the interfacial tension. However, the values of the surface tension do not account for the emulsifying and foaming capacity of the various types of species used in the food industry. In fact, in order to explain the foaming and emulsifying capacity of the different amphiphilic molecules, it is useful to evaluate the response of an interface to a perturbation of its equilibrium state. The response of an interface to a perturbation of its equilibrium state causes a gradient in interfacial area that results in a gradient in the interfacial tension, which tends to compensate the perturbation (Lucassen-Reynders, 1981). This feature is quantified by Gibbs definition of the surface elasticity as the change in surface tension per relative change in surface area: dg e = _____. d ln A

(10.2)

This is a measurement of the resistance that an interface shows to the creation of new regions with higher surface tensions (higher surface area). In a more general case, the response of the interface would have also a viscous response due to relaxation phenomena at the interface. In the case of a sinusoidal perturbation to the interfacial area of frequency n (n = 2pw) and small amplitude, the response of the interface is a complex magnitude: the dilatational elastic modulus. E = E¢ + iE≤.

(10.3)

The dilatational elastic modulus contains important structural information about adsorbed interfacial layers (Benjamins et al., 2006; Maldonado-Valderrama et al., 2005a). Moreover, the elastic

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions

221

FIGURE 10.1 Schematic structure of a liquid foam stabilized by amphiphilic molecules.

modulus measures the ability of the adsorbed layer to resist external disturbances, thus preventing the film from rupturing. Therefore, the surface elasticity of the adsorbed layers appears as a key parameter toward the stabilization of foams and emulsions (Langevin, 2000; Murray, 2007; Wilde, 2000). We have seen how the properties of the interfaces are intimately related with the stability of foams and emulsions. However, another system that plays a crucial role in formation and stability of foams and emulsions are the so-called thin liquid films. These are the liquid regions remaining between two interfaces that approach each other so that the two interfaces can interact. For example, the liquid film remaining between two air bubbles (foam film) or between two oil droplets (emulsion film). The drainage of horizontal thin liquid films can be examined by the porous plate technique in which the drainage of the liquid in the film can be visualized and its thickness measured as a function of the disjoining pressure (Bergeron and Radke, 1992). The thin film thickness contains important structural information of the interfacial layers and depends on the interaction between molecules (Langevin, 2000, 2008). In summary, the interfacial tension, the interfacial dilatational modulus, and the drainage of thin liquid films are fundamental principles for understanding foams and emulsions (Figure 10.1). The correlation between these magnitudes and the behavior of the resulting colloidal dispersion provides a technological application to fundamental interfacial science.

10.1.2 STABILITY OF FOAMS AND EMULSIONS The stability of foams is governed by two processes: Ostwald ripening and coalescence. The former consists in diffusion of gas from small bubbles to larger ones due to differences in capillary pressure. As a result, the overall size of the bubbles in the foam increases with time and the solubility of the gas in the liquid, and the polydispersity of the system plays an important role in Ostwald ripening on foams (Langevin, 2000). The second destabilizing mechanism, coalescence, is the rupture of the liquid film separating two bubbles to form a bigger one. According to literature, this phenomenon is intimately related to the surface elasticity of the surface film since the surface elasticity of an adsorbed layer measures the resistance of the film to deformation (Langevin, 2000; Maldonado-Valderrama et al., 2008; Wilde, 2000). Finally, the drainage of foam films also plays a key role in foam stability since when the thin films thin, Ostwald ripening and coalescence are facilitated (Langevin, 2008). The main mechanisms affecting emulsion stability are similar to those affecting foam stability; Ostwald ripening, creaming phenomena, flocculation, and coalescence. Creaming is the equivalent of liquid drainage in foams: As the drops/bubbles rise because of gravity, they become closer and might coalesce together or with the oil–water/air–water interface, respectively. For dilute emulsions, the velocity of a drop of radius R and density r1 in a fluid of density r2 and viscosity h is 2gR2(r2 - r1) n = ___________ . 9h

(10.4)

222

Structure and Functional Properties of Colloidal Systems

Hence, the creaming rate depends on the size of the droplets, the density difference, and the viscosity of the liquid phase (Langevin, 2000; Robins, 2000). The flocculation of oil droplets is a process in which droplets cluster without loosing their individuality since the liquid film separating them does not rupture. This process leads to faster creaming. Finally, coalescence implies the rupture of the interfacial film and may lead to larger droplets in the emulsion also leading to faster creaming (coalesce-enhanced creaming). Coalescence, as in foams, is expected to depend on the elasticity of the interfacial film, that is, its resistance to deformation and rupture (Carrera-Sánchez and Rodriguez-Patino, 2005; Langevin, 2000; Wilde, 2000). We have now stated clearly the main features characterizing the stability of foams and emulsions in terms of their correlation with interfacial phenomena. The following section illustrates such correlation.

10.2

FOAMS AND EMULSIONS EXAMINED WITH INTERFACIAL STUDIES

In this section, we present some examples of studies connecting the behavior of foams and emulsions with several interfacial magnitudes. The interrelation between the interfacial properties of proteins and their foaming/emulsifying capacity or the stability of the foam/emulsion formed is one of the main challenges of the present food technology. In this sense, there is an increasing interest in the literature toward the understanding of such relationship (Langevin, 2000; Murray, 2007; Wilde, 2000). Davis and Foeding underscore the importance of utilizing the same components in the interfacial and foaming/emulsifying experiments to perform a reliable comparison of interfacial and dispersion properties. Accordingly, the results presented in this chapter have been all obtained under the same experimental conditions. Hence, the aqueous subphase used in all the solutions is a buffer solution made with 6.61 g/L Trizma hydrochloride and 0.97 g/L Trizma base (Sigma-Aldrich®). In order to prevent bacterial contamination, 0.5 g/L NaN3 was added to the buffered solvent. The buffer solution has a final pH 7.4 and an ionic strength (I) of 16.4 mM. Also, all the concentrations of proteins and surfactants used in the dispersions and in the interfacial studies are kept similar so as to ensure a reliable comparison between both phenomena. Finally, all the measurements are carried out at 20°C. Accordingly, the following studies present the stability of foams and emulsions, and use the same components to evaluate the drainage of thin liquid films, the interfacial rheology, and the surface behavior of such systems. The behavior of foams and emulsions is discussed in terms of the interfacial studies, and the different stabilizing mechanisms are evaluated separately. In our opinion, this investigation provides a very accurate description of the behavior of the dispersions and highlights the fundamental role of the surface properties toward the understanding of such a major phenomenon in food technology.

10.2.1 FOAMS STABILIZED BY WHOLE CASEIN AND β-CASEIN As stated in the introduction, differences in protein structure can lead to large differences in the properties of the food dispersions, which is a major concern in the food industry. This section is dedicated to the investigation of foams formed by two different food proteins. Foam stability is firstly described on the basis of the time evolution of the volume of foam formed. Secondly, the different behavior found is discussed in detail in terms of several surface magnitudes. As a result, we show that the surface properties of the proteins contain key information toward the understanding of their foaming properties. Two major food proteins are used in this section. Whole casein (or caseins) is a family of phosphorylated proteins that constitute 80% of milk proteins. It contains four different monomers: a1-casein, a2-casein, b-casein, and k-casein (Turhan et al., 2003). Among them, b-casein is the major component of whole casein. It presents a random coil, an asymmetric configuration, and has an extremely flexible structure. It is made up of 209 amino acids, and it has a molecular weight of 23.8 kDa. The first 50 amino acid residues contain a negative charge at pH 7 so that this part of the molecule is hydrophilic. The rest of the molecule is basically made up of neutral residues that are

223

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions

responsible for the hydrophobic character. As a result, b-casein has a marked amphiphilic character that makes it very surface-active. The surface properties of b-casein have been studied widely in the literature by using different experimental devices such as monolayers (Maldonado-Valderrama et al., 2005b; Rodríguez Patino et al., 1999), dilatational (Benjamins et al., 2006; MaldonadoValderrama et al., 2005a; Noskov et al., 2007) and shear rheology (Krägel et al., 1999), and thin liquid films (Cascao-Pereira et al., 2003). However, the literature studies concerning behavior of this model protein in foams are scarcer in the literature. Regarding the studies carried out with whole casein, it is the contrary; due to its heterogeneous nature, the surface behavior has been less studied but there are some recent works regarding the behavior of foams and emulsions (Carrera-Sánchez and Rodriguez-Patino, 2005). Accordingly, the objective of this section is double. On the one hand, we search for a connection between surface behavior and foam stability of two different proteins. On the other hand, we aim to differentiate between a commercial protein (whole casein) and its major component, commonly used as a model system (b-casein). 10.2.1.1 Foam Stability Let us first evaluate the stability of foams formed by whole casein and b-casein under similar experimental conditions (Figure 10.2). It can be clearly seen in the figure that the foam of b-casein is more stable at this bulk concentration. The stability of the foam formed is often estimated by the half-lifetime of the foam (t1/2), the time taken by the foam to decay to half the original height after the air flow is stopped. Hence, the foam of b-casein (0.1 g/L) has a half-lifetime of 35 min whereas the foam of whole casein (0.1 g/L) has a half-lifetime of 15 min. 10.2.1.2 Surface Tension, Surface Rheology, and Thin Liquid Films In order to look for the origin of this large difference, it is interesting to evaluate some surface properties of these proteins. Table 10.1 shows the final surface tension, the surface elasticity (E¢ in Equation 10.3), the surface viscosity (E≤/w in Equation 10.3), and the final foam film thickness of both proteins under similar conditions to the foam stability experiments. Let us evaluate in detail each of them and their implications for foam stability. The surface tensions of b-casein and whole casein at this bulk concentration are very similar the former being just slightly more surface-active (Maldonado-Valderrama and Langevin, 2008). This

0.8

Vfoam (I)

0.6

0.4

0.2

0.0 0

20

40 t (min)

60

80

FIGURE 10.2 Time evolution of volume of foam formed by b-casein (squares) and whole casein (triangles) 0.1 g/L in a buffer solution at pH 7.4, I = 16.4 mM, T = 20°C. Porosity: 40–100 mm.

224

Structure and Functional Properties of Colloidal Systems

TABLE 10.1 Surface Tension, Surface Rheological Properties (Oscillation Frequency = 0.1 Hz), and Thin Film Thickness of Whole Casein and β-Casein in Buffer Solutions at pH 7.4, I = 16.4 mM, T = 20°C Protein b-Casein Whole casein

Surface Pressure (mJ/m2)

Surface Elasticity (mJ/m2)

Surface Viscosity (mJ/m)

Thin Film Thickness (nm)

23

25

15

26

22

23

5

16

could be due to the heterogeneous nature of whole casein in which the different monomers compete for the surface area possibly resulting in a reduced surface activity of the system. Although higher surface activity has been linked to higher foam stability (Martin et al., 2002), the difference found between whole casein and b-casein is too small to explain the large difference in foam stability shown in Figure 10.2. Accordingly, let us evaluate the differences found in surface rheology. Dilatational rheology is a powerful tool in the surface characterization of proteins (Benjamins et al., 2006). It provides key information about interfacial structure of proteins through simple experiments (MaldonadoValderrama et al., 2005a). Table 10.1 reports the dilatational surface elasticity (E¢ in Equation 10.3) and surface viscosity (E≤/w in Equation 10.3) of b-casein and whole casein solutions of 0.1 g/L at 0.1 Hz measured with an oscillating bubble tensiometer (©TECLIS). Again, the surface rheological parameters of b-casein and whole casein show very little differences (Table 10.1). On the one hand, the surface elasticity of b-casein adsorbed layers is somehow larger than that displayed by whole casein. The difference is more noticeable for the viscosity of the surface layer. According to literature, surface viscosity of adsorbed protein layers is a key parameter in foamability (Langevin, 2000; Wilde, 2000; Maldonado-Valderrama et al., 2008; Langevin, 2008). Namely, the process of formation of foam implies the creation of surface area that should be rapidly covered by protein to prevent rupture of the film. The surface viscosity is a measure of the ability of the film to adapt to a change in surface area, and therefore, it is expected to play a crucial role in the foamability of the protein. Accordingly, the slightly larger surface viscosity shown by b-casein adsorbed film should play a role in the higher stability of the foam. However, again the difference is still too small to fully explain the increased foam stability of b-casein as compared to whole casein. Table 10.1 shows finally the final film thickness of foam films stabilized by b-casein (CascaoPereira et al., 2003) and whole casein (Maldonado-Valderrama and Langevin, 2008). It can be seen that contrary to the other interfacial magnitudes, the large difference between foam films stabilized by b-casein and whole casein certainly implies differences between the surface structure of whole casein and b-casein that might explain their different foam stability. Thus, foam films stabilized by whole casein and b-casein under these experimental conditions show a continuous drainage due to an electrostatic repulsion between the protein layers at the surfaces of the foam film. According to literature results, the thickness of a monomolecular layer of b-casein is of 10 nm (Cascao-Pereira et al., 2003). Cascao-Pereira et al. report a value of 26 nm for the b-casein foam film at pH 9 and 10 mM NaCl, stating that the final film is formed by two layers of b-casein separated by a layer of water, the thickness of which is determined by the electrostatic repulsion between protein layers. Accordingly, the final thickness shown by the whole casein film in Table 10.1 at pH 7.4 and I = 16.4 mM clearly indicates the existence of a different structure within the foam film. This is described in detail in (Maldonado-Valderrama and Langevin, 2008). Briefly, at pH 7.4, the electrostatic repulsion might by slightly screened as compared to pH 9. Also, due to the differences between fractions, the foam film formed by whole casein might present heterogeneities and higher intermolecular interactions. As a result, the two layers of whole casein present in the film would tend to

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions

225

come nearer than those reported for b-casein. The difference in film thickness will definitely have a strong effect in foam stability. Accordingly, the larger thickness of foam films stabilized by b-casein would protect the foam bubbles against collapse, playing a crucial role in the stability of the subsequent foam. Conversely, the lower surface activity, lower surface viscoelasticity, and crucially, the lower foam film thickness of whole casein provide a plausible explanation for the less stable foam formed by whole casein as compared to its major fraction. In summary, we have shown in this section the effect of the nature of protein on the stability of foams. Moreover, these results show the importance of choosing the right experiments in order to enlighten the different behavior. In this case, a clear conclusion arises correlating the thickness of the foam film with the foam stability. Namely, the thinner the film, the more stable the foam. This is an outstanding result that links dispersion and surface magnitudes, and we will come back to this conclusion in the following section.

10.2.2 FOAMS STABILIZED BY MIXTURES OF PROTEIN AND SURFACTANTS The previous section was dedicated to the properties of pure protein foams. However, in real systems, proteins, surfactants, lipids, and other foaming agents coexist and compete for the available interfacial area. The competition between different molecules can result in a very complex behavior of the mixed adsorbed layers at the air–water interface. Proteins form viscoelastic structures that stabilize dispersions, owing to a strong and immobile interfacial network, whereas low molecular weight surfactants form mobile interfacial structures that stabilize the resulting dispersion because of the Gibbs–Marangoni mechanism (Mackie and Wilde, 2005). Once these two kinds of molecules coexist at the interfaces, the two mechanisms also compete and it could happen that none of them can operate correctly. Therefore, understanding the interactions between components and the structure of the mixed interfacial layers is crucial to the correct elaboration of foams. Accordingly, the main objective of this research work is to perform a comprehensive analysis of the foam stability of two mixed protein/surfactant systems over different length scales with the aim of highlighting the basic mechanisms underlying the stability of foams. Regarding the systems used in this study, we use the same proteins as in the previous section (whole casein and b-casein) and they are mixed with Tween 20, respectively. This is a low molecular weight surfactant used in the food industry, which is water soluble and nonionic. The different behavior of these two mixed systems is again discussed on the basis of fundamental magnitudes such as surface tension and foam film thickness. 10.2.2.1 Foam Stability Let us first evaluate the behavior of the foam formed by the surfactant and the different effect caused by adding each protein into the bulk solution. The stability of the foams formed by each of the systems presented in this work is characterized by their half-lifetime. Figure 10.3 shows the halflifetime of the foam formed by the pure Tween 20 along with that of the mixed systems. These were obtained with a fixed concentration of protein of 0.1 g/L for both cases and increasing the concentration of Tween 20 between 10-6 and 10-3 M. Accordingly, the behavior foam formed by pure whole casein and b-casein has been described in detail in the previous section. The stability of the foam formed by pure Tween 20 under these experimental conditions is described in detail by Maldonado-Valderrama et al. (2007). In short, Tween 20 produces poorly stable foam only above a certain concentration in bulk. Regarding whole casein and b-casein, their behavior at this concentration is discussed in detail in the previous section, being the foam of the latter more stable. Concerning the stability of foams formed by mixed systems, it can be seen in Figure 10.3 that the surfactant affects very differently the stability of the foams formed by whole casein and b-casein, respectively. On the one hand, the stability of foam of b-casein/Tween 20 system decreases monotonously with increasing concentration of surfactant. This continues until practically reaching the

226

Structure and Functional Properties of Colloidal Systems 40

t1/2 (min)

30

20

10

0 1E – 6

1E – 5

1E – 4 ctw20 (M)

1E – 3

FIGURE 10.3 Half-lifetime of the foams formed by Tween 20 (straight line), b-casein + Tween 20 (dashed line), and whole casein + Tween 20 (dash-dotted line). The concentration of protein in the mixtures is of 0.1 g/L, and the solutions are prepared in a buffer solution at pH 7.4, I = 16.4 mM, T = 20°C.

value of the half-lifetime of the foam of Tween 20 at sufficiently high concentrations of surfactant in the mixture. Differently, the effect of addition of Tween 20 to the whole casein solution is nonmonotonous. Hence, at low concentration of Tween 20 in the mixture, the stability of the foam decreases. Then, further increasing the concentration of Tween 20 in the system results in a restabilization of the foam reaching a half-lifetime which is practically equal to that corresponding to the pure caseins solution (Figure 10.3). 10.2.2.2 β-Casein/Tween 20 System Let us evaluate first the static surface properties of the adsorbed layer of these same systems used in the formation of foams. Hence, Figure 10.4 shows the measured surface pressure isotherms for Tween 20 and for the mixtures with whole casein and b-casein. For each of the concentrations

40

(mJ/m2)

30

20

10

0 1E – 9

1E – 7

ctw20 (M)

1E – 5

1E – 3

FIGURE 10.4 Surface pressure-concentration isotherms of Tween 20 (closed circles), b-casein + Tween 20 (open squares), and whole casein + Tween 20 (open triangles). The concentration of protein in the mixtures is of 0.1 g/L, and the solutions are prepared in a buffer solution at pH 7.4, I = 16.4 mM, T = 20°C.

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions

227

considered, the adsorption behavior is recorded by measuring the surface tension of the surfactant or protein/surfactant solution with a pendant drop apparatus (Cabrerizo-Vilchez et al., 1999). Figure 10.4 shows the final value of surface tension after 10 h of adsorption at constant surface area; after this time, the rate of change of surface tension was negligible. The experiments are described in detail by Maldonado-Valderrama and Langevin (2008) and Maldonado-Valderrama et al. (2007) and the isotherms are presented here in terms of the surface pressure p. The solutions evaluated in these experiments are the same as that used in the foaming characterization. Hence, the concentration of protein used in the mixtures is fixed to 0.1 g/L and the pure protein solutions provide an already saturated surface and its surface tension is displayed in Table 10.1. Then, the mixtures are formed by adding increasing amounts of Tween 20 to the protein solution, varying the surfactant content between 10-6 and 10-3 M. Regarding the isotherm of pure Tween 20, it can be seen that the surface pressure corresponding to a saturated surface with Tween 20 is larger than that saturated with protein (38 mJ/m2). This is due to a higher compactation of the surfactant monolayer (Wilde et al., 2004). Also, the concentration that provides a saturated interface; that is, constant surface pressure, is a good indication of the critical micelle concentration (cmc). According to Figure 10.4, the cmc of Tween 20 is of 2 × 10-5 M. Concerning the surface pressure isotherms obtained for the mixed systems, it is very interesting to note the large differences appearing between the surface pressure isotherms of whole casein/ Tween 20 and that of b-casein/Tween 20. This feature already envisages a different interaction occurring between both types of molecules in agreement with the behavior seen in the foam stability of the systems. Let us comment on the curve obtained for the mixture b-casein/Tween 20. Figure 10.4 shows how the system behaves either like the sole protein or like the sole surfactant. Namely, at low surfactant concentrations, the surface pressure remains practically constant and coincides with that of the sole protein (Table 10.1). Accordingly, within this interval, it might be assumed that the final adsorbed layer is composed basically of protein and that the presence of surfactant does not significantly affect the surface tension. However, at a well-defined surfactant concentration, 10-5 M, the behavior of the mixed system changes abruptly. From this concentration on, the surfactant seems to control the adsorption process, and the isotherm recorded for the mixed system practically coincides with that of the sole surfactant. The surface pressure increases very steeply in a narrow range of surfactant concentration and the saturation of the surface for the mixed system (critical aggregation concentration, cac) coincides with the cmc of Tween 20. These experimental results suggest that the surfactant displaces the protein from the surface when the surface pressure becomes larger than that for the protein alone (Maldonado-Valderrama et al., 2007). It is noteworthy that the stability of the foam formed by b-casein/Tween 20 mixtures (Figure 10.3) completely reproduces the tendency of the surface pressure isotherms. That is, the stability of the foam formed by the mixture of b-casein with low concentrations of Tween 20 corresponds to that of the sole protein, whereas the half-lifetime of the foam formed by the mixture approaches that of the sole surfactant upon increasing the concentration of Tween 20 in the mixture. Therefore, it may be safely inferred from the data that the surfactant has fully displaced the protein from the surface and the adsorbed layer formed by a mixed solution of b-casein and Tween 20 is composed essentially of Tween 20, at high enough concentrations of Tween 20 in the mixture. This conclusion agrees with the literature results that show the displacement of b-casein by Tween 20 at the air– water interface with atomic force microscopy (Wilde et al., 2004). The protein seems to place itself underneath the surface layer so that the properties of the surface layer do not seem to be affected by the presence of the protein. This observation again underlines the importance of fundamental surface properties for understanding the stability of foams. Even the simplest experiments contain key information toward understanding and prediction of the foamability of a system. 10.2.2.3 Whole Casein/Tween 20 System Let us now evaluate the surface properties of the mixed system formed by whole casein and Tween 20 also shown in Figure 10.4. Despite having similar surface tensions (Table 10.1), the structural

228

Structure and Functional Properties of Colloidal Systems

differences between whole casein and its major component (b-casein) appear clearly in the surface behavior of their mixtures with Tween 20 (Figures 10.3 and 10.4). At first sight it appears that, contrary to the other system, the isotherm of the mixture of whole casein/Tween 20 mixture does not converge to that of Tween 20 in Figure 10.4. In fact, the surface pressure of the whole casein/Tween 20 system rises at lower concentration of Tween 20 in the mixture, 10-6 M, compared to the mixture with b-casein. Also, this surface pressure isotherm shows a cac of 10-5 M, that is, slightly lower that the cmc of Tween 20. This behavior might indicate the formation of mixed protein/surfactant layers in the whole casein/Tween 20 system (Alahverdjieva et al., 2008). Moreover, taking into account the behavior of the b-casein/Tween 20 system, whole casein/Tween 20 mixed adsorbed layers should contain the other fractions contained in the whole casein. Contrary to the case of b-casein/Tween 20 systems, although the surface pressure isotherms shed some light on the surface structure of whole casein/Tween 20 systems, the information extracted does not provide a full explanation to the behavior of foams stabilized by this system. Taking into account the key information provided by the confinement in thin liquid films as regards the foam stability of whole casein and b-casein solutions, which was examined in the previous section, let us evaluate the properties of the foam films stabilized with whole casein/Tween 20 mixtures. Table 10.2 shows the thickness of foam films stabilized by pure whole casein, pure Tween 20 and two mixed systems under similar conditions to the foam stability, and the surface pressure experiments. The film thickness is measured by using Scheludko’s microinterferometric method (Maldonado-Valderrama and Langevin, 2008). A single foam film stabilized by Tween 20 shows a stratified drainage behavior (MaldonadoValderrama and Langevin, 2008). Namely, just after film formation, an interference pattern appears, accounting for the inhomogeneous thickness of the film. Then starts the nucleation of dark holes that gradually grow and provide films with coexistence of different thickness. Smaller thickness domains nucleate and expand until the thinner domain fully covers the film. Drainage then stops; the disjoining pressure is the pressure corresponding to the equilibrium thickness reached. The maximum pressure is limited by rupture of the film. The final thickness of the foam film contains information about the structure and intermolecular forces of the foam film. Accordingly, the final foam film of Tween 20 equals the diameter of a Tween 20 micelle (Dimitrova et al., 2001) The stratification observed reflects the sequential expulsion of micelles of Tween 20 trapped in the foam film and the final thickness of the film suggests the existence of a Newton black film, covered by two surfactant monolayers (Table 10.2). Regarding the drainage of a foam film stabilized by sole whole casein, this has been discussed in the previous section. The thickness of the whole casein film diminishes in a continuous manner until reaching a finite final thickness that corresponds to a bilayer of caseins with somehow screened electrostatic repulsion between layers and a higher intermolecular interaction between the different monomers composing the whole casein (Maldonado-Valderrama and Langevin, 2008). TABLE 10.2 Thickness of the Foam Films Formed by the Pure and Mixed Solutions of Whole Casein and Tween 20 Solution Whole casein (0.1 g/L) Tween 20 (50 g/L)

Thin Film Thickness (nm)

Whole casein + Tween 20 (10–5 M)

16 11 10

Whole casein + Tween 20 (10–3 M)

15

Note: The concentration of whole casein in the mixtures is of 0.1 g/L and the solutions are prepared in a buffer solution at pH 7.4, I = 16.4 mM, T = 20°C.

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions

229

As regards the drainage of thin films formed by mixed systems, as a general observation, the drainage behavior appears to be very dependent on the concentration of surfactant in the mixture. In general, the films made from mixed solutions show an intermediate behavior compared to that of pure systems in agreement with similar studies with other proteins (Wilde et al., 1997). Table 10.2 shows the final film thickness for two specific Tween 20 concentrations: one of the same order of magnitude but lower than the cmc of the surfactant and the other much higher. In the first case, the film undergoes a monotonous drainage, and the final thickness of the film has a smaller value than that of the pure protein film but close to that of the foam film of pure Tween 20 (Table 10. 2). At this concentration of surfactant, there are no micelles in the solution. Since the mixed monolayers contain both surfactant and protein, this suggests that the film is made of a double layer of surfactant molecules with incorporated proteins. Accordingly, the presence of Tween 20 within the foam film seems to compact the protein, possibly preventing bridging between protein molecules or suppressing steric repulsive forces. If we now increase the concentration of Tween 20 in the mixed foam film, well above the cmc, a stratified drainage appears, probably due to the sequential expulsion of layers of micelles (Maldonado-Valderrama and Langevin, 2008). However, the final thickness of films stabilized by this mixture is similar to that corresponding to pure whole casein films. This might indicate that the film consists mainly of a bilayer of protein; but in view of the surface tension data, surfactant is certainly also present in the film. In summary, increasing the concentration of surfactant in the whole casein/Tween 20 system, first decreases the final thickness of the foam film and subsequently recovers the thickness of a pure whole casein film. This observation certainly resembles the tendency found for the stability of the foam of whole casein/ Tween 20 systems in Figure 10.3. The study of thin liquid films of the pure and mixed systems proves to be extremely useful in further comprehension of the stability of the foam. The disappearance of the bridging between protein molecules due to the presence of Tween 20 might well be responsible for the decrease in the stability of the foam shown in Figure 10.3. Furthermore, Table 10.2 shows how the thickness of the film formed by whole casein and Tween 20 at a concentration below its cmc decreases substantially as compared to that of films of pure caseins. Moreover, further increasing the concentration of Tween 20 in the mixture yields an increase in the final film thickness that even approaches the value of pure caseins’ single foam film. The evolution of the stability of the foam with increasing concentrations of Tween 20 in the whole casein solutions shows a completely similar tendency. Accordingly, the thickness of the foam films seems to play a fundamental role in foam stability, and the correlation found in the previous section is verified here for a different system. Again, the thinner the final foam film, the less stable the foam.

10.2.3 FOAMS AND EMULSIONS STABILIZED BY β-CASEIN This final section aims to extend the above reasoning to the other main colloidal dispersion found in food. Hence, this section deals with the properties of foams and emulsions stabilized by a protein, b-casein. The objective is to show the large parallelisms existing between both types of dispersions and, again, to prove how useful the interfacial studies can be in comprehending the stabilizing mechanisms of the dispersion. In this case, the tool used to explain the behavior of foams and emulsions is interfacial rheology. The reason for this is due to the experimental difficulties added by the presence of an oil phase in the system that complicate substantially the interfacial measurements. Analysis of thin emulsion films (confined in an oil phase instead of air) is possible (Dimitrova et al., 2001), but clearly complicated to carry out. However, the interfacial rheology of adsorbed protein layers can be easily studied with the pendant drop technique and contain crucial information as regards the interfacial structure of the protein (Benjamins et al., 2006; Maldonado-Valderrama et al., 2005a). As has been carried out in the previous sections, in order to facilitate the correlation between different phenomena, dispersions were prepared so that the interfacial coverage of bubbles and drops was similar to that of the interfacial rheology experiments. Also, the materials used are

230

Structure and Functional Properties of Colloidal Systems

TABLE 10.3 Half-Lifetime of the Foam Formed by Different Solutions of β-Casein in a Buffer Solution at pH 7.4, I = 16.4 mM, T = 20°C (Flow Rate: 0.5 l/s, Porosity: 40–100 μm) Protein

Concentration (g/L)

Half-Lifetime (min)

b-Casein

0.01

0

b-Casein

0.05

20

b-Casein

0.1

35

the same. Accordingly, let us evaluate first the stability of foams made by b-casein solutions in terms of the time evolution of the volume of foam along with the properties of tetradecane-in-water emulsions stabilized by b-casein from measurements with light-scattering techniques. Next, we will show the dilatational elastic modulus of b-casein adsorbed layer at the air–water and tertradecane–water interfaces as a function of interfacial coverage. As a result, we reveal a direct correlation between the elasticity of the interfacial layers and the stability of foams and emulsions (MaldonadoValderrama et al., 2008). This is a fascinating result that proves the key role played by the interfacial rheology in stabilizing food dispersions and links fundamental and technological issues. 10.2.3.1 Stability of Foams and Emulsions of β-Casein Table 10.3 shows the half-lifetime of the foam formed by solutions b-casein as a function of bulk concentration. The foamability of b-casein is very dependant on the bulk concentration of protein. Stable foams of b-casein were only found above a concentration of protein of 0.05 g/L. Then, the stability of the foam increased very steeply with the bulk concentration having a much larger halflifetime for the foam formed with 0.1 g/L b-casein. The properties of emulsions of tetradecane in aqueous solutions of b-casein are displayed in Table 10.4. Solutions were formed using a constant concentration of protein of 1 g/L and varying the oil content between 5% and 20%. Emulsions were mainly unstable in agreement with similar emulsions found in the literature (Husband et al., 1997). Emulsions are described here by means of the mean droplet size measured with an ALV®-NIBS/HPPS. The size of the droplets increases as the oil content increases in the emulsion (Table 10.4). This feature might be attributable to the low protein concentration: As the oil content increases, there is not enough protein to cover the oil droplets resulting in an increase of the droplet volume in order to decrease the total interfacial area. This tendency was observed by Carrera-Sánchez et al. who reported an even higher diameter for an emulsion containing a 30% of oil in similar emulsion (Carrera-Sánchez and Rodriguez-Patino, 2005). The table also displays the creaming rate of the different emulsions as measured with a Turbiscan Classic MA2000. Interestingly, the creaming velocity decreases with increasing oil content, that is, with increasing droplet size. This feature should be further commented. Theoretically, in dilute emulsions, the creaming rate increases with the size of the droplets (see Equation 10.4). Here, the

TABLE 10.4 Properties of the Emulsions of Tetradecane in Aqueous Solution of β-Casein in a Buffer Solution at pH 7.4, I = 16.4 mM, T = 20°C; Concentration of Protein = 1 g/L Tetradecane (%) 5 10 20

Drop Diameter (μm)

Creaming Rate (mm/min)

1.1 1.4 3.4

4.1 2.6 1.3

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions

231

droplet volume fraction is large and the hydrodynamic interactions between droplets might be important, also slowing down the creaming process (for an oil content of 10%, the distance between droplets surfaces is of the order of their diameter). In this sense, although the numerical values given in Table 10.4 might be slightly inaccurate, both the increasing tendency of the drop’s diameters and decreasing tendency of the creaming rate can be safely inferred from the data. 10.2.3.2 Interfacial Rheology of β-Casein As stated above, one of the main factors affecting foam stability is coalescence of air bubbles (Langevin, 2000, 2008). This de stabilizing mechanism implies the rupture of the liquid film separating two bubbles to form a bigger. The interfacial elasticity of an adsorbed layer measures the resistance of the film to deformation (Langevin, 2000; Wilde, 2000). Therefore, the surface elasticity is expected to determine ultimately the rupture of the foam film, and hence, the stability of the foam. Figure 10.5 shows the surface elasticity (E¢ in Equation 10.3) of b-casein adsorbed at the air–water and the tetradecane–water interfaces for an oscillating frequency of 0.1 Hz. Since the relaxation times for proteins are very long, even slow oscillation frequencies provide convenient time-scales for a reliable comparison between dilatational and dispersion behavior (Langevin, 2000). The experimental conditions are completely similar to that used in the foaming solutions, and the surface pressures corresponding to the bulk concentrations used in the foams are indicated by the arrow. For the b-casein adsorbed at the air–water interface, the elasticity of the surface layer shows a very steep increase as the surface pressure increases and hence as the surface coverage increases (Figure 10.5). This increase perfectly matches the tendency shown by the foam stability, which is significantly improved by increasing the protein concentration in bulk (Table 10.3). This experimental result corroborates the above reasoning and provides a direct experimental evidence of the connection between both phenomena. Moreover, these results show also a link between formation and stability of foams. In the case of proteins, efficient foaming is only achieved when good stability is obtained; otherwise, the foam collapses while forming (Langevin, 2000). As stated above, a major de stabilizing mechanism affecting emulsions is the creaming phenomena (Robins, 2000). The creaming velocity obtained for emulsions of b-casein is reported in Table 10.4 and it is found to decrease as the amount of tetradecane increases. Hence, emulsions with higher oil content display larger droplet sizes but are slightly more stable. This feature appears at

30

E (mJ/m2)

20

10

0 0

5

10

15 20 (mJ/m2)

25

30

FIGURE 10.5 Dilatational elasticity of b-casein adsorbed layers at the air–water (dashed line) and the tetradecane–water (straight line) interfaces. Solutions were prepared in a buffer at pH 7.4, I = 16.4 mM, T = 20°C, and the oscillation frequency is 0.1 Hz.

232

Structure and Functional Properties of Colloidal Systems

first sight surprising since the creaming rate was expected to increase as with droplet size as stated in Equation 10.4 (Robins, 2000). Let us examine the interfacial properties of b-casein absorbed at the tetradecane–water interface under the same experimental conditions as the emulsions. Figure 10.5 shows the interfacial elasticity of b-casein adsorbed layers onto the tetradecane–water interface. Here, it can be seen that, contrary to the findings at the air–water interface, the interfacial elasticity of b-casein at the tetradecane–water interface decreases with interfacial coverage (i.e., interfacial pressure) within the range of concentrations considered here and indicated by the arrow (Figure 10.5). This feature provides a plausible explanation to the properties displayed by emulsions in Table 10.4 (MaldonadoValderrama et al., 2008). Since the protein concentration in the emulsion is fixed, smaller droplets have higher interfacial coverage. Thus, the interfacial elastic modulus corresponding to smaller droplets is that corresponding to higher interfacial coverage. Accordingly, smaller droplets have smaller interfacial elasticity. As a result, the interfacial film is more susceptible to rupture, favoring coalesce of oil droplets. This could result in a coalescence-enhanced creaming of the emulsion in which larger droplets are rapidly formed so that the measured creaming velocity corresponds to very large coalesced droplets. Conversely, for high oil content emulsions, the higher interfacial elasticity of the adsorbed b-casein layer for slightly lower interfacial coverage (Figure 10.5) prevents such coalescence giving rise to the slightly higher stability encountered in the emulsion with higher oil content (Table 10.4). Nonetheless, this does not necessarily means that by further increasing the interfacial coverage, the elasticity would become lower and the emulsions less stable since more stable emulsions with higher concentrations of b-casein are reported in the literature (Husband et al., 1997). In this sense, it is probable that the interfacial elasticity of b-casein adsorbed layers at the tetradecane–water interface increases again as the surface coverage is further increased. In fact, a theoretical model predicts such reincrease in Maldonado-Valderrama et al. (2005a). The analysis of the experimental results performed in this work proves the direct correlation existing between dispersion stability and interfacial elasticity of the adsorbed protein layer. This is a very important issue that brings together surface science and food technology. However, in order to fully understand this issue, further studies that take into consideration the hydrodynamic interactions within the emulsion are needed (Robins, 2000).

10.3 CONCLUSIONS The ultimate aim of this research work is to understand the stability of colloidal dispersions on the basis of the fundamental surface properties displayed by the same system. To this end, we show several situations that illustrate the existing relationship between the different phenomena. In particular, we show the differences between the foam stability of two food proteins; the complex behavior of foams formed with mixed systems protein/surfactant, and finally, the stability of foams and emulsions of a model protein. In all the cases, the systems are evaluated over different length scales ranging from structural to functional properties. Firstly, the stability of the foam formed by b-casein and whole casein appears very different, the former being more stable. In order to further investigate this issue, we evaluate several surface properties of these two proteins. The surface tension and surface rheology do not seem to be accurate enough to account for this large difference in foam stability, since they show very similar values. However, the thickness of the foam films stabilized by the two proteins respectively seems to determine the ultimate behavior of the foam. Hence, the thicker foam film measured for b-casein probably prevents coalesce of air bubbles resulting in more stable foam formed by this protein as compared to whole casein. Once the behavior of these two proteins is understood, the second section deals with more realistic systems. Accordingly, we study the stability of foams formed by the mixtures of these two proteins with a low molecular weight surfactant and again we discuss the differences by means of fundamental surface studies of the same system. Important differences arise upon comparison of

Interfacial Phenomena Underlying the Behavior of Foams and Emulsions

233

the stability of the foams formed by b-casein/Tween 20 and whole casein/Tween 20. The effect of adding Tween 20 to a b-casein solution is a monotonous decrease of the foam stability from that of b-casein to that of Tween 20. Differently, the effect of adding Tween 20 to a whole casein solution is more complex. Hence, the stability of the foam first decreases and after a certain concentration of surfactant in the mixture, the foam stability recovers the value given for pure whole casein. Looking at the surface pressure isotherms, we conclude that the Tween 20 displaces b-casein from the surface since the cac of b-casein/Tween 20 equals the cmc of the surfactant. Regarding the whole casein/Tween 20 mixed system, it displays a different surface behavior characterized by a cac lower than the cmc of the surfactant. However, this feature does not account for the foam stability of these mixtures. Further analysis of the foam films of these systems provides a plausible explanation to the foam stability showing a very direct correlation. Hence, the foam film thickness is significantly diminished upon addition of low concentrations of Tween 20. Then, as the concentration of surfactant in the mixture increases, the foam film thickness is higher than that found for the sole surfactant at concentrations above its cmc, suggesting the presence of whole casein in the foam film. Accordingly, the half-lifetime of the mixed foam displays a minimum located precisely at the concentration that provides the thinner foam film, giving a unique direct correlation between film thickness and foam stability. Whereas Tween 20 displaces completely b-casein from the surface, whole casein appears to be more resistant to the displacement. This feature is reflected in the foam stability and in the drainage of thin liquid films of whole casein/Tween 20 mixtures. The reason for this might be related to the more compact structure of k-casein and its resistance against displacement (Maldonado-Valderrama and Langevin, 2008), also suggesting that this fraction governs the foam stability of whole casein. This section illustrates the important effect of the nature of the components on foam stability of mixed protein/surfactants systems and the important relationship with surface behavior. Finally, we show a connection between stability of foams and emulsions and interfacial rheology of adsorbed protein layers. Accordingly, the elasticity of the interfacial layer of b-casein adsorbed at the air–water and the tetradecane–water interface appears as a key parameter in determining the stability of foams and emulsions, respectively. The stability of the foam clearly increases with the elasticity of the surface layer, within the range of concentrations considered. As regards the properties of the emulsions, the size of the oil droplets increased with the oil content. However, the creaming velocity was lower for the emulsion formed with slightly larger droplets, which have lower interfacial coverage. This behavior is fully understood in terms of a coalescence-enhanced creaming phenomenon due to the lower interfacial elasticity displayed for smaller droplets, which hence tend to coalesce. This whole chapter deals with the correlation of fundamental surface magnitudes with dispersion properties, that is, the connection between structural and functional properties of colloidal dispersions. This is a fascinating issue that provides the fundamental studies with a practical technological application. The conclusions were reached by linking the behavior of the same system over various length scales. This procedure appears as a valuable approach for the understanding of these complex systems.

REFERENCES Alahverdjieva, V. S., Khr. Khristov, D. Exerowa, and R. Miller. 2008. Correlation between adsorption isotherms, thin liquid films and foam properties of protein/surfactant mixtures: Lysozyme/C10DMPO and lysozyme/SDS. Colloids Surf. A: Physicochem. Eng. Aspects 323 (1–3): 132–138. Benjamins, J., J. Lyklema, and E. H. Lucassen-Reynders. 2006. Compression/expansion rheology of oil/ water interfaces with adsorbed proteins. Comparison with the air/water surface. Langmuir 22 (14): 6181–6188. Bergeron, V. and C. J. Radke. 1992. Equilibrium measurements of oscillatory disjoining pressures in aqueous foam films. Langmuir 8 (12): 3020–3026. Cabrerizo-Vilchez, M. A., H. A. Wege, J. A. Holgado-Terriza, and A. W. Neumann. 1999. Axisymmetric drop shape analysis as penetration Langmuir balance. Rev. Sci. Instrum. 70 (5): 2438–2444.

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Carrera-Sánchez, C. and J. M. Rodriguez-Patino. 2005. Interfacial, foaming and emulsifying characteristics of sodium caseinate as influenced by protein concentration in solution. Food Hydrocolloids 19 (3): 407–416. Cascao-Pereira, L. G., C. Johansson, C. J. Radke, and H. W. Blanch. 2003. Surface forces and drainage kinetics of protein-stabilized aqueous films. Langmuir 19 (18): 7503–7513. Dimitrova, T. D., F. Leal-Calderon, T. D. Gurkov, and B. Campbell. 2001. Disjoining pressure vs thickness isotherms of thin emulsion films stabilized by proteins. Langmuir 17 (26): 8069–8077. Husband, F. A., P. J. Wilde, A. R. Mackie, and M. J. Garrood. 1997. A comparison of the functional and interfacial properties of b-casein and dephosphorylated b-casein. J. Colloid Interface Sci. 195 (1): 77–85. Krägel, J., R. Wüstneck, F. A. Husband, P. J. Wilde, A. V. Makievski, D. O. Grigoriev, and J. B. Li. 1999. Properties of mixed protein/surfactant adsorption layers. Colloids Surf. B: Biointerfaces 12 (3–6): 399–407. Langevin, D. 2000. Influence of interfacial rheology on foam and emulsion properties. Adv. Colloid Interface Sci. 88 (1–2): 209–222. Langevin, D. 2008. Aqueous foams: A field of investigation at the frontier between chemistry and physics. Chem. Phys. Chem. 9 (4): 510–522. Lucassen-Reynders, E. H. 1981. Anionic surfactants: Physical chemistry of surfactant. In: E. H. LucasseReynders (Ed.), Surfactant Science Series, Vol. 11. New York/Basel: Marcel Dekker. Mackie, A. R. and P. J. Wilde. 2005. The role of interactions in defining the structure of mixed protein-surfactant interfaces. Adv. Colloid Interface Sci. 117 (1–3): 3–13. Maldonado-Valderrama, J., V. B. Fainerman, M. J. Galvez-Ruiz, A. Martin-Rodriguez, M. A. CabrerizoVilchez, and R. Miller. 2005a. Dilatational rheology of b-casein adsorbed layers at liquid–fluid interfaces. J. Phys. Chem. B 109 (37): 17608–17616. Maldonado-Valderrama, J., M. J. Galvez-Ruiz, A. Martin-Rodriguez, and M. A. Cabrerizo-Vilchez. 2005b. A scaling analysis of b-casein monolayers at liquid–fluid interfaces. Colloids Surf. A: Physicochem. Eng. Aspects 270: 323–328. Maldonado-Valderrama, J. and D. Langevin. 2008. On the difference between foams stabilized by surfactants and whole casein or b-casein. Comparison of foams, foam films, and liquid surfaces studies. J. Phys. Chem. B 112 (13): 3989–3996. Maldonado-Valderrama, J., A. Martin-Molina, A. Martin-Rodriguez, M. A. Cabrerizo-Vilchez, M. J. GalvezRuiz, and D. Langevin. 2007. Surface properties and foam stability of protein/surfactant mixtures: Theory and experiment. J. Phys. Chem. C 111 (6): 2715–2723. Maldonado-Valderrama, J., A. Martín-Rodriguez, M. J. Gálvez-Ruiz, R. Miller, D. Langevin, and M. A. Cabrerizo-Vílchez. 2008. Foams and emulsions of b-casein examined by interfacial rheology. Colloids Surf. A: Physicochem. Eng. Aspects 323 (1–3): 116–122. Martin, A. H., K. Grolle, M. A. Bos, M. A. Cohen-Stuart, and T. van Vliet. 2002. Network forming properties of various proteins adsorbed at the air/water interface in relation to foam stability. J. Colloid Interface Sci. 254 (1): 175–183. Murray, B. S. 2007. Stabilization of bubbles and foams. Curr. Opin. Colloid Interface Sci. 12 (4–5): 232–241. Noskov, B. A., A. V. Latnikova, S. Y. Lin, G. Loglio, and R. Miller. 2007. Dynamic surface elasticity of b-casein solutions during adsorption. J. Phys. Chem. C 111 (45): 16895–16901. Robins, M. M. 2000. Emulsions and creaming phenomena. Curr. Opin. Colloid Interface Sci. 5 (5–6): 265–272. Rodríguez Patino, J. M., C. C. Sánchez, and M. R. Rodríguez Niño. 1999. Structural and morphological characteristics of b-casein monolayers at the air-water interface. Food Hydrocolloids 13 (5): 401–408. Turhan, K. N., D. N. Barbano, and M. R. Etzel. 2003. Fractionation of caseins by anion-exchange chromatography using food-grade buffers. J. Food Sci. 68 (5): 1578–1583. Wilde, P. J. 2000. Interfaces: Their role in foam and emulsion behaviour. Curr. Opin. Colloid Interface Sci. 5 (3–4): 176–181. Wilde, P. J., A. R. Mackie, F. A. Husband, A. P. Gunning, and V. J. Morris. 2004. Proteins and emulsifiers at liquid interfaces. Adv. Colloid Interface Sci. 108–109: 63–71. Wilde, P. J., M. R. Rodriguez-Nino, D. C. Clark, and J. M. Rodriguez-Patino. 1997. Molecular diffusion and drainage of thin liquid films stabilized by bovine serum albumin-tween 20 mixtures in aqueous solutions of ethanol and sucrose. Langmuir 13 (26): 7151–7157.

11

Rheological Models for Structured Fluids Juan de Vicente

CONTENTS 11.1 Introduction ...................................................................................................................... 11.2 Newtonian Fluid Mechanics ............................................................................................. 11.3 Rheological Functions ...................................................................................................... 11.3.1 Standard Flows ..................................................................................................... 11.3.2 Material Functions ................................................................................................ 11.4 Theories on the Mechanical Properties of Structured Fluids ........................................... 11.4.1 Polymeric Liquids ................................................................................................. 11.4.2 Dispersed Systems ................................................................................................ 11.5 Experimental Techniques ................................................................................................. 11.5.1 Measurement Systems .......................................................................................... 11.5.2 Rotational and Oscillatory Tests ........................................................................... 11.5.2.1 Rotational Tests ...................................................................................... 11.5.2.2 Oscillatory Tests .................................................................................... 11.6 Comparison between Theory and Experiment ................................................................. 11.6.1 Polymeric Liquids ................................................................................................. 11.6.2 Dispersed Systems ................................................................................................ Acknowledgments ...................................................................................................................... References ..................................................................................................................................

11.1

235 236 237 237 238 238 238 242 249 249 251 251 252 253 253 255 258 258

INTRODUCTION

In simple Newtonian fluids, stresses (apart from an isotropic pressure) are linearly proportional to the rate of deformation. The equivalent for a solid would be the case of a perfectly elastic material, for which the force depends linearly on the deformation and which is also known as Hooke’s law. Viscoelastic fluids combine the properties of a simple fluid and an elastic solid. They react as a solid to fast deformation and as a fluid to slow deformation, where fast and slow are defined with respect to a characteristic time scale of the material. These fluids are also characterized as structured or complex in contrast to simple fluids. Other examples of structured fluids are fluids, where forces depend nonlinearly on the rate of deformation, such as shear-thinning or shear-thickening fluids. The mechanical behavior of these structured fluids is studied in the field of rheology. This chapter deals with the study of the rheological properties of structured fluids in order to establish a connection between measured material rheological functions and structural properties. The chapter is organized as follows: firstly, a brief introduction on fluid mechanics is given. Secondly, 235

236

Structure and Functional Properties of Colloidal Systems

rheological standard flows and material functions are defined. Thirdly, theories on mechanical properties of structured fluids are presented. Two kinds of structured fluids are investigated: polymeric liquids and dispersed systems. In the fifth section, experimental techniques are reported in terms of measurement systems and both rotational and oscillatory tests. Finally, theories and experiments are compared for both polymeric and dispersed systems.

11.2

NEWTONIAN FLUID MECHANICS

In Newtonian fluid mechanics, the macroscopic behavior of fluids is the same as if they were continuous in structure. This is usually called the continuum hypothesis. Traditionally, Newtonian fluid mechanics can be studied using one of the three basic approaches: control volume, differential, and experimental. Following a differential approach, three basic equations commonly used in Newtonian fluid mechanics are as follows [1]: i. Equation of conservation of mass (Continuity equation): ∂r + — ◊ (rv) = 0, ∂t

(11.1)

where r is the density and v is the velocity of the fluid. ii. Equation of conservation of linear momentum (Cauchy’s equation or equation of motion): r

dv (b) = rF + — ◊ P, dt

(11.2)

where F (b) are body forces per unit mass acting on the fluid and — · = P is the divergence of the total stress tensor. The total stress tensor has two contributions, one associated to the isotropic thermodynamic pressure and a second one called extra stress tensor, deviatoric stress tensor, or viscous tensor t= . Both Continuity and Cauchy’s equation are usually called change equations. iii. Equation of conservation of energy (Energy equation): r

du , = - p—v + DT + S + Q dt

(11.3)

where. u~ is the internal energy, T is the absolute temperature, S is the dissipation function, and Q is the rate of energy production. In addition to density, temperature, and velocity vector, there are six additional unknowns for the stress tensor, for a total of 11 unknowns. For the problem to be mathematically solvable, the number of equations must equal the number of unknowns, and thus we need six more equations. These equations are called constitutive or rheological equations. The constitutive equations enable us to write the components of the stress tensor in terms of the velocity and pressure fields. The simplest constitutive equations provide the following three equations for the conservation of linear momentum: 1. Euler equation (equivalently the Lamb–Gromeka equation): r where p is the pressure in the fluid.

dv (b) = rF - —p, dt

(11.4)

237

Rheological Models for Structured Fluids

2. Navier equation: ds l+m m (b) =F + —— ◊ s + D s, dt r r

(11.5)

where s is the displacement, and l, m are the Lamme constants. These constants are related to Young’s modulus, E, and Poisson’s coefficient, s, through: sE , (1 + s )(1 - 2s )

E . 2(1 + s )

(11.6)

dv 1 l+m m (b) = F - —p + —— ◊ v + Dv. dt r r r

(11.7)

l=

m=

3. Navier–Stokes equation:

Several approximations that eliminate terms reducing the Navier–Stokes equation to solvable equations are as follows: a. Inviscid flow approximation: Valid for high Reynolds number where net viscous forces are negligible. Under this approximation, Equation 11.7 reduces to Equation 11.4. b. Creeping flow approximation: Valid for low Reynolds number when density, velocity, or typical size is very small or viscosity is very large. c. Boundary layer approximation: Valid in a thin region of flow near a solid wall where viscous forces and rotationality cannot be ignored. d. Potential flow approximation: Valid for irrotational flows. e. Lubrication approximation: Valid at narrow gaps.

11.3

RHEOLOGICAL FUNCTIONS

Generally speaking, any material that exhibits behavior not predicted by Equation 11.7 is nonNewtonian. Three well-known reasons for non-Newtonian behavior are shear rate-dependent viscosity, viscoelasticity, and nonelastic time-dependent behavior. In order to establish a background on non-Newtonian fluid mechanics, it is important to understand standard flows that are frequently used to investigate them and specify which material functions are relevant.

11.3.1 STANDARD FLOWS The choice of the kinematics used for probing non-Newtonian behavior is arbitrary, and hence, the number of choices is infinite. However, the rheological community has settled on a small number of standard flows. These standard flows are simple and realizable by experimentalists. Classic flows are simple shear and simple shear-free elongation [2]. Simple shear flows are described by the following kinematics: t)x ˆ Ê z( 2 Á ˜ v = Á 0 ˜, ÁË 0 ˜¯ . . where z (t) is equal to the 21-component of the shear rate tensor g.

(11.8)

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Structure and Functional Properties of Colloidal Systems

Simple shear-free elongational flows are described by t )(1 + b) x1 ˆ Ê - 12 e( t )(1 - b) x2 ˜ , v = Á - 12 e( Á ˜ ÁË ˜¯ t ) x3 e(

(11.9)

. where e(t) is the elongation rate of the flow, and parameter b is associated to the way that the streamlines of the flow change with rotations around the flow direction. Depending on their values, it is possible to classify the elongational flows in: . i. Uniaxial elongational flow (e.g., fiber-spinning): b = 0, e(t) > 0. . ii. Biaxial stretching flow (e.g., lubricated squeezing): b = 0, e(t) < 0. . iii. Planar elongational flow (e.g., cross-channel dies): b = 1, e(t) > 0.

11.3.2 MATERIAL FUNCTIONS For incompressible Newtonian fluids, flow properties are governed by continuity equation (Equation 11.1) and equation of motion (Equation 11.7). The values of only two material parameters are just what are needed to predict the behavior of incompressible Newtonian fluids. However, for non-Newtonian fluids, even though continuity equation and the equation of motion written as Equation 11.2 remain valid, the Newtonian constitutive equation is not correct and a different constitutive equation is needed. To find constitutive equations, experiments are performed on materials using standard flows described above. The functions of kinematic parameters that characterize the rheological behavior of fluids are called rheological material functions. Standardized material functions are shown in Table 11.1 [2–4].

11.4

THEORIES ON THE MECHANICAL PROPERTIES OF STRUCTURED FLUIDS

A detailed theoretical description of structured fluids is beyond the scope of this chapter. Here, only a selection of well-established theories on polymeric and dispersed systems is summarized. More details on these and other structured fluids can be found elsewhere [2,5–7].

11.4.1 POLYMERIC LIQUIDS Continuum mechanics is the approach most frequently used to develop constitutive equations and predict material functions. This approach to constitutive modeling has established the framework for all constitutive modeling, including molecular modeling approaches and thermodynamic and stochastic methods. The simplest non-Newtonian constitutive equation is the so-called generalized Newtonian fluid: t = h( g )g ,

(11.10)

. where h(g ) is the viscosity of the non-Newtonian fluid. Many functional forms have been pro. posed in the literature for h(g ) and related quantities (Table 11.2). For more detailed information see reference [8]. Among all, power-law, Carreau-Yasuda, and Bingham models are the most frequently used.

239

Rheological Models for Structured Fluids

TABLE 11.1 Rheological Material Functions Flow Type

Material Functions

Steady-shear flow

Viscosity h, first normal-stress coefficient y1, second normal-stress coefficient y2

Shear stress growth

Shear stress growth coefficient h+, first normal-stress growth coefficient y1+, second normal-stress growth coefficient y 2+

Shear stress decay

Shear stress decay coefficient h-, first normal-stress decay coefficient y 1-, second normal-stress decay coefficient y 2Shear creep compliance J

Shear creep Step shear strain

Relaxation modulus G, first normal-stress step shear-relaxation-modulus Gy1, second normal-stress step shear-relaxation-modulus Gy2

Small-amplitude oscillatory shear Uniaxial steady elongation

Storage modulus G¢, loss modulus G≤ – Uniaxial elongational viscosity h

Biaxial steady elongation

– Biaxial elongational viscosity h B

Planar steady elongation

– , second planar elongational viscosity h – First planar elongational viscosity h P1 P2

Elongational stress growth

Uniaxial elongational stress growth coefficient h+, biaxial elongational stress growth coefficient hB+, planar elongational stress growth coefficients h+P1, h+P2 Elongational creep compliance D Uniaxial step elongational relaxation modulus E, biaxial step elongational relaxation modulus EB, planar step elongational relaxation moduli EP1, EP2

Elongational creep Step elongational strain Small-amplitude oscillatory elongation

Elongational storage modulus E¢, elongational loss modulus E≤

The most important limitation to the generalized Newtonian fluid is that memory effects (viscoelasticity) in the material are not predicted. In this sense, a step forward is the generalized linear viscoelastic fluid. In integral form, it can be written as follows: t

t=

Ú G(t − t ¢)g (t ¢)dt ¢,

(11.11)

-•

where G(t - t¢) is the shear-relaxation-modulus function and describes the material response to the flow.

TABLE 11.2 Definitions of Viscosity Related Quantities Viscosity Kinematic viscosity

h hK = h/r

Relative viscosity

hr = h/hc

Specific viscosity

hsp = hr - 1

Reduced viscosity

hred = hsp/f

Inherent viscosity

hinh = ln hr /f

Intrinsic viscosity

[h] = lim hred = lim hinh jÆ0

jÆ0

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Structure and Functional Properties of Colloidal Systems

Using G(t - t¢) = (h0/l2) exp[(-t - t¢)/l], with zero shear viscosity h0, Equation 11.11 can be written in differential form as follows (differential Maxwell model):

t+l

∂t = h0 g . ∂t

(11.12)

Even though viscoelasticity is predicted using the generalized linear viscoelastic model, an important limitation to this model is that it is not frame invariant. The two models discussed so far are just extensions of the Newtonian constitutive model and thus formulated in terms of shear rate. A proper formulation of strain is needed to fix the frame invariance problem. This question is closely linked to the types of derivatives that are permitted in the constitutive equations. The Lodge equation satisfactorily passes the objectivity test using the Finger tensor = C -1 as strain measure [9]: t

t=

È h0

Ú ÍÎ l

2

-•

Ê t - t ¢ ˆ ˘ -1 C (t ¢, t )dt ¢ (integral form) exp Á l ˜¯ ˙˚ Ë —

t + lt = h0 g

(differential form)

(11.13)

(11.14)

with the upper convected time derivative defined as —

T ∫

DT - (—v)T ◊ T - T ◊ —v. Dt

(11.15)

Unfortunately, the Lodge equation does not predict shear-thinning phenomena or nonzero normal stress coefficients. Other possibilities exist to solve the frame invariant problem; Cauchy–Maxwell equation uses the Cauchy tensor, = C , which is also independent of the system of reference, the Lodge rubber-like liquid model uses the Finger tensor; but contrarily to the Lodge model, it uses a generalized memory function: N

M (t - t ¢ ) =

È hk

Â ÍÎ l k =1

2 k

Ê t - t¢ ˆ ˘ exp Á ˙. Ë l k ˜¯ ˚

(11.16)

So far, constitutive models do not predict neither shear thinning nor normal forces simultaneously. In this sense, two methods that are commonly used to overcome this problem are continuum and molecular modeling approaches. Most common constitutive equations obtained from continuum approaches are as follows: 1. Jeffrey’s model [3]: t

t( t ) =

È h0 Ê l 2 h0 l2 ˆ Ê t - t¢ ˆ ˘ ÁË 1 - l ˜¯ exp ÁË - l ˜¯ ˙ g (t ¢ )dt ¢ + l g (t ), 1 1 1 1 ˚

Ú ÍÎ l

-•

(11.17)

241

Rheological Models for Structured Fluids

where l1 is the relaxation time and l2 is the retardation time. This model is the equivalent of the sum of a Newtonian model and a generalized linear viscoelastic model. 2. Oldroyd B fluid [10]: —

t + lt = h0 g .

(11.18)

D D Ê ˆ t + l1 t = h0 Á g + l 2 g ˜ , Ë ¯

(11.19)

3. Oldroyd A fluid [3]:

where the lower convected time derivative is defined as D

T ∫

DT + —v ◊ T + T ◊ (—v)T . Dt

(11.20)

h( g ) — , t = h(g)g G0

(11.21)

4. White–Metzner model [11]: t+

where G 0 is a constant modulus parameter associated with the model. For this model, the . linear viscoelastic limit depends on the function h(g). 5. Oldroyd 8-constant model [12]: —

t + l1 t + È = h0 Í g Î

1 1 1 (l - m1 )( g ◊ t - t ◊ g ) + m 0 (tr t)g + n1 (t : g )I 2 1 2 2 — 1 ˘ + l 2 g + (l 2 - m 2 )( g ◊ g ) + n2 ( g : g )I ˙ , 2 ˚

(11.22)

where m0, m1, m2, n1, and n2 are five parameters associated to nonlinear terms in the model. This model contains many of the constitutive equations we have already discussed as special cases [2]. 6. Giesekus model [12]: —

t + lt +

al t ◊ t = h0 g , h0

(11.23)

where a is a parameter that relates to the anisotropy of the drag encountered by flowing polymer segments. Equation 11.23 constitutes an improvement to Equation 11.22 by including second order in stress terms. 7. Factorized Rivlin–Sawyers model [3]: t

t(t ) =

Ú M (t - t ¢) ÈÎF (I 1

-•

1

- I 2 )C

-1

- F 2 ( I1 - I 2 )C ˘˚ dt ¢,

(11.24)

242

Structure and Functional Properties of Colloidal Systems

where M(t - t¢) is the memory function, I1 and I2 are the first and second invariants of C = -1, respectively, and F and F are scalar functions that must be specified. It is a and C 1 2 = nonlinear model in which class of equations includes many successful molecular-modelbased constitutive equations. 8. Factorized K-BKZ model [13,14]: t

t(t ) =

È ∂U

Ú M (t - t ¢) ÍÎ ∂I

-•

-1

C ( t ¢, t ) - 2

1

∂U ˘ C (t , t ¢ )˙ dt ¢, ∂I 2 ˚

(11.25)

where U(I1, I2) is the K-BKZ potential function. This is a subset of Rivlin–Sawyers class of equations. Molecular modeling approach has been traditionally used for polymeric systems. Most common models are as follows: 1. 2. 3. 4.

Rubber elasticity theory [15–18]. Temporary network model [19]. Reptation theory [20,21]. Elastic dumbbell model [4].

11.4.2 DISPERSED SYSTEMS For dilute dispersed systems, many rigorous theoretical results are available and a number of them have been confirmed experimentally. For a review on semidilute or concentrated dispersions read [22] and [23]. The bulk volume-average stress tensor in dispersed systems can be written as follows [6]: P =

1 1 1 P dV = P dV + V V V

Ú

Ú

V

Vc

Ú P dV ,

(11.26)

Vd

where V is a large enough volume containing statistically significant number of particles, Vc is the volume occupied by the continuous phase, and Vd is the volume occupied by the dispersed particles. Assuming a Newtonian continuous phase, force-free particles, and no interfacial film are present at the surface of the particles, we obtain P = P0 +

1 V

N

ÂS ,

(11.27)

i

i =1

. where ·P Ò = -· pÒd= + hc· =g Ò is the stress tensor in pure. matrix material under conditions correspond=0 ing to the imposed macroscopic shear rate tensor · =g Ò on the dispersion and =Si = ÚAi [P · nˆx - (1/3) = (x · P · n ˆ)d h (n ˆu + un ˆ)] dA is the so-called force dipole strength of the ith particle. Here, ·pÒ is the c = = average pressure, = d is the unit tensor, hc is the viscosity of the continuous phase, nˆ is the unit outward normal to the particle surface, x is the position vector, and u is the velocity. It should be noticed that the total stress tensor in dispersion is symmetric only in the absence of externally applied torque (couple) on the particles. In the case of rigid particles, the dipole strength is given by S=

È

1

˘

ˆ - ( x ◊ P ◊ nˆ )d ˙ dA. Ú ÍÎP ◊ nx 3 ˚ Ai

(11.28)

243

Rheological Models for Structured Fluids

On the one hand, for dilute dispersed systems of identical spherical particles (same size and shape), the constitutive equation (Equation 11.27) can be written as P = P0 +

N 3f 0 S = P0 + S , V 4 pR 3

(11.29)

where N/V is the number density of the particles, · =S Ò is the average value of =S for a reference particle, f is the volume fraction of the particles, R is the particle radius, and =S 0 is the dipole strength of a single spherical particle located in an infinite matrix. Then, the constitutive equation for a dilute dispersion of identical spherical particles is given by P = - p d + hc g +

3f 0 S . 4 pR 3

(11.30)

On the other hand, in the case of dilute dispersions of nonspherical particles P = P0 +

N 0 S V

with

S

0

=

0

Ú S ( p)Y( p) d p,

(11.31)

where =S 0( –p) is the value of the dipole strength of the reference particle and Y( –p) is the probability density function for particle orientation –p. Up to now, we have made no mention of nonhydrodynamic (thermodynamic) forces acting on the particles. In this case, the bulk stress in the dispersion can be written as p P = - p d + hc g + P ,

(11.32)

where the particle contribution to the bulk stress is given by P

p

=-

N N N N H Br H kBT d + ( S + S ) = - kBT d + ÈÍ S - ( xF + AA ◊ F )˘˙ , ˚ V V V V Î

(

(

)

)

(11.33)

where k B is the Boltzman constant, · SHÒ is the dipole strength of a particle, which is undisturbed by Brownian motion, · SBrÒ is the direct Brownian motion contribution arising from particle Brownian rotation, F is the force acting on each sphere, and A is a third-order hydrodynamic tensor that relates stresses to forces [6]. Some typical examples where the particle contribution to the bulk stress is easily calculated are as follows. In any case, since a comprehensive study of all material functions is out of the scope of this chapter, only steady shear viscosity results will be presented. In the case of dilute dispersions of rigid solid spherical particles, it can be demonstrated that the constitutive equation can be written as P = - p d + hc g +

5 h f g . 2 c

(11.34)

Hence, a dilute dispersion of rigid solid particles behaves as a Newtonian fluid with a volume fraction-dependent viscosity (Einstein equation): 5 ˆ Ê h = hc Á 1 + f˜ . 2 ¯ Ë

(11.35)

244

Structure and Functional Properties of Colloidal Systems

In the case of nonspherical axisymmetric particles, Equation 11.35 is no longer valid and the intrinsic viscosity, defined as the low volume fraction limit of the reduced viscosity, is no longer 2.5 due to the energy dissipation increase. The intrinsic viscosity has been calculated under specific conditions for a given aspect ratio r. Limiting values are shown in Table 11.3. For more details, read references [7] and [24]. In the case of dilute dispersions of rigid porous particles, it can be demonstrated that [6] P = - p d + hc g +

C h φ g , 2 c

(11.36)

where È ˘ 1 + 3b -2 - 3b -1 coth b C = 5Í ˙, -2 -2 -1 Î 1 + 10b (1 + 3b - 3b coth b) ˚

(11.37)

and b=

R , K

(11.38)

where K is the particle permeability. Equation 11.36 suggests that a dilute dispersion of rigid porous particles behaves as a Newtonian fluid of viscosity: C ˆ Ê h = hc Á 1 + f˜ . 2 ¯ Ë

(11.39)

In the case of dilute dispersions of rigid electrically charged particles at low Peclet number, Pe, [Pe = (Ru 0 e)/(m0 k BT), where u 0 is a characteristic fluid velocity, e is the elementary electronic charge, and m0 is the ion mobility] it can be shown that [25] 5 È ˘ h = hc Í1 + (1 + q )f ˙ , 2 Î ˚

(11.40)

~ where q is the so-called primary electroviscous coefficient. At low surface potential [z (ez)/(k BT) 1, where z is the zeta potential] and low Hartmann number [NHa = (ez2 R2k 2) /(Rhcu 0) 1,

TABLE 11.3 Limiting Values of the Intrinsic Viscosity for Dilute Dispersions of Rigid Axisymmetric Particles rÆ0

rÆ• 4r2 ______

Low shear

32 _____ 15pr

15 ln r

High shear

3.13

0.315r ______ ln r

245

Rheological Models for Structured Fluids

where k -1 is the Debye length and e is the permittivity], the electroviscous coefficient is written as follows: Ê 4 pekBT Á q= hc e Á Ë

N

Â nZm Â nZ • i =1 i N

• i =1 i

2 i

2 i

-1 i

ˆ ˜ z 2 (1 + Rk )2 f ( Rk ), ˜ ¯

(11.41)

where ni• is the number density of ionic species i far away from the particle, Zi is the valence of ionic species i, and f(Rk) is a power series function of Rk with two limiting forms as follows:

f ( Rk ) =

1 11( Rk ) + 200 p( Rk ) 3200 p

f ( Rk ) =

p( Rk )4 2

if Rk 1,

if Rk 1.

(11.42)

(11.43)

Russel [26] developed a model for the shear-thinning behavior at arbitrary Peclet number for the case of Rk 1. An equation for the viscosity–volume fraction dependence up to second order can be found in reference [27]. In the case of dispersions of elastic particles having an elastic modulus Gp, the steady shear viscosity is found to be [28]: 5 È 1 - (3 2) N se2 ˘ Ô¸ ÔÏ h = hc Ì1 + Í f , 2 Î 1 + (3 N se / 2)2 ˙˚ ˝Ô ÔÓ ˛

(11.44)

. where Nse = hcg/Gp is a dimensionless group that introduces a shear-thinning behavior. If particles are viscoelastic having Gp elastic modulus and hp viscosity, the steady shear viscosity of the dispersion is written as follows [29]: È 5Ê 5 t 1 t 2 g 2 ˆ ˘ h = hc Í1 + Á 1 f˙ , 2Ë 3 1 + t 22 g 2 ˜¯ ˙ ÍÎ ˚

(11.45)

where t~1 = (3hc)/(2Gp) and t~2 = (3hc + 2hp)/(2Gp). The rheology of dispersions of liquid particles (emulsions) has also been largely investigated. If the droplets retain their spherical shape under shear, the continuous medium is Newtonian and the dispersion is diluted enough; the emulsion behaves as Newtonian with viscosity given by Taylor equation [30] 5hdc + 2 ˆ Ê h = hc Á 1 + f , 2 hdc + 2 ˜¯ Ë

(11.46)

where hdc is the viscosity ratio (ratio of the droplet fluid viscosity hd, to the continuous medium viscosity hc). The factor multiplying the volume fraction allows for internal currents within the particles and hence reduces the distortion of the flow lines around them.

246

Structure and Functional Properties of Colloidal Systems

Frankel and Acrivos [31] have presented a more general constitutive equation based on a firstorder deformation of droplets where the viscosity is found to be

h=

hc 1 + l 02 g 2

Ï 5hdc + 2 5hdc + 2 19 hdc + 16 ˘¸ 2 2 È Ì1 + 2 h + 2 f + l 0 g Í1 + 2 h + 2 f - (2 h + 2)(2 h + 3) f ˙ ˝ , dc dc dc dc Î ˚ Ô˛ Ó

(11.47)

where a characteristic time constant of the emulsion is given by l0 = [(19hdc + 16)(2hdc + 3)hcR]/ [40 G(hdc + 1)], with interfacial tension G. Second-order deformation of droplets has been included by other authors [32]. Neglecting Marangoni stresses, Oldroyd [33] proposed the following equation for the shear viscosity in emulsions having surface dilational viscosity hsk and surface shear viscosity hs: ÏÔ h = hc Ì1 + ÓÔ

h k È 1 + (5 / 2) hdc + 2 N Bo ˘ ¸Ô + 3 N Bo Í ˙ f˝ h k Î 1 + hdc + (2/5)(2 N Bo + 3 N Bo ) ˚ ˛Ô

(11.48)

k = (hk)/(h R) and shear Boussinesq number N h = (h )/(h R). with dilational Boussinesq number N Bo s c s c Bo Later on, Danov [34] obtained an expression for the zero shear viscosity including Marangoni effects. Unlike the liquid/liquid emulsions, dispersions of bubbles cannot be treated as incompressible. The viscosity of a dispersion of bubbles can be written as [6]:

2 È 1 - (12 5) N Ca ˘ Ô¸ ÔÏ h = hc Ì1 + Í f , 2 ˙ ˝ ÔÓ Î 1 + (6 N Ca 5) ˚ Ô˛

(11.49)

. where NCa = hcgR/G is the capillary number that represents the relative effect of viscous forces versus surface tension acting across the interface. This equation is valid if the presence of surfactant film on the surface of the bubbles is neglected. In the case of dispersions of capsules several theories exist. Here, capsules refer to a particle consisting of a drop of liquid surrounded by a thin deformable elastic-solid membrane. For a Mooney–Rivlin-type membrane the intrinsic viscosity can be written as follows: [ h] = a0 ( hie - 1) -

ˆ a02 Ê b 2 a02 1 + - I, Á ˜ 20b 1 Ë 3d 2 ¯ 4d 2

(11.50)

where hie is the ratio of the internal fluid viscosity to the external fluid viscosity, and I is given as follows:

I =

a 02 [b 1 - b 3 + (1/4d 2 )(d 1 + e 2 /8)][2d 1 (15b 1 - b 2 ) + (3d 2 - b 2 + 15b 1 )(e 2 /4)] , (11.51) 30b 1[(d 22 e 2 /4) + (d 1 + e 2 /8)2 ]

~ ~ ~ ~ ~ where a~0, b1, b2, b3, d 1, d 2, and e~ are given in references [35,36]. For an RBC-type membrane, the intrinsic viscosity is given in references [35,36]:

[ h] =

5(23hie - 16) 7680 . + 2(23hie + 32) (23hie + 32)[64 + (23hie + 32)2 e 2 ]

(11.52)

247

Rheological Models for Structured Fluids

In order to improve stability, particles are usually coated to generate a steric hindrance. Hence the rheology of dispersions of core–shell particles needs to be investigated. Let us assume that the core has a radius a and outer shell of thickness b - a (Figure 11.1). a. Solid core–hairy shell [37]: (11.53) h = hc(1 + _52 X1f) where Χ1 = 1 -

3 [20a 6 b 4 + A¢ sinh(a - b) + B¢ cosh(a - b)], J (ab)3

(11.54)

b a (a)

S

H

S

L

L3

L2

(b)

(c)

L1

FIGURE 11.1 Core–shell structure for (a) solid core–hairy shell, (b) solid core–liquid shell, and (c) double emulsions. S = solid, H = hairy shell, and L = liquid.

248

Structure and Functional Properties of Colloidal Systems

a=

a , K

(11.55)

b=

b , K

(11.56)

A¢ = -30a 6 b - 12a 8b + 30a 3b 4 + 7a 3b6 + 30a 7b2 + 2a 9b2 + 3a 4b7 - 30a 4b 5 , (11.57)

B ' = -30a 6 b2 - 12a 8b2 - 3a 3b7 - 30a 4b 4 - 7a 4 b6 + 30a 7b + 2a 9b + 30a 3b5 , (11.58) J = 60a 3b + 3(10a 3 + 4a 5 + 30b + b5 + 10b3 - 30ab2 )sinh(a - b) - (90ab + 30ab3 + 30a 4 + 2a 6 + 3ab5 - 90b2 )cosh(a - b),

(11.59)

where K its hairy shell permeability. b. Solid core–liquid shell [38]: (11.60)

h = hc(1 + _52 X2f) where -1

2 3 ÏÔ 4 - 25(a/b)3 + 42(a/b)5 - 25(a/b)7 + 4(a/b)10 ¸Ô Χ2 = + Ì1 + ˝ , 5 5 2 hsc ÈÎ2 - 5(a/b)3 + 5(a/b)7 - 2(a/b)10 ˘˚ ÔÓ Ô˛

(11.61)

where hsc is the ratio of the shell viscosity to the continuous medium viscosity. c. Double emulsions [39]: Let us assume that the continuous phase has a viscosity h1, the outer shell has a viscosity h2, and the core has a viscosity h3. In this case: h = hc(1 + _52 X3f),

(11.62)

where Χ3 = 1 -

d1 + (a /b)3(1 + l 32 )-1 d 2 3 , 5 d1 + l 21d 3 + (a /b)3 (1 + l 32 )-1 (d 2 + l 21d 4 )

(11.63)

l 21 =

h2 , h1

(11.64)

l 32 =

h3 , h2

(11.65)

249

Rheological Models for Structured Fluids

11.5

d1 = 4 - 25(a/b)3 + 42(a/b)5 - 25(a/b)7 + 4(a/b)10 ,

(11.66)

d 2 = 15 - 42(a/b)2 + 35(a/b)4 - 8(a/b)7 ,

(11.67)

d 3 = 4 - 10(a/b)3 + 10(a/b)7 - 4(a/b)10 ,

(11.68)

d 4 = 6 - 14(a/b)4 + 8(a/b)7 .

(11.69)

EXPERIMENTAL TECHNIQUES

11.5.1 MEASUREMENT SYSTEMS In order to measure a material function, it is necessary to design an experiment to produce the kinematics prescribed in its definition. Most rheological measurements are performed in one of the following four shear geometries: capillary flow, parallel-plate and cone-plate torsional flow, and Couette flow. Excellent books on this topic to find more detailed information are [40] and [8]. Capillary flow is a unidirectional flow in which cylindrical surfaces slide past each other as in a collapsible telescope. The viscosity is calculated as the ratio of the rz component of the wall shear stress and the wall shear rate. On the one hand, in order to calculate the wall shear stress, it is assumed that the fluid is incompressible, the flow is unidirectional and steady state, and the tube is long enough to neglect any variation in the z direction. With these assumptions, the shear stress is written as follows:

t rz =

(Ρ 0 - Ρ L )r r ∫ tR , 2L Rt

(11.70)

t

where P0 and PL are boundary pressures, L is the tube length, and Rt is the internal radius of the tube. On the other hand, wall shear rate is calculated from the velocity field according to:

g R = t

∂vz ∂r

.

(11.71)

r = Rt

. In the case of a Newtonian fluid gRt = (4Q)/(pR 3t ), where Q is the flow rate. However, for a power. law fluid of exponent n - 1, gRt = [4Q(1 + 3n)]/(4pR 3t n). A general expression for viscosity from capillary data is due to Weissenberg and Rabinowitsch: È1 Ê d ln g a R ) = g R = g a Í Á 3 + g(t 4 d ln t R ÍÎ Ë t

t

t

. where ga is the so-called apparent shear rate.

ˆ˘ ˜ ˙, ¯ ˙˚

(11.72)

250

Structure and Functional Properties of Colloidal Systems

Equation 11.72 allows us to calculate the shear rate at the wall without assuming any form for the velocity profile. Hence, the viscosity equation for any fluid in capillary flow is given as follows: -1

4t R Ê d ln g a ˆ 3+ . d ln t R ˜¯ g a ÁË

h(g R ) =

t

t

(11.73)

t

As a consequence, the viscosity may be determined by measuring the flow rate, needed to calculate the wall shear rate, the difference of pressure, needed to calculate the wall shear stress, and geometric constants. At this point, it is important to remark that there are needed corrections for end effects and wall slip. A discussion on Bagley and Mooney correction techniques is given in reference [2]. Parallel-plate torsional flow is a second choice. Assuming incompressible flow, the viscosity can be calculated from the total torque needed to turn one disk while keeping the other immobile. Following a derivation similar to that used for the Weissenberg–Rabinowitsch equation and using Leibnitz rule, it is straightforward to get the viscosity at the rim of the disk:

h(g R ) = d

T /2 pRd3 È d ln(T /2 pRd3 ) ˘ 3+ Í ˙, d ln g R g R Î ˚ d

(11.74)

d

where T is the torque and Rd is the radius of the disk. If we substitute now one of the flat surfaces by a cone, we get a cone-plate torsional flow. Since both shear rate and stress are constant throughout the flow domain in the limit of small cone angle, q0, the calculation of the viscosity is direct:

h=

t qf 3T q0 = , g 2 pRd3 W

(11.75)

where W is the angular velocity. The cone and plate geometry has the advantage that the first normal stress coefficient, y1 (see Table 11.1), can be easily calculated by measuring the thrust on the cone, F, according to

y1 =

2 F q20 . pR 2 W2

(11.76)

Finally, the viscosity in concentric cylinder Couette flow (bob turning) can be calculated from

h=

T ( K - 1) , 2 pRo3 LK 3W

(11.77)

where K is the ratio between the radius of the bob and the radius of the cup, and Ro is the outer radius. The choice of which geometry to use is dictated by convenience. In Table 11.4, some advantages and disadvantages are summarized.

251

Rheological Models for Structured Fluids

TABLE 11.4 Comparative Analysis of Some Common Rheometers Advantages Capillary Very small viscosities Parallel plate Wall slip control Cone-plate Constant shear rate for small angle Small amounts of samples Concentric cylinders Sedimentation is not necessarily a big problem Possibility of coupling optical techniques

Disadvantages Difficult flow Nonconstant shear rate Large amounts of sample Gap adjustment

Large amounts of sample

The second part of this section deals with a brief introduction of the most typical tests used to get materials functions in structured materials. Both continuous and oscillatory shear rheological techniques are briefly described.

11.5.2 ROTATIONAL AND OSCILLATORY TESTS When carrying out a rheological test, two options are possible. Either to control the shear rate (CSR or CR test) or the shear stress (CSS or CS test). Which operating mode is more convenient to use depends on the test. 11.5.2.1 Rotational Tests 11.5.2.1.1 Flow Curves Flow curves are usually measured under isothermal conditions. The simplest test consists in the application of a shear stress ramp with CSS (shear rate with CSR) and measuring the shear rate (shear stress with CSR) in linear or logarithmic presets. In this case, the value of the measuring point duration should be selected to be at least as long as the value of the reciprocal shear rate. In some cases, rest and preshear intervals are used to improve reproducibility. Under some circumstances, a certain amount of force must be applied before the sample starts to flow. These samples are said to have a yield stress. The yield stress is closely associated to the internal structural forces in the system; dispersions and gels show a yield point due to intermolecular forces. There are different ways to measure the yield stress: i. Using CSR devices by curve fitting to Bingham, Casson, or Herschel–Bulkley models ii. Using CSS devices as the limit value of the applied shear stress when the sample starts to flow iii. Using the “one tangent” or “tangent crossover” methods iv. Using oscillatory techniques (see the section below) It is important to note that the yield stress is not a material constant since the value is always dependent on the options of the measuring instrument used. 11.5.2.1.2 Time-Dependent Curves In this case, we apply a constant shear rate with CSR (shear stress with CSS) and monitor the shear stress (shear rate with CSS) dependence as a function of time. When presetting low shear rate, or

252

Structure and Functional Properties of Colloidal Systems

shear stresses, the duration of each point must be long enough to avoid transient effects. Again, several intervals may be connected in series; rest intervals and preshear intervals. Viscosity time dependence gives an insight on the structure of the material. In general, rheologists distinguish between thixotropic, nonthixotropic, or rheopectic materials. – A material showing thixotropic behavior means a reduction of the structural strength during a shear load phase and a more or less rapid but complete structural regeneration during the subsequent period of rest. This is a reversible process. Almost all structured fluids, such as pastes, creams, gels, ketchups, paints, coatings, printing inks, and sealants are thixotropic. – Nonthixotropic materials are those whose initial structural strength is not fully recovered after an infinitely long period of time at rest (e.g., yogurt). – Rheopectic materials typically increase their structural strength when performing a high shear process. During a subsequent period of rest, there is a complete decomposition of the increased structural strength. Highly concentrated dispersions are typical examples of rheopectic materials. There are two widely used protocols to investigate the viscosity versus time dependence: a. Step test with three intervals: Reference, high shear, and regeneration interval. In this case, CSR tests are preferable to CSS tests. b. Flow curves: Upward ramp, hold time, and downward ramp. It is an outdated method still used in many industrial laboratories. The area between the upward and downward flow curves is called “hysteresis area.” An important point of this method is that it does not reveal any information about structural regeneration behavior during a period of rest. It only deals with structural decomposition. 11.5.2.1.3 Temperature-Dependent Curves In this case, the viscosity is examined as a function of temperature. A constant shear stress (in CSS) or shear rate (in CSR) is applied at a temperature/time profile. Materials showing hardening are typically investigated using this technique. From this, it is possible to infer the so-called softening temperature or melting temperature since it is associated to a viscosity minimum. Thermo-rheologically simple fluids (which do not undergo a change in the structural character in the observed temperature range) are usually modeled using Arrhenius relation. 11.5.2.2 Oscillatory Tests 11.5.2.2.1 Amplitude Sweep Curves Amplitude sweeps are oscillatory tests performed at variable amplitudes, keeping the frequency and also the measuring temperature constant. A typical frequency value used is 10 rad/s. Nonetheless, it is important to stress that some samples show a remarkable frequency dependence. There are two possibilities: either a strain sweep or a stress sweep. At low amplitude values, in the so-called linear viscoelastic region, both storage and loss moduli show a constant plateau. In the case of gel-like samples (G¢ > G≤ in the viscoelastic linear region), this test is frequently used to determine the yield point (yield stress) and flow point (flow stress). The yield point corresponds to the limiting value of the linear viscoelastic region. The flow point corresponds to the stress where G¢ = G≤. 11.5.2.2.2 Frequency Sweep Curves Frequency sweeps are oscillatory tests performed at variables frequencies, keeping the amplitude and temperature at a constant value. For controlled shear strain tests, a sinusoidal strain is fixed with an amplitude in the viscoelastic linear region. These tests are used to investigate the time-dependent shear behavior.

Rheological Models for Structured Fluids

253

Unlinked polymers having narrow molecular mass distribution (MMD) show a Maxwellian frequency dependence. At low frequency, G¢ shows the slope 2 : 1 and G≤ shows the slope 1 : 1. In the case of unlinked polymers having a wide MMD, two levels of the G¢ curve are observed. A total of four ranges can be observed giving information on the structure of a polymer: Initial range, rubber elastic range, transition range, and glassy range. Structural networks coming from dispersions and gels (physical networks) or cross-linked polymers (chemical networks) show a constant G¢ as a function of frequency. Usually, when performing frequency sweeps, one measuring point is measured after the other, each at a single frequency. However, the test period might be reduced if the rheometer is set to measure several frequencies at once. This results in a multiwave function, which is a multiple wave produced from several superimposed single oscillations. 11.5.2.2.3 Time-Dependent Curves Using this type of oscillation test, both the frequency and the amplitude are kept at a constant value in each test interval. Therefore, constant dynamic-mechanical shear conditions are preset [dynamic mechanical analysis (DMA) test]. Thixotropic/rheopectic behavior can be also investigated using a time-dependent step oscillatory test with three intervals methods (reference interval, high-shear interval, and regeneration interval). For practical users, the decisive factor to evaluate structural regeneration is the behavior in the time frame which is related to practice. Chemical cross-linking reactions and sol/gel transitions can also be monitored using a timedependent test. 11.5.2.2.4 Temperature-Dependent Curves Both frequency and amplitude are kept constant. The only variable parameter is the temperature [dynamic mechanical thermal analysis (DMTA) test]. This type of test is used to examine the temperature dependence of materials in terms of phase changes or other structural modification. Three kinds of polymer groups are usually investigated using this technique: a. Amorphous polymers: The molecule chains are chemically unlinked showing no regular superstructures. b. Partially crystalline polymers: Molecule chains are chemically unlinked showing a partially regular superstructure. c. Cross-linked polymers: Molecule chains are connected by chemical primary bonds. Typically, glass transition temperatures can be determined from this kind of tests. 11.5.2.2.5 Superposition of Rotation and Oscillation In some cases, it is possible to superimpose rotation and oscillation at the same time. Examples, where this technique has found to be useful, are testing leveling behavior of an emulsion and emulsion paints.

11.6

COMPARISON BETWEEN THEORY AND EXPERIMENT

The only way to test the validity of a constitutive model is by using experiments and measuring material functions. The ways in which structured fluids fail to follow the Newtonian constitutive equation vary enormously. Two typical examples are polymeric liquids and dispersed systems.

11.6.1 POLYMERIC LIQUIDS Macromolecules constituting a polymer melt or a solution entangle with others many times. At rest, each coil shows approximately spherical shape. However, during shearing process, constituents are oriented in the shear direction. In doing this, molecules disentangle to a certain extent. This lowers

254

Structure and Functional Properties of Colloidal Systems Cross model

Log viscosity

Williamson model Sisko model Power law model

Levelling sedimentation

Dip coating mixing and stirring

Lubrication spraying Log shear rate

FIGURE 11.2 Typical viscosity curve for a polymeric liquid. Includes relationship with physical operations and constitutive models.

their flow resistance, and, as a consequence, the shear viscosity decreases. Typical measurements of the viscosity functions of highly concentrated polymeric liquids show three ranges: A first Newtonian range, a shear-thinning region, and a second Newtonian range (Figure 11.2). Actually, nonconstant viscosities as a function of shear rate are usually the first sign that a fluid is structured. i. In the first range, at low shear, disentanglements and re-entanglements occur simultaneously. As a consequence, there is no significant change in flow resistance and viscosity is constant. In this region, at low concentration/molecular-weight, the viscosity is directly proportional to the concentration. However, at large concentrations/molecular-weight, entanglements appear and viscosity increases more rapidly with the concentration. The existence of the two regimes of concentration/molecular-weight dependence of shear viscosity is one of the strongest pieces of evidence for the existence of entanglements in polymers. ii. In the second range, the number of disentanglements becomes greater than those of reentanglements. In this case, the polymer shows a shear-thinning behavior and viscosity decreases. iii. In the third range, at high shear conditions, all macromolecules are fully oriented and disentangled. In this region viscosity is constant. Even though viscosity measurements and hence rotational techniques provide relevant information on the structure, oscillatory regime investigations are more frequently carried out since, in this case, the structure is only slightly perturbed. The most general overall G¢, G≤ response of structured polymeric liquids is shown in Figure 11.3. Terminal region

Rubbery plateau region

Transition region

log G′ ; log”G″

Glassy region

Viscous

Elastic

1 2

Zones relevant to polymer melt processing

Storage modulus, G′ Loss modulus, G″ Log frequency

FIGURE 11.3 Typical mechanical spectrum of a polymeric liquid. Shadow region corresponds to the most frequently observed one.

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A number of specific regions can often be differentiated; namely i. Terminal region: G≤ is linear with increasing frequency and G¢ is quadratic. The longest relaxation time is given by tmax = G¢/G≤w. ii. Rubbery or plateau region: Here elastic behavior dominates and we see what appears to be a flat plateau. For polymeric systems, the value of the molecular-weight of the chain segments between temporary entanglements, M, can be evaluated, since M = rRT/G¢ where R = 8.314 J mol-1 K-1 is the universal gas constant, r is the polymer density, and T is absolute temperature. iii. Transition region: Due to high frequency relaxation and dissipation mechanisms, the value of G≤ again rises this time faster than G¢. iv. Glassy region: At the highest frequencies, a glassy region is observed where G¢ predominates. In general, Maxwell model usually provides a good fit at low frequency; meanwhile, Kelvin– Voigt model is satisfactorily used at high frequencies. Figure 11.3 is general for most materials. However, depending on the longest relaxation time, only one or two regions are observed. Hence, for a given frequency range, the particular oscillatory region observed for polymer systems depends on their concentration and molecular-weight. In the case of dilute and/or low-molecular-weight polymer solutions (wtmax < 1), only the terminal region is observed. A selection of constitutive equations that have been compared to experiments are as follows: Oldroyd B model has been satisfactorily used for Boger fluids [41]. Giesekus model has been validated in reference [42]. K-BKZ has been used to explain rheological behavior of low-density polyethylene in reference [43] and bread-dough in reference [44]. White–Metzner model has been used in reference [45] to predict the nonlinear rheological behavior of asphalt. Temporary network model has been used to explain the elongational stress growth in low-density polyethylene [46]. Reptation theories have been used in reference [47] to investigate the oscillatory regime of polybutadiene.

11.6.2 DISPERSED SYSTEMS In this section, we will show some relevant experimental results for dispersions and compare them mainly to viscosity theoretical predictions. More detailed information on experimental results in the oscillatory regime as well is given in reference [5]. In the limit of infinite dilution (f < 0.01), Einstein derived the relationship between the viscosity of a dispersion of rigid solid particles, the volume fraction, and the viscosity of the continuous phase (Equation 11.35). Einstein’s result can be verified experimentally in the limit as f Æ 0. However, doing so is not trivial: Settling, migration, wall effects, and particle inertia can cause serious problems. Criterium for neglecting settling, inertia, and migration phenomena can be found in reference [7]. For dilute polymer lattices, good agreement is found between experiments and Einstein prediction [48–50]. As the volume fraction increases beyond 0.15, a rigorous hydrodynamic theory is not available as multiparticle interactions become important. The problem is therefore mostly treated empirically or using cell models, and there are a great many equations. Krieger and Dougherty [51] gave the following equation: fˆ Ê h = hc Á 1 fm ˜¯ Ë

-[ h]fm

. where fm = fm(g, f) is the “maximum packing fraction.”

, (11.78)

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Structure and Functional Properties of Colloidal Systems

In 1977, Quemada [52] derived Equation 11.78 with exponent -2 by applying the minimum energy dissipation principle. The exponent -2 was verified by several experiments and confirmed by the theoretical work of Brady [53] on the basis of statistical mechanics. Good agreement has been obtained as well when comparing theoretical results with experiments on intrinsic viscosity on rigid rod polymers and biological macromolecules [4,54]. Phan-Thien and Graham [55] measured the low shear viscosity for a range of spheroids and cylinders at different particle volume fractions. Shear viscosity of dispersions of rigid electrically charged particles is governed by Equation 11.40. This equation agrees with most experimental data. However, studies with sulfonated polystyrene latex particles gave results higher than theoretical ones [56]. In the case of electrostatically charged particles, Krieger and Eguiluz [57] studied the effect of ionic strength on the rheological properties of latex particles. The rheology of charge-stabilized silica suspensions is studied in reference [58] as a function of volume fraction, ionic strength, and continuous phase composition. If repulsion is strong enough, a lattice structure can be obtained resulting in a colloidal crystal. These materials are characterized by a frequency-independent storage modulus that strongly depends on repulsion forces and a yield stress. There are not many rheological investigations on diluted elastic or viscoelastic particle-based colloids. Most of the works are related to microgel particles that in some cases are also responsive to external stimuli. The degree of cross-linking in microgels is expected to be central in how closely the solution properties resemble those of rigid solid particles versus those of linear polymers. For very low crosslink densities, the microgels would be expected to behave like very high-molecularweight linear polymers, while at high crosslink densities they would be expected to be rigid and impenetrable, similar to hard sphere dispersions. In 1989, Wolfe and Scopazzi [59] studied the effect of degree of cross-linking in polymethylmethacrylate microgels on rheological performance. A year later, Evans and Lips [60] reported measurements on several grades of Sephadex beadlets with varying degrees of cross-linking. Swellable microgels were investigated by Wolfe [61]. A rheological characterization of thermo-responsive microgels was carried out by Kiminta et al. [62]. The rheological investigation of suspensions of spherical microgel particles is carried out over a wide concentration range in reference [63]. The low shear viscosity and dynamic moduli undergo a strong transition when the concentration is such that the particles are closely packed. Snabre and Mills [64] proposed a microrheological model to estimate the steady shear viscosity of concentrated suspensions of viscoelastic particles using a Kelvin–Voigt model. Experimental results on red cell suspensions were in agreement with theoretical predictions. The low-shear viscosity of polyelectrolyte microgels is studied as a function of concentration, crosslink density, and ionic strength in reference [65]. A careful investigation of the cross-linked density on the rheology of methacrylic acid-ethyl acrylate cross linked with ai-allyl phthalate is examined in reference [66]. The size and structural characteristics of polyacrylamide microgels are investigated using rheological methods in reference [67]. Measurements of the first normal stress difference show that increasing the microgel crosslinked density affects the viscosity more than its elasticity. A comprehensive swelling model accounting for the effects of added salt has been recently applied to predict water fraction profiles in COOH-functionalized microgels based on PNIPAM [68]. In the case of emulsions, Equation 11.46 has been satisfactorily tested in reference [69] using a capillary viscometer on dilute O/W emulsions. Vinckier et al. [70] focused on normal stress difference investigations in semidilute emulsions using rather viscous phases. Choi–Schowalter model satisfactorily explains their experimental results [71]. Recently, the viscous behavior of multiple emulsions has been investigated by Pal [72]. Expressions are derived for the viscosity of dilute and concentrated multiple emulsions. The dynamic oscillatory regime of concentrated emulsions having controllable deformability has been recently investigated in reference [73]. Steady shear rheology of a dilute emulsion with viscoelastic inclusions has been numerically investigated using direct numerical simulations in reference [74]. Viscoelasticity is modeled using Oldroyd-B

257

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constitutive equation. For a recent review on emulsion rheology including theoretical and experimental studies read [75]. In 2002, Rust and Manga [76] investigated dilute and surfactant-free bubble suspensions in simple shear using a rotating cylinder Couette rheometer at a Capillary number close to unity. The rheological behavior of dilute polyol-based bubble suspensions is investigated in reference [77]. At high capillary number, viscosity increases as the gas volume fraction increases, while at low capillary number, viscosity decreases as the gas volume fraction increases. Ichihara et al. [78] investigated how the acoustic properties of a liquid-bubble mixture depend on liquid rheology. Muller et al. [79] showed that multilamellar vesicles with stiff shells can generate yield stresses in the fluid and hence improve stability. The main classic theory on the oscillatory rheology of dispersions of thin-walled capsules is due to Oldroyd [80]. In a latter paper, apart from the interfacial tension, Oldroyd introduced surface shear viscosity, surface shear elasticity, dilatational viscosity, and dilatational elasticity [33]. Kattige and Rowley investigated the capsule-filling properties of lactose/poloxamer dispersions in hard gelatin capsules using rheological techniques [81,82]. Recently, Zhang and coworkers [83] developed an immersed boundary lattice Boltzmann approach to simulate deformable capsules in flows. As a first approximation, theories for hard spheres have been frequently adapted in the case of core–shell particle dispersions. Results are found to be in good agreement with experiments. The hydrodynamic volume of each particle is increased by the volume of the adsorbed layer so that the effective volume fraction f¢ is 3

dˆ Ê f¢ = f Á 1 + ˜ , a¯ Ë

(11.79)

where d is the thickness of the layer. This equation has been tested for polymeric stabilized particles in reference [84]. Polydimethylsiloxane (PDMS) coated silica particles are investigated in oscillatory shear in reference [85]. In particular, the high frequency elastic modulus was determined. Deike and Ballauff [86] studied the flow properties of core–shell particles -poly(styrene)/poly(N-isopropylacrylamide) (PNIPA). The shear viscosity in the limit of dilute dispersions is modeled in terms of an effective hydrodynamic radius. The phenomenon of shear thickening has been investigated in sterically stabilized colloidal suspensions in reference [87]. A detailed investigation on the interaction forces between particles containing grafted or absorbed polymer layers have been investigated using rheological and surface force apparatus (SFA) measurements in reference [88]. A comprehensive experimental study of the dynamics and rheology of concentrated aqueous dispersions of poly(ethylene glycol)-grafted colloidal spheres is reported in reference [89]. At high concentration, a glass transition is observed. Recently, Nakamura and Tachi [90] studied the rheological behavior and microstructure of shear-thinning suspensions of core–shell structured carboxylated latex particles. The steady shear viscosity of the suspension was found to increase with increasing dissociation of the carboxyl groups or increasing particle concentration. Hybrid magnetic particles of carbonyl iron/ poly(vinyl butyral) with core/shell microstructure were prepared in order to enhance the dispersion stability of the magnetorheological fluids in reference [91]. To conclude, a brief review on the state-of-art of the rheology of structured fluids has been carried out giving special emphasis on the structure-rheology correlation. Two systems have been investigated: polymeric liquids and dispersions. After a brief introduction on fluid mechanics, material functions and standard flows are explained. Then, existing theories are summarized and compared with experimental results. Even though current researchers continue searching for new constitutive equations, those reported in this chapter serve as a base for future developments.

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ACKNOWLEDGMENTS This work was supported by MEC MAT 2006-13646-C03-03 project (Spain) by the European Regional Development Fund (ERDF) and by Junta de Andalucía P07-FQM-2496, P07-FQM-03099, and P07-FQM-02517 projects (Spain).

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Part III Functional Materials

12

Surface Functionalization of Latex Particles Ainara Imaz, Jose Ramos, and Jacqueline Forcada

CONTENTS 12.1 Introduction ...................................................................................................................... 12.2 Physical Adsorption .......................................................................................................... 12.2.1 In situ Physical Surface Functionalization of Latex Particles .............................. 12.2.1.1 Modification of the Colloidal Stability of the Latex Particles ............... 12.2.1.2 Avoiding the Adsorption of Biological Compounds to Hydrophobic Surfaces ............................................................................ 12.2.2 Posttreatment for Physical Surface Functionalization of Latex Particles ............. 12.3 “Attaching to” Surface Functionalization ......................................................................... 12.3.1 Emulsion Homopolymerization, Emulsion Copolymerization, and Other Polymerizations in Dispersed Media ................................................... 12.3.2 Seeded Emulsion Copolymerizations to Produce Functionalized Latexes .......... 12.3.3 Surface Modification of Preformed Latexes ......................................................... 12.3.4 “Click Chemistry” for Surface Functionalization ................................................ 12.4 “Attaching From” Surface Functionalization ................................................................... 12.4.1 “Attaching From” by Conventional RP ................................................................ 12.4.2 “Attaching From” by CRP .................................................................................... 12.4.2.1 Nitroxide-Mediated Radical Polymerization ......................................... 12.4.2.2 Atom Transfer Radical Polymerization ................................................. 12.4.2.3 Reversible Addition-Fragmentation Chain Transfer Polymerization ......................................................................... Acknowledgment ....................................................................................................................... References ..................................................................................................................................

263 265 265 265 268 269 270 270 273 275 276 277 277 277 277 278 279 279 280

12.1 INTRODUCTION In the last two decades, there has been a huge amount of work reporting the synthesis and characterization of polymer latex particles having different functionalized surface groups useful in a large variety of applications, from biomedicals to photonic crystals. As can be observed in specialized literature, there are increasing investigations focused on the modification of polymer surfaces with the aim of imparting special properties, such as size, shape, and surface functionalization. The selection of the functionality on the surface of the latex particle depends on the final application (solid-phase support, delivery, recognition, transport, etc.) and properties of the latex particles required. By surface modification of nanoparticles, it is possible to promote the stability of the particles in dispersed media, to couple biomaterials for biological purpose, to alter hydrophilicity of 263

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particle surface to hydrophobicity and the other way around, and to obtain functional nanoparticles for biological, biochemical, and medical applications and for membrane treatment. Among the huge possibilities, Pichot [1] gathered a summary of some available functionalities and related properties, which can be installed to latex particles. As mentioned before, there are three main possibilities or ways to obtain the surface functionality: physical adsorption, “attaching to” surface, and “attaching from” surface. Physical adsorption refers to attraction and bonding of the adsorbate onto a surface and may result from physical interaction with the surface. In physical adsorption, the forces involved are intermolecular ones, such as van der Waals forces. Certain specific molecular interactions arise from particular geometrical or electronic properties of the adsorbent and physical adsorption of long polymer chains to the surface can help in attaching polymer chains of well-defined length, molecular weight distribution, and composition. Taking into account the former classification, the second way to confer functionality is known as “attaching to” surface; this option consists of covalently bonded surface groups attached to the surface of the latex particles by means of a radical-free initiated polymerization reaction, carried out in one or various steps, between a main monomer and a comonomer that confers the end-functionalized surface group. This functionalization procedure is a powerful tool in the synthesis of latex particles useful in biomedical applications. In this case, the synthesis strategies are directed to the production of polymeric colloids having surface-functional groups able to form oriented bonds with different biomolecules. As an example related to this, various reports [2–4] indicate that immunoreagents produced by using functionalized latex particles are more stable if they are able to bind proteins covalently due to the stability of the chemical attachment over time. Some of the surface groups more widely used to attach covalently amino acid protein residues are chloromethyl, acetal, aldehyde, carboxyl, and amino groups. Different approaches are used to prepare polymer particles with “attaching to” surface-functionalized groups. In majority of the cases, they consist of step-batch or -semibatch polymerizations in dispersed media, being among them emulsion polymerization (emulsifier-free or not) the most used polymerization process: (i) emulsion homopolymerization of a monomer containing the desired functional group (functionalized monomer), (ii) emulsion copolymerization of styrene (usually) with the functionalized monomer, (iii) seeded copolymerization to produce composite functionalized latexes, and (iv) surface modification of preformed latexes. Cases (i) and (ii) can be analyzed together with other polymerization techniques in dispersed media (miniemulsion, microemulsion, dispersion, etc.) used to produce the desired functionalized latex particles and taking into consideration the reactor type used to carry out the different polymerization reactions. The third way to functionalize the surface of latex particles is the “attaching from” procedure. In this case, grafting polymer chains with single points of attachment at one end to a surface can produce “polymer brushes,” chains of which are extended by virtue of being separated by less than their unconstrained diameter. The “attaching from” technique, in which polymerization is initiated from initiators coupled covalently to the surface, has been used extensively for the controlled synthesis of high-density polymer brushes. It offers several advantages over the “attaching to” approach, in which end-functionalized polymer molecules react with an appropriately treated surface to form tethered chains: First, the reduction of preparative steps, excluding the ex situ preparation and isolation of the macromolecular material. A second advantage is the small influence of dimensions and mutual sterical constrains of the grafting material on the polymer surface density; this on the contrary depends on the initial surface density of the initiating groups. On the other hand, “attaching on” structures are generally better defined and characterized, because they can be first isolated and purified, then grafted. This is a potential advantage that can fade by the application of “attaching from” living polymerization techniques, which should provide structures with low-molecular-weight dispersion and end-functionalization (hence the possibility of block copolymerization).

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The chemistry of “attaching from” latex particles is a relatively new but developing area. Conventional radical polymerization (RP) is inefficient in this application due to the unwanted radical coupling or disproportionation, chain termination that occurs at diffusion-controlled rates, and difficulty in controlling film thickness and grafting density. However, the alternative techniques of controlled/living radical polymerizations (CRP) ought to allow for more precise manipulation of surface properties of grafted latex particles. Using an “attaching from” approach with CRP enables the preparation of particles with high grafting density, controlled graft structure/ composition, and applicability to different monomers. Most recent research activities in this area have been focused on surface modification of latex particles via atom transfer radical polymerization (ATRP), reversible addition-fragmentation transfer (RAFT), and nitroxide-mediated radical polymerization (NMRP).

12.2 PHYSICAL ADSORPTION Adsorption is a consequence of surface energy; atoms in the surface of adsorbent are not wholly surrounded by other adsorbent atoms and can attract adsorbates. The physical adsorption can be described through adsorption isotherms, that is, the amount of adsorbate on the adsorbent as a function of concentration at constant temperature. The equilibrium is established between the adsorbate and the fluid phase. There are some common adsorption isotherms, such as linear, Freundlich, Langmuir, and Brunauer–Emmett–Teller (BET) isotherms. In linear isotherms, the linear relationship between adsorbed amount and concentration of adsorbate at equilibrium is observed, whereas Freunlich isotherm exhibits increasing adsorption with increasing concentration. In Langmuir isotherm, the adsorption increases to a maximum value. It is based on four assumptions; they are the surface of the adsorbent is uniform, adsorbed molecules do not interact, all adsorption occurs through the same mechanism, and at the maximum adsorption only a monolayer is formed. The last assumption is seldom true and is addressed by the BET isotherm, which describes multilayer adsorption. There are two synthesis methods for physical surface functionalization of latex particles. In the first approach, the surface modifier is directly added to the reaction mixture (in situ). This approach is used to extend the latex particle application by using a predesigned molecule or macromolecule that plays a role in the particle-formation process and also in the particle surface functionalization. The second method includes two steps (the synthesis of the latex particles and a posttreatment) since the latex particle is functionalized once the nanoparticle is formed.

12.2.1 IN sITU PHYSICAL SURFACE FUNCTIONALIZATION OF LATEX PARTICLES 12.2.1.1 Modification of the Colloidal Stability of the Latex Particles Attaching molecular chains onto the surface may alter the properties of the latex particles. One of the key properties for the performance and the storage of the latex present in paints, food product, cosmetics, medicines, and so on, is their colloidal stability. There are huge amounts of molecular chains that can control the particle stability and sometimes allow for further functionalization. These molecular chains can be termed as surfactants and they are classified as ionic (anionic or cationic) or nonionic surfactants depending on whether they have or have not ionic groups in the structure and as low-molecular-weight or polymeric surfactants as a function of the molecular weight. Surfactants have widespread industrial and technological applications. Most of their applications in the colloidal science are related to the adsorption of them at the solid–liquid interface, and with their effect on the colloidal stability. When the surfactant is adsorbed in the appropriate amount and orientation, it can produce the aggregation or stabilization of the system. Important factors to maintain the stability of nanoparticles are electrostatic interaction, steric repulsion, interfacial tension, and viscosity of the outer phase. Under equilibrium conditions, the surfactant concentrations in the aqueous phase and at the polymer–water interface are uniquely related by the adsorption isotherm [5]. Various methods

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have been proposed to obtain adsorption isotherms of surfactants on latex particles. Usually, such methods rely on the monitoring of the equilibrium concentration of free surfactant in the aqueous phase. For example, Paxton [6] measured the adsorption of sodium dodecylbenzenesulfonate (SDBS) on polystyrene (PS) and poly(methyl methacrylate) latexes from surface tension titration curves. By using the same method, Vale and Mckenna [7] calculated the adsorption of sodium dodecylsulphate (SDS) and SDBS on poly(vinyl chloride) latexes. Turner et al. [8] performed the SDS adsorption on flat PS surfaces by neutron reflection and attenuated total reflection infrared spectroscopy, whereas Sefcik et al. [9] via ion chromatography analyzed the same system. A different technique was used by Romero-Cano et al. [10], determining the adsorption experiments by UV spectrophotometry. Apart from the analysis of adsorption isotherms, surfactant adsorption onto particle surfaces has been investigated through particle electrophoresis [11,12], dynamic light scattering [13], atomic force microscopy [14], neutron reflection experiments [15], and sedimentation field flow fractionation (SdFFF) [16]. For full characterization of the process of adsorption, it is necessary to know the amount of polymer adsorbed per unit area of the surface, the fraction of segments in close contact with the surface, and the distribution of polymer segments. Nonionic surfactants or polymers adsorbed onto latex particles provide a layer to the particles that act as steric stabilizers, whereas ionic surfactants adsorbed cause electrostatic stabilization through repulsion of the charged surfaces. One advantage of using nonionic surfactants is minimized sensitivity to screening electrolyte. Depending on the amphiphilicity of the low-molecular-weight surfactants, which is controlled by variation of the length of the alkyl chains, the supramolecular order can be controlled. The amphiphilic character guaranteed their adsorption onto surfaces and their stabilizing properties. Attaching molecular chains onto the surface may alter the properties of the colloidal dispersions. An example of a low-molecular-weight surfactant is a nonionic surfactant C12E7 and Zhao and Brown [17] investigated its adsorption to carboxylated styrene-butadiene copolymer latex particles at different ionic strengths. At high ionic strength, hydrophobic interactions dominate the process due to the “salting-out” role of the added salt. Cosgrove et al. [18] analyzed the effect of the SDS on the adsorption properties of the homopolymer poly(ethylene oxide) (PEO) at the solid– liquid interface using photon correlation spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, and small-angle neutron scattering (SANS). Porcel et al. [19] tried to study the aggregation of colloidal particles when two surfactants (Triton X-100 and SDS) are adsorbed below the critical micelle concentration (CMC). It is commonly accepted in the field of surfactant science that mixtures of surfactants often perform better than the individual components. Later, Jódar-Reyes et al. [20] studied the adsorption of low-molecular-weight molecules onto PS latex [an anionic (SDBS), a cationic (dodecyldimethyl-2-phenoxyethyl ammonium bromide, DB), and two nonionic surfactants (Triton X-100 and Triton X-405)]. They concluded that the hydrophobic attraction between the nonpolar part of the molecule and the apolar regions of the surface was the main mechanism involved in the adsorption. Lee and Harris [21] modified the surface of monodisperse magnetic nanoparticles with oleic acid by coordinating their carboxyl end groups on the magnetic nanoparticle surface. The hydrophobic chains on the surface produced stable suspensions only in apolar organic solvents. Recently, Klapper et al. [22] found that the potassium salts of alkoxyisophthalic acids substituted with alkyl chains longer than dodecyl showed improved properties as anionic surfactants in comparison with the industrially applied SDS. The improved stabilization is attributed to the two-carboxylate groups on each surfactant molecule, which increases the surface charge of the final polymer particles. However, these low-molecular-weight surfactants are only efficient at high concentrations. For this reason, polymeric surfactants (homopolymers and copolymers) have been widely used for stabilization of dispersions and they usually have hydrophilic and hydrophobic regions so that they can act as low-molecular-weight surfactants. The process of polymer adsorption involves a number of various interactions, which are the interaction of the solvent molecules with the oil in the case of O/W emulsions, the interaction

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Surface Functionalization of Latex Particles Tail

Train

Loop Homopolymer

Small loops

Tail

Diblock copolymer A–B

FIGURE 12.1

Random copolymer

Triblock copolymer A–B–A

Graft copolymer BAn

Typical configuration of an adsorbed polymer at the surface of the latex particle.

between the chains and the solvent, and the interaction between the polymer and the surface. Apart from these interactions, one of the most fundamental considerations is the conformation of the polymer molecules at the interface. A typical configuration (Figure 12.1) of an adsorbed polymer at the surface presents trains (polymer parts that are bound to the substrate), loops (chain sections that are not in direct contact with the surface), and tails (the dangling ends of the chains). Polymeric surfactants can be homopolymers, random amphiphilic copolymers, and of the A–B (diblock), A–B–A (triblock), and BAn (graft) types [23]. The A chain is referred to as the stabilizing chain (soluble in the medium), and the B chain is referred to as the anchor chain (insoluble in the medium with strong affinity to the surface). The simplest type of a polymeric surfactant is a homopolymer, such as PEO and poly(vinyl pyrrolidone) (PVP). Homopolymers are not the most suitable surfactants and it is better to use polymers with some groups that have affinity to the surface. The most employed copolymers are random amphiphilic copolymers, like poly(vinyl alcohol), diblocks of polystyrene-block-poly(vinyl alcohol) (PS-b-PVA), poly(ethylene oxide)-block-polystyrene (PEOb-PS), and triblocks of poly(ethylene oxide)-block-poly(propylene oxide)-block-poly(ethylene oxide) (PEO-b-PPO-b-PEO, Pluronic) (PPO resides at the hydrophobic surface, leaving the two PEO chains dangling in aqueous solution), and poly(ethylene oxide)-block-polystyrene-block-poly(ethylene oxide) (PEO-b-PS-b-PEO). The graft copolymer is referred to as a comb stabilizer: Atlox 4913,

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Hypermer CF-6, and Inulin. In general, the abilities of polymeric surfactants to decrease surface and interfacial tension are much lower than those of the low-molecular-weight surfactants. Block copolymers exhibit low CMC and lower diffusion coefficient with respect to classical surfactant. Triblock copolymers are much more efficient than the diblock copolymers with the same composition and molecular weight [24]. Adsorption of carbohydrate-based polymers has also been investigated. Usually, these polymers are modified before using as surfactant to provide them with an amphiphilic character. Karlberg et al. [25] presented hydrophobically modified ethyl(hydroxyethyl)cellulose (HM-EHEC), which is used to stabilize O/W macroemulsions with olive oil as the dispersed phase. Akiyama et al. [26] developed the water-soluble amphiphilic polymer hydrophobically hydrophilically modified hydroxyethylcellulose (HHM-HEC): stearyl alkyl chain-bearing and sulfonic acid saltbearing HEC. More examples of water-soluble amphiphilic polymers (hydrophobic chains grafted to a hydrophilic backbone polymer) include cholesteryl-bearing pullulan and polysaccharides with sulfate groups. Moreover, Tadros et al. [23] studied the stabilization of emulsions using hydrophobically modified inulin. In this case, the alkyl groups provide the anchor points leaving loops of polyfructose dangling in solution enhancing the steric stabilization. In addition, the polyfructose loops are strongly hydrated and remain so in high electrolyte concentrations and temperature. 12.2.1.2 Avoiding the Adsorption of Biological Compounds to Hydrophobic Surfaces Nanospheres, nanocapsules, liposomes, micelles, and other nanoparticles can frequently act as carriers for delivery of therapeutic and diagnostic agents [27]. It is a common observation that in a variety of blood handling procedures, biological compounds [28,29] (proteins, peptides, cells, and nucleic acids) adsorb to the surfaces of hydrophobic nanocarriers, which is undesirable. As a result of this consideration, surface modification of these carriers is often used in order to increase longevity and stability of nanocarriers in the circulation, to change their biodistribution, and/or to achieve targeting effect. A nanocarrier, to achieve an increase in quality of life, must be present in the bloodstream long enough to reach or recognize their target. However, the opsonization or removal of nanocarriers from the body by the mononuclear phagocytic system (MPS) is a major obstacle to the realization of these goals. The opsonization of hydrophobic particles, as compared to hydrophilic particles, has been shown to occur more quickly due to the enhanced adsorption of blood serum proteins on these surfaces. Therefore, a widely used method to slow opsonization is the use of small particles with surface adsorbed or grafted shielding groups that can block the electrostatic and hydrophobic interactions. These groups tend to be long hydrophilic polymer chains and nonionic surfactants having a desirable hydrophilic-lypophilic balance number with at least one hydrophilic element that extends into an aqueous phase [27,30]: polysaccharides, polyacrylamides, poly(vinyl alcohol), poly(N-vinyl-2-pyrrolidone), poly(ethylene glycol) (PEG), and PEG-containing copolymers. For example, Simon et al. [31] studied the adsorption characteristics of a series of model amphiphilic polysaccharides (hydrophobically modified carboxymethylpullulans, HM-CMP) onto PS latex particles. They concluded that the difference between the level of adsorption and the layer thickness of HM-CMP differing by their degree of modification of octyl groups was due to the competition between the number of anchors and the size of the loops. Blunk et al. [32] used PS beads as model carriers and their surfaces were modified by adsorption of amphiphilic block copolymers of ethylene oxide and propylene oxide (Poloxamers 184, 188, and 407). Esmaeili et al. [33] modified the surface of rhodamine B isothiocyanate (RBITC)-loaded particles using either PEG or block copolymer of ethylene oxide and propylene oxide (Poloxamer 407). Dellacherie et al. used a hydrophobically modified dextrans (phenoxy groups on dextran, DexP) [34] to functionalize biocompatible poly(lactic acid) (PLA) nanoparticles [35] to improve plasma lifetime. PLA nanospheres have very low plasma lifetime because of their rapid capture by the MPS cells.

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12.2.2 POSTTREATMENT FOR PHYSICAL SURFACE FUNCTIONALIZATION OF LATEX PARTICLES Surface modification should provide good colloidal stability, biocompatibility, and functionality for the potential attachment of biorecognizable molecules. Among numerous applications of nanoparticles in medicine, one of the most important is cancer treatment. Cancer cells express specific antigens and also have surface folate receptors accessible from the circulatory system. As no healthy cells have these characteristic, active targeting can be achieved by immobilizing antibodies on the nanoparticle surface that interact with these macromolecules [36]. Hyperthermia is also a promising approach to cancer therapy, which can be achieved by using magnetic nanoparticles. An increase of temperature to 41–46ºC due to the generation of heat by applying external alternating magnetic field kills tumor cells. Wong et al. [37] combined the magnetic property of magnetic nanoparticles with the thermoresponsive property of some polymers [poly(N-isopropylacrylamide), PNIPAM] to obtain dual-stimuli hybrid core–shell structures. In the synthesis, the surface of the microgel was first modified via the layer-by-layer (LbL) assembly [38,39] of polyelectrolyte multilayer. Electrostatic interactions and the secondary or cooperative interactions such as hydrogen bonding and hydrophobic interactions play a role in polyelectrolyte multilayer. Then, positively charged magnetic particles were added to the positively terminated surface-modified microgels. The LbL technique is a different way to modify the surface of latex particles. The materials produced by this method can be used in different biotechnological applications. Goldenberg et al. [40] reported a controlled consecutive LbL deposition with polyallylamine hydrochloride (positively charged) and PS sulfonate (negatively charged) polyelectrolytes with the objective of a fast and easy modification of particle surface properties in order to prepare ordered arrays with better array building properties as well as to introduce additional particle functionality. On the other hand, Trimaille et al. [41] described the synthesis of an antigen delivery system, that is poly(d,l-lactic acid) (PA)-based DNA carriers by the emulsification-diffusion method in the presence of Pluronic F68 surfactant. The surface functionalization of PLA by the LbL approach was carried out to produce PLA-based DNA vectors for vaccination purposes. In this case, PLA particles were surface modified by adsorbing cationic poly(ethylenimine) (PEI) through electrostatic interactions. Highly cationized latex particles can interact with plasmid DNA under a compacted conformation, and the amount of plasmid immobilized depended on the pH of the medium and on the conformation of the adsorbed PEI. Reb et al. [42] presented the synthesis of latex particles by using isophthalic acid derivatives as anionic surfactants. The bifunctionality present at the surface of the particles allows for a surface functionalization of the final latex particles with dyes or oligopeptides for different medical or biological applications. As can be seen, a huge amount of surface modifiers can be used for physical functionalization, such as molecules, synthetic homopolymers and copolymers, natural polymers, and so on. Table 12.1 includes examples of molecules and macromolecules that can be used for this purpose. The use of physical adsorption for surface modification has limitations of steric hindrance due to the long polymer chains comprising the final surface density of the grafts and end groups. Moreover, in this case, polymer chains are only physically adsorbed onto the surface and freely desorbed from the surface of the latex particle, decreasing the stability of the dispersion. In the case of surfactants, their migration tendency leads to an accumulation at the surface during film formation, which results in deterioration of the film properties (loss of transparency and brittleness). Moreover, when practical industrial emulsions contain high amounts of electrolyte, ionic surfactants may not be strongly adsorbed on the droplets surface and desorption may take place. For these reasons, methods have been proposed to improve the final latex stability, such as covalent binding of the emulsifier to the polymer chain during the polymerization process [43,44]. Another disadvantage of physical adsorption of polymers is their polydispersity in molecular weight and composition; it is difficult to establish correlation between the molecular characteristics of these polymers and their behavior in emulsion polymerization. The rapid progress in the field of nanotechnology has focused on elaboration of various alternative approaches to physical adsorption for surface functionalization of latex particles, such as “attaching to” surface or “attaching from” surface procedures.

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TABLE 12.1 Examples of Molecules and Macromolecules Used for Surface Modification Molecule/Macromolecule Anionic

Alkoxyisophthalic acids substituted with alkyl chains Isophthalic acid Oleic acid SDBS SDS

Cationic

DB Dodecyltrimethylammonium bromide (DTAB) PEI

Nonionic

C12E7 Ethylhydroxyethyl cellulose Hypermer Inulin PEO PVP PEO-PPO PEO-PPO-PEO (Pluronic) PEO-PS-PEO PS-PVOH Triton X-100 Triton X-405 Poly(vinyl alcohol) Polyglycerol Polysaccharides Polyacrylamides PEG

Modified

Dextrans HM-CMP HM-EHEC HEC

Notes: PEO-PPO = Polyethylene oxide-polypropylene oxide; PEO-PPO-PEO = Polyethylene oxidepolypropylene oxide-polyethylene oxide; PEO-PS-PEO = Polyethylene oxide-polystyrenepolyethylene oxide; PS-PVOH = Polystyrene-polyvinylalcohol.

12.3

“ATTACHING TO” SURFACE FUNCTIONALIZATION

12.3.1 EMULSION HOMOPOLYMERIZATION, EMULSION COPOLYMERIZATION, AND OTHER POLYMERIZATIONS IN DISPERSED MEDIA There are two polymerization processes widely used to produce “attaching to” functionalized latex particles: emulsion homopolymerization of a monomer containing the desired functional group (functionalized monomer) and emulsion copolymerization of styrene (usually) with the functionalized monomer. The following paragraphs present some of the more relevant contributions in this field.

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El-Aasser et al. [45] presented the preparation and characterization of PS latexes with the ionic comonomer 2-acrylamido-2-methyl propane sulfonic acid, reporting the presence of strong acid and carboxylic groups when the pH was not controlled. Moreover, the surface charge and the colloidal stability increased with increasing functional monomer concentration. In 1983, the group of Lehigh reported [46] the synthesis latex particles based on poly(vinyl acetate), poly(butyl acrylate), and the corresponding copolymer by using batch and semicontinuous reactors. The surface and colloidal properties of the latexes were compared. They found that the total weak and strong acid surface groups for the semicontinuous latexes were higher, and more dependent on the composition than in the case of the batch latexes. Acid-induced hydrolysis affected the type and concentration of the surface groups of the latex particles produced in a semicontinuous reactor. The colloidal stability against electrolytes was analyzed in terms of electrostatic (due to the surface acid groups) and steric contributions (due to surface poly(vinyl alcohol). The different reactivity ratios and water solubilities of the comonomers were used to justify the results obtained. Pichot presented a review [1] on surface-functionalized latexes for biotechnological applications in which revise the most widely used techniques to produce them in heterogeneous media, especially in aqueous medium, allowing to synthesize a wide variety of polymer particles having different particle size, composition, and morphology together with a broad selection of functionalities that can be installed to the particle interface due to both the availability of a large amount of functional molecular and macromolecular species and versatility of manufacturing protocols. In this work, basic considerations and requirements on surface-functionalized latex particles for biotechnical applications are revised and the more outstanding work of Pichot’s and Elaissari’s group on functionalized latex particles is cited. Several functional groups were installed on the surface of polymeric particles by homo- and/or copolymerization processes, and using the “attaching to” procedure and different investigations were reported. The list of works is large. The following but not excluding ones are commented as an example of the variety and interest on explaining colloidal behaviors, stability mechanisms, applications, and so on. The latex stability characteristics related to surface chemistry were analyzed by Polatajko-Lobos and Xanthopoulo [47] by studying the relationships between the concentration of surface-bound functional groups on carboxylated styrene-butadiene copolymer latex particles and the mechanical and chemical stability of the latexes synthesized. The control of the properties of polymers obtained by emulsion polymerization of ethyl acrylate by incorporating small amounts of functional groups or varying the conditions of polymerization is reported by Eliseeva [48] in 1979. The neutrophilic response to PS microspheres bearing defined surface groups was analyzed [49] by using chemiluminiscence and phagocytosis parameters for particles having carboxyl, hydroxyl, and amino groups. The results show that in protein-free systems, hydrophobic particles are more readily phagocytosed. Elimelech and O’Melia [50] reported electrophoretic mobility studies for surfactant-free PS latex particles carrying sulfate functional groups and covering a wide range of surface charge in various types of inorganic electrolytes. The results suggested that the increase of mobility with salt concentration might be attributed to the approach of coions close to the hydrophobic surface of the particles. Syntheses of different types of latex particles by emulsion polymerization that differ in particle size, polymer hydrophilicity, and surface coverage with functional groups were presented by Paulke et al. [51]. The particles were equipped with intensive fluorescence. Concentrated particle suspensions were injected into the brain tissue of mice and the effect of two kinds of beads is shown in brain sections. The same research group [52] presented a very different work on electrophoretic three-dimensional (3D)-mobility profiles of latex particles with different surface groups. In particular, hydroxyl functions were studied in different surroundings. The latexes gave model colloids with different electrophoretic behavior in comparison with classical anionic monodisperse PS latex particles.

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Surface charge on functionalized latex particles having carboxyl groups and additionally ionizable groups were used [53] to study the behavior of uniformly charged and zwitterionic colloids in electrolytes of different ionic strength. The swelling equilibrium of small polymer colloids (diameters of 30–100 nm) with respect to the surface structure by analyzing different types of covalently bound surface stabilizing groups was studied by Antonietti et al. [54] in 1996. The emulsion copolymerization of styrene and methacrylic acid in the presence of a new polymerizable macromonomer based on PEG as stabilizer is proposed by Tuncel and Serpen [55] to obtain larger and more surface-carboxyl-charged monodisperse particles relative to those obtained by the same emulsifier-free emulsion copolymerization. Related to works devoted to the surface functionalization by emulsion polymerization, nowadays, the manufacturing of surface-modified latex particles useful for self-assembling into thin films exhibiting properties of photonic crystals is of great interest. The ability of monodisperse carboxylated particles of poly(methyl methacrylate) and of styrene copolymers with glycidyl methacrylate or with methacrylic acid for self-assembling forming close-packed 3D-ordered arrays on glass slides, was analyzed recently by Menshikova et al. [56]. By using an emulsifier-free emulsion process, Reese and Asher [57] produced also highly charged monodisperse particles with sulfate and carboxylic acid groups for readily self-assemble into robust 3D-ordered crystalline colloidal array photonic crystals that Bragg diffract light in the near infrared spectral region: photonic crystals. The influence on protein adsorption on PS model nanoparticles to be intravenously administrated was analyzed [58] with respect to the surface characteristics. For that, latex nanoparticles were synthesized with different basic and acidic functional groups. Possible correlations between the surface characteristics and the protein adsorption from human serum are shown and discussed. Novel thymine-functionalized PS particles for potential applications in biotechnology were presented by Dahman et al. [59] in 2003. The structure of the functional group coming from the 1-(vinylbenzyl)thymine monomer with a benzene ring spacer group between the thymine and the polymer backbone gives more flexibility to the functional group. With the same idea referred to the potential of functionalized latex particles to bind biologically active species, D’Agosto et al. [60] presented the synthesis of latexes bearing hydrophilic grafted hairs RAFT-synthesized (see later in Section 12.4 the characteristics of this living (controlled) radical polymerization) with controlled chain length and functionality. Amino-functionalized latexes were synthesized by free-radical emulsion copolymerization of styrene and aminoethylmethacrylate hydrochloride comonomer in a surfactant-free process with a water-soluble azo initiator (V-50). The grafting of the hydrophilic hair onto latexes was implemented by the reaction of the carboxylic acid chain end of poly(acryloylmorpholine) with the amine surface groups. The use of other polymerization techniques in dispersed media, but different from emulsion polymerization, is also introduced in the recent literature about functionalized polymer particles. Among them there are miniemulsion, microemulsion, and dispersion polymerizations. Nanosized PS latexes in the range of 1.0–3.0 mm were prepared [61] by dispersion polymerization of styrene in isopropanol water media using poly(acrylic acid) (PAA) as a steric stabilizer and 2,2-azobis isobutyronitrile (AIBN) as initiator. Styrene/acrylate monomers, acrylic acid, 2-hydroxyethyl methacrylate (HEMA), and dimethylaminoethyl methacrylate were copolymerized onto one of the previously synthesized latex particles to obtain the different surface functionalities. Microemulsion polymerization was used to synthesize ultrafine (diameters of 10–100 and 20–120 nm) latex particles with narrow particle size distribution and controlled size and surface [62,63]. Using relative amounts of polymeric surfactant with respect to styrene controlled the particle size. The particle surface was easily modified by addition of functional comonomers or additives incorporated in the interface. The particles synthesized can be used to prepare a material with the ability of selective ion binding. Miniemulsion polymerization of styrene and n-butyl methacrylate was used by Pich et al. [64] to prepare polymeric particles by using a fluorinated comonomer that acts as a surfmer providing

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efficient stability and narrow particle size distribution. Blends of fluorinated latexes with styrenebutadiene copolymer latex provided more hydrophobic surfaces than similar latex films, where particles prepared by polymerization of expensive fluorinated monomer have been applied. Fluorescent carboxyl and amino-functionalized PS particles useful as markers for cells were prepared by miniemulsion polymerization [65]. A fluorescent dye was incorporated into the copolymer nanoparticles based on styrene and acrylic acid or styrene and amino-ethylmethacrylate hydrochloride. The cell uptake was visualized using fluorescence microscopy. Silanol-functionalized latex nanoparticles were prepared [66] by means of miniemulsion coplymerization of styrene and gamma-methacryloxypropyltrimethoxysilane as the functional comonomer and AIBN as the initiator at neutral conditions. The results of the surface characterization show the silanol groups enriched at the surfaces of the latex particles and could be tailored by changing the functional comonomer concentration. Silica or other inorganic compounds to prepare novel hybrid particles could easily coat these silanol-functionalized latex particles. Carboxyl and amino-functionalized latex particles were synthesized [67] by the miniemulsion polymerization of styrene and acrylic acid or 2-aminoethyl methacrylate hydrochloride, and the effect of hydrophilic comonomer and surfactant type (nonionic versus ionic) on the colloidal stability, particle size, and particle size distribution was analyzed. The reaction mechanisms of particle formation in the presence of nonionic and ionic surfactants were proposed. Recently, functionalized latexes have been obtained by means of CRP in an ab initio batch emulsion and miniemulsion polymerization process. Even though these works may not always intentionally be making surface-functional particles, it is a very efficient way of doing so. The approach consists in adding a hydrophilic and water-soluble control agent and in situ building up a polymeric surfactant, which after micellization causes the control agent to be inside the latex particle. The pioneers in developing this strategy were Ferguson et al. [68–70], who used an amphipatic RAFT agent to produce an initial diblock with hydrophilic and hydrophobic components of degrees of polymerization chosen so that these can self-assemble into micelles. By this process, core–shell latex particles composed largely of PAA-b-poly(butyl acrylate)-b-PS triblocks were synthesized. A novel polymer colloid architecture wherein only a single type of polymer is present in the particle and each individual chain stretches from aqueous phase through the shell and to the core. More recent papers [71,72] used this approach using other kind of hydrophilic macro-RAFT agents to obtain stable latex particles thanks to the in situ created amphiphilic diblock copolymer. The functional macro-RAFT agents were hydrophilic (co)polymers carrying a thiocarbonyl thio end group such as poly(dimethylaminoethyl methacrylate) [71], poly(ethylene oxide) [71,72], and poly(ethylene oxide)-block-poly(dimethylaminoethyl methacrylate) [71]. Furthermore, this approach was also translated very effectively to nitroxide-mediated emulsion and miniemulsion polymerization [73–76] to prepare in situ PAA-based hairy nanoparticles using a water-soluble alkoxyamine initiator.

12.3.2 SEEDED EMULSION COPOLYMERIZATIONS TO PRODUCE FUNCTIONALIZED LATEXES The use of seeded emulsion copolymerizations to produce composite-functionalized latexes is widely proposed in the literature. In some cases, the authors prefer to use the terms core–shell latex particles or two- or multistep polymerization processes to name the different possibilities to produce composite-functionalized latex particles by using previously formed latex particles and polymerize onto their surfaces homopolymers or copolymers containing the desired functionality. From 1994 to now, monodisperse polymers colloids with aldehyde [77,78], acetal [79–83], chloromethyl [84–88], amino [89–95], and macromonomer [96–100] functionalities useful for immunoassays are being synthesized in our research group by a two-step and even multistep emulsion polymerization processes carried out in batch and/or semibatch reactors. In the first step, the seeds were produced by batch emulsion polymerization of styrene, and in the second step, onto the previously formed PS latex particles, the functional monomers were co- and/or terpolymerized.

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Some of the synthesized latexes were chosen as the polymeric support to carry out the covalent coupling with a protein and to test the utility of the latex-protein complexes formed in immunoassays. The group of Pichot and Elaissari has also been active during the last decade in the synthesis and characterization of functionalized latex particles by using the “attaching to” seeded multistep emulsion polymerization technique. They prepared monodisperse copolymer latex particles having aldehyde surface groups in the first step, and in the second step, a surface functionalization of the seed particles by copolymerizing p-formylstyrene [101], emulsifier-free PS latexes covered by disaccharide species by using a seed copolymerization method [102], cationic PS latex particles with different aminated surface charges by using seed particle functionalization, or by the shot-growth procedure [103]. They also reported [104] the production of cationic amino-functionalized poly(styrene-N-isopropylacrylamide) core–shell latex particles to study the adsorption/desorption behavior and covalent bonding of an antibody. The synthesis of thermally sensitive cationic poly(methyl methacrylate)-poly(N-isopropylacrylamide) core–shell latexes by using two-stage emulsion copolymerization [105] was reported in 2005. The variation of the particle hydrodynamic diameter with temperature confirmed the core–shell morphology of the particles with a PNIPAM shell around the poly(methyl methacrylate) (PMMA) core. The covalent binding of proteins to acetal-functionalized latexes produced by means of a twostep emulsion terpolymerization process was reported by Peula et al. [106,107]. In these works, the physical and chemical adsorption, electrokinetic and colloidal characterizations as well as immnoreactivity of the latex–protein complexes were analyzed. One year before, the group of Granada used the same polymerization procedure to obtain core–shell chloromethyl-functionalized latex particles to perform the covalent coupling of antihuman serum albumin IgG protein [108]. There are a series of articles in which several syntheses of core–shell particles useful for a wide variety of applications are presented. The different particles synthesized are used to carry out the covalent coupling of antibodies [109], to analyze the properties of films prepared by using reactive latexes with epoxy and carboxylic surface functional groups [110]; to obtain peroxy-functionalized PS core using these functionalized groups to graft the shell monomers in the second stage [111], by using polyperoxide; to study the adsorption of immunoglobulin G on core–shell chloromethyl latex particles precoated with 3-(3-cholamidopropyl)dymethylammonio-1-propanesulfonate (Chaps) [112]. On the other hand, two-step processes in which a shot of acrylic acid was performed in the last stage of the emulsion polymerization reaction were analyzed as a strategy to increase the surface incorporation efficiency [113]. The two-stage emulsion polymerization technique was also used to produce polymer latexes bearing saccharide moieties on the particle surface; for that a water-soluble monosaccharide monomer was used to functionalize styrene/butyl acrylate latexes [114]. Poly(methyl methacrylate-ethyl acrylate-methacrylic acid) particles were synthesized by using a seeded soap-free emulsion polymerization process to obtain clean surfaces and surface carboxylic groups [115]. Authors found that in this case dropwise addition process was better than batchswelling process to produce large particles with narrow size distribution. Core–shell latex particles prepared through two-stage semicontinuous-starved emulsion polymerization with PS as a seed and butyl acrylate as a second stage monomer were functionalized with two different acrylamides [116]. The effect of functional groups with different hydrophilicity and the locations in core–shell particles on the main colloid characteristics was investigated. The last case to illustrate the use of a two-stage emulsion polymerization process to produce “attaching to” functionalized latex particles is the following: Imidazole-functionalized latex microspheres prepared by a two-stage emulsion copolymerization process by using styrene and 1-vinyl imidazole [117]. Macromolecules incorporating 1-vinyl imidazole and its derivatives are of great interest for the preparation of new materials with ion-exchange properties and carrier agents for protein separations. The results show that the concentration of imidazole groups on the surface of

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the latex particles could be varied by varying the second-stage monomer addition process such as the use of monomer-swollen seed particles or a shot addition of monomers.

12.3.3

SURFACE MODIFICATION OF PREFORMED LATEXES

There is another possibility to obtain functionalized latex particles that can be included in the “attaching to” procedure. It consists of the modification of the surface of previously synthesized latex particles. This modification can be made via chemical or photochemical procedures. In the recent specialized literature, there is a considerable amount of articles reporting different possibilities to carry out the surface modification of latexes. Kawaguchi et al. [118] proposed the modification by hydrolysis of the surface groups of monodisperse acrylamide-styrene copolymer latex obtained by an emulsifier-free aqueous polymerization. In this way they obtained a series of polymer latexes having the same particle size but different kinds and amounts of functional groups on the particles surfaces. Using heterofunctional polymeric peroxides (HFPP), the surface [119] of various polymeric colloidal systems such as emulsions, latexes, polymer-polymer mixtures, and so on, was modified. HFPP are carbon chain polymers, which have statistically located peroxidic and highly polar functional groups such as carboxylic, anhydride, pyridine, and others along the main chain. The reactions of the functional groups provide chemical bonding of the macromolecules to the interfacial surface. The activation of latex particles surface by surface-active HFPP is of special interest. Li et al. [120–124] presented a series of articles on surface modification of aldehyde-functional poly(methylstyrene) latex particles. The synthesis of the different latexes was carried out by emulsifier-free emulsion polymerizations or by using anionic or cationic surfactants, initiated by different initiators, and followed by an in situ surface oxidation catalyzed by copper (II) chloride and tert-butyl hydroperoxyde. Changing the oxidation degree alters the surface morphologies of the functional latexes. The end groups of an amphiphilic comb copolymer used to create a robust hydrophilic coating on the final particles were selectively functionalized to obtain latex particles with a controlled density of ligands tethered to their surfaces [125]. The final latexes were used to form cell-resistant and cell-interactive surfaces. Surface modification of the surface of latexes carried out via photochemistry is also presented by Nakashima et al. [126]. In this work, two examples of photochemical reactions on the surface of PS latex particles were reported with the objective of understanding various phenomena that take place in practical coating systems (e.g., photodegradation of pigments and binders). Bousalem et al. [127–129] presented a series of articles focused on the synthesis of N-succinimidyl ester-functionalized, polypyrrole-coated PS latex particles with potential biomedical applications. The latex particles were prepared by the in situ copolymerization pyrrole and the active esterfunctionalized pyrrole in the presence of bare PS particles as substrates. The final functionalized latex particles were evaluated as bioadsorbents of human serum albumin used as test protein. As an example of surface-modified latex particles via chemical reactions, there is the work of Imbert-Laurenceau et al. [130] PS particles functionalized by various amino acids were prepared in three steps by modifying chemically the surface of cross-linked PS particles. First, phenyl groups of PS were chlorosulphonated by reaction with an excess of monochlorosulfonic acid. Second, The PS-SO2Cl was condensed with various amino acid methyl esters in order to obtain the different functionalized particles. Third, ester groups were hydrolyzed by successive washing of the particles with different aqueous solutions of NaOH. The final particles are able to strongly interact with antiviral antibodies. Due to these biospecific interactions, the functionalized particles are candidates to be used in vaccinal approach as “virus-like” polymers or as affinity matrix for the purification of antiviral antibodies.

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Cell-polymer interactions of fluorescent polystyrene (FPS) latex particles coated with thermosensitive PNIPAM and poly(N-vinylcaprolactam) or grafted with poly(ethylene oxide)-macromonomer were analyzed by Vihola et al. [131] by modifying the surface of the FPS particles with the thermosensitive polymer gels or with poly(ethylene oxide)-macromonomer grafts. In all the cases, the FPS were surface-modified by polymerization of both thermosensitive monomers and macromonomer onto the surface of the fluorescent particles. The final surface-coated particles have potential biotechnological applications in the form of either stealth-carrier behavior or enhanced cellular contact.

12.3.4

“CLICK CHEMISTRY” FOR SURFACE FUNCTIONALIZATION

The “click chemistry” concept was introduced in 2001 by Sharpless and coworkers [132] with the aim of expanding a set of selective and modular “blocks” that work in small- and large-scale applications. “Click chemistry” is focused on the power of a very few reactions that form desired bonds under simple reaction conditions. It is modular, wide in scope and easy to perform, uses only readily available reagents, and it is insensitive to oxygen and water. In some cases, water is the ideal reaction solvent, providing the best yields and highest rates. Purification methods use benign solvents and avoid chromatography. The best example of “click chemistry” is the copper (I)-catalyzed Huisgen 1,3-dipolar cycloaddition between azides and alkynes (CuAAC) [133]. Since 2005, “click chemistry” has attracted increasing attention in polymer science [134] to synthesize different polymeric architectures [135] (linear polymers, linear copolymers, branched polymers, polymeric assemblies, and cross-linked polymers). Another application of “click chemistry” is the surface functionalization of latex particles. Different groups are making a lot of effort to use azide/alkyne coupling as a valuable new synthetic technique in the synthesis chemistry of polymers [136,137]. Following, some examples to show that “click chemistry” as a powerful tool in the design and synthesis of functionalized particles are shown. Lovell and coworkers presented proof of attaching molecules to latex particles in which core– shell latexes with alkyne surface-functionality were prepared and reacted with water-soluble polymers and dye compounds that were modified to possess the required azide moiety [138]. Van Hest and coworkers [139] prepared polymeric blocks (PMMA, PS, and PEG) bearing functional end groups [140] separately and linked them covalently via their end groups. This approach enables full analysis of the separate blocks prior to coupling. The same authors recently reported the modular synthesis of ABC-type triblock terpolymers by performing two successive “click” coupling of polymeric building blocks onto a central block (B) [141]. Agut et al. [142] have also demonstrated “click chemistry” application in the preparation of block copolymers. Other relevant workers in this field are Hawker and coworkers. For example, they reported the synthesis of a family of functionalized 4-vinyl-1,2,3-triazole monomers that combine into a single structure many of the desirable features found in established monomers. By using this new family of vinyl monomers functional materials can be prepared [143]. Wooley and coworkers utilized “click chemistry” to prepare block copolymer micelles and shell cross-linked nanoparticles (SCKs) presenting click-reactive functional groups on their surfaces [144,145]. Moreover, they presented the preparation of well-defined core cross-linked polymeric nanoparticles, utilizing multifunctional dendritic cross-linkers that allow for the effective stabilization of supramolecular polymer assemblies and the simultaneous introduction of reactive groups within the core domain [146]. As can be easily concluded, “attaching to” surface functionalization is a very useful procedure to produce latex particles with surface-functionalized groups useful in a huge number of technological applications. Following, the third way for surface functionalization that is a relative new but interesting area will be commented. “Attaching from” surface functionalization is nowadays a field under continuous development.

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“ATTACHING FROM” SURFACE FUNCTIONALIZATION

12.4.1 “ATTACHING FROM” BY CONVENTIONAL RP Prucker and Rühe [147,148] reported the conventional free RP of styrene from the surface of silica gel coated with a monolayer of covalently bound azo initiator. Graft polymers with high, controlled graft density could be obtained. However, conventional free RP, especially when confined to a thin layer, leads to a wide molecular weight distribution of the grafted polymers, largely due to the termination reactions. Moreover, this approach is no suitable for preparing block copolymers. Hritcu et al. [149] synthesized first cationic PS latexes covered with a shell-containing poly(styrene-co-2-hydroxyethyl acrylate) by a seed copolymerization procedure using an azo initiator. In the second step, grafted chains anchored to the surface were produced by polymerization of N-(2-methoxyethyl) acrylamide (MEA), in the presence of Ce(IV) as a redox initiator. However, the percentage of MEA found covalently attached to the surface was very low because bulk MEA polymerization forming soluble polymers occurred to a much greater extent than the actual grafting reaction on the particles. Guo et al. [150] described the synthesis and characterization of latex particles consisting of a PS core and a shell of linear PAA chains. In the first step, PS cores were covered by a thin layer of a photoinitiator 2-[p-(2-hydroxy-2-methylpropiophenone)] ethylene glycol-methacrylate (HMEM) by means of a seeded emulsion polymerization process. The polymer formed by HMEM on the surface acted as a photoinitiator in the next step, in which acrylic acid was used as a water-soluble monomer. The PAA chains were affixed on the surface by means of an “attaching from” technique that led to particles with well-defined morphology and narrow size distribution.

12.4.2 “ATTACHING FROM” BY CRP Out of all the polymerization techniques used to attach polymer brushes, controlled radical polymerization techniques [151] such as NMRP, ATRP, and RAFT have received widespread attention. The growth of living polymer chains from surfaces ensures better control over the molecular weight distribution and the amount of attached polymer as compared to the conventional RP, which suffers from the unwanted bimolecular terminations. 12.4.2.1 Nitroxide-Mediated Radical Polymerization Fewer reports have been published applying NMRP for the grafting step. NMRP has the advantage of being a relatively simple process, requires no catalyst, and involves no bimolecular exchanges, although it is not as versatile as ATRP or RAFT in the range of monomers suitable for use. Control in NMRP is achieved with dynamic equilibration between dormant alkoxamines and actively propagating radicals. In order to effectively mediate polymerization, alkoxamines should neither react with itself nor with monomer to initiate the growth of new chains, and it should not participate in side reactions such as the abstraction of b-H atoms. The “attaching from” technique is carried out through NMRP attaching the alkoxamine to the surface of polymer particles. Hodges et al. [152] and Bian and Cunningham [153] reported the grafting of PS, poly(acetoxystyrene), poly[styrene-b-(methyl methacrylate-co-styrene)], poly(acetoxystyrene-costyrene), and poly(styrene-co-2-HEMA) copolymers onto 2,2,6,6-tetramethyl-1-piperidinyloxy nitroxide (TEMPO) bound Merrifield resin. Merrifield resin is a PS resin based on a copolymer of styrene and chloromethylstyrene cross-linked with divinylbenzene. In these works, a pronounced increase of particle size was observed, which was attributed to the formation of chains both at the surface and within the microspheres. The polymerization control was enhanced both on the surface and in solution by the addition of sacrificial nitroxide. In a following work, Bian and Cunningham [154] reported the attaching from the surface of alkoxamine-functionalized poly(styrene-cochloromethylstyrene) microspheres of 2-(dimethylamino)ethyl acrylate (DMAEA) by NMRP. Latex particles bearing chloromethyl groups were prepared

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by emulsion polymerization. N-tert-butyl-N-(1-diethyl phosphono-2,2-dimethylpropyl)nitroxide (SG1) was then immobilized on the particle surface. Latex particles attached with the homopolymer polyDMAEA, as well as block copolymers poly(styrene-b-DMAEA) and poly(butyl-b-DMAEA), were prepared by surface-initiated NMRP in N,N-dimethylformamide at 112ºC, with the addition of free SG1 to ensure that control is maintained. Particles sizes increased with the molecular weight of free polyDMAEA in solution. 12.4.2.2 Atom Transfer Radical Polymerization ATRP in organic systems tolerates a variety of reactions conditions and monomers and has been reported for a number of surfaces. Recently, there has been growing interest in aqueous ATRP [155,156] which is able to produce water-soluble polymers at room temperature. Its attraction lies in its simplicity, robustness, its ability to produce narrowly distributed polymer chains, and the possibility of synthesizing controlled block copolymers. However, ATRP is difficult to apply successfully to surface grafting reactions. Standard conditions used successfully in ATRP solution reactions do not provide good control over polymerization from surfaces. The lack of control is evidenced by a large increase in graft thickness over a short time and poor initiating efficiency. Usually, addition of a deactivator or free external initiator is used to control the polymerization. Typically, ATRP initiators are halide compounds, such as benzyl halides and a-halo carboxylates, which convert into stabilized radicals upon halide transfer. The growth of polymer chains from the initiator is activated by metal complexes. An important concern is the uniform functionalization of the particle surface with ATRP initiator, which differentiates latex particle systems from flat surfaces. This represents in fact the basic requirement for the production of polymer brushes uniformly distributed on the surface. Depending on the nature of the initiator and its ability to polymerize on preformed polymer latex particles, the resulting morphology in terms of initiator surface concentration and distribution can drastically change [157]. Based on the kinetic and thermodynamic factors affecting the course of the polymerization, it is often challenging to find the optimum conditions leading to the uniform distribution of ATRP initiator on the latex particles. ATRP-functionalized polymer particles are generally synthesized by emulsion polymerization, where the functionalizing monomer (e.g., benzyl halides and a-halo carboxylates carrying the terminal acrylic or methacrylic groups) is polymerized onto polymer–seed polymer latex particles [157–166]. The process is formally achieved in two different steps, where the polymer seed is formed first with the desired characteristics and the functionalizing monomer is polymerized afterward on the surface. On the other hand, ATRP initiator can be introduced onto particle surface by a chemical reaction between the ATRP initiator and a reactive functional group attached previously at the latex particle surface [167–172]. Guerrini et al. [163] were the first to report the preparation of hydrophobic core-hydrophilic shell particles with a well-defined shell and possible chain-end functionalization by ATRP. Haddleton and coworkers [168,169] attached ATRP initiators to Merrifields and Wang-type microbeads and grafted methyl methacrylate (MMA), benzyl methacrylamide, and N,Ndimethylacrylamide (DMA) from these beads. Botempo et al. [173] grew a variety of brushes from PS latexes with aqueous ATRP, including PNIPAM, poly(2-hydroxyethyl methacrylate) (PHEMA), poly[PEG-1100 monomethacrylate], and block copolymers thereof. Zheng and Stöver [170,171] grew PS, poly(methyl methacrylate), PHEMA, poly(methyl acrylate), poly[2-(dimethylamino)ethyl methacrylate] (PDMAEMA), and block copolymers brushes from functionalized divinylbenzene (DVB)/2-HEMA copolymer microspheres by ATRP. Brooks and coworkers [159,160] used PS particles to initiate the aqueous ATRP of DMA. They observed very high graft molecular weights greater than 600,000 g/mol and as high as 1,200,000 g/mol. In addition, there is the controlled synthesis of PNIPAM homopolymer and block copolymer brushes on the surface of latex particles by aqueous ATRP by Brooks et al. [161,162,167] and Mittal et al. [158]. These studies demonstrated that the thickness of PNIPAM brushes was sensitive to temperature and salt concentration.

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Zhang et al. [165,166] prepared well-defined double-responsive polymer brushes of PDMAEMA with a high density of brushes and low polydispersity (PDI 1.21). 12.4.2.3 Reversible Addition-Fragmentation Chain Transfer Polymerization RAFT polymerization is among the most versatile of all the CRP techniques used to date due to its compatibility with a wide range of monomers and reaction conditions [151]. The RAFT process and the chain transfer agents (CTAs) utilized to mediate the polymerization have been extensively studied over the past few years. The main drawback of this system is the presence of impurities trapped in the final polymer product, including dead polymer chains, monomer, and unrecoverable CTA. Furthermore, the RAFT process requires the use of a free radical source that generates uncontrolled, dead polymeric chains. The use of latex particles in RAFT polymerization has already been reported [174–176]. The CTA can be attached through its leaving and reinitiating group (R group) [174], which results in the final polymer attached to the latex particle, in a similar manner as what is observed in ATRP mediated by a supported initiator. With the particle as a part of the leaving group, radicals are generated on the multifunctional RAFT agents that can grow, transfer to another RAFT agent, or terminate with a second radical. The last possibility broads the molecular weight distribution. On the other hand, if the RAFT group is attached to the surface through the Z group [175,176], latex particles are always bonded to CTA, whereas the growing macroradical is detached. To undergo transfer to the RAFT agent, the radical has to reach the RAFT group close to particle surface. With increasing conversion and, therefore, increasing length of the brushes, the RAFT process is increasingly hindered because of the shielding effect of the polymer brushes. Under certain conditions, the radical will rather terminate with another radical, instead of reacting with the RAFT agent, to generate a dead polymer. Although molecular weight evolution does not necessary follow the theoretical values, the RAFT polymerization with Z group attached onto latex particles leads to a unimodal molecular weight distribution and, therefore, to a better defined polymer. Barner et al. [174] recently reported the synthesis of core–shell poly(divinylbenzene) (PDVB) microspheres via the RAFT graft polymerization of styrene. Cross-linked PDVB core microspheres containing double bonds on the particle surface were used directly to attach polymers from the surface by RAFT without prior modification of the core microspheres. The RAFT agent 1-phenylethyl dithiobenzoate (PEDB) was used. PEDB controlled the particle weight gain, the particle volume, and the molecular weight of the soluble polymer. Jesberger et al. [175] modified polydisperse hyperbranched polyesters for use as novel multifunctional RAFT agent. The polyester-core-based RAFT agent was subsequently employed to synthesize star polymers of n-butyl acrylate and styrene with low polydispersity (PDI < 1.3) in a living free-radical process. Perrier et al. [176] attached a CTA to a solid support (Merrifield resin) by its Z group to control the polymerization of methyl acrylate. Preliminary results revealed that the reaction led to wellcontrolled polymers, and the supported nature of the CTA allows its easy recovery after reaction. The use of free CTAs in solution helped to increase the control over the molecular weight and polydispersity of the product. The main conclusion after reviewing the specialized literature is that surface properties of latex particles are in the majority of occasions responsible in determining their final application characteristics. Many approaches are reported to functionalize the surface of latex particles, among them, physical adsorption, “attaching to” surface, and “attaching from” surface are the more relevant procedures used. Each of them has advantages and disadvantages depending on the latex particles final application.

ACKNOWLEDGMENT This work was supported by the Spanish Ministerio de Ciencia e Innovación/Programa Nacional de Materiales (MAT 2006-12918-CO5-03).

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13

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles María Tirado-Miranda, Miguel A. Rodríguez-Valverde, Artur Schmitt, José Callejas-Fernández, and Antonio Fernández-Barbero

CONTENTS 13.1 Introduction ...................................................................................................................... 13.2 Scattering Functions for Fractal Structures ...................................................................... 13.3 Monitoring of Cluster Growth .......................................................................................... 13.3.1 Long-Time Behavior ............................................................................................. 13.3.2 Short-Time Behavior ............................................................................................. 13.4 Aggregation Mechanisms ................................................................................................. 13.5 Flocculation of Protein-Functionalized Colloidal Particles ............................................. 13.5.1 Influence of the Protein Net Charge ..................................................................... 13.5.2 Influence of Short-Range Interactions and Protein Size ....................................... 13.6 Structural Coefficient for Fractal Aggregates of Protein-Functionalized Colloidal Particles ............................................................................................................. Acknowledgments ...................................................................................................................... References ..................................................................................................................................

13.1

289 291 291 292 293 293 294 297 302 306 310 311

INTRODUCTION

Flocculation and stability of protein-functionalized colloidal particles play an important role especially in life sciences. Examples thereof are immunoassays, where antibodies act as flocculating agents for antigen-covered particles and vice versa [1], and aggregation of micelles that may contain in their membranes different proteins of biological interest [2,3]. Even when these functionalized particles are dispersed in physiological fluid, they may aggregate due to the formation of protein “bridges” [4]. Since other aggregation mechanisms, such as electrolyte-induced coagulation or weak flocculation, may also take place, cluster formation becomes the result of the interplay of these effects. It is, however, still not completely clear how the different experimental conditions affect the aggregation mechanisms and the structure of the resulting clusters. Hence, flocculation of protein-functionalized particles is a highly complex process that shows some analogies with what is known as “bridging flocculation” [5,6]. Nevertheless, it cannot be classified as such in a stricter sense. Moreover, pure 289

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bridging flocculation is not as thoroughly understood as electrolyte-induced coagulation and there are still several aspects that are of indubitable interest for modern research such as the aggregation kinetics [7,8], the influence of the flocculating agent’s nature [9,10], the structure of the aggregates formed [11,12], and the forces that lead to polymer adsorption and their final conformation [13]. For this reason, we focus our attention on the kinetic aspects of the flocculation process and the structural properties of the clusters formed. Both aspects, aggregation kinetics and cluster morphology, are mainly a consequence of the interplay between particle diffusion and interparticle interaction. As several authors have stated, very little work has been reported on this subject so far [8–11]. Currently, it is a known fact that irreversible colloidal particle aggregation leads to clusters exhibiting fractal structures that are directly related to the aggregation mechanism and the interparticle interactions. The purely diffusive movement of noninteracting particles is a consequence of random collisions with the solvent molecules. Brownian diffusion may, of course, be affected by particle correlations that are induced by hydrodynamic interactions. These interactions reduce the aggregation rate by a factor of two with regard to that of pure Brownian motion when the particles are aggregating in a primary energy minimum. Aggregation dominated by this mechanism is characterized by a linear increase of the average cluster mass with time and by fractal dimension of approximately 1.8. When long-range repulsive forces are present, a large number of cluster–cluster collisions become necessary before a new aggregate is formed. In this case, the cluster mass initially grows exponentially with time and later crosses over to a power law behavior with a growth exponent greater than one. Since repulsion allows the colliding particles to reach positions in the inner parts of the clusters, the clusters become more compact having fractal dimension of about 2.1. The two limiting regimes describing irreversible processes are usually referred to as diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA). Aggregation of electrically stabilized bare particles aggregating at low and high electrolyte concentration is an example where these limiting regimes have been successfully obtained. Both regimes, as well as the crossover between them have been widely studied during the past two decades [14–16]. For functionalized colloidal particles, however, the mechanisms governing the aggregation kinetics and cluster morphology are by far more complex, and so it is not very surprising that little experimental work has been published on this topic. Aggregation of functionalized particles has, so far, been mainly addressed by means of simulations that turned out to be a quite useful tool to throw light on this question. In particular, a relationship between the particle interaction energy, Et(r), and the corresponding cluster structure could be established when the Et(r) curves show only one finite minimum [17,18]. For this case, Shih et al. found that (i) the cluster size may become saturated at finite in time, (ii) cluster restructuring and densification are allowed, and (iii) the cluster fractal dimension decreases with increasing depth of the energy minimum. Haw et al. simulated aggregation in two and three dimensions for finite binding energies. Therefore, they considered a certain bond break-up probability [19]. Cluster rupture was found to depend not only on the nearestneighbor binding energy, but also on the number of nearest neighbors. Later, Jin et al. obtained similar conclusions studying gel formation through reversible cluster–cluster aggregation. They also found that the cluster fractal dimension decreases with increasing binding strength, that is, for increasing depth of the energy minimum. So far, only a limited number of experiments on this topic have appeared in the literature. About 10 years ago, Liu et al. aggregated gold particles using a surfactant as coagulating agent [20]. Their results agree with the simulations of Shih et al. Burns et al. studied the structure of flocks that were obtained by adding a nonadsorbing polymer to a stable colloidal latex dispersion giving rise to depletion-induced aggregation [21]. Although they do not state this explicitly, it may also be deduced from their work that the cluster fractal dimension decreases for increasing energy minimum depth. As we will show in this work, the interaction potential develops a minimum of restricted depth when macromolecules are adsorbed. The limited depth of the energy minimum arises from a steric term in the energy balance, which depends mainly on the size of the adsorbed molecules. The smaller binding strength weakens the clusters that subsequently restructure and become more compact. Evidently,

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a shallower energy minimum is also responsible for the reversibility of the aggregation processes. Note that reversibility is allowed neither for pure DLCA nor for RLCA processes.

13.2 SCATTERING FUNCTIONS FOR FRACTAL STRUCTURES Colloidal clusters show self-similar branched structures characterized by a scaling law, V(R) ~ Rdf, which relates the increasing radius R to the cluster volume V(R) through the fractal dimension df. Static light scattering (SLS) allows the cluster fractal dimension, df, to be determined from the angular dependence of the mean scattered intensity. For elastic scattering, the scattered light intensity from a system of clusters may be expressed in a factorized form as [22]: I(q) ~ P(q)S(q),

(13.1)

where q = 4p/l sin(q/2) is the scattering wave number, with l being the wavelength of the light in the solvent and q is the scattering angle. The form factor, P(q), is related to the particle size and shape. The structure factor, S(q), depends on the relative positions of the particles within the clusters and hence contains the information about the cluster morphology. This factor is essentially a Fourier transform of the pair correlation function g(r): sin(qr) S(q) ~ Ú r 2[g(r) - 1] ______ qr dr.

(13.2)

In the case of fractal structures growing in three-dimensional space, the pair correlation function is related to the fractal dimension according to g(r)~ rdf - 3. Integrating Equation 13.2 leads to [23] S (q ) ~ G (df - 1)

sin[( df - 1)arctan(qR)] ~ q-d qR[1 + (qR )2 ]( d -1)/2

f

f

(qR 1),

(13.3)

where R is the mean aggregate size and G the gamma function. In the qR 1 limit, a power law behavior is expected from which the fractal dimension may be determined. The structure factor is defined only for distances larger than the particle size and thus Equation 13.3 is only valid for qR0 < 1, where R0 is the monomer size. In this scattering region, the influence of the particle form factor can be neglected and the angular variation of the intensity is related only to the cluster structure factor, that is, I(q) ~ q-df. For higher q-values, the length scale corresponds to individual spheres within the clusters and the intensity is related to the particle form factor. In lower q regions, topological length correlations between clusters could be studied.

13.3 MONITORING OF CLUSTER GROWTH The aggregation kinetics arising in a colloidal suspensions can be described by the time evolution of the cluster size distribution, Nn(t). For dilute systems, where only binary collisions are relevant, von Smoluchowski proposed the following differential equation for the time evolution of Nn(t) [24]: dN n 1 = k N N - Nn dt 2 i + j = n ij i j

Â

•

Âk

nk

Nk .

(13.4)

k =1

The aggregation kernel, kij, quantifies the rate at which two clusters of size i and j react and form a cluster of size i + j. kij contains all the physical information about the aggregating system and kijNiNj may be interpreted in terms of a collision frequency and a sticking probability for two clusters diffusing toward one another. Analytical solutions of Equation 13.4 exist only for a reduced number

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of kernels such as the constant kernel, the sum kernel, the product kernel, and combinations thereof [25]. The constant kernel, that is, kij = constant, implies that the aggregation rate is size independent. It has been used as a reasonable approximation for the DLCA regime since there all collisions are effective. This is the fastest possible aggregation mode in the absence of attractive forces between particles.

13.3.1

LONG-TIME BEHAVIOR

Most coagulation kernels used in literature are homogeneous functions of i and j, at least for large i and j. Van Dongen and Ernst introduced a classification scheme for these type of kernels according to the relationship, kai,aj μ a l kij where a is a positive constant [26]. The homogeneity parameter l describes the tendency of a large cluster to bind to another large cluster and so governs the overall rate of aggregation. In the absence of cluster breakup, this parameter should become 0 for DLCA and 1 for RLCA. For a reversible aggregation reaction, l > 0 still means a large overall large–large cluster formation, which may be due to a small large–large cluster breakup. Hence, l will be used to characterize the aggregation mechanism. It may be determined from the time evolution of the number-average mean cluster size, ·nnÒ = M1/M0, where Mi = ÂniNn is the i-order moment of the size distribution. For DLCA, a linear increase with time is expected for long aggregation times, ·nn(t)Ò ~ t. In the case of RLCA, an exponential behavior is predicted, ·nn(t)Ò ~ e at, where a is a fitting constant. In the intermediate regime, the aggregation processes are controlled neither by diffusion nor by repulsion. In this case, the number-averaged mean cluster size increases describing a power law in time according to ·nn(t)Ò ~ tz.

(13.5)

For nongelling systems (0 £ l < 1), the kinetic exponent is given by z = 1/(1–l). It is worthy to indicate that the term RLCA–DLCA crossover has been widely employed in the literature, but it may refer to different phenomena. In this work, the crossover refers to the region where the electrolyte concentration leads to an aggregation process, which is neither a pure DLCA nor a pure RLCA. The average mass for fractal clusters may be derived from the mean hydrodynamic radius, ·Rh(t)Ò, once the fractal dimension is known. Therefrom, the number-average mean cluster size is easily calculated by dividing the average cluster mass by the monomer mass [27]: df

·M Ò Ê ·R Ò ˆ · nn Ò = = k0 Á h ˜ , m0 Ë R0 ¯

(13.6)

where m 0 and R0 are the monomer mass and radius, respectively. k0 is the known as structural coefficient. Equations 13.5 and 13.6 are employed in the present chapter for obtaining l. Dynamic light scattering (DLS) is used to measure the mean cluster size, ·RhÒ. The scattered intensity autocorrelation function is determined electronically from the photomultiplier output and converted into the scattered field autocorrelation function using the Siegert relationship [28]. Information about the cluster size distribution is obtained using cumulant analysis, that is, by fitting the logarithm of the field autocorrelation function according to ln gfield(t) = -m1t + m2(t2/2) + m3(t3/3!) + º .

(13.7)

The first cumulant m1 is related to the mean particle diffusion coefficient by m1 = ·DÒq2. Once the mean diffusion coefficient is determined, the average hydrodynamic size may be calculated using the Stokes–Einstein relationship. The homogeneity exponent, l, is then determined from the kinetic

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exponent obtained from the long-time asymptotic behavior of the average cluster size according to Equation 13.5.

13.3.2

SHORT-TIME BEHAVIOR

Olivier and Sorensen obtained the following equation for the first cumulant of the intensity autocorrelation function for short aggregation times [29]: m1(t) = m1(0)(1 + t/tc) -1/df(1 - l) ,

(13.8)

where tc = 2/c0 ks is the characteristic aggregation time. This time is expressed as a function of the initial particle concentration c0 and the Smoluchowski aggregation rate constant, ks. Equation 13.8 allows the characteristic aggregation time, tc, to be obtained, once df and l are known. Then, ks may be determined from tc using the known initial particle concentration. This method is based on the fact that the fractal cluster structure and the dynamic scaling start to develop almost from the very beginning of the aggregation process [30,31].

13.4 AGGREGATION MECHANISMS In the present work, three aggregation mechanisms have been considered to explain the experimental results at short aggregation times: (1) coagulation, where bonds are formed between two uncovered surface patches of the colliding particles; (2) weak flocculation, that is, two covered patches; and (3) pure bridging flocculation, where the collision of an uncovered part of one particle with the covered part of another particle occurs. In this configuration, a “protein bridge” will form between the particles. Several theoretical models have been developed to explain the relation between the aggregation rate and the degree of coverage. The pioneer La Mer model considered only bridging aggregation, assuming that the particle collisions are completely controlled by diffusion and have a binding probability equal to unity [32]. For pure bridging flocculation, ks must be proportional to the fraction q of protein-covered patches on one particle and the fraction (1 - q) of protein-free surface patches on the other particle involved in the collision. Therefore, ks ~ q(1 - q). This relation implies a maximum aggregation rate at half surface coverage and no flocculation at all for uncovered and totally covered particles. Based on this approach, more detailed models were proposed [33–35]. All of them account for additional aggregation mechanisms and, therefore, may be considered as an extension of the La Mer model. Nevertheless, all of them show the common feature that bridging flocculation is always considered to be completely efficient. Hence, an extended model was proposed, which considers an independent collision probability for each aggregation mechanisms mentioned at the beginning of this section [36]. We will apply this model to our experimental system. The probability of finding a covered surface patch is given by the fractional surface coverage, q, and the probability of finding a bare part by (1 - q). Hence, (1 - q)2 gives the probability that two particles collide in coagulation configuration, that is, with the bare parts of their respective surfaces. The aggregation rate for this configuration will be denoted kc. Weak flocculation, that is, the aggregation of two covered surface patches, corresponds to q2, and in this case, the aggregation rate is denoted kwf. For bridging flocculation, stable bonds are formed between a covered surface patch of one particle and an uncovered patch of another one. Thus, the aggregation rate should be proportional to the number of free sites on one particle and also to the number of occupied sites on the other one. Consequently, the corresponding aggregation rate, kbf, should be multiplied by q(1 - q). According to the model proposed by Schmitt et al., the total aggregation rate may be written as the sum of all contributions: ks (q) = kc (1 - q)2 + kwfq2 + 2k bf q(1 - q),

(13.9)

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Structure and Functional Properties of Colloidal Systems

where the factor of 2 is introduced in order to account for the two possible bridging configurations: The collision of an uncovered part of one particle with the covered part of another particle and the reverse case. This relationship allows the contribution of the different aggregation mechanisms to be quantified if the aggregation rate, ks, is known as a function of the degree of surface coverage,q. Equation 13.9 is just valid for binary collisions.

13.5 FLOCCULATION OF PROTEIN-FUNCTIONALIZED COLLOIDAL PARTICLES In this section, we study flocculation processes arising in protein-functionalized colloidal particle suspensions. It is divided into two parts. Part (1) is devoted to the influence of the electrical state of the protein–particle complex and part (2) addresses the importance of short-range interactions and the protein size. Before we expose the experimental results, a short description of the experimental details will be given. The flocculation experiments were carried out using aqueous suspensions of polystyrene microspheres with different degrees of surface coverage. The bare particle diameter was (99 ± 1) nm and the polydispersity index was (0.09 ± 0.02), as determined by DLS. The particle surface charge density s0 was measured by conductometric titration and found to be weakly pH dependent. For example, s0 = -3.3 μC cm-2 was found at pH 5. The negative particle charge arises from dissociated sulfate groups on the particle surface. The colloidal stability was estimated by determining the critical coagulation concentration (c.c.c.) from the time evolution of the scattered light intensity. The obtained value was (0.495 ± 0.007) M KCl. The water used for sample preparation was purified by inverse osmosis using millipore equipment. Prior to aggregation, the samples were sonicated for 15 min in order to break up any initial clusters and to guarantee monomeric initial conditions. Flocculation was induced by mixing equal amounts of buffered electrolyte solution and particle suspension through a Y-shaped mixing device. The initial particle concentration in the reaction vessel was 1.6 × 1010 cm-3 and the temperature was stabilized at (25 ± 1)°C. Aggregation was monitored simultaneously by DLS and SLS. The particle concentration was chosen so that the scattered light intensity was well above the noise level but did not surpass the limit for multiple scattering. At the selected particle concentration, the aggregation kinetics was fast enough to ensure a reasonable run time. Bovine serum albumin (BSA) was chosen as an adsorbing macromolecule. BSA was selected because it is one of the most abundant blood proteins, comprising about 55% of total blood proteins [37]. BSA has its isoelectric point at pH 4.8 and is an acidic protein with a net charge of -18e at pH 7. This means that BSA carries a slight positive charge at a pH below 4.8, close to zero at pH 4.8, and a negative charge above its isoelectric point. The BSA solutions were cleaned only by means of dialysis. BSA is a globular protein of ellipsoidal shape with a molecular weight of 66,411 g mol-1. Its size is approximately 1.6 × 2.7 × 2.7 nm3. This protein has the ability to form covalent dimers through the SH groups contained in its polypeptidic chain. In this work, monomeric (BSAm) and polymeric (BSAp) proteins were employed. The protein’s state of aggregation was checked by native page electrophoresis (Figure 13.1). BSA dimers were found as the main unit of the sample BSAp, although a smaller fraction of high molecular weight aggregates was also detected. Different amounts of protein molecules were adsorbed onto the particle surface in order to study the influence of the degree of surface coverage. Adsorption of globular proteins from aqueous solution on a solid surface is the net result of several subprocesses [38]: (i) electrostatic interaction due to overlap of the electrical double layer around the charged protein molecule and the charged sorbent surface; (ii) steric interaction due to the polymeric components at the sorbent surface that extend into the surrounding aqueous phase; (iii) changes in the state of hydration; and (iv) rearrangements in the protein structure. For the protein adsorption experiment, different amounts of BSA were added to a fixed quantity of buffered colloidal suspension. In order to facilitate adsorption, the pH of the suspension was established near the isoelectric point of the BSA molecules, that is, at pH 4.8. BSA molecules possess a compact structure when the medium pH coincides with their isoelectric point. This

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles

FIGURE 13.1

295

Electrophoresis in polyacrilamide gel with silver and blue dyeing for BSAm and BSAp.

structural organization is partially lost when the protein spreads on the sorbent surface, leading to a net increase of the entropy of the system [38]. This is why maximum adsorption is usually achieved near the isoelectric point of BSA [39,40]. Far from the isoelectric point, lateral intermolecular interactions become important and tend to reduce the adsorbed amounts. The corresponding adsorption isotherms show the high affinity of BSA. At high BSA concentration, a final plateau was reached, which indicates the adsorption of a complete monolayer [39,41]. This gave us the possibility to obtain particles with a known degree of surface coverage by simply controlling the amount of added protein. Figure 13.2 shows the adsorption isotherms of BSAm and BSAp for the polystyrene particles used for this study. Samples with 0%, 25%, 50%, 75%, and 100% of their surface covered by BSAm and BSAp molecules were selected. In order to study the reversibility of adsorption, the BSA-latex complexes were centrifuged. The supernatants were filtered, using a filter of extremely low protein affinity, and measured spectrophotometrically. Since no protein was detected in the supernatant, the initially adsorbed amount of BSA remains invariant. The electrical state of the samples was controlled by changing the pH of the medium. The pH was set to 3.2, 4.8, and 9.0 by means of low ionic strength acetate and borate buffers. The ionic strength of the buffers was approximately 2 mM, which is sufficient to ensure the desired pH. In any case, the exact pH values were always checked experimentally. Prior to the aggregation experiments with functionalized particles, bare polystyrene microspheres were coagulated under well-known conditions, that is, for KCl concentrations ranging from 0.125 to 0.700 M. In this range, aggregation is mainly slowed down by repulsive Coulombian interactions. Using SLS and Equation 13.3, the cluster fractal dimension, df, was obtained by measuring the average scattered light intensity, I, as a function of the scattering vector, q. Figure 13.3 shows in the inset a typical plot for the time evolution of the average scattered intensity versus qR0. It can be seen clearly that this function tends toward a limiting long-time behavior, from which the cluster fractal dimension can be easily obtained. Obviously, a longer time is required for reaching this asymptotic curve in the low q-range, since this region corresponds to larger structures. For higher q-values, the final curve is observed almost from the beginning, when only small clusters are present.

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Structure and Functional Properties of Colloidal Systems 4

BSAp

(BSA)ads (mg/m2)

3

BSAm 2 100% 75% 1 50% 25% 0 0

4 (BSA)add (mg/m2)

2

6

8

FIGURE 13.2 Adsorption isotherms of BSAm (open circles) and BSAp (solid circles) on polystyrene particles. BSAadd and BSAads give the added and adsorbed amount of BSA molecules normalized by the total available particles surface. The arrow-marked points correspond to the samples employed for the aggregation experiments.

Examining the curves in more detail, one might think that the slope increases from about -1.4 to a final value of approximately -2, and so df seems to be time dependent. Nevertheless, this apparent increase is still due to the decreasing curvature of the I(q) curves rather than a change in the fractal structure of the aggregates. As can be seen, the curves with a slope smaller than approximately -2 are still slightly bent and it is not possible to fit a well-defined straight line to them. Hence, the

2.2

20 min 50 min

105

RLCA

100 min 250 min

I (u.a.)

q–2.03

2.1

104

df

103

2.0

0.1

1

qR0

1.9

1.8 DLCA 1.7 0.0

0.1

0.2

0.3

0.4 0.5 [KCl]/M

0.6

0.7

0.8

FIGURE 13.3 Fractal dimensions for bare particles aggregating at different electrolyte concentrations. Inset: Time evolution of the scattered light intensity as a function of qR0 at a concentration of 0.250 M of KCl. The fractal dimension was calculated from the exponent a power law fit at very large aggregation times.

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297

0.6 103

RLCA

t1/(1–λ)

0.4

102

λ

101

1

10 t (min)

100

0.2

DLCA

0.0 0.0

0.2

0.4 [KCl]/M

0.6

0.8

FIGURE 13.4 l parameter as a function of the electrolyte concentration for bare particles. This parameter was obtained from the slopes of the long-time evolution of the number-average mean cluster size.

curves are still affected by the limited cluster size, that is, the clusters are still so small that the lack of self-similarity at large length scales (small q-values) affects the I(q) curves at short length scales (large q-values). Consequently, the fractal dimension df can be reliably obtained only at very large aggregation times when the I(q) curves show the expected linear behavior over a sufficiently large q-range. This, however, does not imply that the fractal structure of the smaller aggregates is changing in time or not well defined. It is only not yet observable. The same procedure was employed for all the other electrolyte concentrations. In all cases, a decreasing long-time asymptotic power law behavior was obtained for the mean scattering intensities, indicating the formation of self-similar fractal structures. The obtained fractal dimensions are plotted in Figure 13.3, as a function of the electrolyte concentration, observing a crossover from 1.75 to 2.1, that is, from diffusion-limited conditions to a situation where the repulsive interactions between particles become especially relevant. The average particle size, ·RhÒ, was measured by means of DLS as a function of the electrolyte concentration. Considering a structural coefficient equal to 1, the mean particle size and the number-average mean cluster size was obtained using the fractal dimensions reported before (see Equation 13.6). As example, the result is plotted in the inset of Figure 13.4. For all experimental conditions, the homogeneity parameter l is shown in Figure 13.4. It was calculated from the kinetic exponent of the long-time power law behavior (Equation 13.5). It should be noted that the time evolution of the average cluster size crosses, as expected, from DLCA to RLCA. At high electrolyte concentration, l ~ 0 was obtained and so the cluster aggregation activity seems to be size independent. Such a behavior is typical for DLCA. When a significant energy barrier is present, larger aggregates are more reactive than smaller ones and l increases. In the literature, a wide range of experimental values is reported for l. This fact is associated with the experimental difficulty to reach the reaction-controlled aggregation limit [27].

13.5.1

INFLUENCE OF THE PROTEIN NET CHARGE

As was stated in the introduction, the electrical state of the protein–particle complexes can give rise to differences in the aggregation mechanisms and the cluster structure. Since this is a major concern for industrial applications, we dedicate the current section to the investigation of aggregates formed

298

Structure and Functional Properties of Colloidal Systems

TABLE 13.1 Fractal Dimension df and Homogeneity Parameter l for Different Degrees of Surface Coverage and pH of the Aqueous Phase pH 3.2 q(%)

pH 4.8

df

l

df

pH 9.0 l

df

l

0

2.17 ± 0.03

0.43 ± 0.02

2.03 ± 0.03

0.33 ± 0.02

2.19 ± 0.03

0.42 ± 0.02

25

1.92 ± 0.03

–0.15 ± 0.02

2.23 ± 0.03

–0.10 ± 0.01

50

–0.31 ± 0.03 —

2.27 ± 0.03

–0.26 ± 0.03

0.28 ± 0.03 —

75

2.24 ± 0.03 —

2.29 ± 0.03 —

2.33 ± 0.04

–0.18 ± 0.02

—

—

100

—

—

2.39 ± 0.04

–0.12 ± 0.01

—

—

kc (cm3 s-1)

(1.8 ± 2.3)10-12

(1.2 ± 0.8)10-12

(0.8 ± 0.3)10-12

kwf (cm3 s-1)

(0.0 ± 2.6)10-12

(4.9 ± 0.8)10-12

(0.0 ± 0.3)10-12

kbf (cm3 s-1)

(9.4 ± 3.6)10-12

(10.5 ± 1.5)10-12

(0.6 ± 0.4)10-12

Note: The rate constants were obtained from the fit shown in figure 13.8.

from BSAp-covered polystyrene microspheres at different protein net charge. For our study, the electrolyte concentration was 0.250 M KCl. This value lies slightly above the one normally found in human blood serum. It is sufficiently low to impede flocculation of bare particles, while aggregation of covered particles can be enhanced or biased depending on the pH of the aqueous phase. Table 13.1 contains the obtained fractal dimensions at all experimental conditions. Excluding the sample for q = 25% at pH 3.2, where df = 1.9, all observed values are larger than 2 and relatively close to 2.1, which is the commonly accepted value for RLCA processes. This indicates that relatively compact structures are formed. Similar values were obtained for processes involving pure bridging flocculation [12] or bare protein aggregation [42]. At pH 4.8, we observe that the fractal dimension rises for increasing surface coverage, that is, the aggregates tend to become more compact. Such a behavior has been observed before for other experimental systems [43]. In that case, the authors found that the fractal dimension rose from df = 1.74 for clusters formed by aggregating bare particles up to 2.7 for sodium dodecyl sulfate (SDS)-covered particles. The dramatic increase in df was explained taking into account that osmotic and elastic–steric interactions lower the binding strength and, thus, allow for internal rearrangement processes once the clusters are formed. In our experimental systems, however, the bare particles aggregate at an electrolyte concentration of 0.250 M, which is not sufficient for ensuring a diffusion-limited aggregation regime. Nevertheless, even in this case, the fractal dimensions tend to increase for increasing surface coverage. This indicates that the adsorbed protein layer plays a similar role as the SDS layer in the aforementioned experiments, and so osmotic and elastic–steric interactions may be evoked for explaining the observed behavior. The aggregation kinetics will be discussed mainly in terms of two parameters, the homogeneity exponent, l, and the Smoluchowski aggregation rate constant, ks. Time-resolved DLS was employed for monitoring the average diffusion coefficient of the aggregates. Therefrom, the average hydrodynamic radius, ·RhÒ, was calculated using the Einstein–Stokes equation. In Figure 13.5, ·RhÒ is plotted in logarithmic scale for two different samples. In all cases, the data follow a straight line, even for quite small clusters. The observed power law is characteristic for dynamic scaling and in good agreement with the theoretical prediction given by Equation 13.5. It should be pointed out that the exponential behavior, predicted for RLCA and l = 1, could not be observed. The curves always rise, describing a power law almost from the beginning. This implies that the cluster size distribution may be described using a dynamic scaling approach practically for the entire aggregation process. Even for the early aggregation stages, the dynamic scaling solution is at least a good

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles

299

pH = 3.2 q = 50% l < 0 pH = 9.0 q = 25% l > 0

(m)

10–6

~t1/(1–l)df 10–7

1

100

10 t (min)

FIGURE 13.5 Time evolution of the average hydrodynamic radius of BSAp-functionalized particle clusters for two different net charges.

approximation. Using the fractal dimension as obtained from the independent SLS experiments, l was determined by fitting the experimental data according to Equation 13.5. The obtained results are summarized in Table 13.1. Prior to a more detailed discussion, however, it should be pointed out that negative values were found for l. Although the Van Dongen–Ernst theory does not forbid those values, it is difficult to find them in the literature [44]. According to its definition, the physical meaning of negative l values is that the small cluster–small cluster union is favored and more likely than the aggregation of larger clusters. This means that the aggregation rate should slow down as the aggregation processes go on. Such a behavior was clearly observed in our data (comparing the two curves showed in Figure 13.5) and thus supports the negative l values found. Before we enter the detailed discussion for different coverage degrees at different pH, it is convenient to compare briefly the results obtained for bare particles. According to Table 13.1, similar values for df and l were obtained and so the aggregation processes seem to be pH independent in all cases. The relatively small l values between 0.3 and 0.4 point toward a slow aggregation kinetics, which implies that the clusters have to collide more than once prior to aggregation. The values found for the fractal dimension, df, are very close to the ones found for RLCA processes indicating that relatively compact structures are formed. Consequently, the fast aggregating DLCA regime is not achieved and the screening of the particle surface charge is not complete. From a point of view of the interaction potential, we conclude that a repulsive energy barrier does still exist, although it is lowered considerably. At pH 3, the latex particles are negatively charged whereas the BSAp molecules bear a positive net charge. Here, aggregation takes place at intermediate degrees of surface coverage, that is, for q = 25–50%. At q = 25%, the fractal dimension of df = 1.92 is smaller than the corresponding one at q = 50%. Furthermore, the homogeneity exponent l is negative and its absolute value is also smaller than the one obtained at q = 50%. This implies that rather small and ramified aggregates are formed. For relatively low degrees of coverage, we may assume that there are surface patches carrying a positive charge arising from the BSAp molecules and areas of unoccupied surface that still bear the original particle charge. So, from a mesoscopic point of view, two particles may collide with positive and negative areas of their respective surfaces, giving rise to bond formation due to electrical attraction.

300

Structure and Functional Properties of Colloidal Systems

[ m1(t)/ m1(0)] –df (1–l)

103

102

101

100 100

101

102

103

1 + t / tc

FIGURE 13.6 q = 50%.

Time evolution of the normalized first cumulant as a function of the scaled time for pH 9.0 and

This process can be seen as a kind of charge-mediated flocculation in which the protein molecules act as “bridges” between the aggregating particles. Nevertheless, we cannot exclude the possibility that the particles join through unoccupied patches of their respective surfaces, that is, that coagulation also takes place. The protein–particle complexes may be considered as homogeneously charged spheres. From this point of view, aggregation is generally favored since the net charge of the covered particles and, consequently, the electrostatic repulsion terms for the total Derjaguin–Landau–Verwey– Overbeeck (DLVO) interaction energy are smaller than the ones for the uncovered particles. The situation is quite different at high degrees of surface coverage, that is, for q = 75–100%, where aggregation practically does not take place. Here, two mechanisms prevent aggregate formation. The first one is related to the fact that practically all surface patches carry locally a positive net charge, which will always give rise to an energy barrier between two colliding particles. The second one is due to repulsive steric interactions that arise when two layers of adsorbed proteins come into contact [43]. Once df and l were known, Equation 13.8 was employed for obtaining the characteristic aggregation time, tc, and there from the Smoluchowski rate constant, ks. For this purpose, the first cumulant, m1, was plotted as a function of the scaled time, 1 + t/tc. The characteristic aggregation was then varied until all experimental points aligned on a straight line with a slope of unity, as we can see in Figure 13.6. The aggregation rate constant, ks, was calculated from the obtained tc values using the known initial particle concentration. Figure 13.7 shows the experimentally obtained ks values as a function of the degree of surface coverage at the three different pH values used in this study. As can be seen, the corresponding curve is roughly “bell shaped,” which indicates that bridging flocculation is the main aggregation mechanism. The fit according to Equation 13.9 is included in the plots. It allows us to quantify the weight of the different aggregation mechanisms. The obtained fitting parameters are summarized in Table 13.1. They show that weak flocculation is not observed whereas protein bridging is about five times more effective than pure coagulation. This means that the aggregation kinetics at pH 3.2 is almost completely controlled by “protein bridging.” This can be illustrated at the relatively low degree of surface coverage of q = 25%. In this case, the probability for two approaching particles to collide with their bare surface patches is given by (1 - q)2 = 56%, while the probability for particle encounters in bridging configuration is only 2q(1 - q) = 38%. Hence, one expects the aggregation process to be mainly governed by pure

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles

6

301

pH 9.0

4 2

ks(10–12 cm3 s–1)

0

6

pH 4.8

4 2 0 pH 3.2 6 4 2 0 0

25

50 q (%)

75

100

FIGURE 13.7 Aggregation rate constant as a function of the surface coverage at three different pH values. The data were obtained averaging over three independent measurements. The error bars give the statistical error of the average value. The lines show the best fit according to Equation 13.9. The corresponding fitting parameters are summarized in Table 13.1. (From Tirado-Miranda, M. et al., 2003. European Biophysics Journal 32: 128–136. With premission.)

coagulation. However, the contribution of pure coagulation to the overall aggregation rate, kc(1 - q)2 = 1.0 × 10-12 cm3 s-1, is more than three times smaller than the contribution of bridging flocculation, 2kbfq(1 - q) = 3.5 × 10-12 cm3 s-1. This means that aggregation occurs mainly in bridging configuration and the probability for bridging flocculation is so high that it overcompensates the lower collision probability for this mechanism at low degrees of surface coverage. At pH = 4.8, aggregation is observed to a greater or lesser extent at all degrees of surface coverage (see Figure 13.7). Here, the latex particles are negatively charged whereas the BSAp molecules are at their isoelectric point. For low surface coverage, there should be negative and electrically uncharged areas on the particle surface. Hence, bridging flocculation becomes possible when uncharged and negative areas of different particles come into contact. This mechanism is based on the high affinity of electrically uncharged BSAp molecules for the negatively charged surface, that is, the binding forces for this aggregation mechanism seem to be very similar to the interaction that enables the isolated protein molecules to adsorb onto the bare particle surface. As can be seen in Table 13.1, coagulation between uncoated surface patches of colliding particles still occurs, although at a relatively low rate, which is about 10 times smaller than the rate for pure bridging flocculation. The explanation for the nonvanishing aggregation rate at low degrees of surface coverage is quite similar to the one given for pH 3.2, that is, the electrolyte concentration is high enough to decrease the height

302

Structure and Functional Properties of Colloidal Systems

of the repulsive energy barrier so that pure aggregation becomes possible. It is, however, insufficient for screening the particle net charge completely and thus only a relatively reduced aggregation rate is achieved. At high degrees of surface coverage, one expects steric repulsion to impede weak flocculation since the covered surface patches of one particle will not find free space on another one. Nevertheless, the aggregation rate for completely covered particles reaches almost half the value observed for pure bridging flocculation (see Table 13.1). This may be understood if one supposes that the adsorbed layer of neutral protein molecules reduces the long-range repulsive electrostatic interactions between two approaching particles. According to Elgersma et al. this is possible since adsorption of neutral protein molecules may lead to entropically driven counterion rearrangement, and thus lower the particle net charge [45]. This idea was corroborated by Ortega-Vinuesa et al. [40], who measured the electrophoretic mobility of protein-covered latex particles. Their results showed that the electrophoretic mobility decreases for an increasing amount of adsorbed protein. At pH = 9.0, slow but significant aggregation is observed only for q = 0% and 25%. In both cases, the experimental data indicate that the small cluster–small cluster union is biased (l > 0; see Table 13.1). As was explained before, the latex particles and the BSAp molecules bear negative net charge at high pH. This means that the electrostatic repulsion is enhanced for increasing degrees of surface coverage and finally becomes so high that it impedes flocculation almost completely. This explains why only pure coagulation and some bridging flocculation at low degrees of surface coverage are still possible.

13.5.2

INFLUENCE OF SHORT-RANGE INTERACTIONS AND PROTEIN SIZE

The aim of the following section is to gain further insight into the mechanism governing the kinetics and cluster structure of protein-functionalized colloidal particles when short-range interactions are dominant. In order to study the effect of short-range interaction on the cluster structure, a high salt concentration of 0.700 M was employed for the experiments. In this case, the long-range repulsive electrostatic interactions are sufficiently screened, so that morphological changes should be due to short-range interactions only. The pH was set close to the isoelectric point of BSA (pH 4.8) where the protein molecules’ net charge vanishes. According to the former section, the clusters restructure and form more compact structures, when macromolecules are adsorbed. This implies that the interaction potential should have a minimum of restricted depth, which is responsible for the reversibility of the aggregation processes. Such a minimum appears when a steric term is included in the energy balance. Since the steric term is molecule size dependent, the presence of such a term can be checked when similar molecules with different sizes are employed. As we will see later, short-range interactions do not significantly affect the aggregation kinetics, and so the cluster aggregation processes at long times are expected to be independent of the size of the adsorbed macromolecules. In order to modify the range of the repulsive barrier without changing the character of the adsorbed macromolecules, BSAm and BSAp protein molecules were employed. Figure 13.8 (inset) shows the average scattered light intensity as a function of the scattering angle for aggregating particles functionalized with BSAm and BSAp. A decreasing power law behavior is observed independently of the degree of protein coverage and so regular fractal structures form in any case. The obtained fractal dimensions are plotted versus the degree of protein coverage in Figure 13.8. For the BSAm-functionalized particles, the slope is much less pronounced than for the BSAp-covered samples. In both cases, however, the fractal dimensions grow starting from the expected DLCA value of 1.75. The fractal dimensions observed for the BSAm-covered samples rise slightly until 75% of coverage, indicating that the cluster structure became more and more compact. For even more added protein, the fractal dimension increases dramatically reaching a value of 2.1. The latter experiment was repeated three times to ensure the validity of the result. For the BSApfunctionalized particles, the cluster fractal dimension reaches 2.1 already for a surface coverage of 25%. Above that, very compact clusters form having a fractal dimension of the order of the typical RLCA values. Vincent et al. also found that adsorbed polyelectrolytes strongly influence colloidal

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles

10

303

BSAp

6

I (u.a.)

2.4 10 10

2.2

10

5

BSAp

4

0% 25% 50% 75%

3

RLCA

0.1

1

df

qR0

BSAm

2.0

BSAm 10

DLCA

I (u.a.)

1.8

1.6

10

10

5

4

3

0% 25% 50% 75% 100%

0.1

0

25

50 q (%)

75

qR0

1

100

FIGURE 13.8 Fractal dimension versus protein surface coverage for BSAp- and BSAm-functionalized particles. Inset: Scattered light intensity versus qR0 for aggregating functionalized particles with different amounts of adsorbed polymeric and monomeric protein.

aggregation due to steric effects [46]. They proposed that, for a proper description, a steric repulsive potential term should be added to the classical DLVO contributions, that is, to the repulsive electrostatic and the attractive London–van der Waals potentials. Now, not only electrostatic but also steric forces compete against the attractive van der Waals force. This gives rise to a strong repulsive barrier at very short distances, which impedes tight bonds. For macromolecules that form an adsorbed layer of average thickness d, an osmotic effect appears when the particles surfaces come closer than 2d. The osmotic pressure of the solvent in the overlap zone will be smaller than in the bulk. This causes a spontaneous flow of solvent into the overlap zone that pushes the particle apart. The osmotic repulsive potential can be expressed as [43] Eosm =

4 pa 2 Ê 1 ˆ ÈÊ r ˆ 1 Ê rˆ˘ f Á - c˜ d 2 ÍÁ ˜ - - ln Á ˜ ˙ , v Ë2 ¯ ÎË 2d ¯ 4 Ë d¯˚

(13.10)

where v is the molecular volume of the solvent (vwater = 1.8011 × 10-5 m3 mol-1), f is the effective volume fraction of molecules in the adsorbed layer, c is the Flory–Huggins solvency parameter, r is the distance between the particles surfaces, and a is the radius of the monomeric particles. The osmotic term overcomes the other contributions quite strongly when the particle surfaces come closer than a distance d. This implies that the minimum is the deeper the smaller the distance d becomes. Figure 13.9 shows the potential energy as a function of the particle surface-to-surface distance for coated particles with a diameter of 100 nm at high electrolyte concentration. Due to the osmotic term, the total interaction potential is always repulsive at short distances (smaller than approximately 5 nm). At large distances, the interaction energy curves are attractive due to the longrange van der Waals terms. Hence a minimum appears at intermediate distances. The clusters formed at this finite interaction potential minimum present a weaker internal structure than those growing in the absence of any steric and osmotic contributions. Thus, monomer rearrangement within the clusters is possible for BSA functionalized particles.

304

Structure and Functional Properties of Colloidal Systems ~d

25%

2

Et (r)/kBT

d

1

75%

0

d 100%

–1

0

5

10 r (nm)

15

20

d

FIGURE 13.9 Normalized DLVO particle–particle interaction potential for functionalized particles as a function of the distance between the particle surfaces. The sketches on the right side illustrate the increase of the monomer–monomer separation within the clusters for increasing surface coverage.

For both samples, the fractal dimensions are larger in the presence of adsorbed protein molecules. This may be understood taking into account that the thermal energy may give rise to particle rearrangement when the depth of the energy minimum is of the order of a few k BT. According to sketches on the right side of Figure 13.9, the distance between the particle surfaces depends on the degree of surface coverage. At low surface coverage, the distance between bond forming particles is very short and so the relatively deep energy minimum impedes internal cluster rearrangement. Hence, DLCA-like clusters are formed. For higher degrees of surface coverage, the particle to particle distance increases and the depth of the minimum becomes smaller. Consequently, cluster rearrangement becomes possible and the cluster fractal dimension increases. The large rise of the fractal dimension at a surface coverage of 100% could be related to the fact that the protein molecules might be adsorbed with an end-side orientation giving rise to relatively large particle separation. The fractal dimensions for BSAp-covered particles follow basically the trend observed for the BSAm-functionalized samples. The fractal dimensions, however, are much larger and the clusters more compact. This finding may be understood quite easily if one takes into account that polymeric protein molecules increase the particle separation d even further. Moreover, also the effective volume fraction f of molecules in the adsorbed layer increases while the solvency parameter c remains constant. The latter depends on the nature of the adsorbed molecules and the solvent and so does not change. Note that the dependence of c on the volume fraction that is necessary to explain phase transitions in some types of polymer gels is insignificant in the present model. Consequently, the contribution due to the repulsive steric potential increases and the potential energy minimum shifts toward larger distances. Hence, cluster restructuring is favored mainly due to the larger size of the BSAp molecules. The aggregation kinetics at long times was monitored by means of DLS. The number-average mean cluster size was calculated from the mean hydrodynamic radius and the fractal dimension was obtained by simultaneous SLS measurements (Equation 13.6). In this way, changes in the clustering mechanism may be detected from the long-time asymptotic scaling behavior. Figure 13.10 (inset) shows ·RhÒ as a function of time for different degrees of BSAm and BSAp coverage. All curves exhibit the predicted power law behavior. The corresponding values of l were calculated from the exponents of the fitted curves and they are shown in Figure 13.10. They are close to zero showing a slight rise for increasing surface coverage. This implies that the clusters form mainly after a pure

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles 0.4

–5

(m)

10

0% 25% 50% 75% 100%

–6

10

0.3

0% 25% 50% 75% 100%

BSAm

BSAp

RLCA

BSAm

–7

10

1

l

305

0.2

10 t (min)

100

1

10 t (min)

100

0.1 BSAp DLCA

0.0 0

25

50

75

100

q (%)

FIGURE 13.10 Influence of the protein surface coverage on the l parameter. The two series shown correspond to monomeric and polymeric protein-functionalized samples. Inset: Time evolution of the average hydrodynamic radii at different amounts of adsorbed BSAm and BSAp. From the slopes at long time, the l parameter is obtained.

diffusion process (DLCA). Such a result is expected since the BSA molecules modify only the interaction between the particles at short distances and do not affect the particle trajectories at larger distances. Even though the steric barrier shifts to larger distances, no significant differences are observed for the aggregation kinetics of polymeric and monomeric protein-functionalized particles. Notwithstanding, the cluster structures exhibit important changes as described before. The slight rise in l with regard to the surface coverage indicates an increasing tendency of large clusters to aggregate with other large clusters. Short-range interactions must be responsible for this result since long-range forces are not present. Aggregation between uncovered sites on the particles cannot be responsible for this trend since this effect increases as the uncovered surface diminishes. The explanation could be based on the interaction between the protein molecules adsorbed on different particles. Despite the fact that the protein molecules do not bear an electrical net charge at their isolectric point, local charge fluctuations may still be present and be responsible for the bonds between two large clusters as soon as they come sufficiently close. Since large clusters have a large mass, a few unions between them play a dominant role on the time evolution of the average cluster mass evolution. In order to confirm this mechanism, the aggregation rate constants were determined from the short-time behavior. The characteristic aggregation time, tc = 2/c0 ks, was determined, and the aggregation rate constant, ks, was calculated. Figure 13.11 shows the ks as a function of the BSAm and BSAp coverage. Within the range from 0% to 50% no important variations have to be pointed out. This result was expected since the high salt concentration and the protein molecules at their isoelectric point guarantee almost perfect conditions for unhindered diffusion. Moreover, the steric term that tends to reduce the collision efficiency does not dominate due to the low protein coverage. Thus, the aggregation kinetics is protein independent. At a surface coverage of 75%, however, an important reduction of the aggregation rate is observed, and the system becomes completely stable for completely covered particles. Despite the fact that the particles can diffuse almost freely and come very close, they cannot establish bonds due to the presence of steric stabilizing forces. This mechanism should

306

Structure and Functional Properties of Colloidal Systems

ks (10–12 cm3 s–1)

6

4

2 kwf kbf kc (10–12 cm3 s–1) (10–12 cm3 s–1) (10–12 cm3 s–1) 0

0

BSAp

6.3 ± 0.2

0.2 ± 0.1

7.2 ± 0.3

BSAm

6.1 ± 0.5

0.5 ± 0.4

8.1 ± 0.8

25

50 q(%)

75

100

FIGURE 13.11 Aggregation rate constant versus degree of surface coverage for BSAm- and BSAp-covered samples at pH 4.8 and 0.700 M KCl. The lines show the best fit according to Equation 13.9. The corresponding fitting parameters are also included in the plot.

be independent of the size of the adsorbed molecules, as long as the protein molecules surpass a minimum size that guarantees that the binding forces cannot act. In fact, Figure 13.11 shows no relevant differences when polymeric instead of the monomeric protein is employed and thus confirms this conclusion. The relationship in Equation 13.9 allows the contribution of the different aggregation mechanisms to be quantified. Therefore, the ks have to be plotted as a function of the surface coverage and fitted accordingly. For our data, the experimental aggregation rate constants are two times higher than the La Mer model prediction. On the other hand, they lie very close to the value for diffusionlimited aggregation, that is, to 6 × 10-12 cm3 s-1. This indicates that all cluster–cluster collisions in coagulation, bridging flocculation, and weak flocculation configuration, are effective. For the fits, the experimental aggregation rate constant at q = 0 was employed as the rate constant for coagulation kc. The best fit and the corresponding fitting parameters are given in Figure 13.11. The results indicate that weak flocculation does not play an important role, although it could be responsible for the slight trend of l observed in Figure 13.10. Bridging flocculation, however, is the predominant aggregation mechanism. The coagulation mechanism is also important but becomes apparent only at low surface coverage. Since this series of experiments was performed at the isoelectric point of the adsorbed BSA molecules, the net charge of the functionalized particles remains practically unaltered. Thus, it is not surprising that the aggregation rate constants for coagulation and bridging flocculation are quite similar.

13.6

STRUCTURAL COEFFICIENT FOR FRACTAL AGGREGATES OF PROTEIN-FUNCTIONALIZED COLLOIDAL PARTICLES

Although colloidal aggregates are of very complex nature, the description of their internal structure can be significantly simplified using fractal geometry [47]. The fractal dimension, df, links the number of primary particles per cluster to the hydrodynamic radius, according to Equation 13.6, where k0 is the structural coefficient. The latter parameter is often neglected and simply set to one. It is,

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles

307

however, an important factor for a complete quantitative characterization of fractal aggregates. In fact, clusters with identical df, Rh, and R0 contain less primary particles when they have a smaller structural coefficient. A larger distance should then exist among the monomeric particles contained within the clusters. Consequently, the structural coefficient k0 must be related to that distance. Therefore, the term “cluster structure” refers to the spatial distribution and distance between the constituent particles of an aggregate. The former is frequently determined by means of scattering techniques and is usually expressed in terms of a cluster fractal dimension df. The latter, however, is quite difficult to determine experimentally. To the best of our knowledge, only small-angle neutron scattering has been applied for assessing the interparticle distance in silica aggregates [48]. In this section, the structural coefficient for aggregates of bare and BSAp-covered particles (50%) will be determined. Therefore, the electrolyte concentration will be set to 0.700 and 0.250 M. The average diffusion coefficient will be employed as experimental parameter for monitoring the aggregation kinetics. As before, the fractal dimension of the formed aggregates and the homogeneity parameter l will be determined by means of SLS and DLS. The results will then be used to calculate the separation between adjacent monomers contained in the clusters and to estimate the adsorbed protein layer thickness. Oh and Sorensen deduced the dependence of the structural coefficient on the monomer–monomer overlap in a cluster [49]: k0 = k0(1)Ddf,

(13.11)

where k0(1) is the structural coefficient of the aggregates formed by touching but nonoverlapping particles. The overlap parameter D is defined as D = 2R0/(2R0 + S), where S is the surface-to-surface distance of two adjacent monomers. In this relationship, S is positive for separated particles and negative for overlapping monomers. Thus k0 should increase as overlap increases. This description agrees well with simulations and experiments from stereoviews of three-dimensional aggregates for which high k0 values were reported [50]. Hence, S may be obtained once k0 is known. For this purpose, the number-average cluster size ·nnÒ has to be determined from the experimental data. This can be achieved by fitting the time evolution of the average cluster diffusion coefficient [30]. According to light scattering theory, the average diffusion coefficient for an aggregating colloidal system may be expressed as [14] nc

m (q ) · D ( q, t ) Ò = 1 2 = q

Â Â

n =1 nc

N n (t )n2 S (qRg ) Dn

n =1

,

(13.12)

2

N n (t )n S (qRg )

where Nn(t) is the cluster size distribution, S(qRg) is the structure factor that accounts for the spatial distribution of the individual particles within an aggregate, and Dn is the diffusion coefficient of a cluster formed by n monomeric particles. The finite character of the aggregation processes has been considered introducing a cut-off size, nc. This cut-off size corresponds to the largest aggregates present in the system. Evidently, nc rises as the clusters grow. Equation 13.12 means that the average diffusion coefficient, obtained from experiments as m1/q2, may be calculated theoretically as the average of the diffusion coefficients, Dn, of individual aggregates weighted by the corresponding scattering intensity and cluster mass distribution at any given time. The cluster diffusion coefficient Dn depends on the translational and rotational diffusion coefficients Dnt and Dnr, respectively. The total diffusion coefficient is usually written as Dn = Dnt + Dnr, where coupling effects have been neglected. Assuming the aggregates to be fractal objects, the average translational diffusion coefficient ·Dnt Ò can be expressed as a function of the number-average mean cluster size ·nnÒ as [48] ·Dnt Ò = B·nnÒ -1/df,

(13.13)

308

Structure and Functional Properties of Colloidal Systems 1/d

where B = D 0 k 0 f is a constant [49] containing not only df and k0, but also the diffusion coefficient, D 0, of free monomeric particles. The rotational contribution is negligible for these experimental systems and so the overall diffusion coefficient may be approximated using only the translational coefficient [30]. In the literature, several functional forms for S(qRg) can be found [51,52]. The difficulty lies in obtaining an expression for the structure factor that is valid for the whole range qRg > 1. Lin et al. [53] calculated the aggregate structure factor directly from computer-generated clusters obtained under diffusion and reaction-limited conditions. They parameterized their result by fitting the polynomial expression: Ê S (qRg ) = Á 1 + Ë

m

ˆ Cs (qRg ) ˜ ¯ s =1

Â

- df /2 m

2s

.

(13.14)

For DLCA aggregates, they obtained m = 4, C1 = 2m/3df, C2 = 2.50, C3 = -1.52, C4 = 1.02, and df =1.8. For RLCA aggregates, the best fit yielded m = 4, C1 = 2m/3df, C2 = 3.13, C3 = -2.58, C4 = 0.95, and df = 2.1. Since the experimental cluster fractal dimensions generally differ from 1.8 to 2.1, the structure factors may be approximated by interpolating the values given by both polynomials. This structure factor provides a good description of aggregates of finite size. Furthermore, Equation 13.14 is in good agreement with experimental I(q) curves in the range qR0 £ 1 £ qRg and has the expected long-time asymptotic power law form: S(qRg) ~ (qRg)-1/df,. The time evolution of the cluster size distribution, Nn(t), arising in an aggregating system may be obtained in the framework of Smoluchowski’s equation once the aggregation kernel, that is, the set of aggregation rate constants kij, is known. However, no valid kernel for the description of aggregation processes of functionalized particle suspensions is known [54]. In order to overcome this difficulty, we use the kernel k11 a a (a) kij = ___ (i + j ) 2

aŒ

(13.15)

as a simple but versatile approximation. This kernel was employed before for describing coagulation– fragmentation processes and its applicability was confirmed by computer simulations [55]. It should be pointed out that the kernel given by Equation 13.15 is a relatively simple analytical kernel that was not derived for any specific aggregation process or physical situation. Hence, we can only expect it to be an approximation that might fit our data reasonably well. In order to keep things as simple as possible, we tried to use it directly for fitting all our experimental data. Doing so, the exponent a can be identified immediately as the homogeneity exponent l and the dimer formation rate constant k11 may be approximated by ks. Figure 13.12 shows the obtained results using the values of l and ks given in Table 13.2. In Figure 13.12, the experimental data for the normalized average diffusion coefficient ·DÒ/D 0 are plotted as points and the solid lines represent the best fits using the kernel given by Equation 13.15. As can be observed, the fitted curves agree quite well with the experimental data except for the bare particles aggregating at 0.250 M. A c2-method was used to select the most adequate fit using B/D0 as fitting parameter. Table 13.3 summarizes the minimized c and the corresponding B/D0 values. For bare particles aggregating at 0.250 M, the c2 value shows the behavior mentioned before. The structural coefficient k0 and the surface-to-surface distance S were obtained therefrom and are also included in Table 13.3. The fitting error was calculated as the standard deviation of three repeated aggregation experiments. For the bare particles at high electrolyte concentration, the structural coefficient k0 is very close to unity and the surface-to-surface distance between the particles contained in the aggregates is approximately zero. The expected result is in good agreement with other experiments [30,56] and simulations [57]. The obtained value is, however, significantly smaller than the values reported in [58] for carbonaceous soot aggregates formed in laminar diffusion flames. This is not surprising

309

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles

/D0

1

0.1

0.01 [KC1]= 0.700 M q = 0%

[KC1]= 0.700 M q = 50%

[KC1]= 0.250 M q = 0%

[KC1]= 0.250 M q = 50%

/D0

1

0.1

0.01

0.1

1

10 t/min

100

0.1

1

10 t/min

100

1000

FIGURE 13.12 Average diffusion coefficient normalized by D 0 for bare (0%) and coated particles (50%) aggregating at pH 4.8 and two different electrolyte concentrations. The solid lines show the best fits using the kernel given by Equation 13.15.

TABLE 13.2 Structural and Kinetic Parameters for Bare (0%) and Functionalized Particles (50%) at Different Electrolyte Concentrations 0.700 M 0%

0.250 M 50%

0%

50%

df

1.75 ± 0.03

2.15 ± 0.03

2.03 ± 0.03

2.27 ± 0.03

l

0.018 ± 0.014

0.044 ± 0.033

0.330 ± 0.018

-0.260 ± 0.028

6.33 ± 0.18

5.57 ± 0.28

0.82 ± 0.36

6.48 ± 0.32

ks (10-12 cm3 s-1)

310

Structure and Functional Properties of Colloidal Systems

TABLE 13.3 Fitting Parameters B/D0 and the Corresponding Minimized c2 Values Obtained from the Best Fits Using the Kernel Given by Equation 13.15 0.700 M

B/D0 c2 k0 S (nm)

0.250 M

0%

50%

0%

50%

0.997 ± 0.003 0.001

0.915 ± 0.003 0.004

0.859 ± 0.005 0.020

0.946 ± 0.003 0.005

0.995 ± 0.005

0.826 ± 0.004

—±—

0.883 ± 0.004

0.18 ± 0.18

9.13 ± 0.81

—±—

5.50 ± 0.55

Note: The corresponding structural coefficient k0 and surface-to-surface distance s are also given.

since overlap between the soot monomers and surface growth of the formed aggregates is possible in the latter case. For the coated particles, smaller structural coefficients were found. This means that, in this case, the distance between the surfaces of two adjacent primary particles is not vanishing. As can be seen in Table 13.3, the corresponding surface-to-surface separations were S 9 and 5.5 nm at 0.700 and 0.250 M, respectively. This means that the presence of protein molecules on the particle surface impedes a close surface-to-surface contact. Consequently, the found surfaceto-surface distances should be related with the adsorbed protein layer thickness. It seems reasonable to assume that the BSA molecules adsorb onto a not completely covered surface in a flat configuration. Hence, the surface-to-surface distance should lay around 2.7 nm, if only one flat layer of BSA molecules was present in the space separating the particles. The results obtained in this work, however, indicate that the gap between two adjacent particles contains between two and four layers of individual BSA molecules or between one and two layers of BSA dimers. In our opinion, the latter case seems to be more likely since gel electrophoresis measurements of the protein composition revealed that the BSAp sample used for adsorption contained mostly BSA dimers and some larger BSA aggregates. This finding confirms that the data reported in this work for the structural coefficient are reasonable and that the analysis of k0 is an adequate tool for estimating the thickness of adsorbed macromolecule layers contained in growing functionalized particle aggregates. It shows furthermore that k0 is directly related to the particle packing within the formed clusters and so contains additional information on the inner cluster structure. Hence, the fractal dimension alone is insufficient for characterizing cluster morphology completely. In our case, for example, the coated particles aggregating at 0.700 M and at 0.250 M have the similar aggregation rate ks and cluster fractal dimension df (see Table 13.2). This seems to indicate that there are no significant differences in either the aggregation kinetics or the cluster morphology. Nevertheless, the detailed analysis of the homogeneity exponent l and the structural coefficient k0 gives clear evidence that this is not the case. The changing sign of l indicates a strong change in the aggregation kinetics and the varying interparticle distance S shows that the formed clusters differ also in their internal structure.

ACKNOWLEDGMENTS Financial support from the Spanish Ministerio de Educación y Ciencia (Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I + D + i), Projects No. MAT200612918-C05-01, MAT2006-13646-C03-02, and MAT2006-13646-C03-03, Contract “Ramón y Cajal” RYC-2005-000983), the European Social Fund (ESF), the Junta de Andalucía (Excellency Project P07-FQM-02517), and the Acción Integrada (Project No. HF2007-0007) are gratefully acknowledged.

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14

Advances in the Preparation and Biomedical Applications of Magnetic Colloids A. Elaissari, J. Chatterjee, M. Hamoudeh, and H. Fessi

CONTENTS 14.1 Introduction ...................................................................................................................... 14.2 State of the Art in the Advancement of Magnetic Particles ............................................. 14.2.1 Principle in the Synthesis of Iron Oxide-Based Magnetic Nanoparticles ............ 14.2.2 Advancement of Modifications of Magnetic Nanoparticles ................................. 14.2.3 Embedding in Inorganic Matrix ........................................................................... 14.2.4 Embedding in an Organic Matrix ......................................................................... 14.2.4.1 Magnetic Latex Particles via Polymerization in Dispersed Media ........ 14.2.4.2 Magnetic Latex Particles from Preformed Polymers ............................. 14.3 Biomedical Applications of Magnetic-Based Particles .................................................... 14.3.1 Magnetic Nanoparticles in MRI (in vivo Diagnostic) ........................................... 14.3.2 In vitro Applications of Magnetic Particles .......................................................... 14.3.2.1 Conventional Biomedical Diagnostic Applications ............................... 14.3.3 Magnetic Particles in Microfluidic-Based Systems .............................................. 14.3.3.1 Magnetic Separation-Based Microsystems ............................................ 14.3.3.2 Magnetic Particles in Biosensing Devices ............................................. 14.3.4 Magnetic Particles as Labels for Detection .......................................................... 14.3.4.1 In Association with Magnetoresistive Detectors ................................... 14.3.4.2 In Association with a Magnetic Transducer .......................................... 14.3.5 Magnetic Gradient as a Dipstick-Like Approach ................................................. 14.4 Conclusions ....................................................................................................................... References ..................................................................................................................................

315 316 316 316 317 317 319 321 323 323 324 324 324 324 326 327 327 331 332 333 333

14.1 INTRODUCTION Nowadays, biomedical diagnostic, clinical analysis, and nanomedicine need tools, devices, and systems with highly automated operations, fast analysis, low volume analysis, and sensitivity similar to existing large-scale analysis equipment. Small size and robust mechanics are important to design cost-effective and easy-to-use portable devices for routine applications. Such systems are called micro-total analysis systems (m-TAS) (all steps are concentrated in one system) and were introduced in 1990 by Manz et al.1 The major approach is based on continuous flow methods. Among them, most involve liquids pumped through tubing, while others use centrifugal force or gravitation for liquid displacement. An alternative to a heavy process such as centrifugation is to manipulate 315

316

Structure and Functional Properties of Colloidal Systems

and control reagent-coated paramagnetic particles to magnetically induced chemical analysis (MICA). In this case, the magnetic particles act as a magnetic field stimuli responsive carrier. The major advantages of using microsystems can be described as follows: (i) small volume analysis (from a few nanoliters to microliters), (ii) a large surface-to-volume ratio of the microsystem offers an intrinsic compatibility between microfluidics and surface-based assays, and (iii) a faster reaction rate when molecular diffusion matches the dimension of the microchannel. Magnetic particles are not always advantageous in m-TAS applications. Externally, magnetic systems, which have to be used to manipulate particles within microchannels, complicate precision handling and result in bulky systems. Incorporation of magnetic component on the wafer level is also a complicated process.2,3 Moreover, there are difficulties associated with handling beads in etched channels. However, magnetic particles nonetheless present several advantages in m-TAS. In microfluidics, colloidal particles represent ideal reagent delivery vehicles and provide large reactive surface areas and a large binding surface capacity per unit volume. An additional advantage of using magnetic particles is related to the intrinsic magnetic property of the particles. In fact, a permanent magnet or an electromagnet can be used to manipulate, transport, and extract magnetic particles irrespective of the type of microfluidics or biological process used in the microsystem. Thus, the use of magnetic particles in the different automated systems and m-TAS (continuous flow or magnetically manipulated reagent) has been considered for a number of applications and principally in the bionanotechnologies domain.4

14.2

STATE OF THE ART IN THE ADVANCEMENT OF MAGNETIC PARTICLES

14.2.1 PRINCIPLE IN THE SYNTHESIS OF IRON OXIDE-BASED MAGNETIC NANOPARTICLES During the last few years, considerable research has been devoted to the synthesis of magnetic nanoparticles. Considerable data from the literature have described efficient synthetic methods to prepare reactive, stable, and monodisperse magnetic nanoparticles. There are various techniques for obtaining magnetic materials such as magnetite as a microcrystalline powder. Among these techniques, applied physical methods such as gas-phase deposition and electron beam lithography are known to form nanoparticles without any control in sizes.5,6 Thus, the wet chemical methods remain simpler and more effective, with an overall good control of the magnetic properties, size distribution, and chemical composition of the nanoparticles.7,8 In this context, coprecipitation is an easy and suitable method to synthesize iron oxide-based nanoparticles, either magnetite (Fe3O4) or maghemite (g-Fe2O3) from aqueous and ferric and ferrous salt solutions by the simple addition of a concentrated base solution under an inert atmosphere with or without heating.9 The final different characteristics of the synthesized clusters depend mainly on the used salts, the ratio Fe2+ /Fe3+, the pH, and the ionic strength of the medium and the incubation temperature.10,11 In this widely applied method, iron oxide-based particles, magnetite (Fe3O4), are prepared by using a molar ratio of 1 : 2 from a mixture of aqueous chloride of Fe2+ and Fe3+. The magnetic saturation values of these magnetite nanoparticles are experimentally determined to be generally lower than that of the bulk material. The chemical reaction of the precipitation of ferric and ferrous salts leading to magnetite is given below: Fe2+ + 2Fe3+ + 8OH- Æ Fe3O4 + 4H2O. The magnetite (Fe3O4) nanoparticles obtained are not very stable under ambient conditions, and they are oxidized to maghemite (g-Fe2O3) or hematite with different magnetic properties.

14.2.2 ADVANCEMENT OF MODIFICATIONS OF MAGNETIC NANOPARTICLES Although there has been significant progress in the synthesis of magnetic materials and principally magnetic nanoparticles, the long-term stability of their colloidal preparations is still an important issue toward optimizing their potential utilizations in new nanotechnologies. Ferrofluids, being

Advances in the Preparation and Biomedical Applications of Magnetic Colloids

317

defined as colloidal suspensions of magnetic nanoparticles (as maghemite or magnetite) and forming magnetizable fluids (i.e., ferrofluid), have been the focus of extensive researches approaching their preparation with higher stability. Ferrofluids remain liquid (monophasic liquid) even in very intensive magnetic fields, which have several potential applications. From this point of view, these magnetic fluids have very interesting properties due to their fluidity and ability to respond to an applied magnetic field.12–15 However, without any surface coating, magnetite or maghemite nanoparticles tend to aggregate, forming clusters, which resulted in an increase in their size. These clusters show a strong magnetic dipole–dipole attraction among them, thus inducing ferromagnetic behavior. Consequently, the induced aggregation of magnetic nanoparticles causes a remnant mutual magnetization16 of the dispersion (or dried material), which results in the irreversible aggregation of the particles. As is understood, the modification of particles’ surfaces by an adequate coating is essential to counteract the attraction forces between magnetic nanoparticles and to confer good stability to the particles in the dispersed medium. Indeed, the surface coating of magnetic nanoparticles usually has a drastic influence on the magnetic properties. These coating strategies generally result in a core–shell-like structure in which the naked magnetic nanoparticle is coated by a shell isolating the magnetic core from environmental influence. The coating (i.e., the shell part) can generally be carried out using inorganic or organic substances.

14.2.3 EMBEDDING IN INORGANIC MATRIX Various inorganic matrices such as silica,17,18 yttrium,19,20 and precious metals such as gold or silver21,22 have been used. For instance, Graf et al.17 described embedding of maghemite nanoparticles in silica spheres in order to protect the embedded nanoparticles against chemical degradation. In the first step, the authors use the amphiphilic polymer poly(vinylpyrrolidone) (PVP) to be adsorbed on iron oxide nanoparticles; then, the PVP-containing maghemite nanoparticles formed are transferred in an alcohol, such as an ethanol or a butanol solution (Figure 14.1). Subsequently, the nanoparticles are adsorbed on amino-functionalized silica seed particles. Finally, after addition of ammonia, a silica shell is grown on the maghemite nanoparticle-containing silica seed particles via consecutive additions of tetraethoxysilane. This process leads to a silica core and a silica meghemite-containing silica shell. It has been shown in the literature that silica coating confers to the magnetic nanoparticles owing to a rich surface in silanol groups that can easily react with alcohols and silane coupling agents to produce dispersions that are not only stable in organic solvents but also represent an ideal anchorage for covalent bonding of specific ligands in many biomedical applications.23

14.2.4 EMBEDDING IN AN ORGANIC MATRIX Iron oxides can be embedded in organic matrixes such as surfactant layers or soft or rigid polymers. Indeed, surfactants and polymers are generally used in order to passivate the iron oxide nanoparticles’ surfaces, avoiding consequently their aggregation. Generally, they are chemically anchored or adsorbed on magnetic nanoparticles to form steric or electrostatic repulsions between nanoparticles in order to enhance the colloidal stability of the dispersion. For instance, as mentioned above, ferrofluids, being prepared by the coprecipitation of Fe+2 and Fe+3 in ammonia, have a tendency to agglomerate. The classical example of this coating is the use of oleic acid (C18H34O2).24,25 To achieve stable colloidal preparations, the surface of iron oxide nanoparticles is therefore derivatized by carboxylate surfactants (e.g., lauric acid and oleic acid) or phosphonate and phosphate-based surfactants (e.g., hexadecylphosphonic acid and dodecylphosphonic acid).26 According to the authors, thermogravimetric analysis (TGA) revealed that phosphonate-based surfactants coating on the iron oxide particles’ surface were stronger than those of the carboxylate. Many reports in the literature have described the use of oleic acid-coated iron oxides (Figure 14.2) as a substance for a further encapsulation in polymers.27,28 Polymeric coating of iron oxide nanoparticles has shown several applications. The most explored applications are directly related to in vivo domain. The iron oxide nanoparticles are incorporated, precipitated, or encapsulated in a cross-linked polymer-based matrix of a polymer or a gel network

318

Structure and Functional Properties of Colloidal Systems Nanoparticle PVP*

(1) Adsorption

Aminofunctionalized SiO2-colloids (2)

(3)

Shell growth

(2)

Si(OEt)4 NH3, H2O O

N (CH—CH2)n

*)PVP = polyvinylpyrrolidone

FIGURE 14.1 Diagram of the general procedure for embedding nanoparticles in silica colloids: in the first step, PVP is adsorbed on the colloidal particles (1). After transfer in an alcohol, for example, ethanol or butanol, the particles are adsorbed on amino-functionalized silica colloids (2). Finally, after addition of ammonia, a silica shell is grown on the nanoparticle-decorated colloid by consecutive additions of tetraethoxysilane (3). (From Graf, G. et al. 2006. Langmuir 22: 5604–5610. With permission.)

(a)

(b)

50 nm 30

20 15 10 5 0

(d)

25 Frequency

Frequency

25 (c)

20 15 10 5

2

8 10 4 6 Particle diameter (nm)

12

0

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4 6 8 10 Particle diameter (nm)

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FIGURE 14.2 TEM images of size-selected HexaDecylPhosphonic acid-MP (a) and Oleic acid-MP (b) samples together with the corresponding size distribution histograms (panels (c) and (d), respectively). (From Sahoo, Y. et al. 2001. Langmuir 17: 7907–7911. With permission.)

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to prevent them from both chemical degradation and aggregation phenomena. Suitable polymers for the coating include, among others, poly(aniline), poly(pyrrole), poly(alkylcyanoacrylate), poly(esters) as poly(lactic acid), poly(lactic-co-glycolic acid), and poly(epsilon caprolactone) (PCL).27–31 14.2.4.1 Magnetic Latex Particles via Polymerization in Dispersed Media The advancement in the modification of magnetic colloids has challenged many research groups as evidenced by the numerous published works, and a detailed review of these works can be found in the following references.32–34 In the literature, various interesting approaches have been described regarding the modifications of magnetic latexes. The developed approaches range from classical heterogeneous polymerization processes such as emulsion,13 suspension,14 dispersion,15 miniemulsion,16 inverse emulsion,17 or inverse microemulsion18 to some multistep synthesis procedures.19–22 Briefly, two major strategies have been envisioned: one that uses preformed nonmagnetic particles, and the other incorporating the magnetic material (i.e., iron oxide) during the polymerization process leading to the composite particle formation. The first approach can be considered as a multistep process. In this direction, monodisperse magnetic particles over 2 μm in size were obtained by Ugelstad et al.35 by precipitating ferric and ferrous salt (into iron oxides) within preformed porous polymer microbeads. To avoid the release of incorporated iron oxide nanoparticles, encapsulation is performed using the seed polymerization process. The final particles present 15–20 wt% iron oxide content. Furusawa et al.36 and then Sauzedde et al.37,38 focused on heterocoagulated aspect of iron oxide nanoparticles onto seed polymer particles (latex particles of 500 size). To avoid cluster formation, latex particles were added drop by drop onto a concentrated iron oxide nanoparticle dispersion. The formed heterocoagulates were then encapsulated using seed radical emulsion polymerization to avoid leakage of the magnetic material. The latex particles used as seeds in those studies are polystyrene (PS), P(S/N-isopropylacrylamide (NIPAM)) core–shell, and PNIPAM produced via emulsion and emulsion–precipitation and precipitation polymerizations, respectively. The magnetic material content in the final magnetic latex particles is in between 10 and 30 wt%. The second strategy was pioneered by Daniel et al.39 using radical polymerization in dispersed media. The authors elaborated PS magnetic particles by dispersion of inorganic magnetic materials in an organic mixture of monomer and initiator emulsified in water, and polymerized. Though the magnetic material content was of 40–50 wt%, the resulting particles were fairly polydispersed in the presence of free inorganic nanoparticles and polymer particles. To improve the size distribution, Ramirez et al.40 mixed a miniemulsion of magnetic material with a second miniemulsion of styrene. After miniemulsion polymerization process, particles in the 40–200 nm range were obtained with a maximum magnetic content of 35 wt%. …The magnetic latex particles present attained the following morphologies: a well-defined magnetic core with a polymer shell, polymer particles with a heterogeneous iron oxide nanoparticle distribution in the polymer matrix, and a polymer seed bearing magnetic nanoparticles in the shell as illustrated in Figure 14.3. Indeed, works on the magnetic latex particles preparation via miniemulsion polymerization have shown many limitations to this method such as (1) an inhomogeneous distribution of the magnetic

FIGURE 14.3 Possible morphologies of magnetic latex particles: (a) core–shell, (b) nanoparticles dispersed in a polymer matrix, and (c) nanoparticles immobilized onto seed particles.

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FIGURE 14.4 SEM micrographs of N-isopropylacrylamide microbeads. (From Muller-Schulte, D. and Schmitz-Rode, T. 2006. J. Magn. Magn. Mater. 302: 267–271. With permission.)

nanoparticles inside and among the particles, (2) pure polymer particles (no incorporation of magnetic material), (3) free nanoparticles or magnetic aggregates in the aqueous phase, (4) large particle size distribution, and/or (5) limited loading of the particles with magnetic material. In an attempt to address some of these challenges, Joumaa et al.41 prepared magnetic PS nanoparticles via miniemulsion polymerization. To make the iron oxide nanoparticles easier to encapsulate, the authors modified iron oxide nanoparticles using a nonconventional phosphate-based poly(propylene glycol) methacrylic monomer as a stabilizer. This stabilizer has a phosphate group known to strongly link to an iron oxide surface26 at one extremity, and a polymerizable methacrylic function at the other one, separated by a short spacer poly(propylene glycol) chain. The methacrylic function was used to favor irreversible incorporation of iron oxide inside the PS particles through copolymerization with styrene. The results showed the crucial role of the stabilizer used in the preparation method. Whereas oleic acid-based ferrofluids led to an inhomogeneous distribution of maghemite nanoparticles inside the PS particles, phosphate-based macromonomer/styrene-based ferrofluids yielded magnetic particles with a homogeneous distribution of maghemite inside PS particles. However, the authors report a broad particle size distribution but associated with a high iron oxide loading (~30 wt%). Both hydrophilic and hydrophobic vinyl monomers have been applied in the preparation of magnetic latex particles via water-in-oil (W/O) and oil-in-water (O/W) emulsions, respectively. In this respect, Muller-Schulte et al.42 prepared magnetic thermosensitive polymer microspheres from an NIPAM monomer in a W/O suspension polymerization (Figure 14.4). The authors claimed that the presence of magnetic material inside the obtained microspheres would allow inductive heating of the particles using an alternating magnetic field above the polymer transition temperature (>35°C) and release encapsulated drugs for in vivo applications. More recently, Montagne et al.43 developed a new method related to radical polymerization and transformation of magnetic droplets into magnetic latex particles using hydrophobic monomers (i.e., styrene) and submicronic droplets of a highly stable magnetic emulsion (Figure 14.5a). The particle size distribution was controlled by the size distribution of the initial magnetic emulsion. In more detail, the synthesis of submicronic highly magnetic latex particles exhibiting various morphologies has been reported. Basically, the methodology consists in radical polymerization of hydrophobic monomer swollen O/W ferrofluid emulsion droplets. The influence of several parameters [i.e., nature of initiator, presence of a crosslinker divinylbenzene (DVB), and adsorption of a carboxylic-containing amphiphilic copolymer] has been investigated on the polymerization kinetics (conversion) and type of final latex particle morphology. It was proved that homogeneous encapsulation of the iron oxide nanoparticles was efficiently achieved using the following conditions: preadsorption of the amphiphilic copolymer on the ferrofluid droplets; use of styrene/DVB monomer mixture (60/40 mass ratio); and potassium persulfate as an initiator. Whereas, the polymerization conducted using styrene only leads to or gives rise to asymmetric-like morphology. The elaborated magnetic latexes were characterized both by transmission electron microscopy (TEM) (Figure 14.5b) and by chemical composition. A high iron oxide content of about 60 wt.% and high carboxylic surface charges were reported. Finally, a tentative polymerization mechanism was proposed and discussed. It is interesting to note that all mentioned, synthesized, and modified magnetic latexes are for in vitro biomedical applications only.

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–c

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coo–

Non ionic surfactant coo–

Ionic surfactant Aqueous phase

+c

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FIGURE 14.5 (a) Turn (O/W) magnetic emulsion into magnetic latex via radical polymerization and (b), Transmission microscopy analysis of magnetic latex particles. (From Montagne, F. et al. 2006. J. Polym. Sci. Part A: Polym. Chem. 44: 2642–2656. With permission.)

14.2.4.2 Magnetic Latex Particles from Preformed Polymers Generally, the synthesis methods involve preparing emulsions in which the polymer is dissolved in aqueous or organic solvents according to the properties of the used polymer, which may be hydrophilic or hydrophobic. Both hydrophilic and hydrophobic polymers have been successfully used to prepare magnetic spheres. Grüttner et al.44,45 prepared biodegradable magnetic polymer particles by coating superparamagnetic materials with natural or synthetic hydrophilic polymers such as dextran, starch, chitosan, and PVP in aqueous solutions for intravenous administrations. The particle shape and size distribution were mainly determined by the iron oxide core sizes and influenced by the molecular weight and amount of the polymer. Wang et al.46 reported the preparation of magnetic chitosan nanoparticles by adding the basic precipitant NaOH solution into a W/O microemulsion system. It was found that the diameter of magnetic chitosan nanoparticles was from 10 to 20 nm. Furthermore, other research groups investigated the coating of iron oxide nanoparticles with dextran.47 In their work, dextrancoated magnetic nanoparticles were prepared by a chemical coprecipitation of iron salts method in the presence of a dextran solution. The immediate coating of the formed nanoparticles leads to stable dextran-containing magnetic nanoparticles (30 nm hydrodynamic size). Many works reported in the literature handled the utilization of hydrophobic polymers such as polyesters (e.g., polylactide, PLLA) and polyepsilon caprolactone (PCL)28 in the preparation of magnetic nanoparticles and microparticles via solvent evaporation27 or solvent diffusion48 methods (Table 14.1). In both methods, iron oxide fine nanoparticles are dispersed in the organic phase including the polymer. The organic phase is emulsified in an external aqueous phase containing a stabilizing polymer or a surfactant such as polyvinyl alcohol (PVA) or pluronic to form an O/W emulsion (Figure 14.6). For instance, applying the solvent evaporation method, Hamoudeh et al.,27,28 have shown the ability to incorporate high amounts of magnetite into poly(lactic acid)-based nanoparticles and PCL-based microparticles with high magnetic saturation values. In addition, they have shown the possibility to use those particles as a negative contrast agent in magnetic resonance imaging (MRI). Briefly, the magnetite crystals were dispersed in a dichloromethane phase containing the dissolved polymer. This organic phase was then emulsified in an aqueous phase containing PVA to form the O/W emulsion. The evaporation of dichloromethane thereafter enabled the precipitation of the dissolved polymer into magnetite-loaded polymeric particles. The magnetite nanoparticles are simply mechanically entrapped during the polymer precipitation process. On the other hand, the solvent diffusion method, being used to prepare magnetic nanoparticles, consists of using a partially water-miscible solvent like ethyl acetate as an organic solvent. This

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TABLE 14.1 Magnetic Latexes from Preformed Polymers Polymer or Monomer Poly(d,l-lactide-co-glycolide) (PLGA) Poly(lactide) (PLLA) PCL Polyvinyl alcohol (PVA) Chitosan Poly(glycidyl methacrylate) (PGMA) Dextran Polystyrene-core/poly (N-isopropylacrylamide) (NIPAM) Preformed polystyrene NIPAM Poly(ethyl-2-cyanoacrylate)

Method of Preparation

Particle Sizes

O/W emulsification followed by solvent diffusion O/W emulsification followed by solvent evaporation O/W emulsification followed by solvent evaporation Precipitation in PVA solution NaOH solution into a W/O microemulsion system Emulsion polymerization Chemical coprecipitation method in presence of dextran Emulsion and precipitation polymerizations Precipitation of iron oxides within preformed porous polymer beads In a W/O suspension polymerization Emulsion/polymerization

Reference

90–180 nm

Lee et al.48

320–1500 nm

Hamoudeh et al.27

3–20 mm

Hamoudeh et al.28

4–7 nm 10–20 nm

Lee et al.51 Wang et al.46

72–84 nm 18 nm

Pollert et al.52 Autenshlyus et al.53

400 nm

Sauzedde et al.38,39

2 μm

Ugelstad et al.35,36

10–200 μm 380 nm

Muller-Schulte et al.42 Arias et al.31

solvent can be emulsified in an aqueous solution of a stabilizing agent, followed by diluting the internal phase with an excess of water to induce the precipitation of the polymer. Using this method, Lee et al.48 reported the synthesis of poly(d,l-lactide-co-glycolide) (PLGA) with a spherical shape of 90–180 nm with good magnetic loading. After emulsifying the iron oxide-PLGA-containing saturated ethyl acetate phase in the pluronic-containing aqueous phase at a high speed using a homogenizer, an excess amount of water was added to the O/W emulsion under ultrasound. The subsequent addition of water dilutes the solvent concentration in water and extracts solvent from the organic solution, leading to the nanoprecipitation of the polymer matrix entrapped with iron oxide nanoparticles. Other research groups reported the modification of the surfaces of iron oxides by hydrophilic macromolecules such as PVA48 and proteins.49,50 In this work, Lee et al.51 carried out a coprecipitation of iron salts in an aqueous solution of PVA to form a stabilized dispersion. They reported a decreasing crystallinity of iron oxide particles accompanying the increase of the concentrations of PVA, while the morphology and particles size remained unchanged. Other studied materials to encapsulate individual iron oxide nanoparticles or small clusters via polymerization methods involved natural polymers52,53 or albumins.50 Furthermore, Chatterjee et al.49 prepared cross-linked albumin magnetic microspheres and could use them for red blood cell separation. Another work of Chatterjee et al.50 described the incorporation

Emulsification

Organic phase Aqueous phase containing magnetite containing a containing the surfactant magnetite + polymer

Evaporation

Simple emulsion

Nanoparticle suspension

FIGURE 14.6 Modification of magnetic latex particles via an emulsification and evaporation process.

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of g-Fe2O3 into a biocompatible polymer gel using PVA. The obtained magnetic gel was dried to form a biocompatible magnetic film. The authors reported success in efficiently crosslinking magnetic nanoparticles in the polymer network with superparamagnetic properties.

14.3 BIOMEDICAL APPLICATIONS OF MAGNETIC-BASED PARTICLES 14.3.1 MAGNETIC NANOPARTICLES IN MRI (IN VIVO DIAGNOSTIC) During the last three decades, great progress has been achieved in the field of pharmaceutical technology toward the synthesis of sophisticated systems of nanoparticles for pharmaceutical applications as novel tools for drug delivery. These nanoparticulate dispersions should improve positively the pharmacokinetics of drugs including their rate of absorption, in situ distribution, metabolism effect, toxicity, and their specific targeting. Thereby, these nanoparticulates in drug delivery systems can allow the drug to bind to its target receptor and influence this receptor’s signaling mechanism and activity.54 Generally, the materials used in the design of nanoscale drug delivery systems should be compatible, able to degrade into eliminable fragments after drug release. Among the different researches carried out to design nanoparticulate systems for medical applications, interesting approaches using nanoparticles as imaging tools for in vivo nanodiagnostic by incorporating luminescent quantum dots,55 liquid perfluorocarbons for ultrasonic imaging,56 and a paramagnetic or a superparamagnetic contrast agent for MRI27,57 have been conducted and reported. Among the above-mentioned systems as diagnostic tools, magnetic nanoparticles have specially attracted considerable attention for their current usefulness as contrast agents in MRI. Apart from their utilization as imaging probes, they can also be used for various applications in the field of medicine and biotechnology such as in hyperthermia,58 deoxyribonucleic acid (DNA) separation,59 drug targeting,60 and enzyme purification.61 A great variety of organic materials has been used to prepare magnetic nanoparticles such as dextran,62 poly(vinyl alcohol),63 PS,64 PVP,65 poly(aniline),66 and polyesters such as poly(lactide) (PLLA) and CL.25 Generally, to elaborate high magnetizable nanoparticles, iron oxides with defined saturation magnetization (Ms) such as paramagnetic or superparamagnetic namely magnetite (Fe3O4) and maghemite (g-Fe2O3) are used in imaging-based in vivo diagnostic. For in vivo applications, numerous magnetic nanoparticle preparation strategies have been described in the literature including emulsion polymerization,67 suspension polymerization,68 dispersion polymerization,69 microemulsion polymerization,70 solvent diffusion,48 and solvent evaporation.71 In this context, magnetic nanoparticles have attracted considerable attention for their great usefulness as contrast agents in MRI with different products already being used in clinics. RI is a noninvasive imaging method using nuclear magnetic resonance to enable obtaining images of the internal portions of the human body. In medicine, it is used to demonstrate pathological or other physiological alterations of living tissues. In an MRI examination, the patient is subjected to an electromagnetic field of a defined strength (given by a Tesla unit). Under the magnetic field, the magnetic particles localized on the tumor, for instance, can be detected. Indeed, the magnetic nanoparticles’ effect in the MRI sequence can be attributed to the resulting magnetic field heterogeneity around these particles and through which water molecules diffuse, inducing proton relaxation modification. Generally, iron oxide crystals between 5 and 12 nm in size can be encapsulated within these magnetic nanoparticles to be used in standard or functionalized MRI. Physically, the superparamagnetic behavior of these subdomain magnetic cores is similar to that of paramagnetic substances [gadolinium (Gd) chelates], but the superparamagnetic iron oxide material is accompanied by a much higher value of magnetic moment and consequently stronger relaxivity compared to gadolinium-loaded contrast agents.72 Some promising researches addressed combining drug delivery and MRI approaches in a onedesign system.25,73 For example, in clinical oncology, magnetic MRI-guided delivery of drug-loaded nanoparticles administered by an intratumoral injection, or intravenously, can result in a remarkably better treatment efficiency.73,74 This can be explained by the fact that the possibility to use MRI to

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visualize anticancer agent-loaded nanoparticles distribution in/on tumors and their adjacent healthy tissues is very important because it leads to many advantages such as (i) a scanning step using a tracer dose without drug for a preimaging of the distribution drug-free nanoparticles in order to enable a further good prediction of tumor targeting and less toxicity to surrounding tissues and (ii) monitoring during and after administration of the drug-loaded nanoparticles leading to an optimal treatment.

14.3.2

IN VITRO APPLICATIONS OF MAGNETIC PARTICLES

Colloidal particles are largely used in biomedical applications and they are principally used as solidphase supports of biomolecules or basically as carriers in various technological aspects. The reactive particles needed are elaborated using many heterophase processes (emulsion, dispersion, precipitation, self-assembly, and physical processes). In this direction, various polymer-based colloids (well adapted for automated systems, nanobiotechnologies, and microsystems) have been prepared for in vitro biomedical applications.4,33,75 14.3.2.1 Conventional Biomedical Diagnostic Applications Before the investigation of any targeted biomedical diagnostic application, several aspects related to colloidal particles should be addressed starting basically from particle size, size distribution, surface polarity, and also intrinsic properties. Consequently, the synthesis process should be well adapted in order to prepare structured latex particles bearing a reactive shell with well-defined properties. The colloidal particles are not only under evaluation or being used as a model in various biomedical applications, but actually are in use in different capacities for various biomedical applications. Some examples are as follows: (i) They are used in rapid diagnostic tests based on the agglutination process76,77 (i.e., submiron sulfate and also carboxylic polystyrene latexes). (ii) Cationic particles (i.e., magnetic latexes) are principally used in nucleic acid extraction and concentration.75,78 (iii) Cell sorting and identification using magnetic or fluorescent particles.79 (iv) Virus extraction and detection via hydrophilic and charged magnetic particles.80 (v) General particles (mainly carboxylic on the surface) are used in immunoassays and specific capture of single-stranded DNA fragments.81,82 The specificity and the sensitivity of the targeted application efficiency are directly related to the surface particles’ properties and to the accessibility of the immobilized biomolecules. The interactions between biomolecules and reactive particles are strongly dependent on the colloidal and surface properties of the dispersion and the physicochemical properties of the biomolecules. In this direction, considerable attention has been paid to the preparation of magnetic latex particles for automated microsystems based on nanobiotechnologies applications.34 The main advantage of colloidal magnetic particles is their separation upon applying an external magnetic field.

14.3.3 MAGNETIC PARTICLES IN MICROFLUIDIC-BASED SYSTEMS 14.3.3.1 Magnetic Separation-Based Microsystems To contribute to the elaboration of a m-TAS, Furdui et al.83 developed an immunomagnetic cell separation technique on a microfluidic platform combined with magnetically trapped bead beds, to isolate specific cells from blood samples, in order to make the sample clean (Figure 14.7). Protein A-coated paramagnetic beads (1 to 2 mm of diameter), functionalized with antihuman CD3, are first introduced into the chips and captured in a field generated by an external magnet. A blood sample is then introduced; T cells are captured, rinsed, collected at the chip outlet, and moved to a different module (off-chip in this case) for subsequent analysis. Several different formats for the fluidic design of a

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(a) Magnetic beads

Magnet Syringe pump

(b)

Magnets Blood

Syringe pump

(c) Micropipet

FIGURE 14.7 Immunomagnetic separation of T cells using a Y-intersection device; (a) syringe pump draws Protein A/anti-human CD3 magnetic beads into the channel for capture with a magnet; (b) blood samples are introduced from both channels, T cells are captured with magnets above and below, then washed with RPMI 1640; (c) magnets were removed and captured cells and magnetic beads were transferred from the outlet with a micropipette. The inset shows T cells captured by beads in a magnetic field. Beads show as dark streaks, and cells as translucent circles (cells are about 15mm in diameter). (From Furdui, V. I. and Harrison, D. J. 2004. Lab Chip 4: 614–618. With permission.)

cell capture system were evaluated. Compared to open tubular capture beds, the magnetically trapped bead provide a means to increase the capture surface area for cells and to readily release the captured cells after washing. The author also demonstrated that the use of many narrower channels is more effective than a wide channel in structuring dense magnetic bead beds.83 Nucleic acid extraction and analysis are the targets of bionanotechnologies in order to replace the heavy classical tools. Xie et al.84 developed the utilization of modified magnetic beads to simultaneously enrich target cells and adsorb DNA from saliva, again to perform the sample cleanup that is needed for PCR. First, saliva is incubated with the nanobeads, which are then immobilized using a Promega magnetic stand, and the supernatant is removed. Next, a lysis buffer is added, and after incubation the beads are magnetically separated and washed. Elution of DNA can be performed, but the nanobead complex can directly be used as PCR templates. HLA typing based on an oligonucleotide array could then be conducted by hybridization with the PCR products. The authors envisaged the construction of miniaturized devices for automated biological molecule separation.84 In molecular biology related to cell analysis, Gijs et al.85 reported a microsystem that combines droplet microfluidics and magnetic microparticles for the extraction and purification of DNA from μL-sized lysed cell samples. The DNA is detected on-chip via fluorescent microscopy or via an offchip amplification step, and by this process extraction and detection of DNA is possible even from as few as 10 cells. In this process, the cells are lysed in a solution containing guanidine thiocyanate, which helps to selectively attach the DNA to the magnetic silica particles. After extracting a small droplet containing the magnetic particles and the attached DNA from the immobilized lysis buffer droplet, the particle-and-DNA compound is passed through three stages of washing. As the last step of the on-chip DNA extraction protocol, the purified DNA is eluted from the particles in a buffer of low ionic strength. Subsequently, the eluted DNA can either be detected using fluorescent microscopy or it can be transferred to a step of amplification via a polymerase chain reaction (PCR). The specific extraction of nucleic acid molecules is the only method of molecular biologybased diagnostic. Jiang et al.86 have reported the isolation of messenger ribonucleic acid (mRNA)

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from samples using commercially available paramagnetic oligo-dT beads in microchannels. The beads are PS (polystyrene based particles) less than 3 mm in diameter that are prepared with covalently attached chains of poly(T). Beads and samples are introduced from two arms of a Y-junction and can be mixed rapidly by diffusion. Reacted beads can then be magnetically trapped while the sample is flushed away, and then released too. It was estimated that 34 ng of mRNA could be retained from 10 mg of total RNA in a sample. 14.3.3.2 Magnetic Particles in Biosensing Devices Immobilization of reagents on particles has several advantages: It reduces the natural loss of biological activity, allows for preconcentration of the analyte, and thus increases the sensitivity of the assay.87 The combination of magnetic latex particles and nucleic acid molecules has been largely examined for classical diagnostic applications. Fan et al.88 attached different single-stranded DNA fragments to the poly(thymine) chain (i.e., oligodT25) containing beads. The sensitivity has been examined using a fluorescent DNA probe. Modified particles were held in place with a magnet (i.e., permanent magnetic field) in a microchannel to form a plug and perfused with the probe-containing solution using a pressure-driven flow. Hybridization was reported to be rapid since only a few seconds are needed compared to hours in bulk. For this study, it was concluded that the interaction between the capture probe and the free nucleic acid molecules in the medium is facilitated due to rapid delivery of the target molecules in the vicinity of the capture probe immobilized onto magnetic beads. Magnetic particles’ sorting is a promising technology in Microsystems and lab-on-chip developments. In this domain, Choi et al.89,90 designed a new planar magnetic bead separator on a glass chip, which permits the successful separation of magnetic beads from a suspension medium (Figure 14.8). The planar magnetic bead separator (used instead of a conventional magnet) is integrated into a microfluidic biochemical-based detection system for protein analysis (immunoassay) using magnetic beads.2,89 The separation yield, rate, and capability of the device were first tested using 1 mm superparamagnetic beads.90 Then the beads were used in an immunoassay as a solid support for the capture of antibodies and as biomolecule carriers for the captured target antigens.89 Antibody-coated magnetic beads can be held by the magnetic field, which is created by the planar electromagnet (also called “biofilter”). Upon injection of antigens, only specific antigens bind to the antibodies and are immobilized, whereas other antigens are washed away. Next, enzyme-labeled secondary antibodies are introduced and bound to the immobilized antigens. After washing, substrate solution allows the electrochemical detection using an electrochemical sensor. After the release of the magnetic beads, the bioseparator is ready for another immunoassay. Both the biofilter and the electrochemical sensor are surface-mounted on a microfluidic motherboard, which contains microchannels (400 mm × 100 mm) fabricated by glass etching and a glass-to-glass direct bonding technique. The inlet and the outlet are

Bioﬁlter

Bioﬁlter & immunosensor

Flow sensor

20 mm Microﬂuidic system Microvalves 50 mm 80 mm

FIGURE 14.8 Schematic diagram of a generic microfluidic system for biochemical detection. (From Choi, J. et al. 2002. Lab Chip 2: 27–30. With permission.)

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Enzyme substrate

Antibody – e– e e– Enzyme product

Electrochemical detection

Magnetic bead Target antigen

FIGURE 14.9 Analytical concept based on sandwich immunoassay and electrochemical detection. (From Choi, J. et al. 2002. Lab Chip 2: 27–30. With permission.)

connected to reservoirs containing the different biochemical reagents (buffer, substrate, beads, antigens, and antibodies). This technique allows a very short time of immunoassay (total assay time was less than 20 min), and a very low sample volume was necessary (10 mL per immunoassay).89 Electrochemical detection is well suited for m-TAS: In this case, it uses the detection of p-aminophenol (PAP), which is the product of the conversion of substrate p-aminophenyl phosphate by the labeling enzyme alkaline phosphatase. By applying an oxidizing potential to the interdigitated array microelectrodes, PAP is converted into 4-quinoneimine by a 2-electron oxidation that is recognized as a signal in terms of electrical parameters.89 The beads on which the device was first tested were commercially available Estapor carboxylate-modified superparamagnetic beads of 1 mm diameter (Bang’s Laboratories)90 (Figure 14.9), and, for the immunoassay, Dynabeads M-280 (Dynal Biotech Inc.) coated with biotinylated sheep antimouse IgG were used.89 To perform chemical reactions on nonmagnetic and magnetic beads, Andersson et al.3 designed, manufactured, and characterized a flow through microfluidic device for bead trapping (Figure 14.10). The device has an uncomplicated design and is batch-fabricated by deep reactive ion etching and sealed by anodically bonded Pyrex to enable real-time optical detection. The beads are applied at the inlet and collected in a square reaction chamber. A waste chamber surrounds the reaction chamber and is connected to the outlet. The reaction chamber is defined by pillars, which compose a filter for trapping particles. Magnetic Dynabeads with a diameter of 2.8 mm (Dynal) were tested.

14.3.4 MAGNETIC PARTICLES AS LABELS FOR DETECTION The well-known application of fluorescent magnetic latex particles is cell sorting using a flux cytometry equipment. However, the use of fluorescent particles has been widely explored in various new technologies such as lab-on-chip devices91 and mircofluidic systems.4 It is interesting to note here that the use of fluorescent or photophysics-based measurement is incontestably impossible to avoid in bionanotechnologies. Since the advent of bionanotechnolgies, various detection tools and concepts have been explored in order to use magnetic particles as labels rather than carriers only. 14.3.4.1 In Association with Magnetoresistive Detectors A magnetoresistive detector is a new technology first explored for magnetic particles’ concentration measurement based on a well-established calibration curve and then extended as biomedical

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1300–2450 mm

f 1 mm

50 mm

Inlet

100–1000 mm

2.5 mm

9 mm

Beads Reaction chamber Filter

1785–2250 mm

Waste chamber

Outlet

100 mm

f 1 mm

FIGURE 14.10 A schematic of the micromachined flow-through device. (From Andersson, H. et al. 2000. Sens. Actuat. B 67: 203–208. With permission.)

detection tools92 (Figure 14.11). In this direction, Schotter et al.91,93 have set up a prototype biosensor consisting of a giant magnetoresistive (GMR)-type multilayer, allowing the detection of magnetic. The system was tested to DNA microarray91 labels (Figure 14.12). In this case, the magneto resistive biosensor is based on the same principle as fluorescence detection (molecular recognition between DNA sequences immobilized locally at the sensor surface and DNA sequences to be analyzed, the only difference being the marker, magnetic property instead of fluorescent). The magnetic stray field of the markers is detected as a resistance change in a GMR-based magneto resistive sensor embedded underneath the probe DNA spot. The detection was examined as a function of concentration of double- stranded PCR-amplified DNA sequences of 1 kb, spotted on the surface as a capture probes, onto which biotin-labeled complementary DNA was hybridized, before addition of the streptavidin-coated paramagnetic markers. The feasibility of detection of DNA was demonstrated by detecting a value lower than 16 pg/mL (more sensitive that standard fluorescent detection).91,93 It was also shown that even single markers could be detected and that even a single marker could be manipulated, as demonstrated by optical microscope observation.93 The magnetic beads that can be used in such magnetoresistive biosensor are those commercially available in a wide range of sizes, functionalities, and magnetic properties. However, they have to satisfy a number of requirements: good colloidal stability, ability to bind specifically to biotinlabeled targeted DNA, no chemical degradation, low affinity to the sensor surface, high magnetic

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Y X

M

+

B

–

Si3N4 H

GMR ﬁlm Si substrate Electromagnet

FIGURE 14.11 Cross-section of a GMR sensor, illustrating the method used to detect superparamagnetic beads. A magnetizing field H magnetizes the bead, which produces regions of positive and negative magnetic induction B in the plane of the underlying GMR film. Because the film is only sensitive to the X component of external magnetic fields, the magnetizing field does not affect the GMR resistance. (From Baselt, D. R. et al. 1996. J. Vac. Sci. Technol. B 14: 789–794. With permission.)

moment for good and rapid detection (i.e., high content of magnetite), low sedimentation, and a narrow size distribution of the used magnetic particles. Some submicron magnetic particles have been evaluated as a magnetoresistive biosensor uniform in shape and in size.91,93 A microsystem named BARC (Bead ARray Counter), using such a magnetoresistive biosensor, has been recently described94–96 (Figure 14.13). It is a table-top instrument currently containing a 64-element sensor array. The beads are detected by giant magnetoresistance (GMR) magnetoelectronic sensors embedded in the chip. The chip can be coated with DNA probes. DNA targets that are biotin-labeled and that hybridize to the capture probes are detected using streptavidin-labeled magnetic beads.95 Otherwise, any ligand–receptor interaction in a “sandwich” configuration can be used.94,96 The BARC can be applied, in particular, for the detection of warfare agents.96 The GMR sensors detect the magnetic beads remaining at the surface after washes, as well as the intensity and location, indicating the concentration and the identity of the target.95 Another advantage of magnetic beads as a label is that a magnetic field can be applied to selectively pull off only those beads that are not specifically bound, which leads to a reduction in the background.95,96 The magnetic particles used in the BARC instrument must have as high a magnetization as possible to maximize the sensor response, and yet remain nonremnant to avoid cluster formation. Factors that determine the optimal bead size include the settling time for suspended beads (i.e., their buoyancy), the magnitude of the force that can be applied to the settled beads (to discriminate against the background), and the sensor response.94 Seradyn’s 0.7 mm Sera-MagTM magnetic beads coated with streptavidin can be used for DNA hybridization.96 M-280 Dynabeads (Dynal Inc.), which are 2.8 mm diameter PS spheres impregnated with 15 nm diameter Fe2O3 particles that compose about 6% or the total volume of the bead, were also used,95 but these resulted in higher nonspecific background adhesion.96 The technique could be improved by using higher “magnetic density” beads, like soft ferromagnetic beads, and some efforts have been undertaken to use NiFe beads.94,95 NiFe beads can be produced by a carbonyl process that generates polydispersed, polycrystalline spherical particles ranging from 800 nm to 4 mm in diameter. They have a superior magnetic moment but they

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SiO2

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SiO2

FIGURE 14.12 Principle of the magnetoresistive biosensor: (a) immobilization of the probe DNA; (b) hybridization of the analyte DNA; and (c) binding of the magnetic markers and detection of their stray field by the GMR-sensor. (From Schotter, J. et al. 2004. Biosens. Bioelectron. 19: 1149–1156. With permission.)

BARC

Data acquisition and analysis computer

Electr onic magne s and tics bo x Assay cartridge

FIGURE 14.13 Photograph of the table-top BARC prototype. The BARC chip and fluidics are contained in the assay cartridge, and the electromagnet assembly and electronics in the electronics and magnetics box. A portable computer is used for data acquisition and analysis via a connection with the serial port. (From Edelstein, R. L. et al. 2000. Biosens. Bioelectron. 14: 805–813. With permission.)

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still need to be size-selected and characterized, and the surfaces must be stably functionalized.94 New magnetic latex particles have been elaborated for microsystem- and microfluidic-based applications as reported by Montagne et al.43 These highly magnetic latex particles are submicron in size, narrowly size distributed, with an iron oxide content above 50%, good colloidal and chemical stability, and a moldable surface functionality. Using BARC, the threshold for detection is approximately 10 beads per 200 mm diameter sensor and a BARC GMR sensor strip can detect the presence of less than a single magnetic bead per sensor. The current BARC chip contains 64 elements sensor array; however, it could be increased using advanced magnetoresistive technology. Thermoplastic-molded microscale channels, displacement pumps, membrane valves, and reservoirs have been developed. 14.3.4.2 In Association with a Magnetic Transducer Changes in magnetic permeability can be measured using a coil, and this measure, involving ferromagnetic substances (but not magnetic beads), can be used to detect a bioanalyte.97,98 The relative magnetic permeability (m r) is a constant, specific for a given material, which provides a measure or a material’s ability to contain and contribute to an externally applied magnetic field (Figure 14.14). When m r is high, the material is called ferromagnetic. Three different detection approaches can be proposed: direct (the analyte is ferromagnetic), sandwich (the nonferromagnetic analyte requires the use of a ferromagnetic label), and competitive (utilization of a ferromagnetic competitor that competes with the nonferromagnetic analyte for binding). The feasibility of a competitive model system was tested and demonstrated: concanavalin A (Con A) immobilized to a carrier (sepharose) is chosen as a biorecognition element for detection of glucose, in competition with a ferromagnetic dextran ferrofluid. Subsequently, feasibility studies were conducted to determine whether a “sandwich” configuration, constituted by silica carriers, ConA, and magneto labels, can be used for the detection of ConA in a binding assay based on passive protein adsorption. This method could be useful for various analyte systems such as antibodies/antigens, receptors/peptides, or DNA/DNA98 and is a step toward achieving magneto immunoassays.97 This detection system is interesting but the sensitivity is poor compared to classical photophysics-based technology.

(a)

(b)

Labeled silica << Sandwich >> eﬀect observed in solution Magneto markers Silica carriers

(c)

Con A protein layer Measuring cell

Change in inductance

FIGURE 14.14 “Sandwich” approach used in the magneto-binding assay, where the target analyte is bound between the silica carrier particles and the magneto markers (a). Subsequent sedimentation of the protein/ particle complex (b) allows the magnetic permeability meter to measure the enrichment of magnetic markers at the bottom of the vial (c). (From Kriz, K. et al. 1998. Biosens. Bioelectron. 13: 817–823. With permission.)

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14.3.5 MAGNETIC GRADIENT AS A DIPSTICK-LIKE APPROACH An alternative to continuous flow methods is to manipulate reagent-coated paramagnetic particles (for MICA). The principle of the MICA system is to substitute liquid movement with magnetically induced movement of particles. Additionally, the immobilization of reagents on particles has several advantages: it reduces the natural loss of biological activity, allows for preconcentration of the analyte, and thus increases the sensitivity of the assay. Ostergaard et al.87 reported a novel approach of the automation (MICA) of clinical chemistry by controlled manipulation of magnetic particles (Figure 14.15). For immunoassay-based applications, the principle is as follows: four compartments constitute the system, one for IgG-coated paramagnetic particles, the second for potentially infected blood, the third for fluorescently labeled IgG, and the fourth is dedicated to detection of the magnetic particles (filled with a nonfluorescent buffer). When an electromagnet is turned on, the particles are dragged first in the sample compartment, then in the fluorescent IgG compartment, and then in the detection compartment, allowing binding of the virus, binding of the fluorescent IgG, and detection

Storage compartment

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D

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-IgG

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Magnetic gradient

(c) -IgG-V

D No flow No flow

Magnetic gradient

(d) D

-IgG-V-F-IgG No flow

FIGURE 14.15 Example of an immunoassay performed via MICA. (a) The system is comprised of four compartments: (1) a storage compartment for the immunoglobulin (IgG)-coated particles; (2) a sample compartment; (3) a F-IgG compartment (fluorescent IgG); and (4) a detection compartment. (b) Before the magnetic manipulation is started the sample compartment is filled with the sample. (c) and (d) The IgGs bind the virus in the sample. A magnetic field pulls the particles through the sample compartment and the F-IgG compartment. During the particle movement, both the virus and F-IgG are accumulated on the particle surface. Finally, the particles reach the detection compartment, where the fluorescence intensities are measured from individual particles. The average of the intensities is correlated to the concentration of the virus in the sample. (From Ostergaard, S. et al. 1999. J. Magn. Magn. Mater. 194: 156–162. With permission.)

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of the bound fluorescence. The intended use of the system is a disposable system87 that can be used as a quick test.

14.4 CONCLUSIONS Magnetic polymer nanoparticles have shown great potential in many fields of medicine and biotechnology applications. Different preparation methods of magnetic latex particles have been applied from both natural and synthetic polymers with the goal of incorporating high iron oxide loadings and to obtain a narrow particle size dispersion with defined magnetic properties. These methods involve the polymerization of monomer used in an emulsion, suspension, dispersion, or miniemulsion, or the use of a preformed polymer as in the solvent diffusion and solvent evaporation techniques. Although some of these techniques, such as the miniemulsion polymerization, yielded high magnetic loadings, they still show some limitations like the inhomogeneous distribution of the magnetic nanoparticles inside and among the particles and the large particle size distribution, and therefore need to be better optimized. The use of magnetic particles is expected to be profitable in m-TAS for biological or chemical analysis: beads represent ideal reagent delivery vehicles, provide large reactive surface areas and large binding surface areas, and magnetism can be used to manipulate and trap beads. However, most of their intended applications in microsystems seem to be still under study, including separation, immobilization, labeling, or manipulation, although several authors consider that they are very promising for future development in bioanalysis. Magnetism is now a widely applicable item in the microfluidicist’s toolbox. Although many applications have been investigated, not all of them are competitive with conventional methods. The unique advantages of magnetic manipulation lie in the possibility of externally controlling matters inside a microchannel. One challenge in the microfluidics research is to design integrated devices in which sample pretreatment, isolation, separation, and/or detection are combined. Labeling with magnetic particles for isolation is an elegant option in such devices. Focus is also on cell analysis in microfluidic platforms. Magnetic forces can be used to capture, move, and detect cells and more works are likely to emerge in this field. With increasing insight into the fabrication of stronger and/or smaller magnetic particles, more studies are likely to be undertaken, self-assembly of magnetic objects into complex threedimensional structures or into small machines is only starting to be investigated and may soon be transferred to the micro- or nanoscale. With advances in this field on so many fronts, more sophisticated devices will emerge and be part of integrated and hopefully widely applicable m-TAS.

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15

Colloidal Dispersion of Metallic Nanoparticles: Formation and Functional Properties Shlomo Magdassi, Michael Grouchko, and Alexander Kamyshny

CONTENTS 15.1 Introduction ...................................................................................................................... 15.2 Chemical Synthesis of Metallic NPs in Aqueous Medium .............................................. 15.2.1 Chemical Reduction of Metal Ions ....................................................................... 15.2.2 Nucleation and Growth: Size and Shape Control ................................................. 15.2.2.1 Seed Nucleation and Particle Size Distribution ..................................... 15.2.2.2 Shape Control by Soft Templates and Selective Adsorption ................. 15.3 Functional Properties: Collective Properties .................................................................... 15.3.1 Colloidal Dispersions ............................................................................................ 15.3.1.1 Catalysis ................................................................................................. 15.3.1.2 Surface Plasmon Resonance .................................................................. 15.3.2 Arrays ................................................................................................................... 15.3.2.1 Surface-Enhanced Raman Scattering .................................................... 15.3.2.2 Sensors ................................................................................................... 15.3.2.3 Thermal Properties ................................................................................ 15.4 Summary .......................................................................................................................... References ..................................................................................................................................

15.1

339 340 341 342 344 345 347 347 347 347 348 349 349 351 352 352

INTRODUCTION

Nanoparticles (NPs), such as nanospheres, nanorods, nanowires, and nanocubes, have attracted extensive attention of material scientists because of their application in many areas of fundamental and technical importance. For example, they can be used to experimentally probe the effects of quantum confinement on electronic, optic, and other related properties [1–3]. They have also been widely exploited for use in manufacturing electronic, photonic, and sensing devices [4–6]. The intrinsic properties of a metal NP are mainly determined by its composition, size, and shape [7]. For studying size- and shape-dependent properties, it is highly desirable to synthesize metal nanostructures with well-controlled size and shape [8]. 339

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Structure and Functional Properties of Colloidal Systems

Among NPs and functional nanosized materials of various compositions, metallic NPs attract special attention. NPs such as decorative pigments have actually been known for many centuries. The glass of the famous Lycurgus Chalice of fourth century exhibited at the British Museum contains nanosized particles of silver and gold (~70 nm). This chalice possesses the unique characteristics of changing in color from green when viewed in reflected light to deep red when light is shone from inside and transmitted through the glass. The luster films of the Renaissance period majolica were shown to be formed by a glossy matrix containing dispersed silver and copper NPs with dimensions ranging from about 5 to 100 nm [9]. Since the pioneering paper of Faraday, published in 1857 and reporting on optical properties of colloidal gold [10], numerous papers and patents have been published on synthesis, properties, and application of metallic NPs [9,11–16]. The unique properties of nanosized metals pave the way to new applications and possibilities of making new products such as electronic, optical, and magnetic devices [17–19], nanoelectromechanical systems [20,21], conductive coatings and conductive ink-jet inks [15,22–24], energy conversion and photothermal devices [25], catalysts [9,18,21], biosensors, biolabels, and drug delivery systems [9,17,26–29]. Methods for preparation of metallic NPs can be roughly divided into two groups: top-down and bottom-up. Top-down methods are usually high-energy methods, in which bulk metal or microscopic particles are dispersed in a proper medium with formation of nanosized particles (plasma evaporation, thermal evaporation, laser ablation, etc.) [30–33]. These routes are not very well suited for preparation of small and uniform NPs, and stabilizing the NPs is a matter of some difficulty [18,34]. In bottom-up methods, NPs are formed from precursor atoms, molecules, or ions [9,18,35–38]. These methods include reduction of metal ions with proper reducing agents, thermal decomposition of salts and organometallic complexes, UV-, g-, electronic, and ultrasonic irradiation of a metal precursor, such as salt or organometallic compound, and electrochemical, sonochemical, and sonoelectrochemical syntheses [9,11,18,30,39–55]. The solvents in which the syntheses are performed can vary from water to polar and nonpolar organic solvents and ionic liquids [18,56–64]. The current approaches to the synthesis of NPs in confined nanometric structures (micelles, microemulsions, capsules, dendrimers, pore channels of mesoporous solids, liquid crystals, etc.) have recently been reviewed in references [4,9,10,14–16,21,30,35,39,47,59,65–78]. In order to prevent flocculation of NPs followed by agglomeration and coagulation, the addition of stabilizing agents to the reaction mixture at the early stages of NP formation is required. These agents are adsorbed on the NP surface and create an energy barrier arising from repulsive electrostatic and/or steric interactions [9,11,13,22,37,39,48,79–81]. In this chapter, we survey the current approaches to the synthesis of size, shape, and structurecontrolled metallic colloids by chemical reduction of metal precursors in homogeneous aqueous medium. Functional properties and applications of such colloidal NPs dispersed in an aqueous medium and deposited as an ordered array are also presented.

15.2

CHEMICAL SYNTHESIS OF METALLIC NPS IN AQUEOUS MEDIUM

The wet chemical bottom-up methods based on approaches and tools of colloid chemistry, especially chemical reduction of a metal precursor (e.g., metal salt) by reducing agents in organic and aqueous media, have been successfully applied for preparation of practically all noble and transition metal NPs. Among these methods, synthesis of metal NPs in aqueous medium may be the most convenient, since it allows preparation of dispersions with different particle characteristics (size and its distribution, morphology, and stability) by varying the experimental parameters. In general, chemical synthesis of metallic colloids has two main stages: (a) reduction of a metal salt and (b) stabilization of the obtained NPs. In the first stage, the metal salt is reduced to give zero valent metal atoms. The lifetime of these atoms in solution is short, as they tend to coalesce quickly into larger arrangements—clusters and nuclei—which grow to form particles that aggregate together to form the bulk metal. Preventing this, final aggregation step is arrested by the stabilization of the formed

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particles. The stabilization is based on the adsorption of a capping agent at the particle surface, providing it with an electrostatic potential, a steric barrier, or both.

15.2.1

CHEMICAL REDUCTION OF METAL IONS

The first step in the process of metallic NP formation is the reduction of metal ions with a proper reducing agent resulting in formation of metal atoms: mMn+ + Red = mM0 + Redmn+.

(15.1)

The driving force of this reaction is the electromotive force (EMF), which is the difference in reduction potentials of a metal and a reducer. The standard EMF, DE 0, can be evaluated from the standard reduction potentials, E 0, of the metal and the reducing agent, and is related to the free energy of the redox process: DG 0 = -nFDE 0,

(15.2)

where n is the number of electrons in a reaction equation and F is Faraday’s constant. As follows from Equation 15.2, at standard conditions, the reduction is thermodynamically possible only if DE0 is positive, which means that the reduction potential of the reducing agent is more negative than that of the oxidizer (metal precursor). Practically, this difference should be larger than 0.3–0.4 V; otherwise the reaction may not proceed or proceeds too slowly to be of practical importance [82,83]. Uncomplexed cations of strongly electropositive metals (E 0 > 0.7 V), such as Au3+ (E 0 = 1.5 V), Pt2+ (E 0 = 1.2 V), Ir3+ (E 0 = 1.16 V), Pd2+ (E 0 = 0.99 V), Rh3+ (E 0 = 0.8 V), and Ag+ (E 0 = 0.8 V), can be reduced with relatively mild reducing agents at ambient conditions; cations of moderately electropositive metals (E 0 > 0.3 V), such as Ru2+ (E 0 = 0.46 V), Cu2+ (E 0 = 0.34 V), and Co3+ (E 0 = 0.33 V), require stronger reducing agents, whereas cations of electronegative metals, such as Fe3+ (E 0 = -0.04 V), Fe2+ (E 0 = -0.44 V), Co2+ (E 0 = -0.28 V), and Ni2+ (E 0 = -0.25 V), can be reduced only with strong reducing agents, usually at elevated temperatures (here and below, the E 0 values are taken from references [84,85]. Formation of soluble complexes with organic and inorganic functional groups results in decrease in reduction potential, and this decrease correlates with the stability of a complex cation [47,82,83]. For example, E 0 for Ag(NH3)+2 (Kf = 1.6 × 107) and Ag(CN)2(Kf = 5.6 × 1018) complex ions are 0.57 and -0.31 V, respectively, compared with E 0 = 0.8 V for free Ag+ ions. Complexes of Au, Pt, and Pd cations with inorganic ligands, which are usually used for synthesis of NPs, are also characterized by a lower reduction potential compared to the corresponding uncomplexed cations (AuCl4- + 3e = Au + 3Cl-, E 0 = 1.00 V; PtCl62- + 2e = PtCl42- + 2Cl-, E 0 = 0.68 V; PtCl42- + 2e = Pt + 4Cl-, E 0 = 0.73 V; PdCl62- + 4e = Pd + 6Cl-, E 0 = 0.96 V; and PdCl42- + 2e = Pd + 4Cl-, E 0 = 0.62 V). Another very important factor is the pH. Since H+ and OH- ions are very often involved in redox reactions performed in aqueous medium, change in pH may result in considerable change in DE 0 as follows from the Nernst equation. For example, the negative E0 value for hydrazine as a reducing agent decreases from -1.16 V in alkaline solution (N2 + 4H2O + 4e = N2H4 + 4OH-) to -0.23 V in acidic solution (N2 + 5H+ + 4e = N2H+5 ). Therefore, addition of a complexing agent and/or changing the pH of the reaction mixture is a very powerful tool for tuning DE 0. In some cases, the compound used for pH adjustment is a complexing agent as well (e.g., ammonia). Useful E0/pH diagrams reflecting also the complexation of metal ions are presented in references [82,83]. A wide range of various inorganic and organic reducing agents has been applied in the synthesis of metallic NPs in aqueous medium. Among them, borohydride, BH4-, is one of the strongest reducing agents at ambient conditions with a reduction potential of -1.24 V in alkaline medium (BO2- + 6H2O + 8e = BH4- + 8OH-), which decreases to -0.48 V in acidic medium (HBO2 + 7H+ + 8e = BH4- + 2H2O).

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It plays a dominant role in the synthesis of NPs of various metals: Au [86–101], Ag [41,56,58,69, 93–95,100–110], Cu [96,111,112], Pt [91,93,99,100,113–116], Pd [91,94,117,118], Rh [91,119– 121], Ir [121], Ru [99,122,123], Co [124–126], and Fe [127–129]. Another very strong reducing agent, hydrazine (N2 + 4H2O + 4e = N2H4 + 4OH-, E0 = -1.16 V), is also used for fabrication of Au [130], Ag [107,131–133], Pt [115], Pd [115,134], Ni [135–140], Cu [111,141–143], and Fe [144] NPs. Citrate, starting from the comprehensive studies of Turkevich [145,146], has been shown to be a very convenient reducing agent (E 0 = -0.56 V), which has been widely used for preparation of Au [88,92,145–151] and Ag [22,102,148,152–158] NPs. Fabrication of Pt [114,159,160] and Pd [161] NPs with the use of citrate as a reducing agent is also reported (the reaction is performed at elevated temperatures) [22,88,92,145,154,157]. Ascorbic (or isoascorbic) acid is also a commonly employed reducing agent for producing metallic NPs. Ascorbic acid is a weak reducing agent and therefore it is effective in reactions with ions of electronegative metals. Most reports describe the use of ascorbic acid for synthesis of Au [87,151,162,163] and Ag [51,83,131,164,165] NPs, but there are examples of its use for preparation of Pt [116] and Pd [83] NPs as well. There are also other reducing agents, which are not commonly used in practice; among them, a few are hydroxylamine [92,114,166], tannins [107,167,168], EDTA [102,153,169,170], formamide [171], tartrate [15,172], hydroquinone [173], o-anisidine [174], dimethylamine borane [114], and hypophosphite [107].

15.2.2 NUCLEATION AND GROWTH: SIZE AND SHAPE CONTROL When fabricating metallic NPs, tuning their characteristics (particle size and its distribution, morphology, and stability) is very important, especially for their successful utilization in various applications. Although the synthesis of metallic NPs is still a matter of skill, the basic mechanisms that govern their properties are rather well established [11,34,40,59,82,83,87,145,164,165,175–186]. According to the general mechanism for formation of metallic NPs [82,83,145,175,187,188], reduction of the metal precursor (molecules or ions) results in formation of atoms that aggregate into clusters, or embryos (the size is not clear). Such embryos are in dynamic equilibrium with the metal atoms. When embryos reach a critical size, they represent stable insoluble particles—nuclei (from one to a few nm [11,145,189]). Nuclei grow to primary NPs, which are characterized by large free energy and continue to grow. At least three mechanisms for growth of primary NPs to the final metal NPs are available: (1) growth by atom diffusion and addition (atom-by-atom growth); (2) growth by aggregation (coalescence) of preformed nuclei and/or NPs; and (3) autocatalytic growth (metallic nucleus serves as a catalyst for metal precursor reduction [185,190]; Figure 15.1). In the absence of a hard template, solution-based methods for the synthesis of NPs require precise tuning of nucleation and growth steps to achieve crystallographic control. These reactions are governed by thermodynamic (e.g., temperature and reduction potential) and kinetic (e.g., reactant concentration, diffusion, solubility, and reaction rate) parameters, which are very well linked. Thus, the exact mechanisms for shape-controlled colloidal synthesis are often not well understood or characterized. Surface-energy considerations are crucial in understanding and predicting the morphology of metallic nanocrystals. Surface energy, defined as the excess free energy per unit area for a particular crystallographic face, largely determines the faceting and crystal growth observed for particles at both the nano- and mesoscale levels. For a material with an isotropic surface energy such as an amorphous solid or liquid droplet, the total surface energy can be lowered simply by decreasing the amount of surface area corresponding to a given volume. The resulting particle shape is a perfectly symmetrical sphere. However, noble metals, for example, which adopt a face-centered cubic (fcc) lattice, possess different surface energies for different crystal planes. This anisotropy results in

Colloidal Dispersion of Metallic Nanoparticles

FIGURE 15.1

343

Schematic presentation of the process of metallic NP formation.

stable morphologies where the free energy is minimized by the exposure of low-index crystal planes that exhibit closest atomic packing. Not being completely spherical, most of the chemical reduction syntheses of metallic NPs produce hemispherical morphologies such as decahedron, icosahedron, and various multiple twinned particles [191]. During the last two decades, metallic NPs of different sizes and various morphologies were synthesized. In the case of noble metal NPs such as Ag, Au, Pt, and Pd, properties shape dependence is

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particularly evident. For example, Ag and Au nanocrystals of different shapes possess unique optical properties. Whereas highly symmetric spherical particles exhibit a single-surface plasmon peak, anisotropic shapes such as rods [192], triangular prisms [193], and cubes [7] exhibit multiple surface plasmon peaks in visible wavelengths due to highly localized charge polarizations at corners and edges. Controlling nanocrystal shape thus provides an elegant strategy for optical tuning (see Section 15.3.1.2). Similarly, chemical reactivity is highly dependent on surface morphology. The bounding facets of the nanocrystal, the number of step edges and kink sites, and the surface area-to-volume ratio can dictate unique surface chemistries. For this reason, Pt and Pd nanocrystals exhibit shapeand size-dependent catalytic properties [194] that may prove useful in achieving highly selective catalysis (see Section 15.3.1.1). Hence, optimizing NP morphology has become an emerging issue in the nanoscience and nanotechnology field. Such an optimization is achieved, on the one hand, by controlling the particle size distribution to achieve a narrow distribution, monodispersity, and on the other, by controlling the shape of these monodispersed particles. The templateless approach to obtaining monodispersed NPs is by controlling the nucleation and growth mechanisms by seeded nucleation—homogeneous or heterogeneous (Section 15.2.2.1). Control over the NPs’ shape is achieved by the use of soft templates (micelles, etc.), selective adsorption of molecules, or inherent stability of specific planes in the preformed nucleus (Section 15.2.2.2). 15.2.2.1 Seed Nucleation and Particle Size Distribution The nucleation process is critical for obtaining specifically monodispersed metal NPs. The formation or addition of small seed particles—particles that serve as nucleation sites for metal reduction—can drastically change the kinetics of NP growth. This approach can be carried out via either homogeneous or heterogeneous nucleation. In homogeneous nucleation, seed particles are formed in situ and, typically, nucleation and growth proceed by the same chemical process. This is the more common synthetic strategy due to the practical ease of carrying out a one-pot reaction. Heterogeneous nucleation is carried out by adding preformed seed particles to a reactant mixture, effectively isolating nanocrystal nucleation and growth as separate synthetic steps. 15.2.2.1.1 Homogeneous Nucleation In homogeneous nucleation, seed formation proceeds according to the LaMer model [195,196], where reduction of metal ions takes place to form a critical composition of atomic species in solution. Above this critical concentration, nucleation results in a rapid depletion of the reactants such that all subsequent growth occurs on the preexisting nuclei. As long as the concentration of reactants is kept below the critical level, further nucleation is prevented. This is particularly important for obtaining a monodispersed population of particles; the nuclei are formed during a short period of time, and the growth of the particles, which takes place in the second step, is uniform for all the nuclei. Thus, in principle, in order to obtain monodispersed particles the nucleation and growth should be separated. This can be achieved in homogeneous nucleation, by proper control of the reduction and its concentration, or by changing the conditions that favor either growth or nucleation (such as temperature and pH). 15.2.2.1.2 Heterogeneous Nucleation In heterogeneous nucleation, the reaction conditions to achieve monodispersity are less strict, since seed particles are preformed in a separate synthetic step. In addition, the activation energy for metal reduction on already formed seeds is significantly less than in the case of homogeneous nucleation in solution [196]. The size distribution control by heterogeneous nucleation can be considered as an overgrowth process: seed particles are added to a growth medium to facilitate the reduction of metal ions. Thus, utilizing heterogeneous nucleation allows a wider range of growth conditions that employ milder reducing agents, lower temperatures, or aqueous solutions. Kwon et al. [197] synthesized monodispersed polyhedral gold NPs by reduction of Au3+ ions by ascorbic acid on preformed

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FIGURE 15.2 Transmission electron micrographs of gold nanorods with progressively greater aspect ratios (1–5). (Reproduced from Murphy, C. J. et al. 2008. Chem. Commun. (5): 544–557. With permission of The Royal Society of Chemistry.)

Au seeds. The use of a weak reducing agent effectively isolates the nucleation and growth events, allowing control over the size and distribution of the resulting NPs [198,199]. The addition of an increasing amount of growth solution (HAuCl4 solution) to a fixed amount of gold seed dispersions leads to the formation of monodispersed gold NPs with sizes from 32 to 50 nm with respect to the growth solution quantity. Murphy and colleagues [198–201] used the same technique to obtain monodispersed gold nanorods with various aspect ratios. Small Au seeds (3–5 nm in diameter), added to an Au precursor solution containing ascorbic acid (a weak reducing agent that is only strong enough to induce autocatalytic growth on preexisting nuclei), preferably reduce the Au precursor in the presence of seeds. The resulting structures, rods, and wires, as well as a variety of other shapes presented in Figure 15.2, can be synthesized by carefully controlling the growth stage, which is distinctly separate from the nucleation event. 15.2.2.2 Shape Control by Soft Templates and Selective Adsorption Confined structures, such as microemulsions [124,202], vesicles, [203], micelles, and reverse micelles [204–206], are very often used as templates or nanoreactors for controlled colloidal syntheses. These so-called soft templates may be composed of a variety of molecules such as block copolymers and fatty acids. For example, the reduction of HAuCl4 or AgNO3, by soft templating was typically carried out in aqueous surfactant systems such as cetyltrimethylammonium bromide (CTAB), sodium dodecylsulfate (SDS), or bis(2-ethylhexyl) sulfosuccinate (AOT). As these surfactants possess a hydrophilic head group and a hydrophobic tail, they readily self-assemble into spherical or rod-like micelles in water, depending on concentration and the presence of other additives such as cosurfactants. For example, Au nanorods with aspect ratios of 2–10 were synthesized in the presence of CTAB and a cosurfactant, tetradodecylammonium bromide. It was shown that using concentrated CTAB solution enhances the rod yield, probably due to the CTAB tendency to form elongated rod-like micellar structures [207] that possibly assist in rod formation, as well as stabilizing the rods [208]. Although much research has been performed on these soft-template systems in past decades, it is not clear whether these surfactants really serve as templates or actually as growth-directing adsorbates [209]. Several groups [87,210] claim that surfactants such as CTAB do not serve as soft templates but rather promote the formation of nanorods due to their adsorption onto selective planes on the seed surface. It was found that most of the metallic rods are composed of (100) side planes and (111) end planes, as presented in Figure 15.3. In an fcc lattice, (111) planes have the highest atomic density and fewer open sites to which the hydrophobic ends of the surfactants can attach. On the other hand, (100) planes have lower atomic density, offering more open sites to adsorb surfactant molecules. The high surfactant coverage on the (100) planes, as illustrated in Figure 15.3, constitutes a barrier to further lateral attachment of

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Structure and Functional Properties of Colloidal Systems

FIGURE 15.3 Preferential attachment of surfactant molecules to the lateral (100) planes of the nanorod. (Reproduced from Ni, C. Y. et al. 2005. Langmuir 21 (8): 3334–3337. With permission of The American Chemical Society.)

metal atoms. Kou et al. demonstrated that the use of an additional surfactant, for example, glutathione or cysteine, to block the growth by its selective adsorption to the rod ends leads to the increase in the rods’ diameter [211]. Therefore, the use of surfactants as selective adsorbates to specific facets enables three-dimensional control over the growth of primary particles to form the morphology. A well-known shape-controlling agent, polyvinylpyrrolidon (PVP), was reported to yield Ag [7,63,205], Au [63,205], and Pt NPs [212] with various morphologies. Xia et al. [7,213] suggested that the same mechanism of selective adsorption (of PVP) to specific planes on the seed surface governs the obtained particles’ shape. It was shown that the reduction of silver ions on preformed silver rods in the presence of PVP led to their selective growth along the rod’s elongated axis. PVP interacts more strongly with the (100) side facets than the (111) facets at the ends of the nanorods; thus the side surface of the nanorods are passivated by PVP, whereas the ends remain reactive toward silver atoms, and nanowires are formed. Tao et al. [214] reported the synthesis of monodispersed silver NPs capped by PVP with regular polyhedral shapes with solely (100) and (111) facets on the fcc crystal lattice. Initially, small silver particles (<10 nm) nucleate and develop into nanocubes bound by (100) planes. As the reaction continues, silver deposits selectively onto the (100) nanocrystal facets and various polyhedral shapes capped with (100) and (111) faces can be obtained, depending on the reaction duration (Figure 15.4). It was suggested that the driving force for the selective (100) planes growth is not the adsorption of the capping polymer (PVP) to the (111) planes, but rather the inherent increased stability of the (100) planes [215]. Selective adsorption on different crystal planes is not limited to large surfactants and long-chain polymers. In several cases, metal nanocrystal shape can be effectively modified by the addition of

FIGURE 15.4 Schematic presentation of the nucleation and growth process, in which silver continuously deposits onto the (100) facets to eventually result in a completely (111)-bound octahedron. (Reproduced from Tao, A. et al. 2006. Angew. Chem. Int. Ed. 45 (28): 4597–4601. With permission of Wiley-VCH Verlag GmbH & Co. KGaA.)

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small molecules, which also exhibit preferential binding. For example, during the heterogeneous overgrowth of Pd on small Pt cubic seeds to form a core–shell structure, addition of gaseous NO2 to the reactant solution has been demonstrated to yield cuboctahedra and octahedral nanocrystals with increasing NO2 concentration, in addition to cubes formed with no NO2. A preferential interaction of NO2 with the (111) surfaces acts to partially passivate the (111) surfaces with adsorbed oxygen, limiting growth along the (111) directions, so controlling the NO2 concentration enables control of the (111) facet growth [216].

15.3

FUNCTIONAL PROPERTIES: COLLECTIVE PROPERTIES

Due to their unique properties, metallic NPs serve as catalysts in various reactions, are integrated in optical, chemical, and electrochemical sensors, and are used as conductive inks. In the next section, we present several examples for these functional properties of metallic NPs as colloids in aqueous dispersion and as NPs assembled in arrays.

15.3.1

COLLOIDAL DISPERSIONS

15.3.1.1 Catalysis Two types of nanocatalyses can be distinguished: the “homogeneous” type with catalysis in colloidal solution and the “heterogeneous” type in which the NPs are supported on solid surfaces. Homogeneous nanocatalysts are usually spherical NPs or NPs of undetermined shapes. A very important question is whether the catalytic activity of atoms located on the surfaces of nanocrystals of different shapes, and therefore different facets, is different [217–219]. Furthermore, the NPs have different fractions of atoms located at different corners and edges, and various defects (resulting from the loss of atoms at these locations). Since the catalytic activity of these NPs is determined by the catalytically active sites at the NP surface, one would expect the catalytic activity of NPs with different size and shape to vary in catalyzing the same reaction. Narayanan and El-Sayed [220] compared the activation energies of the electron-transfer reaction between hexacyanoferrate(III) ions and thiosulfate ions in colloidal solution containing dominantly tetrahedral, cubic, or “nearspherical” platinum NPs as catalysts. It has been found that the efficiency of the nanocatalyst decreases in the order: tetrahedral NPs > “near-spherical” NPs > cubic NPs, correlating well with the fraction of surface atoms located on the corners and edges [221]. It has also been demonstrated that not only the number of corners and edges on the surface of NP, but the type of crystal face also plays an essential role in the catalytic properties of metallic NPs. For example, silver nanocubes, which have only (100) faces, were found to be 4 times more catalytically active than spherical silver NPs with (100) and (111) faces, and 14 times more active than silver nanoplates with (111) faces [222]. Since the shape control is usually achieved by using different capping agents at the stage of NP synthesis, the capping molecules should be taken into account when comparing different shapes. “Good” capping ligands (i.e., those that stabilize robust nanocrystals with very narrow size distributions) appear to be poor choices for catalytic applications. However, the nanocrystals must have a good dispersibility in multiple reaction cycles; hence the ligands must be bound to the NPs strongly enough to effectively stabilize the nanocrystals but weakly enough to provide reactant access to the metal surface [218]. 15.3.1.2 Surface Plasmon Resonance The intense color of colloidal metal particles in stained glass windows is caused by the effect of surface plasmon resonance (SPR). The SPR is a result of coherent motion of the conduction-band electrons caused by interaction with the electromagnetic field of incident light [223,224]. The position and the width of the surface plasmon band depend on the size and shape of the metal NP as well as on the dielectric constant of the metal itself and of the surrounding medium

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[223,224]. The plasmon resonance is stronger and shifted into the visible part of the electromagnetic spectrum for silver, gold, and copper NPs. Most of the other metals show only a broad and poorly resolved absorption band in the ultraviolet [225]. This is the reason why Ag, Au, and Cu historically fascinated scientists dating as early as Faraday [10]. The linear optical properties, such as extinction and scattering by small spherical metal particles, were explained theoretically by the groundbreaking work of Mie in 1908 [226]. Within this theory, metallic NPs with a diameter of 2 nm are characterized by the SPR effect as well. However, experimental data [223,227] show that very small particles with a diameter of 1–2 nm do not display this phenomenon, since the electrons are located in discrete energy levels [89,228–230]. Since Mie’s theory was developed for particles of spherical shape only, theoretical methods describing the properties of nonspherical NPs were developed [231]. These approaches were found to rather efficiently predict the absorption spectra of the metal NPs of various shapes. An example of the shape dependence of the SPR peak position for Ag colloidal NPs is presented in Table 15.1.

15.3.2

ARRAYS

The precise control of composition, size, and morphology of metal NPs (“primary structure”) enables fabrication of the “secondary structures” of NPs—the regularly ordered assemblies with well-defined one-dimensional (1D) and two-dimensional (2D) spatial configuration. The forces that drive colloidal assembly depend on the physical characteristics and surface chemistry of the particles as well as on the assembly technique. For the very smallest NPs (<10 nm), the assembly process would be mostly driven by interactions at molecular length scales (e.g., surfactant chain interactions) [204,232–244]. As particle size increases, self-organization is often determined by a more complex balance of van der Waals interactions, electrostatic forces, and/or short-range steric repulsion. Not surprisingly, methods for assembling colloidal particles into ordered superlattices vary considerably, depending on the nature of the particles used and the media in which they are dispersed. Surfactant-coated nanocrystals with weak van der Waals interactions can be deposited at submonolayer concentrations onto air–water interfaces and compressed into close-packed 2D arrays by the Langmuir–Blodgett process [245–250]. Stable dispersions of NPs and colloids with greater long-range interactions (e.g., metallic NPs stabilized by low-molecular-weight polyelectrolytes) can self-assemble into 2D arrays by simple drop evaporation techniques onto wettable surfaces [234,247,251–255].

TABLE 15.1 Shape Dependence of the SPR Peak Position for Ag Colloidal NPs

a a b

a b a b

Colloidal Dispersion of Metallic Nanoparticles

349

The interparticle spacing in these arrays is known to be a function of the stabilizer, surfactant length, or polymer molecular weight [256,257]. Highly ordered metallic NP systems are very interesting from the fundamental point of view, since their optical properties can be adjusted not only by varying the particle size and shape, but also by controlling the array periodicity [258]. It has been shown, for example, that the periodicity (or grating constant) in 2D arrays strongly influences the spectral position and the width of the extinction maximum [259]. A significant red-shift of the SPR peak has been reported for decreasing periodicities regardless of particle shape, material, and array geometry [259–261]. Examples for applications of such superlattices are surface-enhanced Raman scattering (SERS) on metallic films [232,233], optical grating [262], antireflective surface coating [263], and data storage devices [264]. Moreover, the monodispersed metal NPs have the potential to be the building blocks for nanoelectronic devices using the single electron tunneling effect [265,266]. Metal NPs smaller than 2 nm are required for such devices to make use of the phenomenon of Coulomb blockade at room temperature [266–271]. In the next two sections, we focus on the SERS phenomenon (Section 15.3.2.1) and various sensors (Section 15.3.2.2) that are good examples of the use of devices based on NPs arrays. 15.3.2.1 Surface-Enhanced Raman Scattering The strong electromagnetic field produced by the excited plasmons is effective at a distance as long as ~10 nm from the NP surface [272]. Therefore, any molecule located within this distance will be affected by this field. There are a number of “surface-enhanced” spectroscopies that are based on that effect [273,274], but the most widely used is surface-enhanced Raman spectroscopy also known as SERS [275]. The Raman effect itself is a weak one: visible light that is not directly absorbed by the molecule of interest is only weakly inelastically scattered. The fundamental selection rule for Raman spectroscopy is that the polarizability of the molecule must change during the course of the vibration (in contrast, the fundamental selection rule for infrared spectroscopy is that the dipole moment of the molecule must change during the course of the vibration). The resulting Raman spectrum does provide a nice vibrational fingerprint of the molecule, but the required sample concentration to obtain a reasonable Raman spectrum should be very high to be applicable for dilute (less than millimolar) concentrations of an analyte. The intensity of Raman signals depends on the fourth power of the local electric field, and molecules that are located near a nanoscale metal surface are not only affected by this field, but also, through charge-transfer interactions with the metal surface itself, can undergo changes in polarizability. These effects increase the intensity of the Raman signal by a factor of up to 1014 [232,276], which makes possible detection of a single molecule [274,275,277]. Xu and colleagues [278,279] revealed the influence of the analyte adsorption site on the enhancement effect. It was found that extraordinary enhancement of the Raman signal is possible if the analyte is adsorbed at the junctions between aggregated particles. Since SERS is a resonance effect related to the SPR of colloidal particles, its sensitivity is affected by size, shape, and structure of their arrays. The ability to tailor colloid size, size distribution, and shape motivates examination of the influence of these factors on SERS [280]. The optimal particle size for the SERS effect was found to be between 30 and 150 nm, with decreasing enhancement in smaller and larger particles [280,281]. Cotton and coworkers checked the influence of the interparticle distance in silver and gold NP (40 nm) arrays on their SERS activity. It was found that only arrays with aggregates (NPs in contact) showed the Raman enhancement, whereas arrays composed of NPs with any interparticle distance had no SERS activity [282]. These findings are in agreement with other research that found that the strongest enhancement comes from molecules situated between particles [278,283]. 15.3.2.2 Sensors Several new approaches to the design of arrays of metallic NPs for entirely new chemical sensing platforms and/or the capability of an existing sensing technique have been reported. In general,

350

Structure and Functional Properties of Colloidal Systems

Analyte Ag

Ag

Ag

FIGURE 15.5 Generalized scheme for chemical sensing based on silver NP aggregation. Exposure of the modified silver NPs to an analyte leads to their analyte-mediated aggregation.

metallic NPs-based chemical sensors are fabricated by controlled assembly of metallic NPs through hydrogen bonding [284], p–p [285], host–guest [286], van der Waals [287], electrostatic [288], charge-transfer [289], or specific antigen–antibody [290] interactions. Metallic NP arrays for sensing are based on electrochemical, optical, SPR, piezoelectric, and electrical transduction approaches; so when an analyte is interacting with the array, one of these properties is modified and a signal is detected. Sensing applications include chemical and biochemical processing, medical diagnosis, environmental monitoring, transportation, and so on [291–294]. For example, analyte-mediated gold NP selective aggregation has been shown to be particularly useful in the detection of DNA, proteins, antibodies, glucose, toxic metal ions, and other substances [295–298]. The general idea for aggregation-based chemical sensing is shown in Figure 15.5. Sensing is based on the coupling of plasmon resonances during the aggregation process, which takes place only when the particles are exposed to a specific analyte. The surface of the metal (e.g., Ag) NPs needs to be modified with a molecule that recognizes the analyte of interest. For maximal aggregation, the analyte should bind to its partner in a multivalent fashion, so that multiple silver NPs will be brought close to each other upon introduction of the analyte. The plasmon band(s) of aggregated silver NPs will be broadened and red-shifted as a function of aggregation state, and therefore as a function of analyte concentration. Mirkin and colleagues have studied the detection of DNA by gold NPs, where the specific aggregation is due to addition of complimentary target oligonucleotide to various oligonucleotidemodified gold NPs [295,296]. In addition, there are reports on DNA detection using 2D aggregation

NH3

PtNT

1.0

DR/RInt [%]

0.8 0.6 C7H6

0.4 0.2

H2O

CO

0.0 0 (A)

50

0

50

0 Time [s]

100

0

100

FIGURE 15.6 Responses of the Pt NP-based chemiresistor to exposure with 400 ppm toluene vapor, water vapor, ammonia, and carbon monoxide. (Reproduced from Joseph, Y. et al. 2004. Sens. Actuat. B: Chem. 98 (2–3): 188–195. With permission of Elsevier.)

Colloidal Dispersion of Metallic Nanoparticles

351

[297] and the use of peptide nucleic acid-stabilized gold [299]. Such an aggregation sensing mechanism usually takes place in dispersions, thus limiting the sensing to liquid sensors. Utilization of chemiresistors for sensing has attracted considerable attention as well [300]. The advantages of such sensors are their small size and weight, fast response, reliability, low-output impedance, the possibility of automatic packaging at wafer level, on-chip integration of sensor arrays, and the possibility of mass-producing portable microanalysis systems at low cost. In standard chemiresistors, the electrical resistance of a device changes in the presence of a specific chemical species. The use of metallic NP arrays in chemiresistors was demonstrated in the work of Joseph et al. [301] (Figure 15.6). It has been shown that the exposure of a chemiresistor sensor composed of dodecylamine capped-Pt NPs to ammonia, carbon monoxide, and vapors of water and toluene led to an increase in their resistance with a detection limit for ammonia below 100 ppb. It has also been shown by other groups [302–304] that the selectivity of such sensor coatings can be tuned by introducing chemical functionality into the organic ligand shell for selective binding of specific molecules to be detected. Examples of capping ligands that were reported include alkylthiols [301,305], alkylamines, [306], p-thiophenols, [303], carboxylates, [307], and cross-linked polyphenylene dendrimers [304]. 15.3.2.3 Thermal Properties Due to their high surface-to-volume ratio, the thermal properties of metallic NPs noticeably differ from those of the bulk form. The important role of the surface in the melting process has long been suspected [308,309] and has recently been confirmed [310] via surface-sensitive measurements of melting. These recent measurements include structural demonstrations of liquid skin formation on flat surfaces below the melting temperature of the interior crystal [311,312] and a dramatic melting temperature reduction for nanometer-sized particles in proportion to the surface-to-volume ratio [313–315].

FIGURE 15.7 Silver NPs in a printed pattern heated to various temperatures up to 320°C. (Reproduced from Kamyshny, A. et al. 2005. Macromol. Rapid Commun. 26 (4): 281–288. With permission of Wiley-VCH Verlag GmbH & Co. KGaA.)

352

Structure and Functional Properties of Colloidal Systems

Thermodynamically, the proposed models [77,309,316] predict a size-dependent melting temperature, Tm(d), that is directly proportional to the difference in the surface free energy, Dg (between the solid and liquid), and inversely proportional to the crystallite diameter d: Tm(d)aDg/d. Tm(d) was measured by various methods for Pb, Sn, Bi, In, Au, Ag, Cu, Ge, Al, and Na [313,317– 321]. Buffat and Borel determined Tm(d) of gold NPs as a function of d down to 2.5 nm by following their electron diffraction pattern and its broadening with the formation of the liquid phase [77]. Dick et al. [322] found by differential thermal analysis that the melting endothermic peak of 1.5 nm gold NPs is 380°C, a decrease of more than 680° relative to the melting point of the bulk gold. As mentioned, an atom diffusion process (liquid skin formation) at the NPs’ surface takes place at temperatures much lower than their melting point. Due to this diffusion phenomenon, sintering of metallic NPs may occur at temperatures below 150°C [15,323]. This fact is the main motivation for the use of metallic NPs in conductive inks. After deposition of the metallic NP dispersion on a solid substrate, for example, by ink-jet printing, the obtained pattern is composed of closely packed NPs. In order to obtain high conductivities, a sintering process should be carried out. The use of metallic NPs enables taking advantage of this diffusion phenomenon to achieve low sintering temperatures. Silver NPs composing a printed pattern are presented in Figure 15.7. At room temperature, due to a small number of contact points between the NPs, the pattern does not possess useful conductivity. Higher conductivities are achieved by heating the patterns to temperatures between 100°C and 320°C. The electrical conductivity of these patterns sintered at 320°C for 10 min was found to be ~1.5% of that of bulk silver [15]. As seen, interconnections between NPs appear already at 150°C, but the critical changes in morphology with formation of continuous interconnections between the particles are seen at 260°C and become much more significant at 320°C (it should be noted that the melting point of bulk silver is 960°C).

15.4

SUMMARY

It is certain that controlling the structures of metal NPs, that is, size, shape, and composition, by proper selection of stabilizer and reaction conditions is most important to tailoring the physical and chemical properties of metallic colloids. The current view of colloid chemists is that the complete control of these NPs’ growth is achieved by modification of the nuclei crystallographic facets exposed during the growth process. The stability of specific facets relative to others, as well as the adsorption of various molecules to selective facets, determines the growth directions and therefore the obtained particle characteristics. The functional properties of these metallic colloids dispersed in solution, as well as those arranged in 2D and three-dimensional arrays, are reviewed. Control over the size and morphology of the primary structures—the colloids—enables formation of the secondary structures—arrays—with tunable properties.

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Hydrophilic Colloidal Networks (Micro- and Nanogels) in Drug Delivery and Diagnostics Serguei V. Vinogradov

CONTENTS 16.1 Introduction: Colloidal Micro- and Nanogels or What is Their Difference from Other Drug Carriers? ............................................................................................... 16.2 Formation of Hydrophilic Micro- and Nanogel Networks ............................................... 16.2.1 Molecular Association .......................................................................................... 16.2.2 Chemical Cross-Linking ...................................................................................... 16.2.3 Surface Grafting and Vectorization ...................................................................... 16.3 Properties of Micro- and Nanogels ................................................................................... 16.3.1 Swelling ................................................................................................................ 16.3.2 Permeability .......................................................................................................... 16.3.3 Drug Loading ....................................................................................................... 16.3.4 Biocompatibility ................................................................................................... 16.4 Micro- and Nanogels in Drug Delivery ............................................................................ 16.4.1 Biodegradable Micro- and Nanogels .................................................................... 16.4.2 Composite Micro- and Nanogels .......................................................................... 16.4.3 Core–Shell and Multilayer Micro- and Nanogels ................................................. 16.4.4 Stimuli-Responsive Micro- and Nanogels ............................................................ 16.5 Conclusions and Future Prospects .................................................................................... Acknowledgment ....................................................................................................................... References ..................................................................................................................................

367 369 369 369 370 371 371 372 372 373 374 374 376 378 380 381 381 382

16.1 INTRODUCTION: COLLOIDAL MICRO- AND NANOGELS OR WHAT IS THEIR DIFFERENCE FROM OTHER DRUG CARRIERS? The attractive pharmaceutical concept of “magic bullet” introduced by Paul Ehrlich precisely one hundred years ago is finally getting a distinctive shape due to the recent developments in the area of drug delivery systems (DDS). This earlier idea was lately transformed into the concept of “magic pill” based on assisted targeted delivery and controlled release of therapeutic amounts of drugs in the site of disease. Administration of minute drug-filled containers instead of low-molecular-weight (MW) drugs has an obvious advantage in preventing nonspecific drug distribution and toxicity in 367

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healthy tissues. The two principal requirements are for them to be biodegradable and small to pass through microcapillaries without blocking flow and oxygen supply to the tissues. Enormous efforts were recently applied to the rational design and evaluation of various microsized (0.5–5 mm in diameter) and nanosized (<0.5 mm in diameter) drug carriers in vitro and in vivo. Currently, only a small fraction of microencapsulated drugs have found their way into the clinic; these are mostly liposomal drugs employing advantages of the oldest and well-studied DDS. But another time is close when an enormous variety of specialized DDS will come into medical practice to help us in tracing and eradicating malicious cells, delivering drugs across biological barriers, reaching intracellular targets, and enhancing current therapies. Chemical engineering and development of micro- and nano-DDS, such as liposomes, polymer micelles, micro/nanoparticles (NPs), micro/nanocontainers, micro/nanogels, nanotubes, and wires, to name a few, cover a significant segment among biomedical applications of nanotechnology. Most of them are out of the scope of this review, but it does not make them less promising objects to biomedical research. Our subjects, colloidal micro- and nanogels, contrary to macrohydrogels, are very small cross-linked polymer networks encompassing a certain volume of water and easily dispersed in aqueous media. Colloidal microgels are divided generally into two subgroups, nanogel networks with a diameter less than 0.5 mm and microgel networks with a diameter between 0.5 and 5 mm, and characterized by flexibility of polymer chains, large pore size, and easy volume change under the influence of different environmental factors. These networks offer several unique advantages over other DDS: Hydrophilic interior volume easily accessible for incorporation of biomolecules, uniquely high dispersion stability, tunable pore and particle size, large surface well-adopted for multivalent bioconjugation, and intrinsic ability to the stimuli-responsive volume change. Biodegradability and exceptional biocompatibility are also properties characteristic to many micro- and nanogels. Scrupulous reviews on micro- and nanogels in applications to various aspects of drug delivery have been recently published [1–4]. Although, many latest advances in the design and biomedical applications are overviewed here; this review will reserve a special attention to novel advanced forms of these DDS: Metal composite micro- and nanogels, multilayer and core–shell carriers with stimuli-responsive properties for drug delivery, and diagnostics (Figure 16.1). (a) A

(d) A D

– +

–

– +

(c) C

(e) BE

(fCF)

–

+

+

+

–

+

–

+ –

+ –

(b) B

+

+

–

–

FIGURE 16.1 Principal structural types of micro- and nanogels: (a) Cross-linked networks, (b) networks associated with hydrophobic domains (e.g., cholesterol molecules), (c) core-shell structure of two different networks, (d) multilayer networks (microgel covered with two polyelectrolyte layers is shown), (e) composite solid core–soft shell networks containing metal, ceramic, or protein NPs, (f) composite “raisin-in-pie” type of microgel with metal NPs dispersed in the network.

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16.2 FORMATION OF HYDROPHILIC MICRO- AND NANOGEL NETWORKS 16.2.1 MOLECULAR ASSOCIATION Molecular association of hydrophilic synthetic or biopolymers is the simplest way to obtain threedimensional networks that are capable to sterically encapsulate macromolecules presented in the solution during the process of assembly. Self-assembled monodispersed nanogels (diameter ca. 50 nm) formed from cholesteryl-pullulan (CHP) have found broad applications for protein encapsulation and delivery and were recently reviewed [5]. Efficient intracellular protein delivery was reported for these cationic CHP carrying bovine serum albumin (BSA) or beta-galactosidase [6]. Enzymatic activity was mostly retained after internalization. When the protein–nanogel complex was dissociated, protein was released inside the cell. These CHP nanogels were found promising for s.c. injection of HER2 vaccine and generated antibodies after the second and third vaccination inducing HER2-specific humoral responses in patients [7]. Another example of CHP nanogel application to anticancer therapy was incorporation and sustained release of interleukine-12 (IL-12) following s.c. injection [8]. Repetitive administration of the nanogel-encapsulated IL-12, but not IL-12 alone, induced drastic growth retardation of fibrosarcoma without causing any serious toxic effect. Immobilization of the active enzyme beta-galactosidase can be achieved by using highly specific biotin-avidin or DNA-peptide nucleic acid (PNA) interactions. The enzyme was conjugated to avidin and then encapsulated into microgel network formed of DNA and complementary biotinylated PNA [9]. Multiple repeated cycles of the substrate hydrolysis were obtained without inactivation of the beta-galactosidase. This approach of building the microgel network around single protein molecules can be applied for encapsulation of various enzymes and active proteins. In a separate approach, hydrogel association of natural polymer (alginate) molecules could be performed in the presence of Ca2+, for example, using liposomes as nanocontainers [10]. Alginate nanogels of 120–200 nm were obtained after removal of lipid bilayer by addition of surfactant. Nanogel size can be tuned by varying the nanocontainer volume and further controlled by the addition of monovalent salt to the dispersion. The success of this approach depends on the careful balance between self-association and aggregation of hydrophobized polymers during drug encapsulation and the drug release kinetics that is difficult to control and may be positively or negatively affected by various environmental factors.

16.2.2 CHEMICAL CROSS-LINKING A significant number of colloidal microgels was designed and produced by chemical cross-linking of synthetic or natural polyfunctional macromolecules in aqueous solutions. These macromolecules exist in the randomly coiled form that could be stabilized by bifunctional linkers forming nanogels. Their hydrodynamic diameter and pore size will depend on MW of macromolecules and the linker density. For example, cationic polyethylenimine (PEI) nanogels could be prepared by cross-linking using photo-Fenton reaction in aqueous solution [11] or bis-epoxides [12]. When various sizes of PEI-nanogel–plasmid DNA complexes were evaluated by transfection efficacy at uniform doses and surface charges, the highest efficacy (ca. 30% of all cells as determined by flow cytometry) was observed for nanogels of 75–87 nm in diameter irrelevantly to the cell line. The second approach includes cross-linking reactions in liquid microcontainers or microemulsions. In the “inverse emulsion” method hydrophilic polymers and cross-linkers are mixed together in aqueous solution that immediately dispersed in nonmixable organic solvent in the presence of surfactant (Figure 16.2). We successfully formed various poly(ethylene glycol) (PEG)- and Pluronic®cl-polyacrylate nano- and microgels by reaction of bis-amino-PEG/Pluronic® and polyacrylic acid (pAA) in the inverse water-in-dichloroethane emulsion following the addition of water-soluble carbodiimide [13]. Hyaluronic acid (HA) nanogels of 200–500 nm in diameter were fabricated by the inverse water-in-oil emulsion method using the formation of disulfide linkages as a cross-linking [14]. Green fluorescent protein (GFP)-specific small interfering RNA (siRNA) physically entrapped

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Structure and Functional Properties of Colloidal Systems Monomers or polymer molecules, cross-linker and surfactant in water-in-oil emulsion

Formation of nanogel network

Removal of the surfactant

FIGURE 16.2 Preparation of nanogels by the “inverse emulsion” method. Aqueous solution of major components for microgel synthesis (hydrophilic polymer molecules and cross-linkers, or monomers, crosslinkers, and reaction initiators) is dispersed in the presence of surfactant in nonmiscible organic solvent and sonicated to form a fine water-in-solvent dispersion. The cross-linked microgel network was formed inside the dispersed aqueous microcontainers and then purified from the surfactant and low-MW reactants by extraction and dialysis.

within the HA nanogels during the process could efficiently silence GFP expression in cotransfected GFP pDNA/Lipofectamine HCT-116 cells having HA-specific CD44 receptors on the surface. The third approach called the “emulsification–solvent evaporation” method was used to link water-soluble multifunctional polymers to the surface of heterogeneous phase-contained activated macromolecules dissolved in a volatile organic solvent [15]. Following the reaction, the solvent is removed in vacuo and the polymer network is allowed maturating in aqueous solution. Various cationic nanogels of PEG- and Pluronic®-cl-PEI structures have been synthesized using this approach for applications including encapsulation and delivery of oligonucleotides, 5¢-triphosphates of nucleoside analogs, and low-soluble drugs [16,17]. The efficient encapsulation of the anticancer natural spice curcumin in Pluronic® F127-cl-spermine nanogels was achieved that resulted in stable and tumor-targeted (folate) formulations with more than 100 times higher concentration of the drug, increased stability, and noneffaced activity against various cancer cells [18]. In the similar studies with PEG-cl-PEI nanogels, nontoxic carriers were obtained at the PEG/PEI molar ratio near 7 [19]. The anticancer drug AQ10 was incorporated in these carriers and a significant 3–4-fold inhibition of Pan 02 cell growth was obtained compared to the free AQ10. Finally, numerous methods were developed to introduce cross-links during emulsion polymerization processes. Detailed description of these methods is out of the focus of the manuscript; although many of them will be mentioned in the following sections.

16.2.3 SURFACE GRAFTING AND VECTORIZATION Surface grafting and modification by vector molecules is one of the most important ways of modulating the microgel network interactions with proteins and components of the cellular membrane. Serum proteins (opsonins and albumins), when deposited on the network, make it visible for internal macrophages of the reticuloendothelial system (RES) that rapidly remove the protein-covered particles from circulation and prevent their efficient interaction with targeting organs. Surface charge of microgels makes them especially vulnerable to the opsonization. Grafting the surface of microgels with PEG molecules can efficiently shield the charge, increase the circulation time, and reduce their binding with cellular membranes [3]. Decoration of the microgel network with zwitterionic compounds or formation of ionic pairs on its surface may also have a positive effect on systemic properties of these carriers as the modification with PEG. As many other DDS, microgels can be derivatized with receptor-recognizing ligands to enhance the tumor- or specific organ targeting. Surface modification of poly(N-isopropylacrylamide)

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(pNIPAm) microgels with folic acid resulted in rapid internalization by receptor-mediated endocytosis with higher efficacy as compared to nonmodified cationic particles [20]. Particle collapse and aggregation at 37°C following their internalization increased the toxic effect in mouth carcinoma KB cells, causing cell death. Authors suggest application of these microgels as anticancer cytotoxic agents (sic: without any drug!). Simple nanogels with tumor-targeting properties could be obtained by modification of pullulan with folic acid through ester bond [21]. These conjugated formed nanogels with sizes 70 and 270 nm at concentrations less than 10 mg/L and demonstrated doxorubicin (DOX) release kinetics reversibly depending on the substitution rate of pullulan. KB cells efficiently internalized all these nanogels. Formation of nanogels with active tethered PEG molecules on the surface was suggested for attachment of vector ligands [22]. The successful derivatization of the surface of PEG-cl-PEI nanogels with human transferrin was achieved using the following approach [23]. Controlled fraction of amino groups in the nanogel was initially converted into thiol groups using the Trout reagent, 2-iminothiolane. Transferrin was modified by reaction with a succinimidyl4-(N-maleimidomethyl)cyclohexane-1-carboxylate (SMCC) converting a limited number of accessible amino groups into thiol-specific maleimide moieties. When both the activated nanogel and protein were mixed together, the surface-decorated transferrin-nanogel conjugates formed with high yields. Only slight increase in hydrodynamic diameter corresponding to the formation of an outer protein layer on the nanogel was observed. These nanogels encapsulated plasmid DNA and increased delivery and transfection of prostate cancer PC-3 cells more than 50 times [3]. Polypeptidebased microgels have been extensively used for drug delivery and bioimaging, but their surface modification for cell targeting has been problematic [24]. In the novel electrostatic adhesion approach, the surface of serum albumin microgel encapsulating inorganic fluorescent core was modified by integrin-specific K n-RGD peptides carrying cationic poly-l-lysine stretch [25]. These tumor-targeted microgels efficiently accumulated in human colon cancer HT29 cells. Virus-mimetic nanogels can be designed by covering the surface of the core poly(histidine-co-phenylalanine) NPs with BSA via multiple short PEG linkers [26]. The BSA molecules were subsequently modified with folate for tumor targeting. Nanogels were loaded with anticancer drug DOX that could be specifically released at endosomal pH, whereas at cytosolic pH nanogels retained compact conformation with no traces of DOX release and were able to transfect other tumor cells forming at division.

16.3

PROPERTIES OF MICRO- AND NANOGELS

The key properties of microgels that determine their drug delivery characteristics are (i) network parameters such as particle size, swelling/deswelling, and pore size, (ii) surface features such as the charge and surface modifications (e.g., PEG density), and (iii) degradability. Microgel networks’ elasticity and hydrophilicity as well as their hydrodynamic diameter directly depend on chemical nature of polymer molecules and degree of cross-linking. Microgel permeability (pore size) and chemical properties of polymers (presence of ionizable groups, lipophilicity, and degradability) control drug binding, retention, and release. Thus, pore size was one of the decisive factors in the association of bioactive macromolecules (e.g., oligonucleotides) with an interior volume of cationic nanogels for better nucleolytic protection [27]. In addition, drug affinity to hydrophobic clusters or charged network in microgels can additionally slow down the drug release because of the diffusion or network degradation.

16.3.1

SWELLING

The equilibrium swelling degree is one of the most important properties of colloidal networks caused by water sorption or repulsion of molecular charges in the microgel network. Mechanical strength or elasticity of the microgel network, as in case of macrohydrogels, will depend on polymer content/concentration in the aqueous medium encapsulated by this network. Atomic force microscopy demonstrated a significant flattening of microgels on the surface of mica by gravitational

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or ionic forces with a width/height ratio up to 10, directly depending on the microgels cross-linking density [28]. As determined at saturated moist conditions, the equilibrium swelling degree (S) = volume of swollen gel/weight of dry polymer. S may range from 1 to over 1000. In general, S = f(n, m, k), where n is the cross-linking density, m is the number, and k is the ionization degree of charged groups. The cross-linking density is usually fixed at synthesis, depends on the method used, and has a significant effect on the equilibrium swelling degree. The number and ionization degree of a polyelectrolyte microgel are key swelling parameters: the former introduced at synthesis and the latter is a function of pH and ionic strength in the medium. Swelling kinetics for microgels is usually very fast compared to macrohydrogels; therefore microgels may swell and shrink practically immediately following changes in environmental conditions or drug association with the network, for example, resulting in the reduction of total charge of microgels. These processes are easily observed by monitoring hydrodynamic diameter of microgels as a function of ionic strength, pH, temperature, or drug binding using the dynamic light scattering (DLS) method [29].

16.3.2

PERMEABILITY

An adjusted permeability of microgel networks is the major attribute that makes them useful in DDS. The most important parameter that affects the permeability of microgels to a given solute is its swelling degree. Permeability (P) is usually defined as a function of the partition or solubility coefficient, K, and the diffusion coefficient, D. The partition coefficient is defined as the ratio at equilibrium of the solute concentration inside the microgel to that in solution. A value of K > 1 indicates that the solute is preferentially sorbed by the microgel. In pore size exclusion, K is always <1, even for solutes smaller than pores of the microgel. The diffusion coefficient is defined by the equation: F = -D dc/dx, where F is the solute flux (moles per unit area per unit time) and dc/dx is the concentration gradient. Values of D vary widely, from nearly the free solution value in highly swollen microgels (10-6 –10-5 cm2/s for most drugs) to zero. Diffusion coefficient depends primarily on MW and does not vary strongly for most low-MW drugs; however, the partition coefficient can widely vary for different drugs according to their affinity to polymer network (hydrogen bonding), hydrophobicity, size, or electrostatic nature, creating a virtually unlimited range of combinations. The octanol/water partition coefficient, Kow, is commonly used parameter for different drugs. A drug with a larger value of Kow will be low soluble in water and may be efficiently absorbed by microgels with hydrophobic core or regions. For negatively charged polyacrylate microgels, an increased ionic strength resulted in decreased deswelling rate but also in an increased polylysine transport within the microgel network. Peptide MW was responsible for entering in the network, as well as pH, which regulated the pore size of microgels [30].

16.3.3

DRUG LOADING

Soluble drug molecules can be encapsulated in microgel network by various techniques including physical drug entrapment caused by (i) the formation of microgel network or (ii) swelling/deswelling cycle or drug diffusion from aqueous solution and association with (iii) hydrophobic domains, or (iv) the oppositely charged polymer chains in microgel network (Figure 16.3). The incorporation of solutes into microgels also has a substantial effect on the swelling behavior. Charged low-MW solutes freely travel inside the swollen polymer network until they reach and bind oppositely charged moieties and neutralize the microgel. The structure then collapses and pumps out water off the microgel volume forming particles with significantly lower internal volume. For example, nanogels of the PEG-cl-PEI structure readily bind nucleoside 5¢-triphosphates in aqueous dispersion, the process accompanied by 8–10-fold reduction in nanogel volume [31]. Another example is physical entrapment of proteins in cholesterol-modified polysaccharide microgels following simple mixing of both solutions of polymer and protein. The amount of protein encapsulated in microgels depends on MW of protein and can vary from 1 to 9 molecules per microgel [32]. Temperature- or pH-dependent

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FIGURE 16.3 Drug loading of the preformed tumor-targeted cationic nanogel networks dispersed in aqueous solution of small and medium-sized anionic drug molecules (e.g., oligonucleotides, siRNA, or NATP). Negatively charged drug molecules (black-filled circles) rapidly penetrate the nanogel interior through large pores and complex with cationic polymers. Formation of hydrophobic domains results in drastic reduction of the nanogel volume and formation of the core–shell nanogels with the drug safely encapsulated in the inner core. Tumor-targeting ligands (transferrin, folate, or EGF-related peptides) remain on the surface of the drug-loaded nanogels.

microgels are ideal carriers for loading drugs by a swelling–shrinking process. Initially, these microgels swell in the presence of soluble drug at conditions when their swelling degree is at maximum (e.g., at body temperature or acidic pH). Then, a fast change to normal conditions will result in microgel shrinking and entrapment of the drug inside of microgel network. Again, the drug will be released from the microgel as soon as conditions after administration will change its swelling degree. Formation of both impermeable local skin layers and hydrophobic sites in microgels strongly affects drug partitioning between the solution and microgel phases [33]. For example, cationic drugs induce local phase transitions in anionic microgels that can be used to regulate small molecule diffusion in and out of the gel. In another example, the interaction between lysozyme and oppositely charged pAA microgels resulted in nonuniform distribution of lysozyme within the microgel network [34]. For a range of conditions, lysozyme formed a protein shell on the microgel surface. The shell formation was associated both with the increased lysozyme loading and with increased lysozymeinduced microgel compaction. The loaded microgels characterized by a net positive surface charge (zeta potential) and by a relatively fast exchange of lysozyme between microgels and the medium. Colloidal microgels may be used for the encapsulation and controlled release of sensitive biomolecules such as proteins and peptides due to the changes in conformation of the microgel network. For example, monodisperse microgels obtained by copolymerization of NIPAm and methacrylic acid (MAA) in the presence of a cross-linker displayed dramatic conformational changes in response to environmental changes in pH, temperature, or salt concentration [35]. They were capable to initially absorb different biomolecules at one pH and, following a pH change, encapsulate and protect them because of the collapse of the microgel network. In this form, biomolecules can be safely delivered to a target site where microgels again adopt the initial extended conformation and allow the release of encapsulated biomolecules. The insulin for oral delivery may be one example from many similar ones. At the pH of the stomach, the poly(NIPAm-MAA) microgel in its compact conformation protects the protein from denaturation. Passing into gastrointestinal (GI) tract, the microgel changes the conformation to an expanded one and releases the insulin. Thus, colloidal microgels offer an opportunity for administration of sensitive proteins and peptides via an oral route.

16.3.4 BIOCOMPATIBILITY Biocompatibility refers to the inertness of microgels and the absence of undesirable physiological reactions, including reaction with plasma proteins, immune response, toxicity, and vascular damage. The high water content of microgels makes them soft and flexible, and, thus, these networks

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(<5 mm in diameter) usually do not cause vascular damage (embolism). However, some polysaccharide gels may cause serious inflammation due to their antigenic nature when administrated. Permeability to low-MW solutes is another important property that affects biocompatibility, including removal of unreacted monomers, residual initiators, surfactants, or by-products of the microgel synthesis. The rate of thrombosis or blood clotting time is a very important determinant for nanocarriers administered systemically. Since nonadherent clots transporting through the circulation caused the most adverse biological effects, the kidneys are usually examined for signs of damage from these clots. Cytotoxicity of many microgel polymer components, such as polycations (PEI, etc.) or pNIPAm can also cause problems at biomedical applications [36].

16.4 16.4.1

MICRO- AND NANOGELS IN DRUG DELIVERY BIODEGRADABLE MICRO- AND NANOGELS

In most cases, it is desirable that microgel drug-delivery carriers could degrade over time. Biodegradation includes complete or partial enzymatic digestion, or chemical degradation of microgel network at biological conditions. Microgels are degraded because of (1) degradation of crosslinkers, (2) linear polymer degradation, or (3) microbial digestion. The rates of degradation can vary widely, from hours to years. In any case, it is critical that all degradation products can be metabolized or excreted without adverse consequences. There is a wide range of biocompatible materials used in the production of microgels generating linear polymers after degradation of cross-links, or low-MW oligomers after the longest polymer chains are hydrolytically or enzymatically cleaved. Labile bonds are also frequently introduced into nanogels to make them biodegradable and facilitate drug release. Degradable acrylamide (AAm)-based nanogels containing acid-liable acetal crosslinkers prepared by free radical polymerization in the inverse emulsion were reported as advantageous carriers for protein, antigen, and DNA delivery [37]. Acetal cross-linker which is stable at neutral pH (t1/2 = 24 h) resulted in fast degradation of nanogels at acidic endosomal pH (t1/2 = 5 min) facilitating release of the payload [38,39]. Atom transfer radical polymerization (ATRP) was used for the synthesis of stable cross-linked nanogels based on water-soluble polymers in the inverse emulsion system [40]. A disulfide-functionalized cross-linker was used in this work to synthesize biodegradable nanogels. This approach was further extended to synthesize biodegradable, crosslinked poly[oligo(ethylene oxide)-MAA] nanogels [41]. Reducible disulfide-bridged heparin-PEG nanogels were obtained by ultrasonication of thiolated heparin and PEG nanocomplexes in selected organic solvents [42]. They were stable at physiological conditions but rapidly disintegrated and released free heparin inside the cells. This heparin was capable to significantly inhibit proliferation of mouse melanoma cells by inducing caspase-mediated apoptotic cell death. Biodegradable protein microgels encapsulating hydrophobic drugs can be prepared by ultrasonication of drugs with polyglutamate that formed at slightly basic pH core–shell microspheres stabilized by noncovalent crosslinking and hydrogen bond formation [43]. These microgels were smaller and even more stable than protein microgels stabilized by disulfide bonds. Drug release as the result of dissociation of the three-dimensional network holding drug molecules inside the carriers is regulated only by the stability of polymeric conjugates forming the micro- or nanogels and can cover periods from hours to months. Biodegradable microgels can be fabricated from a variety of natural biopolymers undergoing degradation in the presence of specific enzymes, from synthetic polymers containing degradable segments in their structures, or by using degradable cross-linker molecules binding linear polymers in their network. A degradable polymeric nanogel obtained from 2,2-bis(acryloxymethyl)propionic acid, analogous PEG macromonomer, and piperazine and consisted of repeating carboxylic acid moieties with PEG side chains was recently described [44]. Complexation of carboxylic group with cisplatin resulted in nanogels with cross-linked core, PEG corona, and average diameter 220–240 nm (Figure 16.4). These nanogels had similar in vivo anticancer activity to cisplatin tested in nude mice inoculated

Hydrophilic Colloidal Networks (Micro- and Nanogels) in Drug Delivery and Diagnostics

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FIGURE 16.4 Covalent drug binding in anionic core–shell nanogels. The preformed nanogels with the cross-linked pAA in the core can be loaded by direct reaction of cisplatin (black-filled circles) with carboxyl groups in the polymer network. The drug is protected from inactivation in the nanocarrier that can deliver and release the drug into the tumor following the slow hydrolysis.

with SKOV-3 tumors. Another type of biodegradable nanogels was synthesized using ATRP in water-in-cyclohexane inverse emulsion in the presence of disulfide-functionalized dimethacrylate cross-linker [41]. These nanogels degraded into polymer solutions in a reducing environment releasing the encapsulated biomacromolecules, for example, carbohydrates drugs against lectinrich pathogens. Biodegradable dextran nanogels could be also prepared by UV polymerization of dextran and hydroxyethylmethacrylate (HEMA) using liposomes as nanoreactor [45,46]. This method allowed efficient encapsulation of proteins (lysozyme and BSA), which were protected in human serum and demonstrated sustained release over days to weeks. Bis-vinyl-functionalized acetal-based cross-linkers that are cleavable at slightly acidic pH were synthesized and used as building elements to form nanogel particles for controlled drug release applications [47]. These nanogels degraded in less than 1 h but were stable at neutral pH for longer times. HEMA-based cross-linked colloidal nanogels with diameters ranging from 150 to 475 nm and narrow distribution were produced. Model biomacromolecules, that is, rhodamine-labeled dextran and BSA were efficiently loaded into these degradable nanogels. The release of biomolecules from loaded nanogels was controlled by pH along with the particle degradation. The BSA released from nanogels retained its structural stability. We introduced cationic PEG-cl-PEI nanogels as drug carriers in 1999 and initially demonstrated their application for enhancing the intracellular and transcellular transport of anionic polynucleotides [15,16,27]. Later these nanogels were successfully applied to the encapsulation of cytotoxic and antiviral drugs, 5¢-triphosphates of nucleoside analogs (NATP) [48–50]. The dispersed protonated nanogel networks spontaneously interact with phosphate groups of dissolved triphosphates or polynucleotides by simple mixing of their aqueous solutions, immediately forming the compact drug-loaded core–shell particles surrounded by a protective PEG layer (Figure 16.3). The drug-loaded nanogels could be lyophilized and stored for long periods in dry form. Nanogel drug formulations demonstrated attractive properties for tumor-targeted delivery of active therapeutic nucleoside analogs, for example, in chemotherapy of regular or drug-resistant tumors and viral diseases. To initially test this strategy, cytotoxic activities of biodegradable nanogels including fludarabine, zidovudine, cytarabine, floxuridine, or gemcitabine have been studied in sensitive and drug-resistant cancer cells. Azidothymidine-5¢-triphosphate (AZT-TP) drug loading directly depended on the polycation content in nanogels and was equal to 10–30% by weight or up to 0.5 mmol of AZT-TP/g. Lyophilized drug-loaded nanogels easily solubilized without aggregation and formed stable dispersions in aqueous media. Nanogels provided good protection of encapsulated drugs from degradation by ubiquitous extra- and intracellular phosphatases or phosphodiesterases. In model digestion experiments, up to 60% of nanogel-loaded AZT-TP remained intact compared to only 10% of the free AZT-TP. In addition, more than 50% of degraded nucleotide was

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found in the active forms of 5¢-mono- and diphosphates [49]. A sustained in vitro drug release from nanogels was observed at physiological conditions; ca. 2/3 of the initial load could be released during the first 24 h and another 1/3 during the next two days of incubation [31]. Drug release from nanogels in the presence of serum was too slow to cause any serious problem associated with nonspecific drug release. Strong affinity of cationic nanogels to cellular membranes was observed in our experiments with isolated cellular membranes, and by direct visualization of these cellular events using transmission electron and atomic force microscopy. What is interesting is that an accumulation of nanogels on the cellular surface was accompanied by the fast drug release from the nanogel across the cellular membrane and accumulation in the cytoplasm. This putative alternative mechanism of drug release can enhance the antitumor effect of the activated drugs delivered by tumor-targeted nanogels.

16.4.2

COMPOSITE MICRO- AND NANOGELS

More sophisticated types of micro- and nanogels contain complex composites of metal or ceramic NPs, where each component is designated for separate and special functions. Recent review on hydrogel–metal nanocomposites described different synthetic strategies and biomedical applications [51]. Precise control of particle size can be achieved by using solid NPs as matrices for building of thin hydrogel layer on their surface (Figure 16.1e). Core NPs can represent fluorescent nanoprobes, for example, quantum dots (QD), encapsulated enzymes, thermoablating crystalline ferrous oxides, porous silicates, and metal NPs. By adding novel controlled functionalities to the composite microgels, these NPs are capable to add diagnostic capabilities to microgels or to modulate drug release following the effect of an external stimuli (e.g., light, irradiation, magnetic field, etc.). Fabrication of smaller metal NPs inside the microgel network is novel and extremely important direction in the development of future advanced and remotely controlled micro- and nanogel devices (Figure 16.1f). According to the first method, pH-sensitive PEGylated nanogels were synthesized that contained dispersed in the internal volume metal NPs; for example, platinum (<2 nm) or gold (ca. 6 nm) [52,53]. These systems demonstrated pH-dependent shift of the surface plasmon band, the property that could be used for photodynamic therapy. We used a similar approach for preparation of small gold NPs dispersed inside of Pluronic®-cl-PEI nanogels using the reduction of the coordination complex of PEI with hydrochloroauric acid by sodium borohydride. The formation of gold nanoshells can be observed at the nanogel loading with NATP and the following volume collapse (Figure 16.5). Maximum of the plasmon resonance depended on the size of gold NPs forming the nanoshell and increased with the increase of the reduction time. Optimal surface plasmon range at 800–820 nm (the deepest light penetration in the body) was obtained when the diameter of

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FIGURE 16.5 Formation of nanogel-based gold nanoshells. (a) Cationic PEG-cl-PEI nanogels dispersed in the network small gold NPs form gold nanoshells following the loading with anionic drugs (NATP; blackfilled circles). The drug becomes accumulated in the core surrounded by the gold nanoshell as is shown by the transmission electron microscopy with contrast staining by vanadate (b). Bar is 100 nm.

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gold NPs was 10–12 nm. These nanogels demonstrated faster drug release, following irradiation by near-IR light (Vinogradov, unpublished data). Dispersed silver NPs in microgel can be obtained by reduction of silver nitrate with sodium borohydride in the poly(NIPAm-acrylate) microgel [54]. Uniformly dispersed and stabilized small spherical NPs with a narrow size distribution were formed only in the cross-linked microgel network, while interpenetrating polymer network was unable to stabilize the rapidly aggregating metal NPs. The network structure determined the size and even shape of silver NPs. Using similar approach, the silver NPs–microgel composites were obtained from the poly(Am-AA) microgels [55]. NPs of 24–30 nm in diameter were quite uniformly distributed in the polymer network. The silver NPs–microgel composites demonstrated excellent antibacterial effects on E. coli that depended on the size and amount of NPs and the acrylate percentage in the microgel. Further modification of the protocol allowed one to obtain silver NPs–microgel composites based on semiinterpenetrating poly(vinyl pyrrolidone) and poly(acrylamide) (pAm) chains in the microgel network. Silver NPs about 3–5 nm diameter were produced and preliminary evaluated in antibacterial applications [56]. Gold NPs of different shapes were successfully synthesized on supramolecular template of microgel consisted of tryptophan-containing amphiphiles without using reducing agents [57]. Various gold nanocrystals including wires, sheets, and octa- and decahedral shapes were obtained and stabilized in the microgel network. A unique strategy for preparation of composite pNIPAm microgels with solid gold core was developed recently [58]. This method allows precise control of the size of the encapsulated gold cores (tunable between 60 and 150 nm in diameter) and affords composite microgels from 200 to 550 nm in diameter. These microgels can be used as optically modulated DDS. Temperature-sensitive poly(NIPAm-maleate) microgels were loaded with gold nanorods designed to absorb in the near-IR spectral range [59]. Irradiation at 809 nm triggered a large reversible change in the volume of these microgels, which make them promising candidates for controlled drug release. Growing pNIPAm shell onto gold NPs covered by amino groupterminated polymer resulted in thermoresponsive nanogels [60]. Hollow nanogels could be also obtained by etching of the gold core with potassium cyanide. Recently, microgels copolymerized from NIPAm and glycidyl methacrylate (GMA) were employed as templates for in situ synthesis of gold and composite NPs [61]. Dispersed in the microgels, gold NPs could be grown by successive reduction of Au and Ag cations. The interparticle interactions was controlled by thermosensitive swelling/deswelling of microgels. Interestingly, the hybrid microgels exhibited multiple brilliant colors by variations of Au/Ag bimetallic NPs in the microgels, and the color change of each hybrid microgel could be adjusted by the size of NPs. Obtained microgels may find important applications in imaging and biomedical devices. To develop a photothermally modulated DDS, silica–gold nanoshells were fabricated with a peak absorption in the near-IR region of the spectrum, whose light is transmitted through tissue with relatively little attenuation due to the absorption. Irradiation of silica–gold nanoshells at their peak absorption will result in the conversion of light to heat energy that produces a local rise in temperature. The nanoshells can be embedded in thermosensitive poly(NIPAm-Am) microgels. Degree of collapsing of these drug-loaded microgels can be controlled by the laser irradiation, the concentration of nanoshells, and MW of drug molecules (Figure 16.6). Successful modulated drug release was achieved for insulin and lysozyme [62]. A hybrid nanogel has been also developed based on the interpenetrating network of thermosensitive pNIPAm microgels filled with NPs such as silicagel and hydroxyapatite [63,64]. Photochemical emulsion-free polymerization on the surface of superparamagnetic ferrous oxide NPs in aqueous solution resulted in magnetic nanogels [65,66]. These core–shell nanogels can be used for microwave ablation therapy [67]. A successful confinement of positively or negatively charged magnetic NPs between two polyelectrolyte layers can be achieved using layer-by-layer surface modification of pNIPAm microgels [68]. Inductive heat generated by the magnetic NPs was sufficient to cause the microgel collapsing above its volume phase transition temperature (VPTT) (near 50°C). This combination of thermoresponsive microgel with magnetic NPs opens up novel perspectives toward remotely controlled drug release.

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Drug release

FIGURE 16.6 Application of gold NPs-composite thermosensitive nanogels to drug delivery and lightinduced drug release. Core-gold NPs are heated by irradiation with the near-IR light, deeply penetrating into the body. Drug encapsulated in the thermosensitive network can be burst-released during the sharp volume collapse at the VPTT of ca. 50°C.

Another prospective direction is encapsulation of enzymes or bioactive proteins by covering them with nanometer-thick hydrogel layers. The method of direct hydrogel layer growing on the surface of single-enzyme molecules was described recently [69,70]. Cross-linked pAm nanogels encapsulating horse radish peroxidase or bovine carboxyanhydrase maintained enzymatic activity and demonstrated high thermal stability. Similarly, a novel method was developed for preparation of nanogels with lysozyme core and dextran shell [71]. First, lysozyme–dextran conjugate was prepared and, then, it was heated above the denaturation temperature of lysozyme to produce nanogels (diameter ca. 200 nm). Nanogels could be loaded with protonated ibuprofen. This approach can be applied to prepare other globular protein–dextran nanogels. Properties of fluorescent materials can be influenced by the embedding in microgel networks. Patent on nanogel-based contrast agents for optical molecular imaging was recently issued [72]. This nanogel was produced by polymerization of water-soluble monomer containing ionic or H-bonding moieties, macromonomer with repetitive hydrophilic units, and multifunctional cross-linkers. A pHsensitive PEGylated nanogel encapsulating gold NPs was synthesized by the one-pot method through autoreduction of HAuCl4 within nanogels constructed from cross-linked poly(dimethylaminoethyl methacrylate) (DMAE-MAA) core and tethered PEG chains [52]. Average amount of gold NPs with diameter 6 nm was 10 per nanogels with average nanogel diameter <100 nm. The surface plasmon resonance band of these nanogels shifted in response to pH demonstrating potential bioimaging applications. Microgels with functional glycidyl groups were obtained by copolymerization of N-vinylcaprolactam and GMA in aqueous emulsion [73]. These microgels could be modified with 2-aminoethyl phosphate and with photoluminescent europium-doped lanthanum fluoride NPs for analytical applications. Antibody and the DNA aptamer retained their recognition capabilities when coupled to carboxylic groups of pNIPAm microgels [74]. They could be used for biodetection based on paper strips to isolate biosensing molecules from the paper surface and to form well-wetted and easily exposing gel-supported probes interacting with their targets.

16.4.3 CORE–SHELL AND MULTILAYER MICRO- AND NANOGELS Strict control of spatial distribution of polymer chains in nanogels can be achieved by the recently described Ca2+ condensation method [75]. Authors developed a procedure, in which block ionomer complexes were initially prepared by self-assembly of ionic blocks of double hydrophilic block copolymers with an oppositely charged condensing agent, followed by chemical cross-linking of ionic blocks in the core and removal of the condensing agent. Nanogels were prepared using this approach from PEG-b-pMAA (polymethacrylate) copolymers, which swell in water and incorporated hydrophilic drugs [76] (Figure 16.4). Similar technique was also used to prepare core–shell

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nanogels by condensation and cross-linking of PEG-grafted pAA (PEG-g-pAA) [28]. Mixture of the thiol-functionalized six-arm-branched PEG and DNA could produce a dimethylsulfoxide (DMSO)soluble complex with a size of 100 nm and, then, form cross-linked DNA-loaded nanogels following the oxidation [77]. Small nontoxic cross-linked pullulan nanogels with a diameter of 40–45 nm for delivery of polynucleotide drugs and genes could be prepared in the inverse emulsion system [Aerosol-OT (AOT)/hexane] [78,79]. Simultaneous administration of two or more drugs is now considered as potent strategy for enhancing chemotherapeutic treatments and preventing development of drug resistance. We successfully encapsulated two anticancer drugs with different modes of action in core–shell Pluronic®-cl-PEI nanogel particles (Figure 16.7a). Low-soluble anticancer drugs, such as paclitaxel and curcumin, could be encapsulated in the hydrophobic polymer core with efficacy up to 12% of nanogel weight. The solubility of nanoformulated drugs was increased in hundreds times. Additional anticancer drug, 5-fluoro-2¢-deoxyuridine 5¢-triphosphate, can be loaded in the outer cationic shell. Folatedecorated polychemotherapeutic nanogels were obtained that demonstrated increased cytotoxicity to various cancer cells as compared to separate drugs [80]. Core–shell microgels having thermoresponsive pNIPAm network with primary amino groups in the core or shell can be prepared via two-stage free-radical polymerization using 2-aminoethyl MAA as a comonomer. These amino groups were used to initiate ring-opening polymerization of benzyl-l-glutamate N-carboxyanhydride and produce poly(benzyl-l-glutamate) side chains. The core–shell architecture allows obtaining carriers, which are tunable for deswelling or other colloidal properties [81]. Size-controlled synthesis of monodisperse core–shell nanogels by freeradical polymerization was described starting from NIPAm, acrylic acid, and methylene-bisacrylamide (MBAm) as a cross-linker [82]. By varying the surfactant and initiator concentrations, particle size could be controlled with high precision while maintaining excellent monodispersity. Folic acid was used for decoration of these nanogels to obtain tumor-targeted drug carriers. Interesting type of nanogels covered by the lipid bilayer was recently introduced [83,84] (Figure 16.7b). Liposomes containing water-soluble monomer components formed the pAm/ MBAm network by photoinitiation. PEG-based microgels that are made positively or negatively charged through copolymerization with MAA or diethylaminoethyl (DEAE)-MAA could be efficiently lipid-coated by mixing with a suspension of oppositely charged liposomes [85]. The release (a)

(b) +

+

+

+ + + + +

+ +

FIGURE 16.7 Simultaneous encapsulation of hydrophobic and anionic anticancer drugs in core–shell nanogels for polychemotherapy. (a) Hydrophobic polymer core of Pluronic-cl-PEI based nanogels binds significant amounts of low-soluble anticancer drugs (e.g., paclitaxel and curcumin; black-filled circles), while positively charged cross-linked outer layer can load complimentary cytotoxic NATP (white-filled circles). The compact tumor-targeted nanogels can be formed in lyophilized form. (b) Hydrophilic nanogel covered by lipid bilayer can encapsulate separate loads of hydrophilic and hydrophobic drugs.

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of fluorescent molecules from microgels, lasting from hours to months, depended on the type of molecules used. Encapsulated BSA was released over a period of one week. Self-exploding microgels are another example of potential microgel DDS [86]. In the degradable dextran–HEMA microgels coated by a lipid membrane, the surrounding layer could rupture by an increase in swelling pressure of the degrading microgels. The coated microgels were obtained by using oppositely charged liposomes and the fact that swelling pressure can destroy the lipid layer surrounding the microgels was confirmed experimentally. This approach was extended to obtain biodegradable dextran-based microgels surrounded by a polyelectrolyte membrane [87] (Figure 16.7). The membrane surrounding the microgels was deposited using a layer-by-layer technique based on the alternate adsorption of oppositely charged polyelectrolytes onto a charged substrate. Increase in swelling pressure of microgels was sufficiently high to also rupture the surrounding polyelectrolyte membrane. The coating of the microgel surface can dramatically lower the burst drug release from the multilayer particles and, then, result in massive drug release at the membrane rupture. The undesirable burst drug release from microgels could be controlled by coating the drugloaded nanogels (<200 nm in diameter) with alternating layers of poly(allylamine hydrochloride) and poly(sodium 4-styrenesulfonate) [88]. This procedure resulted in nanogels stable at various pHs and drug release reversely depended on the number of polyelectrolyte layers. Creating polyelectrolyte layers on the surface of nanogels would result in multifunctional drug carriers. Poly(NIPAmpropyl-AA) nanogels were prepared by emulsion polymerization and then the second PEI layer was added by grafting their surface [89]. These nanogels demonstrated good thermo- and pH-sensitivity without apparent cytotoxicity. In another published example, these nanogels could be conjugated with horse radish peroxidase or urease and demonstrated an enhanced biocatalytic activity over free enzymes, higher stability, and storage time. These microgels can be used in various bioanalytical applications [90]. Microgels with covalently bound cationic nanolayers could be produced by direct surface modification of activated poly(GMA-Am) microgels with PEI molecules [91]. Folic acid functionalized poly(e-caprolactone)-block-PEI nanogels loaded with DOX formed particles with a diameter of 120 nm [92]. These particles demonstrated pH-triggered surface charge reversal from negative at pH 7.4 to positive at pH 5 existing in lysosomes that allowed them to transfer the drug in nuclei of ovarian SKOV-3 cancer cells very efficiently.

16.4.4 STIMULI-RESPONSIVE MICRO- AND NANOGELS Reversible volume change by the influence of environmental factors such as temperature and pH is the major feature of the soft polymer network of microgels. As in thermosensitive pNIPAm microgels, where the network collapse occurs in narrow temperature interval near 32°C, pHresponsive microgels can be carefully tuned to undergo drastic volume changes in compartments with a required pH, for example, those existing in tumors, stomach, or intestines, to release their drug load. Recently, many applications of pH-responsive nanogels for tumor delivery have been extensively reviewed [93]. Core–shell microgels with pH-sensitive poly(vinylpyridine) core and a temperature-responsive pNIPAm shell have been prepared and the dependence of drug binding on solution pH and temperature was investigated [94]. Electrostatic interactions were found dominant in the binding with small-MW drugs. Another pNIPAm-based microgel was functionalized with aminophenylboronic acid to produce glucose-responsive drug carriers [95]. Amphoteric microgels can be designed to obtain controlled swelling, deswelling, or switching the net charge of the microgels from cationic to anionic in response to the glucose concentration. In preliminary experiments, these microgels demonstrated high capacity for insulin and ability to release it at higher glucose concentrations. Above the microgels VPTT in the collapsed state, they adopt a more spherical form indicating a more compact chain packing on the particle surface. In water, below the VPTT, the microgels adopted flattened pancake-like shapes [96].

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Biocompatible nontoxic pH-sensitive nanogels made of pectin cross-linked with glutaraldehyde were designed for drug delivery [97]. Faster drug release was observed at higher pH and could be further accelerated in the presence of pectinolytic enzyme, indicating that the nanogel may be used for colon-specific drug delivery. A novel chitosan-ethylene diamine tetraacetic acid (EDTA) nanogel responsive to pH changes in the medium was composed the way it could undergo reversible surface switch inside out and back again without compromising the particle integrity [98]. A large class of stimuli-responsive PEGylated nanogels composed of cross-linked poly-2-DEAE-MAA core and PEG-tethered chains with terminal carboxylic group for binding tag molecules was reviewed recently [53]. The core volume phase transition is responsive to change in pH, ionic strength and temperature. Thermally exchangeable cross-linking approach was used to obtain core–shell star-like nanogels from linear diblock copolymers of MAA and alcoxamine methacrylic esters [99]. PEGylated nanogels were prepared by emulsion copolymerization of DMAE-MAA with 4-vinylbenzyl-PEG and ethylene glycol dimethacrylate as cross-linker [100]. DOX-loaded nanogels demonstrated pH-sensitive behavior, efficiently releasing the drug in endosomes at low pH. The antitumor activity of pH-sensitive DOX nanogels against a drug-resistant human hepatoma HuH-7 cell line was superior to that of both free DOX and pH-insensitive DOX nanogels. DOX-loaded nanogels responding to tumor extracellular pH 6.2 were fabricated also by self-association of Boc-l-histidine conjugate with deoxycholate-pollulan (DCP) [101]. Critical aggregation concentration (CAC) of these nanogels dramatically depended on pH and changed from 10 mg/mL at pH 7 to 660 mg/mL at pH 6.2, so that the release rate of DOX increased significantly with reduction in pH. In the MCF-7 cellular model, more than twofold increase in cytotoxicity was observed at pH change from 7.4 to 6.8. Other pH-responsive microgels with narrow size distribution were obtained by microemulsion copolymerization of N-acryloyl-N′-methylpiperazine and MAA. These microgels could swell excessively at acidic pH reaching a diameter of 750 nm and deswell to 350 nm in basic solutions [102]. Dispersions of cross-linked pMAA microgels underwent a drastic change from a fluid to a gel with increasing pH. Ionic cross-linking of these microgels in the presence of Ca2+ resulted in a significant decrease in CAC but had no influence on elasticity of the gel in view of the tissue engineering applications [103]. Monodisperse poly(Am-MAA) microgels with sharp pH–volume transition can be prepared in ethanol [104]. Osmotic pressure in microgels increased at higher content of MAA or cross-linking degree and resulted in moving the onset of pH-dependent volume transition to a higher pH. As a result, microgels demonstrated pH-dependent 12-fold volume transition compared to their original volume within a narrow pH interval of 0.5. Pluronic-based nanocarriers could be useful for enhancement of endosomal drug release; for example, small (120 nm), at physiological conditions, nanogels were capable of swelling drastically at a brief cold shock, reaching the diameter >400 nm, and of disrupting the endosomes [105].

16.5 CONCLUSIONS AND FUTURE PROSPECTS Colloidal micro- and nanogels, in general, form a valuable and extremely diverse platform for development of stimuli-responsive drug carriers, and, in the current preclinical phase, they demonstrate all features of promising DDS. Further studies would determine what structural and surface parameters of these carriers are required in order to enhance systemic-targeted delivery, avoid retention by the RES system, to efficiently cross important biological barriers (GI, BBB, cellular, etc.), and to control drug release in the targeted organ or the site of disease. In future, the composite micro- and nanogel platform will allow creation of silicon chip-based DDS with even higher extent of control over the drug delivery and release.

ACKNOWLEDGMENT The author is grateful for the financial support from National Institutes of Health (RO1 grants CA102791 and NS050660).

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17

Responsive Microgels for Drug Delivery Applications Jeremy P. K. Tan and Kam C. Tam

CONTENTS 17.1 Introduction ...................................................................................................................... 17.2 Synthesis Technique ......................................................................................................... 17.3 Microgel Swelling ............................................................................................................. 17.3.1 Swelling of Stimuli-Responsive Microgel ............................................................ 17.3.2 Microgel Swelling and Deswelling Theory .......................................................... 17.4 Controlled Drug Delivery ................................................................................................. 17.4.1 Passive and Active Targeting ................................................................................ 17.4.2 Intracellular Delivery and Subcellular Distribution ............................................. 17.5 Drug Selective Electrode .................................................................................................. 17.6 Microgels as Drug Delivery Vehicles ............................................................................... 17.7 Mathematical Modeling for pH-Responsive Microgels .................................................... 17.8 Conclusions and Prospects ............................................................................................... References ..................................................................................................................................

17.1

387 390 391 391 395 395 397 398 399 400 405 408 409

INTRODUCTION

The recent history of drug delivery began in the 1950s with the introduction of the first microencapsulated drug particles, and in the 1960s an initial understanding of pharmacokinetics and therapeutic drug monitoring was reported [1]. Polymers were used to deliver drugs, and formulation of drug delivery systems based on the understanding of pharmacokinetics, biological interface, and biocompatibility was evaluated [1]. In the past two decades, there has been a progressive increase in the number of nanoparticulate therapeutic products [2]. Although some progress has been achieved in cancer biology, this has not been translated to successes in clinic trials using nanoparticulate therapeutic products [3]. So far, only 24 nanoparticles-based therapeutic products have been approved for clinical use [2]. However, among these 24, only 20% are polymeric-based nanoparticles, whereas the rest are liposomal drugs and polymer–drug conjugates. As the mortality rate due to cancer continues to rise, two approaches have been adopted to reduce the mortality rate [4,5]. Genomics and proteomics research to identify new tumor-specific targets [6] and the use of innovative drug delivery systems to precisely target drugs to tumor cells with reduced toxicity and, at the same time, maintaining drugs at therapeutic concentration over the treatment duration are being explored [4,7]. With the parallel breakthroughs in molecular understanding of diseases and controlled manipulations of material at the nanometer length scale, nanotechnology offers great promise in disease prevention, diagnosis, and therapy [2,3,5,8,9]. Nanoparticles have been known to offer greater improvements in target to nontarget concentration 387

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Structure and Functional Properties of Colloidal Systems

ratios, increased drug content at diseased sites, and improved cellular uptake and intracellular stability [9]. Polymeric drug delivery systems in the form of beads, pellets, microspheres, and nanospheres have been developed to supplement conventional single or multiple dosage delivery modes [10]. Potential advantages of improved drug delivery include (1) improvement in the solubility of poorly water-soluble drugs; (2) continuous maintenance of drug levels in a therapeutically desirable range or in an environmentally responsive manner; (3) reduction of harmful side effects due to a targeted delivery to a diseased site; (4) reduction of the content of drugs needed to treat the disease; (5) reduction in the number of dosages and less invasive dosing, thus leading to an improvement in patients compliance with the correct dosage regimen; (6) delivery of two or more drugs simultaneously for combination therapy to generate a synergistic effect and suppress drug resistance; and (7) facilitation of drug administration for therapeutic products with short in vivo half-lives [2,3,11,12]. Through this method, optimization of pharmacological activity of drug and reduction in toxicity level can be achieved. These advantages must be weighed against the concerns in the development of polymeric drug delivery systems: (1) toxicity of the material or degraded products; (2) undesirable rapid release or unnecessary release of drugs; (3) discomfort caused by the system or extra pain during intravenous injection of a delivery system; and (4) high cost involved in the manufacturing of the material and during the drug encapsulation process [11]. The criteria used in the design of a drug delivery system are commonly based on the drug’s physicochemical and pharmacokinetic properties. Currently, significant efforts have been devoted to the development of controlled release devices for the delivery of rapidly metabolized drugs. One strategy is to use colloidal drug carriers that can provide site-specific or targeted drug delivery with an optimum drug release profile. Liposomes, polymeric nanoparticles (microgels or nanogels), micelles, and dendrimers are colloidal molecular assemblies (Figure 17.1). Release systems based on the swelling controlled by nanoparticles (e.g., microgels or nanogels) seem to fit the above requirements. The colloidal phenomenon of soft particles is becoming an important field of research due to the growing interest in using a polymeric system in drug delivery. The selection of material for the development of nanodrug delivery carrier depends on the desired diagnostic or therapeutic goal, types of drugs, material toxicity, and route of administration. Depending on the method of preparation, two different types of nanoparticles can be considered, namely, nanospheres and nanocapsules. Nanospheres have a matrix-typed structure in which a drug is dispersed, whereas nanocapsules possess a membrane-like wall structure with a core containing the drugs. New controlled release systems that respond to changes in the external environmental conditions such as temperature, pH, light, electric fields, and certain chemicals have been developed. Such systems, which are useful in pulsed drug delivery, normally undergo changes in their structures or intramolecular interactions brought about by external stimuli. The use of stimuli-responsive release system provides an interesting opportunity in optimizing a drug delivery system, where the system is an active rather than a passive carrier. Different stimuli-responsive polymers with different compositions could be used to design an ideal drug delivery vehicle with desired stimuli properties for treating a specific disease. The use of stimuli is particularly important when the stimuli is/are unique to the disease pathology allowing the vehicle to respond specifically to the pathological “triggers” [9]. In the following sections, an overview of the four main methods (emulsion polymerization, anionic copolymerization, cross-linking of neighboring polymer chains, and microemulsion polymerization) used to synthesize microgels or nanogels and the fundamentals of swelling and deswelling of pH-responsive microgels will be discussed. In addition, the later sections of this chapter provide an overview of drug release from microgel- and nanogel-based delivery vehicles. The use of drug selective electrodes for quantifying the drug release kinetics will be described, and the drug loading mechanism and release behavior of pH-responsive nanogels will be elucidated. Through the use of layer-by-layer (LBL) coating, the burst release from nanoparticles can be controlled.

FIGURE 17.1

Different types of nanodrug delivery carriers.

Microgel or nanogel

Micelle

Encapsulated drug

Cross-linked junctions

Liposome

Hydrophobic tail

Polymeric chain

Hydrophobic drug

Encapsulated drug

Hydrophilic tail

PEO chain

Hydrophilic drug

Phospholipid groups

Dendrimer

Drugs

Responsive Microgels for Drug Delivery Applications 389

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Structure and Functional Properties of Colloidal Systems

17.2 SYNTHESIS TECHNIQUE Gelation is the result of a cross-linking process of polymeric chains to produce a three-dimensional network of infinitely large size [13]. The term “infinitely large,” according to Flory, refers to a molecule having dimensions that approaches the percolation threshold [14]. However, by decreasing the size of the percolation threshold (constrained by emulsion droplets) through the reduction of emulsifier concentration, the size of the microgel produced can be reduced to nano length scale. Since microgels are intramolecularly cross-linked macromolecules, the size of the growing cross-linked chains must be controlled during synthesis. This can be accomplished by performing the polymerization and cross-linking within a defined volume, that is, in a surfactant micelle acting as an emulsifier. Four methods have been reported for the preparation of microgel particles, namely, emulsion polymerization [15,16], anionic copolymerization [17–19], cross-linking of neighboring polymer chains, [20] and microemulsion polymerization [21–24]. Emulsion polymerization is a versatile technique that yields narrow particle size distributions. In emulsion polymerization, each micelle in an emulsion behaves like a separate microcontinuous reactor that contains all the different monomers and radicals in the aqueous phase, where the hydrophobic monomer will be emulsified by surfactants. There are three regimes in emulsion polymerization: In Region I, initiation occurs where particles are nucleated. In Region II, the particles grow by diffusion of monomers from droplets through the aqueous phase into particles. When the monomers in the droplets are consumed, Region III commences, where the residual monomer in the particles and monomers dissolved in the aqueous phase are polymerized. The end of Region III corresponds to the completion of polymerization reaction. This heterogeneous freeradical polymerization involves emulsification of hydrophobic monomers in water by an oil-inwater (O/W) emulsifier followed by an initiation reaction via a water-soluble (sodium persulfate [22–26]) or an oil-soluble initiator (2-20-azobisisobutyronitrile [27–32]). During the polymerization, a large oil–water interfacial area is created as the nuclei are formed and grow in size as the polymerization progresses. Stabilizers, such as ionic or nonionic surfactants and protective colloid (polyvinyl alcohol), can be either physically adsorbed or chemically attached onto the particle surface to prevent coagulation of latex particles. The colloidal stability can be achieved by electrostatic stabilization [33], steric stabilization [34,35], or a combination of both. Emulsion polymerization can be conducted in the absence of added surfactant (surfactant-free emulsion polymerization) [15,36], where the continuous phase will have a high dielectric constant and ionic initiators are used. The charged polymer chains formed during polymerization act as surfactant molecules that stabilize the growing particles. Surfactant-free polymerization will not suffer from residual surfactant contamination unlike conventional emulsion polymerization. Thermal decomposition of the ionic initiator induces the free-radical polymerization process. The oligomers produced are surface-active and form nuclei when the length of the oligomers exceeds the solubility limit of the solvent. The nuclei then undergo limited aggregation, thereby increasing the surface charge until electrostatic stabilization is achieved [37]. The polymer growth is achieved through the absorption of monomers or oligomeric chains. Polymerization continues within the particles until another radical species enters the growing particle and termination occurs. Particle nucleation period is very short (of the order of minutes), which ensures a narrow particle size distribution. The final particle size increases with increasing electrolyte concentration and decreasing initiator concentration [37]. An alternative method to prepare microgels involves anionic polymerization in the presence of a good solvent. However, the poor size uniformity due to the lack of electrostatic stabilization during polymerization limits the utility of this technique. From the work of Okay and Funke [18,19], pendant groups in divinylbenzene (DVB) are able to react with radical sites on neighboring polymeric chains resulting in network growth between neighboring particles. This will increase the polydispersity of the microgel particles. However, it is likely that particles formed using this method have a relatively uniform distribution of comonomers because precipitation of high-molecular-weight-chains does not

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391

occur [38]. When a good solvent is used for both monomers, as demonstrated by Abrol and Solomon [39], a better control of the distribution of size is achieved. An elegant approach to producing microgels is through cross-linking of neighboring particles. Since an increased in dilution during cross-linking increases the probability of intramolecular cross-linking, the growing polymer chains in a highly diluted solution become intramolecularly cross-linked and the structure approaches that of a microgel formed within the micelles, where the initial micelle size determines the microgel particle size. The method discussed above should produce monodispersed particle size distributions since micelles of block copolymers dispersed in hemisolvents are usually narrow. The concept of microemulsion first appeared around 1980s and this is relatively new compared to emulsion polymerization. Briefly, microemulsion polymerization can be divided into O/W and water-in-oil (W/O) (also known as inverse microemulsion) [40–46]. The features of microemulsion are that they are isotropic and thermodynamically stable dispersions containing oil and water, where the stability is ensured by a very low interfacial tension capable of compensating for the dispersion entropy, which is very largely due to the small droplet size. These interesting features of microemulsions result in a unique microenvironment that can be harnessed to produce novel materials with interesting morphologies or polymers with specific properties. The resulting product can be classified as nanogels. A major difference between emulsions and microemulsions comes from the amount of monomers and surfactant needed to stabilize the systems. The amount of monomer is usually restricted to 5–10 wt% with respect to the overall mass and that of surfactants lies within the same range or greater. This could be a drawback that can restrict the potential application of microemulsions since high solid content and low surfactant amounts are usually desirable in most applications. However, there have been studies that deviate from these conditions [47–49]. The main difficulty encountered with using a high monomer concentration is the retention of optical transparency and stability of the microemulsions. Three characteristics distinguish microemulsion from emulsion polymerization [50]: (1) no monomer droplets exist but only micelles or microemulsion droplets; (2) initiators remain in the microemulsion droplets and polymerization occurs only when oil-soluble initiators are used; and (3) reaction mixture is optically transparent and in an equilibrium state.

17.3 MICROGEL SWELLING 17.3.1

SWELLING OF STIMULI-RESPONSIVE MICROGEL

A microgel particle is an intramolecularly cross-linked latex macromolecule, which is dispersed in a solvent and the degree of swelling depends on the degree of cross-linking and the characteristics of the solvent [38,51–54]. Microgel particles are classified as “smart materials” because of their conformational changes in response to changes in environmental conditions (either physical or chemical) such as temperature [55], pH [46,56], ionic strength [25], or a change in solvent types. The extent of swelling and deswelling is an interesting property of microgel [57], and microgels containing ionic comonomers exhibit pH-dependent swelling. The swelling of these particles are governed by the increase in internal osmotic pressure due to the mobile counterions contained within the microgels, which balances the internal electrostatic repulsion [38]. The extent of swelling will depend on the quality of solvent, ionization degree, cross-link density, and ionic strength. In Figure 17.2, the hydrodynamic diameter (nm) of poly(methyl methacrylate-co-methacrylic acid) is plotted against pH [36]. On increasing pH, swelling increases due to the ionization of carboxylic acid groups. The rate of increase levels off at around pH 8 as the ionization attains a maximum value. The increase in ionic strength screens the internal repulsion or reduces the osmotic pressure difference inside and outside the microgels. Similar trends can be observed for methacrylic acidethyl acrylate (MAA-EA) cross-linked with diallyl phthalate (DAP) [25,26], poly(methacrylic acid-co-acrylic acid) (PMAA-co-AA) [58,59], and poly(methacrylic acid-g-ethylene glycol) (PMAA-g-EG) [60].

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Structure and Functional Properties of Colloidal Systems 1000

Hydrodynamic diameter (nm)

900 800 700 600 500 400 300 200

6

7

8

9

10

11

12

13

pH

FIGURE 17.2 Effect of dispersion pH on the hydrodynamic diameter of poly(MMA-MAA) microgel particles. (From Saunders, B. R. et al. 1997. Macromolecules 30: 482–487. With permission.)

Tan et al. investigated the swelling characteristics of pH-responsive MAA-EA microgels cross-linked with DAP [26]. The swelling characteristic at degree of neutralization, a £ 0.3, is roughly similar for all the MAA-EA microgels containing 20 mole percent MAA at varying crosslinked densities as shown in Figure 17.3. The Rh values deviated more when a exceeded 0.3. This suggests a change in the microstructure of microgel particles during the neutralization process. The

220

HASE-20–80–0.5 HASE-20–80–0.75 HASE-20–80–1 HASE-20–80–1.5 HASE-20–80–2 HASE-20–80–3 HASE-20–80–4

Rh (nm)

180

140

100

60

20

0

0.2

0.4 0.6 0.8 Degree of neutralization, a

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1.2

FIGURE 17.3 Dependence of hydrodynamic radius, Rh, on neutralization degree, a, for microgels with 20 mol% MAA cross-linked at different DAP density (wt% DAP) in salt-free solution. The microgels were designated as HASE x-y-z, where HASE is hydrophobically modified alkali-swellable emulsions, x and y correspond to the molar fractions of MAA and EA, respectively, and z denotes the weight percentage of the cross-linker. (From Tan, B. H. et al. 2005. Adv. Colloid Interface Sci. 113: 111–120. With permission.)

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hydrophobic attraction between EA blocks limits the increase of Rh at a £ 0.3 and preserves the internal structure, whereas the osmotic pressure caused by counterions attracted to the carboxylate groups produces a sudden increase in Rh after a = 0.3 and swells the microgel. Therefore, a transition region of microgels caused by the competition of these two types of interactions was observed at a = 0.3. At a £ 0.3, the increment in Rh was smaller due to the osmotic pressure exerted by the counterions and the electrostatic repulsion between the carboxylate groups that are insufficient in overcoming the strong hydrophobic attraction between EA blocks. When a was increased beyond 0.3, the osmotic pressure overcame the hydrophobic attraction, which caused the Rh to increase rapidly. Microgels with lower cross-linked densities exhibited larger swelling behavior. For polyelectrolytes (PEs) [61,62] where ionic charges are present, the polymer is very sensitive to the effect of salt in solution due to the Coulombic repulsive forces between similar ionic charges and the attractive interactions between hydrophobic segments. The addition of inert electrolyte leads to deswelling of the particles. In Figure 17.4, the hydrodynamic radius of MAA-EA-DAP microgel is plotted against degree of neutralization [25]. The free ions distribute themselves inside and outside of the microgels therefore causing a net reduction in the osmotic pressure difference in the microgels. This is an additional screening of the electrostatic repulsion within the particles. Rodriguez and Wolfe studied the changes in the internal polymer segment density and particle swelling during the alkalization of internally cross-linked MAA-EA microgels using static and dynamic light scattering [52]. Upon neutralization with a base, pronounced negative deviations of the ratio of the radius of gyration (Rg) to hydrodynamic radius (Rh) from the known value of between 0.75 and 0.78 for homogeneous spheres along with the curvature in Guinier plots of the angulardependent light scattering intensity that suggested the presence of a nonuniform polymer segment distribution within the microgels [51,63]. Zwitterionic polymers, or polyampholytes, are charged macromolecules carrying both acidic and basic groups. In aqueous solution, these groups will dissociate and produce ions on the polymeric chains with counterions distributed in the bulk solution. After ionization, there are positively and negatively charged groups on the polymer chain, and varying the pH can change the net charge of a polyampholyte in aqueous solution as shown in Figure 17.5. At a particular pH called the isoelectric

120 0.1 mM

100

1 mM

90

10 mM

80

100 mM

Rh (nm)

110

70 60 50 40 30 20

0

0.2

0.4 0.6 0.8 Degree of neutralization, a

1

1.2

FIGURE 17.4 Results from dynamic light scattering illustrating the dependence of Rh on a for HASE 20-80-2 at different salt concentrations [KCl]. (From Tan, B. H. et al. 2004. Langmuir 20:11380–11386. With permission.)

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Structure and Functional Properties of Colloidal Systems

point (IEP), there are equal numbers of positive and negative charges, giving it a net zero charge. In this range, the polyampholyte is usually insoluble in pure water because of the net electrostatic attraction between opposite charge groups along the chain. However, such chains become soluble upon the addition of electrolyte, as a result of the screening of these attractive forces. The viscosity increases with increasing electrolyte concentration and this is referred to as anti-PE behavior. The conformation of a polyampholyte chain with a net charge is the result of competition between two antagonistic effects: (i) the polyampholyte effect, associated with the electrostatic attraction between groups of opposite sign, which tends to lead to chain compaction; and (ii) the PE effect, associated with the electrostatic repulsion between like charges, which tends to stretch the chains. When the net charge for polyampholyte is nearly zero, formation of ion pairs between positive and negative groups leads to an increase in entropy, which drives the association and compaction of polymeric chains. However, if the number of positive and negative groups is not equal, there will be a net positive or negative charge and polyampholytes will behave like a PE. PEs refer to polymers with either positive or negative groups that swell due to two effects: (1) electrostatic repulsion between identical charges that makes the chains become locally more rigid and (2) electrostatic attraction between charges on the polymer backbone and counterions that increases the osmotic pressure, which increases the excluded volume. There are multiple interactions acting in parallel with or competing against each other within the interior of polyampholyte microgels. There is however very few reported studies on the synthesis of polyampholytes microgels [64–67]. Such systems are worth exploring because they are able to maintain a robust internal microstructure under all pH conditions. Kihara et al. demonstrated the preparation of a zwitterionic microgel as a simple application of the methyl thioglycolate modified polychloromethylstyrene microgel [65]. Schulz et al. prepared and characterized a series of spherical latex particles with functionalized polyampholytes on the surface, resulting in a range of IEP. These latexes coagulate around the IEP. However, by adjusting the pH values, their surface charge increases and this allows the latexes to be dispersed [64]. Tan et al. prepared cross-linked poly(methacrylic acid) (MAA) and poly(2-(diethylamino)ethyl methacrylate) (DMA) microgels grafted with poly(ethylene glycol)methacrylate (PEGMA). This grafting provides colloidal stability at IEP [67]. At low and high pH, the microgels are in the swollen state due to the protonation of DMA and neutralization of MAA segments, respectively, and they are at the most compact state near the IEP.

(b) 2 20M–80D 40M–60D 60M–40D

1

0

–1

IEP–9.2 IEP–8.4 IEP–6.8

–2

Mobility (m2 s–1V –1)

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(a) 2

30M–70D 50M–50D 70M–30D

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IEP–8.8 IEP–7.7 IEP–6.2

–2 2

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6 pH

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FIGURE 17.5 Mobility of 0.02 wt% microgel solution of (a) 20M-80D, 40M-60D, and 60M-40D and (b) 30M-70D, 50M-50D, and 70M-30D at different pH values at 25°C. The microgels were designated as xM–yD where x and y correspond to the mole percent of MAA and DMA. (From Ho, B. S. et al. 2008. Langmuir 24: 7698–7703. With permission.)

Responsive Microgels for Drug Delivery Applications

17.3.2

395

MICROGEL SWELLING AND DESWELLING THEORY

The equation for swelling of microgels is due to the balance of the osmotic pressure inside and outside the microgels [68]: Pin + Pel = Pout,

(17.1)

where Pin and Pout are the osmotic pressure of mobile ions inside the microgels and in the solution, respectively. Pel is the elastic pressure of the polymeric network. Pel of a polymeric network is the pressure needed to stretch the polymeric network from the collapsed state (reference state) to the swollen state: RTco Pel = - _______ , 2NxQ1/3

(17.2)

where Q = (Ra /Ra=0)3 is the swelling ratio of the microgels, Nx is the cross-link density, and co is the polymer concentration when the particles are at the collapsed state. Ra represents the hydrodynamic radius of the particle at ionization a and Ra=0 represents the hydrodynamic radius of the unneutralized (a = 0) particle, both measured using the dynamic light-scattering technique. The terms Pin and Pout can be expressed as Pin = RTCin, Pout = RTCout,

(17.3)

where Cin and Cout are the concentrations of ions inside and outside the microgels, respectively. In the absence of salt, all the counterions associated with the ionized units are trapped inside the polymeric network by electrostatic attraction exerted by fixed charges. Thus, Cout = 0 and Cin are simply the concentrations of ionized units inside the microgels. coxa Cin = ____ , Q

(17.4)

where x is the molar fraction of acidic units and a is the degree of neutralization. By combining Equations 17.1 through 17.4, the swelling ratio, Q, of the microgels can be expressed as a function of the ionization degree and cross-linked density: Qμ(Nxa)3/2,

(17.5)

which assumes that all the counterions are trapped inside the microgels. Based on Figure 17.6a, Cloitre and coworkers found that the swelling ratio at maximum swelling, Qm, is proportional to Nx3/2. A plot of Q/Qm as a function of a3/2 as shown in Figure 17.6b shows that when ionization degree is low, Q/Qm is small, indicating that the microgels do not swell. When the ionization degree increases, the swelling ratio rises sharply and all the predicted and experimental data at varying cross-linked densities collapsed onto a master curve.

17.4

CONTROLLED DRUG DELIVERY

Polymeric drug delivery systems have attracted increasing interest during the last two decades [69]. An idealized drug delivery system is one where the required amounts of drug are made available at a desired time and at a specific site of action in the body [70]. The drug delivery systems can be in various types of formulations or dosage forms (tablets, implants, injectables, and transdermal patch). Through the use of such delivery systems, the efficiency of drug therapy is greatly improved

396

Structure and Functional Properties of Colloidal Systems (a) 100

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1500

2000

0

0.2

0.4

0.6

0.8

1

a3/2

FIGURE 17.6 (a) Variation of the swelling ratio at maximum swelling versus the cross-link density. (b) Variation of the swelling ratio with the neutralization degree for Nx = 140 (•), Nx = 70 (䉫), and Nx = 28 (䉱). (From Cloitre, M. et al. 2003. C. R. Physique 4: 221–230. With permission.)

Concentration of drug in biophase

compared to conventional dosage forms. Polymer-based controlled release systems are normally classified into either reservoir (membrane) or matrix (monolithic) devices. In the former type, the release is controlled by a polymeric membrane that surrounds the drug, whereas in the latter type, the drug is either dissolved or dispersed homogeneously as solid particles throughout the polymer matrix. Drugs can be released from the drug delivery system by diffusion, degradation, and swelling followed by diffusion. A continuous input of drug into the body can ensure that the concentration of drug remains above the lowest effective limit (ineffective range) required to provide the desired response but does not exceed the toxic range that will produce an excessive and possibly a dangerous response as shown in Figure 17.7. These situations are difficult to achieve with conventional dosage systems that must be administrated repeatedly at regular intervals, because such administration provides a pulsed input. A conventional dosage system has limitations like minimal synchronization between the time required for therapeutically effective drug concentrations. The ability for a drug delivery system to maintain an optimum drug content that lies within the ineffective and toxic range for an appropriate period is the most readily achievable of the two main aims of controlled release therapy. The second aim is to ensure that at least a substantial proportion of the active drug is released in the correct biophase at a controlled and satisfactory rate or is transported only to that biophase at some distance from the site of administration. The positioning of the delivery system near or adjacent to the appropriate biophase has achieved reasonable success.

Toxic range

Maximum safe concentration B

Therapeutic range Minimum safe concentration A

Ineffective range

Time

FIGURE 17.7 Drug concentration versus time profile of a drug following (A) administration of successive doses in a conventional dosage form (arrows indicate the time of administration) and (B) a single administration of a controlled release dosage form.

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Responsive Microgels for Drug Delivery Applications

However, applications where the drug has to travel a long distance via the blood or lymphatic site before reaching the final destination are still in their infancy [71].

17.4.1

PASSIVE AND ACTIVE TARGETING

For systemic drug delivery, both the passive and active targeting strategies are utilized. Passive targeting makes use of the properties of the delivery system and disease pathology in order to bring the delivery system to the site of interest that eliminates nonspecific distribution. In this way, drugs are selectively accumulated at the site of interest. Due to the enhanced permeability and retention (EPR) effect (illustrated in Figure 17.8), poly(ethylene glycol) (PEG)-modified nanoparticles can preferentially accumulate in the tumor tissue upon intravenous injection. The EPR effect in murine solid tumor was first reported by Maeda and coworkers [72,73]. A 10–100fold increase in concentration could be achieved in tumors when nanoparticles loaded with drugs were used instead of free drugs alone [74]. Stimuli-responsive delivery system that can release the drugs when an external stimulus is present is another form of passive targeting. The pH around tumor and other hypoxic disease tissues in the body is more acidic (~5.5–6.5) compared to physiological pH (7.4). Through the use of pH-responsive poly(beta-amino ester) (PbAE) nanoparticles compared to poly(epsiloncaprolactone) (PCL), a non-pH-responsive polymer, there was a significant enhancement in drug delivery and accumulation in tumor mass [75–77]. We have shown that pH-responsive methacrylic acid-ethyl acetate (MAA-EA) microgels cross-linked with DAP release profile could be controlled by manipulating pH [22,23]. Passive targeting could be achieved by controlling the size of nanoparticles and surface charge. Particles below 200 nm and with a positive surface charge were able to accumulate and reside in tumors for longer duration than neutral or negatively charged particles [74]. Active targeting includes PEG-modified nanoparticles to enhance circulation time and achieve passive targeting coupled with specific ligand on the surface that will be recognized by cells present in the disease site. There are several strategies that can be employed to modify the surface properties of nanoparticles for effective targeted delivery to tumors. Folic acid [78–82] or sugars [83–87] can be attached to nanoparticles for targeting over proliferating tumors, which have certain overexpress receptors for enhanced uptake of nutrients. Folic acid could target tumors with overexpress folate receptors [78–82] and galactose could target ASGP receptors found in liver cancer cells (Figure 17.9) [83–87]. For tumors with integrin receptors such as avb3 or avb5 that can bind to arginine–glycine–aspartic acid (RGD) tripeptide sequence, RGD modification to the nanoparticles

Healthy tissues pH 7.4

Drug molecules

Drug loaded nanoparticles

Tumor tissues pH <6.5

FIGURE 17.8 Schematic diagram for passive targeting due to the EPR effect. (From Ganta, S. et al. 2008. J. Control. Release 126: 187–204. With permission.)

398

Structure and Functional Properties of Colloidal Systems Galactose modified nanoparticle ASGP receptor

Drug molecules Receptor mediated endocytosis

Intracellular pH dependent drug release Nucleus

FIGURE 17.9 Schematic illustration of active targeting together with intracellular and subcellular distribution of nanoparticles.

has been carried out to target such tumors [88–92]. One very useful method to identify specific peptide sequences to target tumors is the phage display method [93]. Schluesener and Tan [94] made use of the M13 phages to select peptides that targets the brain tumor in rats. Farokhzad and coworkers [95] described the use of aptamers, nucleic acid constructs that specifically target prostate membrane antigen on prostate cancer cells. The aptamer technology provides an elegant way of targeting tumor cells. Another active targeting strategy is the use of monoclonal antibodies to target epitopes present on certain tumor cells. For example, Trastuzumab® or Herceptin® could be used to target HER2-expressing tumor cells [96]. Nanocarriers with targeting ability could be combined with stimuli responsiveness to improve the effectiveness to destroy tumors. Oishi and Nagasaki [97] demonstrated the use of pH-responsive and PEGlyated nanogels with lactose groups at the PEG end display endosomolytic abilities.

17.4.2

INTRACELLULAR DELIVERY AND SUBCELLULAR DISTRIBUTION

After the delivery of the nanoparticles to the tumor site, these nanoparticles may need to enter the cells to deliver the contents to subcellular organelles (Figure 17.9). Either passive or targeting strategies can be adopted. Passive or nonspecific targeting uptake of nanoparticles is due to endocytosis, where the cell membrane will engulf the nanoparticles to form an endosome. These endosomes usually travel in a specific direction toward the nuclear membrane. The endocytosis could be achieved by either phagocytosis (cells digest large objects) or pinocytosis (cells digest solutes and single molecules). Effective and efficient delivery could be achieved by buffering the endosomes using cationic carriers. Active or specific targeting can occur through receptor-mediated endocytosis. Once the ligand-modified carrier is bonded to the cells receptor, the cytoplasm membrane folds

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inward to form coated pits. These inward budding vesicles bud to form cytoplasmic vesicles. After dissociation of the nanoparticle–receptor complex, the receptor will be recycled back to the cell membrane. Arginine-rich cell-penetrating peptides (CPP) could be used to enhance cellular uptake [98]. For example, Tat peptide could promote nonspecific intracellular localization upon systemic delivery [99].

17.5 DRUG SELECTIVE ELECTRODE Previous studies on drug release using nanoparticles have focused on using techniques like UV spectrophotometry or high-performance liquid chromatography to monitor the concentration of drugs being released. All these techniques would require the use of a dialysis membrane or centrifugal machine to separate the nanoparticles from free drugs before measurements could be made. Such techniques will affect the measurements as drugs can be adsorbed onto the dialysis membrane and be inhibited by the diffusional barrier of the membrane, or the high centrifugal forces may also force additional drugs to be released from the delivery vehicle [22]. All the above techniques are tedious and cannot be automated compared to the drug electrode, which is simpler and more efficient in determining the drug release profile. However, using the drug selective electrode to directly measure the concentration of free drugs, there is no need for a dialysis membrane or centrifugal machine. A schematic diagram of the drug selective electrode can be seen in Figure 17.10. The development and application of such electrodes continue to be of interest in pharmaceutical research in the past several decades because these sensors offer advantages, such as simpler operation, better selectivity, lower cost, lower detection limit, wider linear concentration range, faster response, better accuracy, and can be applied to color, turbid solutions and possible interfacing with automated and computerized system [100–106].

Rubber sleeve

Screened cable

Electrode body

Inner Ag/AgBr reference electrode

Solder connection

End cap

Drug selective membrane

FIGURE 17.10 Schematic diagram of all the components of a drug ion selective electrode.

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Structure and Functional Properties of Colloidal Systems

The measurement of the EMF of a cell is required for potentiometric studies using drug selective membrane electrodes. The cell consists of two solutions separated by the membrane and the reference electrode in the following order: Inner Ag/AgBr reference electrode (2)

Internal solution (II)

Drug selective membrane

Test solution Ag/AgCl reference (I) electrode (1)

The drug selective electrode should respond ideally according to the Nernst equation: Ê RT ˆ EISE = Á ln c + C , Ë nF ˜¯ Ê 2.303 RT ˆ =Á log c + C , Ë nF ˜¯

(17.6)

where c is the concentration of the drug, R (=8.314) is the universal gas constant in J mol-1 K-1, T is the temperature in K, n is the charge number of the drug, F (= 96485 C) is Faraday’s constant, and C is the y-intercept. The sign + should be taken as positive when the drug is a cation and negative when the drug is an anion.

17.6 MICROGELS AS DRUG DELIVERY VEHICLES In the next few sections, we will discuss the development of microgels or nanogels for drug delivery applications and the evaluation of drug release mechanism in pH-responsive microgels or nanogels. Parenteral route is of utmost importance in the drug development and drug delivery research [107]. This will be the only possible route to deliver hydrophobic drugs such as amphotericin B and paclitaxel, which have very low solubility and cannot be administered through the oral route [108,109]. Parenteral route is the only feasible approach to administer drugs in emergency cases as this ensures quick onset of therapeutic products. From the work of Lee et al. [110], injections made with microgels are less painful as compared to cosolvent-based formulation. For microgels that are below 1 mm, the chances of emboli formation in the blood are insignificant. The small size of microgels will ensure that there is a higher circulation time, which is very useful in cancer treatment. Vinogradov [111] has also reported that hydrophilic and biocompatible microgels could provide a unique mode for the targeted delivery of encapsulated drugs via blood circulation. Microgels have exhibited excellent thermodynamic stability, high solubilization capacity, low viscosity, and ability to withstand sterilization techniques, and these properties will make microgels interesting delivery systems [107]. Most importantly, toxic drugs encapsulated by microgels have shown to reduce the toxicity of drugs like amphotericin B [112]. The microgels can be used for the controlled delivery of hydrophilic therapeutic molecules like aminoglycoside antibiotics [107]. Tan et al. [23] has shown the ability of an MAA-EA nanogel (microgels that are below 1 mm) cross-linked with DAP to encapsulate two different kinds of drugs through different interactions. There are two goals that should be satisfied in the design of a delivery system [113], namely, (1) efficient binding of drugs to the polymeric matrix and the release of drugs in a controlled manner and (2) the ability to release through a local or externally applied trigger by changing the binding affinity between the drug and the polymeric matrix. Therefore, a good understanding of the interactions between the drug and the delivery vehicle is critical for achieving an efficient delivery system. The two drugs that are loaded via different interactions showed remarkable differences in release behavior at the same pH (Figure 17.11). Procaine hydrochloride (PrHy) loaded via hydrophobic and hydrogen bonding showed higher

401

Responsive Microgels for Drug Delivery Applications (a)

1

Drug molecule Cross-linked junction

0.9 0.8

Mt/M•

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

pH 5 0

1

pH 6 2

3

pH 7.4 4

(b) 0.3

6 5 Time (h)

pH 8 7

8

9

10

Drug molecule Cross-linked junction

0.25

Mt/M•

0.2 0.15 0.1 0.05 pH 7.4 0

pH 6

pH 5

pH 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Time (h)

FIGURE 17.11 In vitro release profiles of (a) PrHy from 0.1 wt% HASE 50-50-4 in a changing pH environment and (b) IMI from 0.1 wt% HASE 50-50-4 in a changing pH environment. Mt is the amount of drugs released at time t and M• is the total amount of drugs incorporated in the carrier. (From Tan, J. P. K. et al. 2007. Eur. J. Pharm. Sci. 32: 340–348. With permission.)

release at high pH, whereas imipramine hydrochloride (IMI) showed an opposite trend due to the electrostatic attraction between the IMI and MAA-EA nanogels. These different kinds of interactions will result in a different release profile, which will be important for the design of a delivery carrier, where the drug release can be triggered by changing the pH. These pH-responsive nanogels can protect sensitive drugs against digestion by proteolytic/nucleolytic enzymes in the stomach due to their low swelling ratio at low pH. The pH-responsive property of nanogels in controlling the release of drugs at different pH is illustrated in Figure 17.12, where the concentration of the drug with respect to step changes in pH was measured with a drug selective electrode. Three comments can be made from the results reported in Figure 17.12. Firstly, the results demonstrated the viability of tuning the release profiles of a drug from a pH-responsive nanogel system. The proportion of drug released can be controlled by manipulating the pH of the environment. Hence, it is possible to design a pH-dependent gradient release drug delivery system, such that the active drugs could be released from the carrier in different regions of the physiological environment that possess different pH conditions. This will extend the therapeutical range of active drugs and ensure that the targeted areas receive the right dosage of drugs. Secondly, the interaction between drug and delivery carriers controls the release profiles and the effective therapeutic range of drug carriers. Therefore, it is possible to design a drug delivery system with enhanced interaction in order to have a better control of

402

Structure and Functional Properties of Colloidal Systems 1.2

pH 7.4 pH 5

1

Mt/M•

0.8 0.6 0.4 0.2 0

0

0.5

1

1.5 2 Time (h)

2.5

3

3.5

FIGURE 17.12 Step change in pH to induce variation in drug release using HASE 50-50-4 loaded with 2.44 g of drug/g of polymer. Mt is the amount of drugs released at time t and M• is the total amount of drugs incorporated in the carrier. (From Tan, J. P. K. and Tam, K. C. 2007. J. Control. Release 118: 87–94. With permission.)

release behavior. Thirdly, the monitoring of amounts of drug released cannot be easily performed using the conventional dialysis membrane or centrifugation techniques. A continuous monitoring technique using the drug selective electrode system reported here is the only practical method to measure the concentration of drug released in an environment, where the pH is continuously changed [22,23]. However, when the nanogels are passed through the gastrointestinal (GI) tract, the increase in pH will swell the nanogels thereby releasing the drugs. In the case of IMI, this will be an example of parenteral delivery, while after the endocytosis process the low pH in the endosome will promote the release of drugs. The drug release profile of a pH-responsive microgel could be manipulated by pH and salt concentration, as well as cross-linked density and content of carboxylic groups. We have shown by using the MAA-EA nanogels that the cross-linked density can impact the porosity of nanogels, which in turn affects the swelling ratio of nanogels. At a lower cross-linked density, the network is loose and possess a larger hydrodynamic free volume that can accommodate more solvent molecules resulting in a higher swelling ratio, which promotes a higher fractional release of drugs [23]. The content of carboxylic groups present in the microgel will affect the swelling ratio. We observed that nanogels with higher MAA-EA molar ratio exhibited a higher fractional release rate. Larger MAA-EA molar ratio produced a greater osmotic pressure after neutralization, which yielded a larger porosity within the nanogels. Thus, the overall diffusional barrier for drug molecules is reduced, which enhances the release rate [23]. Based on these studies, it is feasible to design nanogels that will provide optimal therapeutic property. Histological studies showed that particles with a size of 100 nm could diffuse through the submucosal layers, whereas larger-sized particles (500 nm to 10 mm) were found to concentrate within the epithelial tissue linings [114]. Because of the large surface-to-volume ratio, the release of hydrophilic drugs or proteins will be rapid as indicated by the undesirable initial burst release phenomenon. High initial burst release has been observed for poly(DL-lactide-co-glycolide) nanoparticles [115,116], chitosan nanoparticles [117], and poly(N-isopropylacrylamide)-co-acrylic acid hydrogels in the submicron range [118]. Hence, the control of initial burst release of drugs from nanoparticles is one of the major problems confronting the development of polymeric nanostructured systems for drug delivery applications. One strategy in addressing this problem is to develop a double-walled composite particle with a less permeable outer shell comprising a hydrophobic matrix for controlling the initial burst release [114]; however, this may not be the easiest route to limit the

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Responsive Microgels for Drug Delivery Applications

initial burst release. Quantifying the rapid initial burst release requires methodologies that can rapidly monitor drug concentrations at short times, therefore, we have used the drug selective electrode to monitor the release profile from pH-responsive nanogels [22–24]. We have used the LBL coating approach on MAA-EA nanogels to produce coated nanogels with desired release profiles suitable for drug release applications [119]. By adopting this strategy, the initial burst release was alleviated, and this technique offers potential advantages over the double-walled composite particle system because: 1. It is simple since it only involves electrostatic attraction between oppositely charged PEs and the layer thickness can be produced with nanometer precision 2. It is more cost-efficient and practical than chemically modifying the surface of nanoparticles. 3. The release process can be extended by manipulating the number of PE layers. However, the coating must not be too stable as this will impact the effectiveness of the coated particles as a vehicle for drug delivery. Permeability will be reduced if the coating is too stable, resulting in a continuous slow release of the drugs. The LBL technique [120] is a popular method for preparing multilayered films due to its low cost, simplicity, and versatility. This technique makes use of electrostatic attractions between oppositely charged PEs to construct layers with controllable thickness at nanometer length scale onto planar [120,121] or curve [122–125] surfaces. Based on Figure 17.13, initial burst release of PrHy was evident for nanogels coated with zero, one, and two PE layers, which was attributed to the release of PrHy from the surface of nanogels. The burst release phenomenon was reduced and minimized with the addition of more PE layers. As evident from Figure 17.13, the steep slope at short times (signifying burst release) became gentler when the number of PE layers was increased. With more PE layers, the more accessible PrHy molecules have to diffuse through the PE layers, therefore mitigating the burst release phenomenon. Two factors contributed to this trend, that is, (1) differences in the size of swollen nanogel and (2) changes in the permeability of PE layers. Such differences in the swelling altered the porosity that impacted the diffusion of drugs from the coated nanogel, resulting in a different kinetic release

1 0.9 0.8

Mt/M•

0.7 0.6 0.5 0.4

No coating 1st layer coating 2nd layer coating 3rd layer coating 4th layer coating 5th layer coating

0.3 0.2 0.1 0

0

1

2

3

4

5

6

7 8 9 Time (h)

10 11 12 13 14 15

FIGURE 17.13 Experimental release profiles for 0.1 wt% HASE 50-50-4 grafted with PEGMA and PE-coated HASE 50-50-4 grafted with PEGMA nanogels in 100 ml 10 mM NaCl at pH 7.4 (symbols), and theoretical fit of the mathematical model taking into account drug diffusion and chain relaxation (solid lines). Mt is the amount of drugs released at time t and M• is the total amount of drugs incorporated in the carrier. (From Tan, J. P. K. et al. 2008. J. Control. Release 128: 248–254. With permission.)

+

–

–

–

–

–

–

+–

Bare MAA-EA nanogel

–

+

–

Cross-linked junction

+ PrHy

+

–

–

–

–

+

+

+ +

+

+

+

–

–

+

–

–

– – –

+

PSS

PAH

–

+

–

–

3 coated layer nanogel

–

–

+

+

+

+

+

+

+

+

–

–

+

+

–

–

– –

– –

–

4 coated layer nanogel

–

– –

+

–

+

+

+ – – – – – + – – – – – + – – + – + + – –+ + + +– – – – – – + + – – ++ +– –+ + – – + + + – – – + – – – – – + + – – + + – –

+

FIGURE 17.14 Scheme illustrating the drug release process for the bare and coated nanogels. (a) Bare nanogels drug release demonstrated high burst release due to surface-adhered drugs. (b) Three coated layer nanogels demonstrated a lowered burst release and prolonged sustained drug release as compared to the bare nanogels due to the reduced permeability. (c) Four coated layer nanogels demonstrated very low burst release and prolonged sustained drug release as compared to three coated layer nanogels. (From Tan, J. P. K. et al. 2008. J. Control. Release 128: 248–254. With permission.)

+

+

+

+ + + + + – –+ + + + – – – + – + – + + + + + + –+ – – – +– + – – – – – + – – + + + – –

+

404 Structure and Functional Properties of Colloidal Systems

Responsive Microgels for Drug Delivery Applications

405

profiles. Nanogels with more PE layers swelled the least, resulting in a corresponding lower diffusion of drugs. The mechanism for the delayed and controlled release of drug from coated nanogel is depicted in Figure 17.14, where the increased layers of coating increase the diffusional barrier for the drug molecules. There is a linear dependence for the time to attain equilibrium and layer thickness suggesting that the diffusion process was controlled by the increase in layer thickness, and the type of PE within the layer was not particularly important. This simple relationship provided a useful guide in the design of coated nanogels for drug delivery applications as required by the therapeutic specifications of a treatment regime [119].

17.7

MATHEMATICAL MODELING FOR PH-RESPONSIVE MICROGELS

Mathematical modeling plays an important role in elucidating the drug release mechanism, thus facilitating the development of new delivery systems by a systematic rather than trial and error method [114]. Based on the physical or chemical characteristics of the polymer, the drug release mechanism from a polymer matrix can be categorized according to three processes [11,126], namely, (1) diffusion of the drugs from or through the system; (2) swelling or osmosis of the system through solvent extraction; or (3) erosion or degradation controlled because of a chemical or enzymatic reaction. In a swelling controlled system (nanogel), drug release is not only controlled by diffusion of drugs from polymeric matrix, but also by disentanglement of polymeric chains within the matrix resulting in the dissolution (chain relaxation) of the microgel. The “anomalous transport” of drugs is often present in controlled release systems since both diffusion and chain relaxation exist together [114]. Higuchi and the power law models [127,128] are the two most widely used models for predicting drug release kinetics, where only the diffusion contribution was considered. This constraint was introduced because of the limited number of data points obtained in a typical drug release study using the traditional data acquisition methodology. A more comprehensive model that describes both diffusion and relaxation contributions to drug release kinetics has been proposed by Berens and Hopfenberg [129,130]. Since this is a nonlinear model, larger number of data points would be required to generate a statistically meaningful model fitting, and hence this model was not often used. In most drug release kinetic studies, the release of drugs from dialysis membrane and nanoparticles was measured. Hence, the relaxation contribution of drug release from nanoparticle was often not observed. With the use of drug selective electrode, dialysis membrane was not necessary, thus the relaxation contribution of drug release from the nanoparticles can now be quantified. Although Torres-Lugo and Peppas [60] and Soppimath et al. [131] did use the Berens and Hopfenberg model to fit their results, they did not present the relaxation contribution of drug release kinetics. The diffusion behavior of the drug from the interior of microgel to the bulk solution can be mathematically described by [132] ∂C ∂ È ∂C ˘ D = - vC ˙ , ∂t ∂x ÍÎ ∂x ˚

(17.7)

where C is the concentration of solute, x is the diffusional path, D is the diffusion coefficient, v is the velocity of the solvent front, and t is the time. This equation contains both the Fickian behavior, described by D(dC/dt) and the non-Fickian behavior, given by vC. To obtain a better approximation, an exact solution of Equation 17.7 was proposed by Berens and Hopfenberg [115,116] having the form È6 Mt = 1 - FF Í 2 M• ÎÍ p

•

1

Ân n =1

2

˘ exp(-4 p 2 n2 Dt /d 2)˙ - F R exp( - kt ), ˚˙

(17.8)

406

Structure and Functional Properties of Colloidal Systems

where D is the diffusion coefficient for the Fickian portion of the transport, k is the first-order relaxation constant, FF and FR are the fractions of sorption contributed by Fickian diffusion and chain relaxation, respectively, d is the diameter of sphere, t is the time, Mt is the amount of drugs released at time t, and M• is the total amount of drugs incorporated in the carrier. The above model describes the overall release behavior in terms of Fickian and non-Fickian contributions. This analysis can lead to the determination of diffusion coefficient, D, and characteristic relaxation time, t, which is a reciprocal of k. An example of the fitting of Berens and Hopfenberg model on the experimental release data for varying pH studies is shown in Figure 17.15. The solid lines are the Berens and Hopfenberg model fitting using the nonlinear least squares fitting routine of MATLAB. Excellent agreement between the experimental and predicted kinetic profiles was obtained in all cases. We have demonstrated that during the release of drugs from pH-responsive nanogels, both the chain relaxation and Fickian diffusion play an important role (Figure 17.16). For a compact nanogel, chain relaxation should occur before diffusion of drugs into the bulk solution could take place, and the vice versa is true. Another parameter for determining the mechanism of drug release from swellable polymer matrices is the swelling interface number, Sw: vd(t) Sw = ____, D

(17.9)

where v is the velocity of the penetrating swelling front, d(t) is the time-dependent thickness of the swollen phase, and D is the diffusion coefficient in the swollen phase. Sw compares the relative mobilities of penetrating solvent and drug in the presence of macromolecular relaxation in the polymer. For Sw 1, the rate of drug diffusion through the swollen phase is much faster than the rate at which the swelling front advances. The release kinetics is governed by the rate of swelling (polymer relaxation), and the zero-order release kinetics is expected. In the case of Sw 1, the swelling front advances faster than the diffusion of drugs and thus a Fickian diffusion behavior is obtained. When Sw 1, non-Fickian release behavior dominates. As water penetrates into the microgels or nanogels, encapsulated drugs within the matrices produce a larger osmotic pressure, thus the swollen gels expand further while the polymeric chain swells. At the early stage of drug release, the swelling of

1 0.9 0.8

Mt/M•

0.7 0.6 0.5 0.4 0.3 0.2 pH 5

0.1 0

0

1

2

3

4

pH 6

5 6 Time (h)

pH 7.4 7

8

pH 8 9

10

FIGURE 17.15 Experimental in vitro release profile of PrHy from 0.1 wt% HASE 50-50-4 at varying pH: (a) pH 5 (䊐), (b) pH 6 (D), (c) pH 7.4 (䉫), and (d) pH 8 (䊊), and theoretical fit of the mathematical model taking into account drug diffusion and chain relaxation (solid lines). Mt is the amount of drugs released at time t and M• is the total amount of drugs incorporated in the carrier. (From Tan, J. P. K. et al. 2008. Int. J. Pharm. 357: 305–313. With permission.)

407

Responsive Microgels for Drug Delivery Applications

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3 0.2

0.2 FF FR

0.1

FF

(b)

4

0.1 5

6 7 Vary pH

8

9

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3 0.2

0.2 FF FR

0.1 0

FF

(c)

0

0

0.1 0 0.06

0.02 0.04 Vary [PrHy] (M)

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

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0.2

FF FR

0.1 0

FR

0

FR

1

1

0

5 10 15 20 Vary concentration gradient

FR

FF

(a)

0.2 0.1 0 25

FIGURE 17.16 Dependence of FF and FR on (a) pH, (b) [PrHy], and (c) concentration gradient obtained by the Berens and Hopfenberg model. (From Tan, J. P. K. et al. 2008. Int. J. Pharm. 357: 305–313. With permission.)

408

Structure and Functional Properties of Colloidal Systems

the polymer chains is negligible when compared with the osmotic pressure of the encapsulated drugs. However, when the drug release continues, the osmotic pressure decreases, while the configuration of the polymer chains approaches equilibrium.

17.8

CONCLUSIONS AND PROSPECTS

Drug delivery is an interdisciplinary field combining the efforts of materials and pharmaceutical scientists, engineers, and biologists. Nanoparticles have been shown to improve the therapeutic index of drugs by reducing the toxicity and enhancing drug efficacy. Microgels and nanogels have been demonstrated to meet the above requirements, and they are potential novel vehicles for either local or systemic delivery of hydrophobic or hydrophilic drugs. This is due to the unique set of properties that satisfies conditions for the encapsulation of low- or high-molecular-weight drugs for sustained release or prolonged blood circulation. With greater understanding of the physiological differences between normal and diseased tissues and advances in material design and chemical synthetic techniques, better and more efficient microgels or nanogels could be developed for passive or active drug delivery. Nanogels could be the vector-driven carriers with high differential organs distribution ratios [111]. Various mechanisms of drug release could be utilized by using the nanogels, from biodegradation to stimuli-responsive, that is, pH and temperature controlled release. Many reported studies on cancer therapy have shown promising advantages in using stimuli-responsive triggered release for the treatment of tumors. A novel technique in utilizing a drug selective electrode for studying drug release profiles was proposed. This technique is critical for understanding the drug release mechanism in pH-responsive nanogels, where drugs are released due to a combination between chain relaxation and Fickian diffusion can be elucidated. The ability of MAA-EA nanogels to deliver drugs bound by hydrophobic, electrostatic, or hydrogen bonding interactions was demonstrated. These different kinds of interactions will result in different release profiles and hence will be important in the design of a delivery carrier, where the release of drugs can be triggered by pH changes. Besides changes in pH, the amounts of cross-linked density and carboxylic acid content could affect the release profile. A simple and cost-effective method using the LBL technique could reduce the high burst release accompanied by nanoparticles. The future of the next generation of microgels or nanogels will contain better targeting ligands such as peptides, antibodies, sugar, or aptamers, which will further increase the efficacy of drug delivery vehicles or reduce toxicity of drugs. This could be possible with a better understanding of immunology and human genomics, which would lead to better insight of targeting ligands for site-specific targeting and better combinatorial chemistry to create new and better biomaterials. Through the greater knowledge in human genomics, a better understanding of the transport phenomena through various organs in the body, such as the intestine, lung, and skin, will lead to new drug delivery strategies. Multifunctional microgels or nanogels that can concurrently deliver and image the diseased tissue could become a major subject for future research. From the work of Amiji and coworkers [133], significant amounts of effort have been focused on multifunctional carriers, and these carriers are capable in overcoming physiological barriers and delivering therapeutic drugs and provide enhanced imaging of targeted tissues. With the functionality of microgels becoming more complex, there is a need to precisely engineer optimally designed microgels with the required physicochemical and biological properties to achieve the desired functions. Another major issue that requires more effort is the nonspecific uptake by mononuclear phagocytic cells and by nontargeted cells. This will increase the complexity of the microgels without affecting the ability to progress to clinical trials. In the next one to two decades, we hope to see more microgels- or nanogels-based carriers making their entry to clinical trials rather than seeing them only in in vivo studies using established animal models. Although we are still far from creating ideal microgels or nanogels for drug delivery, many researchers believe that one day polymeric-based nanocarriers will be used in therapeutic and imaging oncology, and increasing number of these carriers are now in clinical trials.

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109. Robbie, G., Wu, T. C., and Chiou, W. L. 1999. Poor and unusually prolonged absorption of amphotericin B in rats. Pharm. Res. 16: 455–458. 110. Lee, J. M., Park, K. M., Lim, S. J., Lee, M. K., and Kim, C. K. 2002. Microemulsion formulation of clonixic acid: Solubility enhancement and pain reduction. J. Pharm. Pharmacol. 54: 43–49. 111. Vinogradov, S. V. 2006. Collodial microgels in drug delivery applications. Curr. Pharm. Des. 12: 4703–4712. 112. Moreno, M. A., Ballesteros, M. P., and Frutos, P. 2003. Lyophilized lecithin based oil-water microemulsions as a new and low toxic delivery system for amphotericin B. Pharm. Res. 18: 344–351. 113. Bruce, A., Bray, D., Lewis, J., Raff, M., Roberts, K., and Watson, J. D. 1994. Molecular Biology of the Cell. New York: Garland. 114. Arifin, A. D., Lee, L. Y., and Wang, C. H. 2006. Mathematical modeling and simulation of drug release from microspheres: Implications to drug delivery system. Adv. Drug Deliv. Rev. 58: 1274–1325. 115. Govender, T., Stolnik, S., Garnett, M. C., Illum, L., and Davis, S. S. 1999. PLGA nanoparticles prepared by nanoprecipitation: Drug loading and release studies of a water soluble drug. J. Control. Release 57: 171–185. 116. Lee, W., Park, J., Yang, E. H., Suh, H., Kim, S. H., Chung, D. S., Choi, K., Yang, C. W., and Park, J. 2002. Investigation of the factors influencing the release rates of cyclosporine A loaded micro- and nanoparticles prepared by high pressure homogenizer. J. Control. Release 84: 115–123. 117. Borges, O., Silva, A. C., Romeijn, S. G., Amidi, M., de Sousa, A., Borchard, G., and Junginger, H. E. 2006. Uptake studies rat Payer’s patches, cytotoxicity and release studies of alginated coated chitosan nanoparticles for mucosal vaccination. J. Control. Release 114: 348–358. 118. Gu, J., Xia, F., Wu, Y., Qu, X., Yang, Z., and Jiang, L. 2007. Programmable delivery of hydrophilic drug using dually responsive hydrogel cages. J. Control. Release 117: 396–402. 119. Tan, J. P. K., Wang, Q., and Tam, K. C. 2008. Control of burst release from nanogels via layer by layer assembly. J. Control. Release 128: 248–254. 120. Decher, G. 1997. Fuzzy nanoassemblies: Toward layered polymeric multicomposites. Science 277: 1232–1237. 121. Burke, S. E. and Barrett, C. J. 2004. pH-dependent loading and release behavior of small hydrophilic molecules in weak polyelectrolyte multilayer films. Macromolecules 37: 5375–5384. 122. Caruso, F., Caruso, R. A., and Möhwald, H. 1998. Nanoengineering of inorganic and hybrid hollow spheres by colloidal templating. Science 282: 1111–1114. 123. Hirsjärvi, S., Peltonen, L., and Hirvonen, J. 2006. Layer-by-layer polyelectrolyte coating of low molecular weight poly(lactic acid) nanoparticles. Colloids Surf. B: Biointerfaces 49: 93–99. 124. De Geest, B. G., Déjugnat, C., Verhoeven, E., Sukhorukov, G. B., Jonas, A. M., Plain, J., Demeester, J., and De Smedt, S. C. 2006. Layer-by-layer coating of degradable microgels for pulsed drug delivery. J. Control. Release 116: 159–169. 125. Caruso, F. and Möhwald, H. 1999. Preparation and characterization of ordered nanoparticle and polymer composite multilayers on colloids. Langmuir 15: 8276–8281. 126. Leong, K. W. and Langer, R. 1987. Polymeric controlled drug delivery. Adv. Drug Deliv. Rev. 1: 199–233. 127. Higuchi, T. 1963. Mechanisms of sustained action mediation. Theoretical analysis of rate of release of solid drugs dispersed in solid matrices. J. Pharm. Sci. 52:1145–1149. 128. Ritger, P. L. and Peppas, N. A. 1987. A simple equation for description of solute release I. Fickian and non-Fickian release from non-swellable devices in the form of slabs, spheres, cylinders or discs. J. Control. Release 5: 23–36. 129. Enscore, D. J., Hopfenberg, H. B., and Stannett, V. T. 1977. Effect of particle size on the mechanism controlling n-hexane sorption in glassy polystyrene microspheres. Polymer 18: 793–800. 130. Berens, A. R. and Hopfenberg, H. B. 1978. Diffusion and relaxation in glassy polymer powders: 2. Separation of diffusion and relaxation parameters. Polymer 19: 489–496. 131. Soppimath, K. S., Kulkarni, A. R., and Aminabhavi, T. M. 2001. Chemically modified polyacrylamideg-guar gum-based crosslinked anionic microgels as pH-sensitive drug delivery systems: Preparation and characterization. J. Control. Release 75: 331–345. 132. Frisch, H. L. 1969. Diffusion in glassy polymer. J. Polym. Sci. 7: 879–887. 133. Jabr-Milane, L., van Vlerken, L., Devalapally, H., Shenoy, D., Komareddy, S., Bhavsar, M., and Amiji, M. 2008. Multi-functional nanocarriers for targeted delivery of drugs and genes. J. Control. Release 130: 121–128.

18

Colloidal Photonic Crystals and Laser Applications Seiichi Furumi

CONTENTS 18.1 Introduction ...................................................................................................................... 18.2 Fabrication of Colloidal Crystals ...................................................................................... 18.3 Properties of Colloidal Crystals ....................................................................................... 18.4 Fabrication of Colloidal Crystal Laser Devices ................................................................ 18.5 Properties of Colloidal Crystal Laser Devices ................................................................. 18.6 Conclusions ....................................................................................................................... Acknowledgments ...................................................................................................................... References ..................................................................................................................................

415 416 417 420 421 425 425 425

18.1 INTRODUCTION Interest in photonic crystals (PCs) has recently been increasing from both scientific and technological viewpoints of the photonics research, since the seminal reports independently published by Yablonovitch1 and John2 two decades ago. This is because the PC structures have the prospects of manipulating the flow of light at will.3 The PCs are dielectric materials possessing spatial structures with a periodic scale comparable to that of electromagnetic waves, resulting in the appearance of forbidden regions for photons in the dispersion spectrum. These regions are known as photonic band gaps (PBGs). Novel PC systems could cause some intriguing optical phenomena such as the strong localization of photons, the sharp bending of light propagation, and the suppression or enhancement of light emission from small-size devices. According to the dimensionality of the periodicity, the PC can be divided into three categories; namely, one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) PCs. The 1-D PCs, corresponding to dielectric multilayered structures, are nowadays used as optical filters, mirrors, and so on. Numerous efforts have been devoted to the development of the 3-D PC structures, because there is the possibility to exhibit complete PBGs, meaning that the PC structures can reflect specific light waves with all the polarization characteristics at any incident angle. Many reports have already been published on a wide variety of methodologies with regard to the fabrication of 3-D PC structures by lithographic and selective etching,4,5 multiphoton polymerization,6,7 and holographic techniques.8 For optoelectronic applications, it is important to obtain 3-D PC structures with the PBGs in the visible range. In this case, the periodicity in the PC structures should be designed on a submicrometer scale that is compatible with the wavelength of light. Nevertheless, such top-down fabrication methods cannot be adapted to the visible-range PBG structures in a straightforward way. Because the top-down method might encounter some serious problems, which arise from the relative difficulty in preparing the visible-range PBGs by the limitation of the optical resolution and the high cost of the method. 415

416

Structure and Functional Properties of Colloidal Systems Bragg reflection lmax

Colloidal micro-particle Gentle storing Self-assembly

Colloidal suspension

FIGURE 18.1

“fcc lattice structure” Colloidal crystal

Schematic representation of self-assembly process to form the CC structure.

In this context, colloidal crystals (CCs)—highly ordered 3-D architectures of colloidal particles of silica, polymers, and so forth—have received significant attention as one of the alternative and facile fabrication techniques of the 3-D PC structures with visible-range PBGs,9–11 since the first independent demonstrations of the PCs based on CC structures by the Russian12 and Spanish groups.13 As shown in Figure 18.1, the monodispersed microparticles have an intrinsic capability to assemble the face-centered-cubic (fcc) lattice structures on the substrate from suspension media in a spontaneous manner, which is known as bottom-up processing. The self-assembly to form the CC structures is generally triggered by an electrostatic repulsive interaction among the colloidal microparticles. Interestingly, we do not use any specific instruments to assemble the 3-D PC structures of the microparticles. When the particle diameter corresponds to several hundred nanometers, the PBG of uniform CC structures can be observed as the Bragg reflection of visible light.14 The maximum reflection wavelength (l max) is quantitatively determined by the intrinsic physical parameters comprising the effective refractive index of the CC (n), the lattice spacing along the [hkl] direction (dhkl), and the angle between the surface normal and [hkl] plane (q) according to the following equation:

( )

dhkl 2 2 1/2 l max = 2 ___ m (n - sin q) ,

(18.1)

where m denotes the diffraction order. In addition, the effective refractive index involves the filling ratio ( f ) of the microparticles in the 3-D PC structures by the following equation: n = [n2p f + n2s (1 - f)]1/2,

(18.2)

where np and ns denote the refractive indices of the microparticles and surrounding materials, respectively. As is evident from Equation 18.1, the wavelength of reflected light is directly proportional to the lattice spacing and average refractive index. Furthermore, the Bragg reflection wavelength is shifted when the spacing value or refractive index is changed. Therefore, the 3-D CCs can serve as various optical sensors capable of displaying and monitoring environmental changes. This chapter presents an overview regarding the CC structures as 3-D PCs. The following sections introduce several aspects of the fabrication procedure and optical properties of the CC structures, and furthermore, the potential utility of polymer CCs for laser applications based on the author’s experimental results.15

18.2 FABRICATION OF COLLOIDAL CRYSTALS Significant advancement in the CC research realm has established various strategies to assemble the CC structures by gravitational sedimentation,16,17 physical confinement,18–20 electrophoretic deposition,21 centrifugation,22 withdrawal,23 shear flow,24 and solvent evaporation.25–29 Among them, the solvent evaporation technique seems to be simplest. The CC films can be obtained not only by

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horizontal deposition, but also by vertical deposition. The first demonstration of the horizontal deposition enabled the fabrication of 2-D colloidal arrays on a substrate.26,27 Later, the 3-D CC structures were realized through this method by Kopnov and coworkers.28 Colvin and coworkers developed the vertical deposition technique to assemble high-quality CC films on a substrate surface through capillary force.29 In this technique, a substrate is vertically placed in a colloidal suspension. After gradual evaporation of the solvent, a thin film of colloidal particles is formed on the substrate. This vertical deposition technique produces excellent CC structures without cracks and grain boundaries. However, it is not easy to assemble large-size CC films. This is an important issue for practical manufacturing. Previously, Xia and Fudouzi developed a novel and simple procedure to obtain a uniform and large-size CC films of polystyrene (PS) microparticles embedded in poly(dimethylsiloxane) (PDMS) elastomer.30,31 As depicted in Figure 18.2, the fabrication process is very simple. First, a glass substrate was treated with a plasma cleaner to modify the hydrophilic surface. An aqueous suspension of PS microparticles with a diameter of ca. 200 nm was then loaded on the hydrophilic surface so as to form a liquid film on the glass substrate. Next, silicone oil was placed on the liquid film of the PS suspension. After storing for a few days, a well-ordered CC structure of PS microparticles could be observed as Bragg reflection. As shown in the bottom picture of Figure 18.2, the left part of the substrate exhibited the Bragg reflection, although the color of the right part remained white due to no crystallization of the PS microparticles. After complete assembly of the CC structure, the top cover silicone oil was carefully removed from the surface. The void space among the ordered PS microparticles was occupied by the PDMS precursor solution. By thermal treatment at room temperature for 12 h and at 60°C for 3 h, a flat and stable CC film of PS microparticles could be prepared by polymerization of the PDMS precursor. Although the conventional CCs tend to be fragile, the PS/PDMS CC films are robust, because the void space among the PS microparticles is filled and polymerized with the PDMS elastomer. This procedure enables the fabrication of a large, stable, and homogeneous CC domain even over 75 cm2. As-prepared CC film of PS/PDMS showed a characteristic reflection band as the PBG in the wavelength range from 540 to 580 nm, as shown in Curve 1 of Figure 18.3a. Fabry–Perót fringes were found around the reflection band, indicating the uniformity of the CC film thickness over a large area.32 A scanning electron microscopy (SEM) image viewed from the top surface revealed that this CC film adopts a well-organized fcc lattice structure comprised of the PS microparticles in a large area, as presented in Figure 18.4. As observed from a cross-section, this CC film showed a relatively flat surface and a thickness of ca. 20 mm. The magnified SEM image indicated that each layer of the PS particles is uniformly aligned in parallel to the substrate surface. The theoretical reflection spectrum of this CC film can be simulated by the scalar wave approximation (SWA) technique.33,34 Briefly, this SWA technique is similar to the analytic procedure of electron tunneling behavior in the quantum mechanics field. As simulated with the empirical parameters of the film thickness and lattice spacing, the theoretical reflection spectrum is obtained as Curve 2 in Figure 18.3a. A comparison between Curves 1 and 2 confirmed that the simulated reflection spectrum shows an appearance similar to the corresponding experimental spectrum. In particular, the simulated values of the reflection wavelength and bandwidth were in good agreement with the experimental results, implying the very high-quality CC film over the large area with a controlled thickness. Although there are currently many sophisticated techniques to fabricate the CC structures, this technique by Xia and Fudouzi provides a remarkably simple and convenient route to prepare largesize and uniform polymer CC films.

18.3 PROPERTIES OF COLLOIDAL CRYSTALS Recent progress has led to the versatile applications of the CC structures to optical sensing devices with external stimuli. For instance, Stein and co-workers developed an optical sensor of inverse

418

Structure and Functional Properties of Colloidal Systems 1 Loading PS particle suspension Aqueous suspension of poly(styrene) (PS) micro-particles with 200 nm diamater Hydrophilic glass substrate

2 Covering with silicone oil Poly(dimethylsiloxane) (PDMS) precursor solution PS micro-particles 3 Storing for a few days Gentle drying of water PS particle Silicone oil

Substrate

4 Colloidal crystal by self-assembly

1 cm

FIGURE 18.2 Fabrication procedure of the CC film of PS microparticles embedded in PDMS elastomer.

ceramics CCs that differentiated various organic solvents through the changes in refractive index.35 Asher and coworkers demonstrated the development of a number of smart optical sensors by combining the CC structures with polymer hydrogels.36 When the materials around the colloidal microparticles were made of poly(N-isopropyl acrylamide), the reversible shrinkage of this polymer between 10°C and 35°C changed the lattice space and subsequently brought about a shift in the Bragg reflection wavelength. Similar temperature sensing could be observed by inverse CC hydrogels.37 By using polymer hydrogels tethering crown ether and phenylboronic acid side chain, the CC gels could exhibit spectral changes in the Bragg reflection with respect to Pb2+, Ba2+, and K+

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Reflectance (arbitrary unit)

(a) Curve 2

Curve 4

Curve 1

Curve 3

500

550

600

700

650

Fluorescence (arbitrary unit)

(b) N+ ClO–

o

N

4

COOH Rhodamine 101

500

550

600 Wavelength (nm)

650

700

FIGURE 18.3 (a) Changes in reflection spectra of the CC film of PS/PDMS before (Curves 1, 2) and after swelling treatment with a PDMS oligomer and the subsequent polymerization (Curves 3, 4). The upper (Curves 2, 4) and lower curves (Curves 1, 3) correspond to the theoretical and experimental reflection spectra, respectively. The theoretical reflection spectra are calculated by the SWA technique. (b) Fluorescence spectrum of the film of Rhodamine 101, which is dispersed in poly(ethylene glycol), by excitation with 532 nm light from a Xe lamp. The inset indicates the chemical structure of Rhodamine 101. (Reprinted from Furumi, S. et al. 2007. Adv. Mater. 19: 2067–2072. With permission of Wiley-VCH, Copyright 2007.)

From top-surface

500 nm NONE

SEI

5.0kV x43,000100nm WD 18.1 nm

PDMS elastomer 5 mm NONE

SEI

10.0kv x5,0001mm WD10.0nm

From cross-section

10 mm NONE

Glass substrate

SEI

5.0kV x2, 500 10 mm WD 10.4 nm

500 nm

PS micro-particles

NONE

SEI

5.0kVx2, 500100 nmWD 10.4 nm

FIGURE 18.4 SEM images of the CC film of PS microparticles stabilized with PDMS elastomer observed from top surface and cross-section.

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Structure and Functional Properties of Colloidal Systems

cations38 and glucose molecule,39,40 respectively. Moreover, the reflection band from the other CC structures combined with functional materials could be modulated by humidity,41 mechanical stress,42–48 light irradiation,49–51 pH,52,53 and electrical54,55 and magnetic fields.56 Ozin and coworkers succeeded in the fabrication of tunable inverse CC films by compressive stress that enabled the visualization of colored fingerprints.48 The PS/PDMS CC film fabricated according to Figure 18.2 possesses a reversible tunability of the reflection band by various liquids. This phenomenon can be applied to a new type of paper such as producing colors with colorless inks.30,57 The paper corresponds to the CC films of PS microparticles stabilized with the PDMS elastomer. The ink is a silicone oil or any other liquid capable of swelling the PDMS. As the ink is applied to the surface of the paper, the position of the reflection band will be shifted to another wavelength. The displayed colors arise from the changes in the lattice space between the PS microparticles through swelling or shrinking of the PDMS elastomer. If the two color states are adequately different to be visually distinguishable, one can use their contrast to achieve color writing with colorless liquids. When the ink evaporates, the PDMS elastomer will shrink to its initial state and the colored patterns will be automatically erased. Such an ability to reversibly write and erase the colored patterns represents probably the most interesting feature associated with a new type of paper. The color changes could be repeated over 10 times with no deterioration in the color quality displayed by the PS/PDMS CC films. Moreover, the changed color could be preserved by chemical cross-linking reaction with monomer liquids capable of swelling the PDMS. When the As-prepared CC film of PS/PDMS fabricated according to Figure 18.2 was immersed into a PDMS oligomer solution, the initial reflection band around 560 nm was red-shifted and concomitant with a slight broadening of the bandwidth. Thermal polymerization of the PDMS oligomer was then carried out in order to preserve the red-shifted reflection band. After two rounds of swelling treatment, the resultant CC film exhibited a reflection band centered at 610 nm, as shown in Curve 3 of Figure 18.3b. The red-shift in the reflection band is ascribed to an increase in the lattice spacing from 178 to 209 nm, as revealed by the SEM observation. By recalculating using the SWA technique, the theoretical reflection spectrum of the swollen CC film was substantially identical to that obtained by the experimental measurement. These results are indicated by Curves 3 and 4 of Figure 18.3a. In this way, the CC films of PS/PDMS have unique capabilities for not only the desired tuning of the reflection band by adding monomer liquids, but also the on-demand preservation of the shifted reflection band by the subsequent polymerization.

18.4

FABRICATION OF COLLOIDAL CRYSTAL LASER DEVICES

Recently, more emphasis in the CC research field has focused on the design and introduction of specific defects such as points, lines, and layers within the 3-D CC structures for the creation of localized points and pathways of photons in minute optical systems.58–60 The specific defect structures on a submicrometer scale can be obtained by multiphoton-induced polymerization,61,62 Langmuir–Blodgett technique,63 chemical vapor deposition,64,65 layer-by-layer assembly,66 and photoresist templates.67 Regarding the planar defects, recent investigations have revealed the emergence of a localized state of photons within the PBGs, which is known as the defect mode, by introduction of nonlight-emitting layers between the CC structures.61–67 Ozin and coworkers reported fabricating light-emitting planar defects between the polymer CC structures by spin-coating process. The broad emission band could be partially modified within the reflection band, and be modulated by mechanical stress.68 This situation motivated to demonstrate the optically excited laser action by the introduction of a light-emitting layer as a specific planar defect inside the CC structure. Taking advantage of the prominent properties of the above-mentioned PS/PDMS CC films, the author and coworkers developed a new and potential utility of high-quality CC films for lowthreshold laser applications by optical excitation.15 Figure 18.5 shows a schematic illustration of the designed CC laser (CC-L) device structure. This structure is very simple. When a lightemitting polymer layer as a planar defect is introduced between a pair of CC films, a highly efficient

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Excitation (input)

Laser oscillation Colloidal crystal Emissive media Colloidal crystal

Laser emission (output) Polystyrene microparticle

Rhodamine 101 dye

FIGURE 18.5 Schematic illustration of the CC-L device. A light-emitting planar defect of Rhodamine 101 (Rh) is sandwiched between a pair of CC films of PS particles stabilized in PDMS elastomer. The left-hand picture is a cross-sectional SEM image of the magnified light-emitting planar defect between the PS/PDMS CC films. The white scale bar represents 2 mm. (Reprinted from Furumi, S. et al. 2007. Adv. Mater. 19: 2067–2072. With permission of Wiley-VCH, Copyright 2007.)

laser-feedback occurs due to the PBG effect of the CC films. Most recently, Foulger and his coworkers demonstrated the lasing action by combining a CC film stabilized with a polymer hydrogel on one side of the substrate and a commercial dielectric multilayered mirror on the other.69 As will be discussed below, the author and coworkers successfully demonstrated the first example of flexible CC-L devices of all-polymer materials using the CC films. These findings would provide significant practical advantages for novel photonic devices combined with the polymer CC structures. The CC laser device consists of a light-emitting planar defect between a pair of CC films of PS/ PDMS, as illustrated in Figure 18.5. In order to fabricate the CC-L device, ca. 1 cm2 of the highquality CC was clipped from a large-size PS/PDMS CC film. The light-emitting material in the planar defect was prepared by mixing poly(ethylene glycol) diacrylate (PEG-DA) with a molecular weight of ca. 575 as the fluid oligomer, 2-benzyl-2-(dimethylamino)-4¢-morpholinobutyrophenone (BDMB) as the photopolymerization initiator, and Rhodamine 101 (Rh) as the fluorescent dye at the weight ratio of 99:1.0:0.5, respectively. BDMB was used as the initiator because it has a relatively high molar extinction coefficient at 365 nm. Moreover, the fluorescent Rh dye was chosen as the light-emitting material due to its good solubility in the PEG-DA matrix by its ionic nature. The fluid mixture was injected between a pair of CC films through a capillary force. The thickness between the CC films was adjusted to ca. 2.7 mm by micrometer silica particles. Finally, the PEG-DA was polymerized by exposure to 365 nm light in order to substantially immobilize the intermediate layer of Rh/PEG-DA between the CC films.

18.5 PROPERTIES OF COLLOIDAL CRYSTAL LASER DEVICES Figure 18.6a shows the reflection spectrum of the CC-L device fabricated with a pair of PS/PDMS CC films, as illustrated in Figure 18.5. The reflection band originating from the CC films appeared at 610 nm even after construction of the CC-L device structure. As compared to the reflection spectrum of a single film of the PS/PDMS CC structure (Curve 3 of Figure 18.4a), there was no remarkable shift in the reflection band of this CC-L device. This suggests that the well-ordered CC structure of PS microparticles stabilized in the PDMS matrix remains intact after coming into contact with the Rh/PEG-DA mixture. This phenomenon probably occurred due to the repulsion between the hydrophilic PEG-DA matrix of the defect layer and the hydrophobic PDMS elastomer of the CC films. As a result, the light-emitting planar defect of Rh/PEG-DA could be completely formed between a pair of CC films.

422

Structure and Functional Properties of Colloidal Systems

Reflectance (arbitrary unit)

(a)

600

550

15000

10000

5000

0 550

Curve 2

Curve 1 (× 15)

650

Emission intensity

Emission intensity (arbitrary unit)

(b)

700

0.17nm

610.5 611 611.5 612 Wavelength (nm)

600 650 Wavelength (nm)

700

FIGURE 18.6 (a) Reflection spectrum of the CC-L device, which is comprised of a light-emitting planar defect of Rh/PEG-DA between the PS/PDMS CC films. The longitudinal dotted arrow in this spectrum indicates the theoretical wavelength of the defect mode in the CC-L device. (b) Emission spectra from the CC-L device by optical excitation with the second-harmonic light at 532 nm from an Nd : YAG laser beam with an energy of 70 nJ/pulse (Curve 1) and 210 nJ/pulse (Curve 2). The emission intensity of Curve 1 is magnified 15 times. The inset of this figure represents the high-resolution spectrum of laser emission around 610 nm. (Reprinted from Furumi, S. et al. 2007. Adv. Mater. 19: 2067–2072. With permission of Wiley-VCH, Copyright 2007.)

Subsequently, the laser emission characteristics of this CC-L device were investigated by optical excitation using the second-harmonic light at 532 nm from a Q-switched Nd : yttrium aluminum garnet (Nd : YAG) laser beam. This excitation wavelength of 532 nm almost coincided with the maximum absorption wavelength of the fluorescence Rh dye. The pulse duration was 3 ns and the repetition frequency was 10 Hz. The excitation pulse energy was adjusted using the combination of a l/2 plate, a polarizing prism, and neutral density filters, and was monitored by a pulse energy analyzer equipped with a pyroelectric sensor. The excitation beam propagating along the surface normal was focused onto the CC-L device through an achromatic doublet lens to obtain a spot diameter of ca. 200 mm. The collinearly transmitted emission from the CC-L device was focused onto the entrance of an optical fiber connected to a charge coupled device (CCD) spectrometer. Figure 18.6b shows the emission spectra of the CC-L device obtained by optical excitation below and above the threshold energy for the laser oscillation. From a relatively low excitation energy up to 130 nJ/pulse, the emission intensity was gradually increased in line with the excitation energy. The optical excitation at an energy of 70 nJ/pulse caused a partial modification in the broad fluorescence spectrum of the Rh/PEG-DA layer. As indicated in Curve 1 of Figure 18.6b, the emission was forbidden within the CC reflection band in a range from 600 to 630 nm. Such a modified fluorescence spectrum was evident when compared to the spectrum by excitation with a Xe lamp, as shown in Figure 18.4b. This observation distinctly indicated that the reflection band of the CC films works

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Colloidal Photonic Crystals and Laser Applications

as the PBG even in the CC-L device, resulting in the optical confinement of emission from the Rh dyes in the orthogonal direction of the device, as illustrated in Figure 18.5. When the excitation energy was increased to 210 nJ/pulse, a laser emission peak could be suddenly observed at 611 nm within the reflection band of the CC films, as shown in Curve 2 of Figure 18.6b. It is plausible that the laser-feedback effect in the CC-L device is caused by the defect mode within the PBG of the CC film. However, there was no obvious appearance of the defect mode peak in the reflection spectrum of the CC-L device, as shown in Figure 18.6a. This was probably due to the strong absorption of light by the intermediate Rh/PEG-DA layer.70 Such an observation is consistent with several precedents of 1-D PC structures with light-emitting organic planar defects such that the defect mode peaks in the PBG are not easily detected in the transmission and reflection spectral measurements.71,72 However, by considering the oscillation condition, the theoretical wavelength of the defect mode in the CC-L device could be estimated on the basis of the SWA results as follows. According to the report by Shung and coworkers,33 the light wave in the PBG of the CC film can be described by the following equation: EPBG(x) = CPBG e-qx cos(Gx/2 + d),

(18.3)

where CPBG stands for the amplitude coefficient and the other notations follow the definition of the reference.33 The defect mode is a standing wave formed between the CC films. Therefore, the light wave in the defect layer can be expressed by the following equation: Ed(x) = Cd cos kd x

or C¢d sin kd x,

(18.4)

where Cd and C d¢ are the amplitude coefficients and kd is the wave vector. The defect mode peak is given by the continuity of E(x) and dE(x)/dx at the interface between the defect layer and CC film. For this calculation, the thickness and refractive index of the Rh/PEG-DA defect layer are the experimental values 2.8 and 1.49 mm, respectively. The optical refractive indices of PS and PDMS are 1.59 and 1.43, respectively, and the PS particle diameter is 202 nm. As a result, this theoretical wavelength was 613 nm, as depicted by the dotted arrow in Figure 18.6a. This value was in fairly good agreement with the laser oscillation wavelength experimentally observed in the CC-L device. The inset of Figure 18.6b shows a magnified spectrum of the laser emission for the resolution analysis. At this stage, the spectral linewidth of the laser emission was as narrow as 0.17 nm. The cavity quality factor was estimated to be ca. 3.6 × 103 from the laser spectrum. This value is relatively high compared with that in a previous report on lasing from cavity structures with a dielectric mirror and hydrogel CC film.69,73,74 Such a narrow laser emission from the CC-L device can be realized by the formation of a high-quality laser cavity structure, as shown in the SEM image of Figure 18.5. From a technological viewpoint, the single-mode laser emission with a narrow linewidth is important for the fabrication of high-density photonic devices. In order to determine the threshold excitation energy for the laser oscillation, the emission intensity and its spectral linewidth were evaluated as a function of the excitation energy. It was found that there is a threshold excitation pulse energy around 130 nJ/pulse for the laser oscillation. When the excitation energy exceeded 130 nJ/pulse, the emission intensity was significantly enhanced with a concurrent narrowing of the spectral linewidth from 50 to 0.17 nm. In this experimental setup, the pulse duration and the focused diameter of the excitation beam were set at 3 ns and 200 mm, respectively. Therefore, the lasing-threshold peak power was calculated to be 138 kW/cm 2. More importantly, to the best of our knowledge, this threshold peak power is two orders of magnitude smaller than that in the previous reports on the optically excited laser action from the other CC structures.69,73,74 Taking the overall results into account, the author and coworkers succeeded in the lowthreshold lasing from the CC-L device by well confinement of the photons emitted from the Rh/ PEG-DA defect layer within the PBG of the sandwiching CC films, as shown in Figure 18.5. The present CC-L device is not optimized, however, and tailoring the materials and device structures

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Structure and Functional Properties of Colloidal Systems

would lead to a further reduction in the threshold peak power for realizing continuous wave (CW)excited laser devices having versatile applications.75 Finally, as an extension of the above-mentioned findings, an attempt was made to fabricate flexible polymer CC-L devices using CC films assembled on poly(ethylene terephthalate) (PET) sheets. Although the PS/PDMS CC films fabricated according to Figure 18.2 were formed on glass substrates, this fabrication procedure ensured the formation of the CC structure onto flexible PET sheets. Figure 18.7a shows a photograph of a flexible PS/PDMS CC film on the PET sheet. This fact suggests us the possibility of demonstrating the laser-feedback effect in the flexible polymer CC-L device fabricated on PET sheets. As a result, the laser oscillation could be observed within the PBG region of the PS/PDMS CC films. After the lasing experiment, there was no deterioration of the polymer materials in the device, such as laser ablation, due to low pulse energy of the excitation

(a)

(b) Excitation (532 nm)

Laser emission (610 nm)

Flexibility

Processability

Mechanical stress

FIGURE 18.7 Photograph of the laser action of the flexible polymer CC-L device fabricated on PET sheets by optical excitation above the threshold pulse energy. As can be observed, it was possible to demonstrate an excellent mechanical flexibility of the all-polymer CC-L device. (Reprinted from Furumi, S. et al. 2007. Adv. Mater. 19: 2067–2072. With permission of Wiley-VCH, Copyright 2007.)

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beam. Figure 18.7b is a photograph of the lasing action of the flexible polymer CC-L device by the optical excitation. Interestingly, it was possible to generate the laser action by optical excitation even though the CC-L device became bent by mechanical stress. From the viewpoint of next-generation optoelectronics, it would be significantly advantageous to be able to fabricate all-polymer laser devices with intrinsic properties of flexibility, processability, ultralightweight and low cost. This result provides promising guidelines to not only facilitate the generation of the optically excited laser-feedback effect by utilizing the PBG of CC structures, but also fabricate miniaturized photonic devices for all-optical integrated circuits.

18.6 CONCLUSIONS This chapter presented a brief overview of the fabrication procedure and optical properties of CC structures. Application of the CC as a tool for assembling PCs has been a long-standing subject from fundamental and technological viewpoints. This is because the microparticles can self-organize the 3-D PC structures through a typically electrostatic interaction. Numerous efforts have established a wide variety of strategies to fabricate large area and highly ordered CC films. As the CC structures are combined with functional materials, tuning of the PBG is realized by external stimuli such as chemical components, mechanical forces, photoirradiation, and electrical and magnetic fields. On the basis of the author’s results, this review introduce a novel utility of CC structures as low-threshold laser by optical excitation. The laser cavity structure consists of an intermediate light-emitting layer between two CC films to generate the laser-feedback effect through the PBG effect. The present strategy with CC structures would provide clues to the development of miniaturized photonic devices.

ACKNOWLEDGMENTS Seiichi Furumi expresses sincere thanks to Drs H. Fudouzi, H. T. Miyazaki, Y. Sakka, M. Nishino, and T. Sawada for their helpful discussions. This work was supported in part by the Strategic Information and Communications R&D Promotion Programme (SCOPE) Project from the Ministry of Internal Affairs and Communications (MIC), the Grant-in-Aid for Scientific Research on Priority Area “Strong Photon-Molecule Coupling Fields” from the Ministry of Education, Science, Sports and Culture (MEXT) of Japan, the Kao Foundation for Arts and Sciences, and the Yazaki Memorial Foundation for Science and Technology.

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19

Droplet-Based Microﬂuidics: Picoliter-Sized Reactors for Mesoporous Microparticle Synthesis Nick J. Carroll, Sergio Mendez, Jeremy S. Edwards, David A. Weitz, and Dimiter N. Petsev

CONTENTS 19.1 Introduction ........................................................................................................................ 19.1.1 Motivation to Work with Microfluidics .................................................................. 19.1.2 Methods to Fabricate Microfluidic Devices ........................................................... 19.1.3 Scope of Review ..................................................................................................... 19.2 Transport in Microchannels................................................................................................ 19.2.1 Pressure-Driven Flows............................................................................................ 19.2.2 Capillary Flows....................................................................................................... 19.2.3 Electrokinetic Transport in Microchannels ............................................................ 19.2.4 Droplet Formation in Microchannels ..................................................................... 19.2.5 Lab-on-a-Chip Concept .......................................................................................... 19.3 Droplet Microfluidics for Mesoporous Particle Synthesis .................................................. 19.4 Summary and Outlook ....................................................................................................... Acknowledgments........................................................................................................................ References ....................................................................................................................................

19.1 19.1.1

429 429 430 431 431 431 432 433 434 437 438 441 443 443

INTRODUCTION MOTIVATION TO WORK WITH MICROFLUIDICS

The miniaturization of chemical flow and analysis systems has opened up exciting avenues of scientific and engineering possibilities. Channels with widths in the tens of micrometer range are referred to as microfluidic devices. Fluidic behavior at the microscale may differ from that at larger scales in that interfacial tension, viscous effects, and energy dissipation can dominate the system. Microfluidics has received much attention in the scientific community and many excellent reviews have been published [1,2]. A key advantage of microfluidics is the ability to perform experiments and bioassays using miniscule quantities of solution. This provides an economic benefit and is important for certain biosensing applications, experiments requiring single-molecule interrogation

429

430

Structure and Functional Properties of Colloidal Systems

(e.g., deoxyribonucleic acid (DNA) sequencing [3,4]), or diffusion-limited regimes. Another benefit is that rapid measurements of these minute quantities can be performed with miniaturized analytical systems [5–7]. In some applications slow or minimal mixing is required, and the laminar flows obtained in microchannels become highly desirable. Water-in-oil emulsions can be formed in microfluidic devices to form a steady stream of monodisperse aqueous droplets with volumes as small as picoliters [8]. The drops can be loaded with reactants to perform chemical reactions of interest [9].

19.1.2

METHODS TO FABRICATE MICROFLUIDIC DEVICES

Various methods have been presented in the literature to fabricate microfluidic devices. One of the most widely used are those made from poly(dimethylsiloxane) (PDMS) gels [8,10]. The materials for these devices are relatively inexpensive and they can be made with established soft lithography processes. This approach also has the potential to form complicated and intricate flow patterns. The soft lithography process is robust and reproducible, which allows replicating flows in different devices that have the same design. The first step in soft lithography is the production of a master mold, which is typically a Si wafer with epoxy structures. The epoxy structures are created with a high-resolution transparency as a photomask for generation of the master by photolithography. A negative photoresist is a type of epoxy in which the portion of the photoresist that is exposed to light becomes nearly insoluble to the photoresist developer. The unexposed portion of the photoresist is dissolved by the photoresist developer, leaving behind the desired channel features. Liquid PDMS precursor base along with a curing agent (typically 1 : 10 ratio) is then poured over the master mold. The liquid PDMS precursor conforms to the shape of the master mold, thus replicating the designed features. Vinyl groups present in the base react with silicon hydride groups in the curing agent to form a clear, cross-linked, elastomeric solid. The PDMS is then peeled away from the master mold; the master mold is not destroyed during the removal process, allowing it to be used again for subsequent devices. The PDMS channels are then sealed to a glass slide by exposing both the PDMS device and the glass slide to oxygen plasma and then pressing both together within 1 min after exposure to the plasma. The disadvantages are that PDMS devices typically can only handle low pressure drops before catastrophic rupture, and untreated PDMS can become swollen or chemically react with some liquids, thus making the channels not reusable. Recently, there have been reports demonstrating that the walls of such channels can be coated with material that can improve the flow properties [11]. Other popular fabrication methods involve etching glass, silicon, or plastic substrates. The benefits of such devices are that these materials can be chemically inert and amenable to cleaning, thereby making them reusable. Another advantage is that these substrates with etched channels can subsequently be well bonded to a working platform and allow for high pressure drops. Another microfluidic design that has been demonstrated to show promising applications utilizes very small diameter, commercially available, glass capillaries [12,13]. The advantages for this system are that the geometry of the capillaries can be reconfigured with established glass molding techniques, and the inner glass walls can be chemically modified. Moreover, such concentric glass capillaries can be used to generate droplets composed of double emulsions [12]. However, there is limited complexity of flow patterns and the capillary pulling methods used to fabricate these devices is time consuming and often irreproducible. A method that combines the excellent wetting and surface property control of glass with the simplicity of PDMS-based photolithography fabrication was demonstrated by Harvard researchers [14]. This method involves the use of sol–gel chemistry to coat the channels of lithography-formed PDMS microfluidic devices with a glass layer (Figure 19.1). The glass layer provides a chemical barrier to the PDMS walls, which are permeable to many liquids and gases. Additionally, the glass

431

Droplet-Based Microﬂuidics (b) Coated

(a) Uncoated

10 mm x1500

20 mm

1.00 kv

4 mm

10 mm

x1500

20mm

1.00 kv

4 mm

FIGURE 19.1 Scanning electron micrographs of cross-sections of (a) uncoated and (b) coated PDMS channels. Rectangular PDMS channel dimensions are 50 × 35 μm. This fabrication method combines the ease of soft lithography processing with the benefits of a glass surface. (Reprinted from Abate, A. R. et al. 2008. Lab Chip 8: 516–518. With permission.)

surface chemistry can be controlled allowing for manipulation of the hydrophilicity and wetting properties of the microchannels.

19.1.3

SCOPE OF REVIEW

In this chapter, we present an overview of the transport of fluids in microchannels with a focus on the formation and manipulation of emulsion droplets. The next section deals with defining terminology and describing the physics of fluid flow in small channels. We briefly summarize approaches that are used to drive the fluid flow in the microchannels and dispersion of immiscible phases (oil and water) to create well-defined droplets. In the last section, we focus on the applications of dropletbased microfluidics. In particular, we review microparticle formation and biochemical reactions in small droplet reactors.

19.2 19.2.1

TRANSPORT IN MICROCHANNELS PRESSURE-DRIVEN FLOWS

Microfluidic channels, as suggested by the name, have dimensions that are in the micrometer range. The typical dimensions of the cross-section range are between a micrometer and a few hundreds of micrometers. A typical flow rate might be of the order of mL/minute. The small channel dimensions imply that the Reynolds number (Re) will typically be <1. This results in laminar flow meaning that viscous forces dominate over inertial forces. In such cases, it becomes impossible to stir the fluid by turbulent flow (e.g., shaking). Recall that mixing is an important consideration when using the microfluidic channel as a chemical reactor or when dispersing two liquid phases into each other. Another practical consideration is the pressure drop across such channels. Because the pressure drop is proportional to the square of the equivalent diameter, we can appreciate the fact that small decreases of channel dimensions can result in large increases in pressure drops—so much that the microchannels can rupture catastrophically. More details on the mechanics of fluid flow in microfluidic channels can be found in references [15–17]. An alternative way to move fluids in microfluidic devices is by using electric fields. Most often this is accomplished by exploiting various electrokinetic phenomena [18,19]. The equations of motion for an incompressible Newtonian fluid are [20]

(

)

∂v + v—v = -—p + h—2v + force terms, r ___ ∂t

— ◊ v = 0,

(19.1)

where v is the velocity field, r is the fluid density, p is the pressure, and h is the dynamic viscosity. The momentum balance may also include other body force terms that are due to gravity and electric

432

Structure and Functional Properties of Colloidal Systems

or magnetic fields (for charged or magnetic fluids). The relative importance of the acceleration and viscous terms is expressed by the Reynolds number: Re = rUl/h, where U and l are some characteristic velocity and length scale for the problem. Scaling of (19.1) leads to [20] ~ v+~ ∂~ Re ___ v—~ v = - —~ p + —2~ v, ~ ∂t

(

~

)

~

v=~ v / U, t = tU/l,

—·~ v = 0,

~

p = pl/hU,

~ — = l—.

(19.2)

For viscous-dominated low Re fluid flow, the equations are [20,21] h—2 v - —p = 0,

— ◊ v = 0.

(19.3)

For slit-shaped channels, the momentum balance for fluid flow in the z-direction is ∂2n ∂p h ____2z = ___, ∂y ∂z

(19.4)

whereas for a circular capillary, it is given by

( )

∂nz ___ ∂p ∂ 1 __ ___ h __ r ∂r r ∂r = ∂z .

(19.5)

Integrating Equation 19.5 leads to the well-known Hagen–Poiseuille expression for the flow velocity in cylindrical symmetry: dp 2 in - pout 1 ___ 1 p_______ nz = ___ (r - R2) = ___ (R2 - r2), L 4h dz 4h

(19.6)

where pin and pout are the inlet and outlet pressures, L is the channel length, and R is the capillary radius. Similar expression exists for a slit-shaped channel where the flow profile also has a parabolic shape [20]. Equation 19.6 can be averaged over the circular cross-sectional area to give the mean fluid velocity · nz Ò =

p_______ in - pout ___ R2 . L 8h

(19.7)

19.2.2 CAPILLARY FLOWS Capillary flows occur (e.g., parallel slit or cylindrical) and are driven by surface tension. The flow regime is viscous-dominated (see above) with a possible exception at the very beginning of the flow [22]. The scaling length scale is associated with the channel width or radius. Capillary flows are very convenient for initial loading of micro- and nanofluidic devices. The fluid must wet the surface and then a curved air/solution interface is formed at the flow front (Figure 19.2). Then the pressure drop in a perfectly wetted circular capillary with radius R is [23] 2g pin - pout = ___ - rgL, R

(19.8)

with g being the interfacial tension. The second term accounts for the hydrostatic pressure due to gravity (see Figure 19.2a). Inserting Equation 19.8 into 19.7 leads to gR

rgR2

____ _____ · nz Ò = 4hL - 8h

(19.9)

For a horizontal capillary (Figure 19.2b), the second term on the right-hand side of Equation 19.9 is zero. This equation also allows for tracing the position of the meniscus in time. Introducing ·vz Ò = dL/ dt in Equation 19.9 and integrating over time (in the absence of gravitational effects), one obtains ___

gR L = ___ t. 2h

√

(19.10)

433

Droplet-Based Microﬂuidics

FIGURE 19.2

Fluid in a vertical (a) and horizontal (b) capillaries.

The boundary initial condition is L = 0 for t = 0. Equation 19.10 is known as the Washburn equation. The time change in the average fluid velocity can be obtained by differentiating Equation 19.10 with respect to time, which yields ____

gR 1 ___ . · nz(t) Ò = __

√

(19.11)

2 2ht

Equations 19.9 through 19.11 are very useful to model the flow propagation in narrow capillaries. They are also often used to describe flow in porous media, where all geometrical quantities such as capillary radii and lengths have the meaning of statistical averages that represent the actual structure.

19.2.3

ELECTROKINETIC TRANSPORT IN MICROCHANNELS

Electrokinetic phenomena present a very useful tool to generate and control fluid and solute transport in microchannels. In most cases of practical importance the electric double layer (EDL) (Figure 19.3) that forms at the wall has much smaller thickness than the channel width. The electrokinetic motion of fluid, relative to a solid, in the thin double-layer approximation was analyzed by

Shear plane +

–

–

+ + + + EZ

+ +

neo

+

–

– + X

+ –

Y

z

+

Y0 – +

+

~1/k

+

FIGURE 19.3 EDL in the vicinity of solid wall (electro-osmosis) or particle (electrophoresis). The electrokinetic z-potential is defined at the shear plane where the relative fluid velocity is zero. The EDL thickness is given by 1/k and depends on the concentration of background electrolyte. The electrostatic potential decays with the distance from the wall into the fluid. At the same time, the velocity increases until it reaches a constant value outside the EDL. For channels that are much wider than 1/k, the detailed velocity distribution in the EDL can be ignored and the velocity can be assumed to be constant and given by Equation 19.12.

434

Structure and Functional Properties of Colloidal Systems

Smoluchowski at the beginning of the twentieth century [24]. He used the similarity between the fluid velocity and electrostatic potential field in a thin EDL to obtain (see also references [18,19]) ee0zw neo = - _____ h E,

ee0zp nep = _____ h E.

(19.12)

In Equation 19.12, e0 = 8.854 × 10-12 J-1 C2 m-1 is the dielectric constant in vacuum, e is the relative dielectric permittivity of the solvent (e = 78.5 for water at room temperature 298 K), zw and zp are the electrokinetic zeta potential defined at the shear plane (see Figure 19.3), h is the dynamic viscosity of the solvent (h = 8.91 × 10-3 kg m-1 s-1 for water at room temperature 298 K), and E is the externally applied electric field. The first equation in Equation 19.12 represents the fluid motion in a stationary channel under the action of an externally applied electric field. The motion is called electro-osmosis and the velocity is neo. The second equation in Equation 19.12 gives the velocity vep of charged suspended colloidal particle (or a dissolved molecule) driven by the same electric field. This phenomenon is called electrophoresis. The EDL thickness 1/k depends on the concentration of background electrolyte [18,19,25,26].

(

e2 k = _____ ee0 kT

S z n ), i

2 0 i i

(19.13)

where e is the elementary charge, kT is the thermal energy, zi is the charge of the ith ionic species, and n 0i is its number concentration. Hence, the thickness of the EDL depends on the background electrolyte concentration. Thus, aqueous solutions of monovalent electrolyte at room temperature with concentrations ranging between 0.1 and 10-6 M will lead to an EDL thickness varying from 1 to 300 nm. Since microfluidic channels often have cross-sectional dimensions in tens or even hundreds of micrometers, the EDL can be considered to be much thinner. The thin EDL approximation, used by Smoluchowski [24], implies that the electric field effect on the fluid in a microchannel can be accounted for by means of boundary conditions at the wall instead of a body force term in the momentum balance (Equation 19.1). These boundary conditions correspond to the electro-osmotic velocity given by Equation 19.12. The electro-osmotic effect can be used to design electro-osmotic pumps in microfluidic devices. The applied external electric field with magnitude E translates into a pressure gradient, which for circular capillary with radius R is equal to [19] dp 8ee 0 zwE ___ . = _______ 2 dx

R

(19.14)

Using porous network structures can increase the wall/solution interfacial area that allows for an increase in the electro-osmotic pump efficiency. A number of different pump designs have been suggested [27–33] to accomplish higher pressures and/or better flow rates.

19.2.4 DROPLET FORMATION IN MICROCHANNELS Emulsions are dispersions of two immiscible liquids into each other. They are thermodynamically unstable, but the addition of surfactant molecules can provide significant kinetic stability. Emulsions are extensively used in food, cosmetic, and pharmaceutical industries, just to name a few. Because of the thermodynamic penalty, emulsion formation requires an energy input. In bulk systems, this can most easily be achieved by vigorous stirring or shaking of the whole oil/water/surfactant system. This approach leads to an emulsion with broad droplet size distribution. Microfluidics allows for the minimization of polydispersity and the creation of droplets that are virtually identical in size. The liquid comprising the droplets are referred to as the dispersed or discontinuous phase, while the surrounding fluid is the continuous phase. At constant volume and temperature, the Gibb’s free

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Droplet-Based Microﬂuidics

energy for emulsification of two immiscible fluids is the difference in free energy between the mixed (or emulsified) state (Gm) and the unmixed state (G u) [23]: DG form = Gm - G u = gDA - TDSm,

(19.15)

where g is the interfacial tension, DA is the change in interfacial area, T is the temperature, and DSm is the positive change in entropy gained upon mixing (emulsification). The first term on the righthand side corresponds to the free energy, which is the work required to expand the interfacial area (DW = gDA). Immiscible fluids in contact can produce a large interfacial tension; therefore, the interfacial energy is generally much larger than the favorable entropic contribution, and the emulsification process is not spontaneous (positive free energy term). This is the reason for the required external energy for emulsification. Given enough time, emulsified liquids would phase separate to lower the stored interfacial energy (by decreasing the interfacial area), with the less dense liquid making up the top layer because of gravity. However, when amphiphilic molecules, that is, surfactants, are added to the mixture, the droplets might become metastable making the emulsion last for a much longer amount of time. Equation 19.15 makes clear that lowering of the interfacial tension via surfactant adsorption at the droplet interface will also reduce the interfacial free energy. Rupturing of a single spherical droplet is governed by the balance between interfacial (capillary) and shear forces. The proper conditions can be estimated from the dimensionless “capillary” number (Ca), which is the ratio of viscous stress over the restorative Laplace pressure (due to the interfacial tension, see Equation 19.8): Rdtij hUd Ca = ____ ª ____ g , 2g

(19.16)

where Rd is the droplet radius, h is the continuous phase viscosity, tij is the relevant component of the viscous stress tensor, and Ud is the characteristic velocity. At Ca 1, the energy input is sufficient enough to rupture a droplet. The necessary viscous stress can be generated by applying pressure or electrically driven flow (see Sections 19.2.1 through 19.2.3). Droplet-based microfluidic system behavior is impacted by multiphase flow characteristics, interfacial tension, and the wetting and physical properties of the liquids. At different channel scales and fluid velocities, the system can be influenced by balancing inertial, viscous, or gravitational forces against the interfacial forces. In these cases, other dimensionless numbers are relevant. For example, the ratio of interfacial and gravitational force is given by the Bond number (Bo), Drgd2h Bo = ______ g ,

(19.17)

where Dr is the density difference between the two liquid phases and dh is the characteristic channel dimension. At the dimensions of most droplet-forming microfluidic devices, interfacial tension typically dominates over the influence of gravity (Bo 1). In the special cases when rapid acceleration (entrance effects, etc.) or inertia is present, the important physics is given by the Weber number rU2ddh We = _____ g .

(19.18)

By controlling fluid pressures, flow regimes, fluid viscosities, wetting properties, channel dimensions, and interfacial properties (i.e., surfactant concentration), it is possible to induce and control periodic breakup of a liquid stream to produce microfluidic-generated emulsion drops. The controlled formation of water-in-oil emulsion droplet stream in a T-junction microfluidic system was first demonstrated by Thorsen [34]. Since then, there has been a plethora of articles published in this area. The analysis of the mechanism for droplet formation in a T-junction device was proposed by Garstecki et al. [35]. At low capillary numbers, it was demonstrated that breakup

436

Structure and Functional Properties of Colloidal Systems

FIGURE 19.4 Flow-focusing cross-junction configuration for droplet formation. The center fluid (liquid A) is hydrodynamically focused by the continuous phase (liquid B) within the small orifice. Viscous forces overcome interfacial tension with thread instability in the enclosed orifice producing monodisperse droplets as rapidly as kilohertz frequencies.

of drops or bubbles in a microfluidic T-junction is due to the pressure drop across the emerging bubble or drop and not dependent on shear stress. There exist a few literature review articles that discuss how these microfluidic systems can be exploited to study flow characteristics and for use as picoliter-sized chemical reactors [36–39]. Figure 19.4 shows the design of a cross-junction microfluidic flow-focusing device (MFFD), with typical operation in the capillary regime where viscous forces are dominant. A pressure gradient along the long axis of the device forces two immiscible liquids through the orifice. The continuous phase is infused from two sides. The liquid stream comprising the dispersed phase is supplied from the central channel. The continuous phase focuses the inner, immiscible liquid so that the inner thread becomes unstable and breaks in the narrow orifice in a periodic manner. In some flow regimes, the drop size is effectively set by the size of the orifice. In other cases, the flow-focusing geometry produces threads that break into drops substantially smaller than the orifice. Droplet formation within the orifice is referred to as the dripping regime whereas droplet formation of the liquid thread further downstream of the orifice is known as jetting. In general, a liquid thread in the jetting regime experiences unbound flow capillary instabilities, and thus will typically produce less monodisperse drops than those formed in the dripping regime. Another configuration (Figure 19.5) combines flow focusing cross-junction droplet formation with nozzle shape channel geometry [40]. The droplet breakup occurs at a fixed point due to the focused velocity gradient created by the nozzle shape geometry. As illustrated in Figure 19.5, the normal force balance on the surface of the liquid thread includes pressure and shear stresses differences against interfacial tension: g (pw + hwG) - (po + hoG) = __, Rt

(19.19)

where G is the shear rate, po, ho, pw, and hw are the pressures and viscosities of the continuous and dispersed phase, respectively, and Rt is the radius of the liquid thread. In the expanding nozzle geometry, the droplet always breaks at the orifice due to a designed focused shear gradient. The fluid velocity increases before entering the nozzle and reaches the maximum velocity at the orifice before decreasing as it exits thus creating a focused velocity gradient (and therefore shear). Monodisperse submicron satellite drops were also observed with this system. Link et al. [41] further modified cross-junction flow focusing devices to incorporate forces that result from capacitance-like charging of the discontinuous aqueous phase in an applied electric

437

Droplet-Based Microﬂuidics

FIGURE 19.5 Nozzle shape channel geometry. Droplet breakup occurs at a fixed point due to the focused velocity gradient created by the nozzle shape geometry. The radius of the liquid thread decreases due to the perturbation caused by the extension of the thread. Initial balance of the pressure and shear forces at the interface of the thread determines the initial radius of the thread. The device also produces monodisperse submicron satellite drops. (Reprinted from Tan, Y.-C. et al. 2006. Sens. Actuat. B 114: 350–356. With permission.)

field. This method produced smaller droplets at higher frequencies with more precise control over individual timing than is observed in pressure gradient-driven systems that rely only on viscous forces to overcome surface tension. A voltage was applied to indium tin oxide electrodes that were incorporated into the device. The aqueous stream acts as a conductor whereas the oil acts as an insulator. Electrochemical reactions charge the fluid interface like a capacitor; thus the charge of the interface remains on the droplet. At high electric field strengths, there is an additional force F on the growing drop given by F = qE,

(19.20)

where q is the charge on the droplet and E is the electric field magnitude. Also, the capacitance C is directly related to the applied voltage: q C = __. F

(19.21)

Therefore the charge on the droplet is dependent on the applied voltage F (see Equation 19.20). The additional force applied to the growing drop thus shows a dependence on the applied voltage squared. The size and frequency of the droplets are determined by a simple mass (or volume) balance Qd = fVd,

(19.22)

where Qd is the dispersed phase flow rate, f is the frequency of droplet formation, and Vd is the droplet volume. The droplet size decreases with the field strength (Figure 19.6), thereby increasing droplet formation frequency. Additionally, the authors demonstrated selective splitting, coalescence, and sorting using charged droplet-electric field manipulation.

19.2.5

LAB-ON-A-CHIP CONCEPT

Over the past decade, there has been a broad effort in the scientific community to miniaturize chemical systems via microfluidic platforms. The concept is to construct compact devices with integrated modular functionalities [42]. These micrometer-sized modules can perform functions that are analogous to bench-scale laboratory processes, hence the name, “lab-on-a-chip.” For making droplets, a lab-on-a-chip microfluidic device could be envisioned to have the following modules:

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Structure and Functional Properties of Colloidal Systems

Vd/pL

500 400 300 200 100 0

10

100

1000

V/V

FIGURE 19.6 Droplet size as a function of voltage. The crossover between flow-dominated and fielddominated droplet formation for three different flow rates (Qc) of the continuous phase oil (Qc increases from top to bottom: Qc = 80, 110, and 140 nLs). (Reprinted from Link, D. R. et al. 2006. Angew. Chem. Int. Ed. 45: 2556–2560. With permission.)

A built-in filter, a junction where the streams of immiscible fluids meet and where droplets are consequently formed; a tortuous channel where the contents within the droplets can be well mixed by the method of chaotic advection, which was developed by Song et al. [43]; a very long section of channel to allow time for chemical reactions to occur within the drops; and finally a detection/ analytical feature to discriminate the resulting content inside the droplets.

19.3 DROPLET MICROFLUIDICS FOR MESOPOROUS PARTICLE SYNTHESIS MFFDs can be used as a means to synthesize solid particles. As mentioned earlier, by simply tuning flow rates or other system parameters, the morphology of the “drops” can be tuned to be spheres, spheroids, discs, or long rods. Not surprisingly, if a liquid-to-solid chemical reaction proceeds to completion within these drops, then the resultant solid particles will possess such shapes. Serra et al. performed a parametric study to demonstrate how various operating parameters of a microfluidic system affected the size of the synthesized polymer particles [44]. Production of particles with various shapes and morphologies via microfluidic reactors were reported by several research groups [45–47]. Moreover, by using clever double emulsion microfluidic schemes, Kim et al. formed spherical, hollow microparticles [12]. Furthermore, Zhang et al. exploited this concept to make solid particles with multiple, separate hollow compartments [48]. Another appealing feature to engineer into the particles is porosity. The motivation, then, is to encapsulate a material of interest within the pores. Such loaded particles could be used for drug delivery: monodisperse particles with known porosity can allow for having strict control of the amount of drugs being delivered to a patient. Also, it has been envisioned that biological cells can be kept alive within the pores, and by tuning the porosity, one could control the transport of nutrients and cell byproducts in and out of the particles and therefore optimize the cells’ viability [49]. Finally, microfluidics has also been shown to be viable for making nanoparticles [50]. The authors have recently reported on the fabrication of monodisperse, nanostructured silica microspheres utilizing droplet-based microfluidics [51]. In this process, equally sized emulsion droplets with controlled diameter were produced at a frequency of ~100 Hz. The droplets contained the silica precursor/surfactant solution and were dispersed in hexadecane as the continuous oil phase. The solvent was then expelled from the droplets leading to concentration and micellization of the surfactant. At the same time, the silica was solidified around the surfactant structures forming equally sized mesoporous particles. The procedure could be tuned to produce individual particles or alternatively particles that are chemically bonded together allowing for the creation of threedimensional structures with hierarchical porosity.

Droplet-Based Microﬂuidics

439

The synthesis of mesoporous silicate solids using surfactant templating was developed over a decade ago [52]. An extension of that procedure, referred to as evaporation-induced self-assembly (EISA), involves confining all silica precursor and templating surfactant species within aerosol drops. The solvent progressively evaporates, which leads to an increase of the concentration of templating surfactant that, upon surpassing the critical micelle concentration, assembles into spherical, cylindrical, or lamellar structures [53,54]. After a certain time of reaction, the silica solidifies around the surfactant self-assembled structures. This is followed by surfactant removal through high-temperature calcination, resulting in the formation of a well-ordered mesoporous silica material. EISA has been successfully utilized to fabricate mesoporous silica thin films [53] and particles with diameters ~1 mm [53,54] using a wide range of surfactants and block copolymers. However, such mesoporous silica particles obtained via aerosol spray EISA are usually limited by substantial polydispersity. Andersson et al. [55] recently demonstrated the synthesis of spherical, mesoporous silica particles using an approach that combines emulsion-based precipitation methods [56,57] with the EISA process. This synthetic route, termed the emulsion and solvent evaporation (ESE) method, produced well-ordered two-dimensional (2D) hexagonal mesoporous silica microspheres. Because the emulsions were prepared in bulk using vigorous stirring, the droplets, and therefore the resulting particles, were produced with a broad particle size distribution. The production of monodisperse silica spheres containing highly ordered nanometer-scale pores (mesopores) of controllable size is of practical interest [58]. They can be used for drug delivery and biomolecular [59] and cellular encapsulation [49]. Monodisperse particles have been ordered into 2D ordered arrays [60,61] that allow for the fabrication of catalytic structures with controlled pore hierarchy. Mesoporous particles also have the potential for being used as chemical and biochemical sensors [62]. MFFDs provide a robust approach to form monodisperse emulsion drops [8]. It has been demonstrated that microfluidic-generated drops can function as both morphological templates and chemical reactors for the synthesis of monodisperse polymer [44,45,63] and biomolecular microparticles [48]. Below, we summarize our recent results on a novel procedure for fabrication of well-defined monodisperse, mesoporous silica particles. It is based on MFFD emulsification of an aqueous-based sol [64] with subsequent EISA processing utilizing the aforementioned ESE method [55]. The details of the materials and methods are provided in the paper by Carroll et al. [51]. Briefly, the silica precursor solution was prepared by hydrolyzing tetraethylorthosilicate (TEOS) in ethanol and hydrochloric acid under vigorous stirring at room temperature for 30 min. The amphiphilic, triblock copolymer templating agent (Pluronic P104) was dissolved in deionized water and then mixed with the hydrolyzed TEOS solution to complete the preparation of the aqueous sol. This recipe allowed for the use of Pluronic surfactant as a templating reagent in the presence of a much lower concentration of ethanol than was used by Andersson et al. [55]. Emulsification of the aqueous siliceous precursor was achieved by supplying the aqueous (sol–gel solution) phase and the oil phase to the microfluidic device using two syringe pumps. The continuous phase was prepared by dissolving ABIL EM 90 surfactant in hexadecane, which served as the emulsifier. The SU-8 photoresist-templated PDMS microfluidic device was fabricated using a well-established soft lithography method [10]. The microfluidic devices used in this study is shown in Figure 19.7. The droplets, after being produced through the MFFD, were transferred to a flask, and heated to 80°C at reduced pressure for 2 h. To prevent droplet coalescence before the silica solidification was complete, the emulsion was mildly stirred. The particles were then separated from the oil phase followed by calcination at 500°C for 5 h to remove the templating surfactant. Figure 19.8 shows silica particles obtained from droplets formed in a bulk shaken-emulsion (Figure 19.8a) and the MFFD (Figure 19.8b). The low ambient pressure allowed for the solvent (water and ethanol) to be transported out of the droplets and through the hexadecane oil phase, which led to micellization of the surfactant and subsequent gelation of the silica precursor. Therefore,

440

Structure and Functional Properties of Colloidal Systems

FIGURE 19.7 Optical microscopy image of aqueous droplets formed at a cross-junction in a microfluidic device. The channel dimensions of the orifice are 25 μm (width) by 30 μm (height). The scale bar is 100 μm. (Reprinted from Carroll, N. S. et al. 2008. Langmuir 24: 658–661. With permission.)

50

(a) 45

(b)

40 35 25

%

%

30 20 15 10 5 0

50 45 40 35 30 25 20 15 10 5 0 0 0.3 0.5 0.7 0.8 0.9 1 1.1 1.2 1.3 1.5 2 2.5 3 3.3 3.5

0 0.3 0.5 0.7 0.8 0.9 1 1.1 1.2 1.3 1.5 2 2.5 3 3.3 3.5

Particle diameter/mean particle diameter

Particle diameter/mean particle diameter

(c)

10kV x450 SE

(d)

100mm

FIGURE 19.8 (a) Polydisperse silica microspheres templated from bulk shaken-emulsion. Scale bar is 20 μm. (b) Silica microspheres templated by monodisperse microfluidic device-generated droplets. Scale bar is 100 μm. (c) Scanning electron microscopy image and (d) optical photograph (scale bar is 40 μm) of particles that have fused together in a hexagonal array. The particles are connected by “bridges” that form when the particles come into contact before the completion of the gelation process. (Reprinted from Carroll, N. S. et al. 2008. Langmuir 24: 658–661. With permission.)

Droplet-Based Microﬂuidics

441

FIGURE 19.9 Transmission electron image of silica microspheres containing well-ordered mesostructures templated by P104 surfactant. Scale bar is 100 nm. (Reprinted from Carroll, N. S. et al. 2008. Langmuir 24: 658–661. With permission.)

due to loss of solvent, the solid particles are approximately half the size of the original liquid droplets. The particles obtained in a bulk shaken emulsion are spherical but very polydisperse. Technically, it is very difficult and impractical to narrow the size distribution because shaken emulsions, in general, produce polydisperse droplets, which then result in a broad particle size distribution. Increasing the stirring intensity during the emulsion formation might reduce the average particle size and narrow the size distribution but cannot produce monodisperse droplets. Using the microfluidic device, we formed monodisperse droplets, which is the necessary step to form monodisperse particles. The size of the droplets can depend on the dimensions of the dropletforming region, the flow rates of the oil and aqueous phases, the viscosity of the fluids (water/ethanol and oil), and the stabilizing surfactant. Hence a single device can produce monodisperse droplets (and particles) of different sizes by varying the relative magnitude of the viscous and interfacial forces that are involved [8]. The particles might stick together during the final stages of solvent evaporation unless some preventative actions are taken. The particles can become connected by silica “bridges” that form when the particles come into contact before the completion of the gelation chemistry. This agglomeration could be exploited to obtain arrays of interconnected particles. Such bonding can result in robust layers of monodisperse mesoporous particles that could then perhaps be deposited on other substrates to use in applications such as catalysis and sensing. Such multiparticle monolayers are shown in Figure 19.8c and d. Figure 19.9 depicts a transmission electron microscopy (TEM) image that clearly shows the presence of the nanostructured pores within the microparticles. The diameter of the pores is about 5–6 nm. As mentioned above, there is great interest in encapsulating cells into porous monodisperse solid particles [49]. Biopolymer capsules with diameters in the tens of microns with controlled morphology have been fabricated [48]. It has been demonstrated that droplet-based microfluidics can indeed be used to encapsulate single cells [65,66]. Tan et al. formed lipid vesicles within microfluidic channels with the aim of encapsulating and keeping cells viable [67]. DNA amplification is another application that is well suited for such droplet microfluidics because of the requirement of small volumes, compartmentalization of chemical processes, and the capability of in situ detection [68,69].

19.4 SUMMARY AND OUTLOOK Droplet-based microfluidics has a great potential for implementation in a variety of new technologies and applications. Researchers have successfully designed a variety of droplet-based microfluidic systems for the precise control of droplet formation, selective coalescence, sorting, and

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Structure and Functional Properties of Colloidal Systems

mixing to enable the manipulation of femto- to nanoliter droplet reactors. Particle synthesis in microfluidic-generated droplet reactors has the advantage of producing samples with high monodispersity. This brings another level of structural control in addition to well-defined internal mesostructures. Thus, opportunities are created for the design of new and better catalysts, drug delivery vehicles, and coatings with fine-tuned optical, mechanical, and chemical properties. The utilization of droplet microfluidics is not limited to the synthesis of microparticles. The novel and unique opportunities for processing minute solution samples within micron-sized droplets offers engineers, chemists, and biologists an innovative platform for molecular and biomolecular synthesis in addition to high-throughput analysis. Integration of heating elements into droplet-based microfluidic devices will make it possible to carry out in vitro single-molecule DNA amplification, protein expression, and a host of other biomolecular reactions. Another advantage of droplet-based fluidic systems over continuous flow ones is the sequestering of reagents into isolated droplet compartments surrounded by an immiscible oil phase. Thus, each droplet presents a tiny test tube that allows for large-scale experiments using small amounts of reactants. Emulsion droplet reactors have been utilized in a number of biological applications. However, most of these assays have used emulsion droplets formed with batch methods such as stirring or agitation, which result in an emulsion with a polydisperse size distribution. Droplet-based microfluidics facilitates the production of monodisperse emulsion droplet bioreactors as well as on-chip characterization. Protein expression in microfluidic-generated droplets was demonstrated by Dittrich and coworkers [70]. Transcription and translation reagents were encapsulated along with the proteinencoding gene for green fluorescent protein (GFP). Following incubation at 37°C, protein expression was verified by fluorescence measurements. In another assay, single Escherichia coli cells were encapsulated by Huebner and coworkers to express a yellow fluorescent protein [71]. The number of cells loaded into droplet reactors, and thus the degree of protein expression, was determined by controlling the flow rates of two streams: One containing only buffer and the other containing cells and buffer. Droplet-based microfluidics combined with on-chip fluorescence measurements suggests the potential for the high-throughput study of directed evolution of proteins. Droplet-based microfluidics has also been employed for the amplification of DNA. While most microfluidic DNA amplification assays have been achieved in continuous flow systems, there is great potential for droplet microfluidics as a platform for high-throughput DNA amplification and sequencing. In one study, Zhang et al. [72] demonstrated that microfluidic DNA polymerase chain reaction (PCR) amplification using droplet-based methods was theoretically more efficient than continuous flow methods. Beer et al. demonstrated a real-time single-molecule PCR amplification utilizing Poisson statistical loading of template DNA and an off-chip valving system, which allowed them to thermally cycle through the PCR protocol without droplet motion [73]. Real-time PCR curves showed a cycle threshold of 18–20 fewer cycles than commercially available instruments, which demonstrates the potential of improved sensitivity as well as reaction times. Wang et al. demonstrated successful PCR amplification using a droplet-based microfluidic device that utilized an oscillating flow system and three isolated temperature zones in a single channel [74]. As advancements are realized in cell and molecular biology within microfluidic droplet reactors, there will be a greater need for optical detection systems necessary for on-chip assay characterization. Srisa-Art et al. [75] have integrated confocal fluorescence spectroscopy as a high-efficiency detection method for droplet-based microfluidics fluorescence in a fluorescence resonance energy transfer (FRET)based DNA assay. This detection system utilizes ultrasensitive, high-speed avalanche photodiode detectors, which were shown to have the millisecond time resolution necessary for detecting droplets in a device formed at kHz frequencies. Droplet-based microfluidics have also been employed for a number of unique chemical reactions including precipitation reactions [76,77] synthesis of cobalt [78] and titania [79], titration of formic acid [80] and anticoagulants [38], and crystal growth [81]. Lau et al. [82] examined 40 different conditions within a single device to crystallize catalase, glucose isomerase, thaumatin, and ferritin.

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A better understanding of the droplet microfluidic processes and the underlying physical laws is crucial for further advancement of the research and development of the area. The fundamental challenges presented by these systems are fascinating, and the potential afforded by multiphase droplet microfluidics for engineering, chemical, and biological applications is enormous.

ACKNOWLEDGMENTS This work was supported by grants NSF/PREM (DMR 0611616), NSF/IGERT (DGE 0549500), and NSF (CBET/CHE 0828900), and the W. M. Keck Foundation.

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20

Electro-Optical Properties of Gel–Glass Dispersed Liquid Crystals Devices by Chemical Modiﬁcation of the LC/Matrix Interface Marcos Zayat, Rosario Pardo, and David Levy

CONTENTS 20.1 20.2 20.3 20.4

Introduction ...................................................................................................................... Electro-Optical Coatings .................................................................................................. GDLC Devices .................................................................................................................. Sol–Gel Preparation of GDLCs ........................................................................................ 20.4.1 Phase Separation ................................................................................................... 20.5 Performance of the GDLC Devices .................................................................................. 20.6 Conclusions and Future Trends ........................................................................................ References ..................................................................................................................................

20.1

447 448 448 449 450 451 455 455

INTRODUCTION

The sol–gel chemistry has gained a large number of researchers in the last 25 years. Interesting and sophisticated novel synthetic methods were therefore developed, offering a variety of approaches toward new goals and systems, overcoming many of the synthesis difficulties of the past. A strong argument for using the sol–gel chemistry is found in the high flexibility of the method and the large choice of commercially available “dopants” that can be incorporated in the solid matrices, which might have a specific activity or reactivity to an external signal (i.e., light, magnetic, electrical, etc.). The flexibility is reflected in the possibility to control the chemical composition of the matrices to be prepared, the kind of substrate, or the drying conditions. Generally, this can be achieved either by a chemical adaptation of the pore environment to meet the requirements of the chosen organic dopant or by an adjustment of the chemical precursors and process, in order to make it compatible with the dopants chosen for the specific application. From the point of view of nanotechnology applications, sol–gel materials are being required for critical components embedded in systems such as industrial equipment and scientific instrumentation, imaging and display, medical applications, aerospace and defense, and so on. The rapidly developing sol–gel process has been used for the preparation of materials for a wide range of fields, adapting the chemistry and the novel synthetic routes to the specific systems, in order to achieve 447

448

Structure and Functional Properties of Colloidal Systems

complicated developments oriented to nanotechnology applications. Clear examples can be found in sol–gel optical applications.

20.2

ELECTRO-OPTICAL COATINGS

The electro-optical coatings described in this chapter are based on a dispersion of liquid crystal (LC) droplets in a transparent and solid thin-film matrix. Owing to their intrinsic properties, the birefringent LC molecules in each droplet adopt a director configuration (long-range preferred orientation) induced by the specific anchoring sites on the surface of the pores. The light scattering produced by the LC microdroplets in the so-called OFF state results in a material, which is opaque to the incident light. If an electrical field that overcomes these anchoring forces is applied to this material (ON state), a reorientation of the LC molecules inside the microdroplets takes place, to adopt the direction of the applied field (Figure 20.1). In this situation, the material becomes transparent to the incident light. The maximum light transmission in the ON state will be achieved when the refractive index of the LC droplets (ordinary) is the same as that of the matrix. Upon cessation of the application of the electric field, the molecules in the microdroplets recover their original preferred orientation and the films become opaque again. These materials are of an increasing interest for their application as electronic displays liquid crystal device (LCD), optical switches, or light modulators [1,2]. Among other advantages, they need no polarizers for operation and no surface treatment is required to scatter light in the OFF state. The preparation of this type of materials was first reported by Fergason [3] consisting in a dispersion of LC microdroplets in solid polymer matrices [4–6]. Many efforts have been devoted since then to the development of these coatings, known as polymer dispersed liquid crystal (PDLC) in order to control specific electro-optical properties in a thin film [7–13]. The ability to entrap organic molecules into inorganic matrices via sol–gel procedures [14] opened a wide range of possibilities in the field of optical and electro-optical devices. The sol–gel method allowed the preparation of gel–glass dispersed liquid crystals (GDLC) [15–17]. These new composites offer better transparency and thermal stability as compared to PDLC and a larger difference between the refractive indexes of the matrix and the LC droplets, leading to higher light scattering in the OFF state. Since the first works published on GDLCs [18,19], efforts were oriented to improve the performance of the prototype devices using different sol–gel silica precursors [15,18], amino silicon alkoxides [20], or titanium oxide precursors [17,21].

20.3

GDLC DEVICES

The GDLC electro-optical device consists in an all-solid sandwich-like assembly of a GDLC layer deposited on a conductive substrate with a clean ITO-coated glass as a second transparent electrode.

Glass ITO

n (LC) π nm

LC microdroplets Δ ª 1m Silica-gel glass 20 kHz 10-400 Vp-p

GDLC in ON state Transmitted light

Dispersed light

GDLC in OFF state

Glass ITO

n0 (LC) ª nm

LC microdroplets Δ ª 1m Silica-gel glass 20 kHz 10-400 Vp-p

FIGURE 20.1 Schematic representation of the GDLC devices in OFF and ON states.

Electro-Optical Properties of Gel–Glass Dispersed Liquid Crystals

FIGURE 20.2 Vp-p 90 V).

449

GDLC device switched between the OFF and ON state (thickness 10 mm, 4 2 cm,

A schematic representation of the operation of a GDLC device is given in Figure 20.1. A picture of the device in ON and OFF states is given in Figure 20.2.

20.4 SOL–GEL PREPARATION OF GDLCs The sol–gel method allows the preparation of transparent, solid, and porous inorganic matrices at low temperatures, and the incorporation of organic molecules in its porosity [22,14]. The method involves reactions of hydrolysis and condensation of silicon alkoxides (Figure 20.3) to produce a three-dimensional (3-D), amorphous, porous, and stable silica network. Owing to its high versatility, the sol–gel method offers important advantages for the preparation of this kind of materials [15–19]. There is a large choice of metal-alkoxide precursors that allow the preparation of matrices with different physical and chemical properties. Among these properties, one can modify the index of refraction, thickness, or density of the sol–gel-prepared thin films. One of the most important advantages of using the sol–gel procedure for the preparation of this kind of materials is the possibility to control the chemical environment of the inner pore surface of the embedding matrix, which have a decisive effect on the electro-optical properties of the coatings, and hence on the performance of the resulting device [23,24]. The sol–gel method allows the preparation of hybrid organic–inorganic matrices by using organically modified alkoxide precursors. The nonhydrolyzable organic groups of the alkoxide remain attached to the matrix porosity (Figure 20.4). The usage of these precursors allows controlling the polarity and size of the pores according to the nature and amount of the nonhydrolyzable organic group used for the preparation of the embedding ormosil glassy matrix (pore surface functionalization).

Hydrolysis OR OR

Si OR + 4H2O OR

OH HO Si OH + 4ROH OH

Condensation OH OH OH OH –H O –H O Si(O)2–x(OH)2x HO Si OH + HO Si OH 2 HO Si O Si OH 2 Polycondensation OH OH OH OH

FIGURE 20.3 Sol–gel reactions to amorphous silica matrices.

450

Structure and Functional Properties of Colloidal Systems Hydrolysis OR Si OR + 3H2O

OH

OR

OH

Si OH + 3ROH

Condensation OH

OH

Si OH + HO Si OH OH

FIGURE 20.4

–H2O

OH OH Si O Si OH OH

–H2O

PhSi(O)1.5–x(OH)2x Polycondensation

Sol–gel reactions of phenyl-modified alkoxide precursors.

The sols used for thin-film deposition must be clear and viscosity stable to allow the preparation of homogeneous films and guarantee the reproducibility of the thin-film formation and its properties. The sol–gel method allows the preparation and control over the properties and performance of the GDLC films. One of the most important steps in the preparation of GDLCs is the phase separation that occurs simultaneously with the gelation and solidification of the matrix.

20.4.1

PHASE SEPARATION

After the deposition of the films onto the ITO-coated substrate, the sol must undergo a phase separation, forced by the evaporation of the solvent, to produce the LC microdroplets. This is the most important step in the preparation of the samples. The solvent in the sol is responsible for miscibility of the LC and silica precursor phases, at the same time it delays the gelation of the sol. Once the solvents are evaporated, a fast hydrolysis and condensation of the sol takes place leading to the gelation of the matrix. The LC molecules aggregate to larger droplets until the glass matrix solidifies (gel) and confine the droplets on their pores. The amount of solvent and its evaporation rate as well as the gelation rate of the sol will play a decisive role during the formation of the LC droplets. The faster the glass matrix is formed, the smaller the resulting LC droplets. The process of phase separation and gel formation is illustrated in Figure 20.5.

FIGURE 20.5 Phase separation and droplets grow during GDLC formation.

451

Electro-Optical Properties of Gel–Glass Dispersed Liquid Crystals

Silica O

Si Si HO O OH Si OH O Si

O

Si HO

O

O

Si

Si

OH

O S

O O Si Si O CH 2 OH CH CH Si 2 2 CH3 CH3 CH OH

O

3

Si CH3 CH2 O Si CH3 CH2 O

OH

O Si OH O

OH Si OH CH3 Si OH CH3 OH CH2 O CH3 OH H C CH3 CH O Si 3 2 Si CH Si OH 2 CH2 O CH2 Si Si O Si Polar Si O Si Si O O O O Non polar (LC)

NC

(CH2)4 CH3

Modified silica

FIGURE 20.6 Schematic representation of a pore structure in silica and ormosil matrices.

As mentioned before, the preparation of GDLC in ormosil matrices opens the possibility of functionalizing the inner porosity of the gel–glass matrix, where the LC droplets are located, with different organic groups. This, in term, allows controlling the size of the pores and the polarity of their surface, and hence the anchoring forces of the pore surface, which determines the electro-optical properties of the resulting device. A schematic representation of a typical pore in silica and modified silica matrices is shown in Figure 20.6. The ormosil used for preparation of the GDLC samples, especially the nature of the nonhydrolyzable organic functional group on the alkoxide precursor, plays also an important role in the formation of the LC droplets. It is known that during glass formation, these organic groups are oriented toward the pore cavities where the LC molecules will be allocated. Therefore, the size of the organic substituents and its affinity to the LC molecules will affect the size of the pores and the LC loading admitted by the specific matrix. The introduction of larger organic groups into the matrix (Figure 20.6) allows preparation of samples with much lower droplet-size distribution as observed in Figure 20.7. The organic functional groups in the pores led to a preferential confinement of the LC molecules in the pores rich in organic groups due to chemical affinity and size/volume considerations. In unfunctionalized samples, the aggregation of the droplets occurs without any preferential sites, resulting in a very wide size distribution.

20.5

PERFORMANCE OF THE GDLC DEVICES

GDLC devices are electro-optical windows in which the light transmittance can be controlled by means of an external electric field. Light absorption in these systems is mainly due to light scattering, and depends therefore on the wavelength of the incident light [25]. The visible transmittance spectra of a GDLC device prepared in a silica matrix, given in Figure 20.8, was carried out while switching the device between ON and OFF states. The devices prepared were switched more than 1000 times between ON and OFF states without any losses in electro-optical properties. The performance of the GDLC devices and particularly their dynamic properties are strongly related to the composition of the embedding matrix, especially the nature of the organic modifying group used for film preparation, as it provides the anchoring on the LC molecules in the

452

Structure and Functional Properties of Colloidal Systems

FIGURE 20.7 Optical microscopy photographs of the LC droplets dispersed in gel–glass matrices doped with different organic groups, (a) not doped; (b) Me (methyl); (c) Et (ethyl); and (d) Pr (propyl), taken through crossed polarizers.

droplets responsible for its orientation in the OFF state. The introduction of organic groups in the silica matrix resulted in a reduction of its anchoring forces, which is more remarkable for larger organic groups. This reduction in the anchoring forces can be observed in the lower threshold voltage required for device operation (Figure 20.9), about 25 Vp-p in samples functionalized with

Switching GDLC device 80 ON state

Transmittance (%)

70

60

50

40 OFF state 30

20 400

500

600

700

800

Wavelength (nm)

FIGURE 20.8 Ultra violet-visible light (UV-Vis) transmission spectra of S sample during ON–OFF states switching at an applied electrical field of 200 Vp-p.

453

Electro-Optical Properties of Gel–Glass Dispersed Liquid Crystals

100

90

T(%)

80

70

60 S S/Me S/Et S/Pr

50

40 0

50

100

150

200

250

300

Voltage applied (Vp-p)

FIGURE 20.9 Light transmittance of the GDLC devices as a function of applied voltage, measured at 632 nm.

Propyl groups (S/Pr) versus more than 100 Vp-p in unfunctionalized samples (S). The faster transition from OFF to ON states, when a given external field is applied (seconds in sample S to milliseconds in sample S/Pr), accounts also for the lower anchoring observed in the functionalized sample (Figure 20.10). The recovery of the initial opacity (ON to OFF states), when the LC molecules must reorient to their initial orientation based on the anchoring forces of the matrix surface, was particularly slower in a Pr-modified matrix (more than 10 s) as compared with an unmodified silica matrix (sample S), where it took only several milliseconds to recover the original opacity. The dynamic behavior of the devices, as a function of the applied voltage, shown in Figure 20.10 and measured for samples S, S/Me, and S/Et, shows the strong dependence of the activation time (OFF to ON transition) on the driving voltage. An interesting observation is that the transmittance of the samples can be controlled by means of the applied voltage (Figure 20.10). The usage of ormosil-based matrices to prepare GDLC films resulted in an improvement of many properties of the electro-optical devices, such as the remarked narrowing of the droplet-size distribution and the reduction of the voltage required for operation. At the same time, the weaker anchoring forces observed resulted in devices with much weaker opacities in the OFF state and much slower recovery kinetics, as observed in Figure 20.11. In unfunctionalized silica matrices (sample S), the polar OH groups at the surface of the silica network are responsible for the anchoring of the LC molecules in a GDLC. The anchoring strength of the glass matrix to the LC molecules in the droplets decreased drastically as larger organic groups are incorporated in the pore surface, as they hinder the electrostatic forces between the LC droplets and the OH groups on the silica surface. The very weak anchoring forces of matrices with large organic groups (e.g., butyl groups) cannot induce the molecules in the droplet to adopt a director configuration, resulting in a homeotropic orientation of the molecules inside the droplet; hence the sample is transparent to the incident light.

454

Structure and Functional Properties of Colloidal Systems 90

300 Vp-p

85 200 Vp-p

Transmittance (%)

80 75

Composition GDLC S (silica)

70 65 60 55 100 Vp-p

50

40 Vp-p

45 0.0

0.5

1.5

1.0

2.0

2.5

Time (s) 90

200 Vp-p

85 100 Vp-p

80 Transmittance (%)

75 Composition GDLC S/Me

70 65 60 55

40 Vp-p

50 45 40 0.0

0.5

1.0

1.5

2.0

2.5

Time (s) 100 100 Vp-p 95

Transmittance (%)

40 Vp-p

Composition GDLC S/Et

90 85 80 20 Vp-p

75

70 0.0

0.5

1.0

1.5

2.0

2.5

Time (s)

FIGURE 20.10 Dynamic response of GDLC devices (samples S, S/Me, and S/Et) at different driving voltages.

455

Electro-Optical Properties of Gel–Glass Dispersed Liquid Crystals

Electric field OFF 1.0 Recovery of the opacity at 0 V

Norm. transmittance

0.8 S/Pr 0.6 S/Et 0.4 S/Me 0.2 S (silica) 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time (s)

FIGURE 20.11 Recovery of the original opacity after cessation of the application of the electric field in samples prepared with different organic functionalization.

20.6

CONCLUSIONS AND FUTURE TRENDS

GDLC electro-optical devices were prepared using the sol–gel chemistry. The main step in the preparation of these materials involves a phase separation that must take place simultaneously with the gelation of the embedding matrix. The functionalization of the matrix with organic groups allowed controlling many properties of the electro-optical devices, such as the reduction of the voltage required for operation and the possibility to allocate larger amounts of LC in the matrix. At the same time, the activation and recovery times can be adjusted between milliseconds and several seconds by the appropriate matrix composition. Moreover, the chemical environment of the pores in the matrix can determine the arrangement of the LC molecules in the confined pore. Another advantage of functionalizing the matrices with organic groups is the drastic narrowing of the droplet-size distribution and the ability to control their size. The light transmittance of the GDLC devices can be adjusted by the applied electric field, allowing the control of the transparency of the devices in a continuous gray scale.

REFERENCES 1. Bloisi, F., Ruocchio, C., Terrecuso, P., and Vicari, L. 1996. Optoelectronic polarizer by PDLC. Liq. Cryst. 20: 377–379. 2. Cheng, S.X., Bai, R.K., Zou, Y.F., and Pan, C.Y. 1996. Electro-optical properties of polymer dispersed liquid crystal materials. J. Appl. Phys. 80(4): 1991–1995. 3. Fergason, J.L. 1985. Polymer encapsulated nematic liquid crystals for display and light control applications. SID Int. Symp. Digest Tech. Papers 16: 68–70. 4. Kitzerow, H.S. 1994. Polymer-dispersed liquid-crystals—from the nematic curvilinear aligned phase to ferroelectric films. Liq. Cryst. 16(1): 1–31. 5. Dou, Y., Zhang, W., In, C., and Xu, J. 2007. Polymer dispersed liquid crystal films. Prog. Chem. 19: 1400. 6. Bouteiller, L. and Le Barny, P. 1996. Polymer-dispersed liquid crystals: Preparation, operation and application. Liq. Cryst. 21(2): 157–174.

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7. Kim, E.J. and Park, O.O. 1995. Configuration control of liquid crystal in the droplet dispersed in the polymer with mesogenic side group. Mol. Cryst. Liq. Cryst. 267: 41. 8. Ferrari, J.A., Dalchiele, E.A., Frins, E.M., Gentilini, J.A., Perciante, C.D., and Scherschener, E. 2008. Effect of size polydispersity in polymer-dispersed liquid-crystal films. J. Appl. Phys. 103: 084505. 9. Cupelli, D., De Filpo, G., Chidichimo, G., and Nicoletta, F.P. 2006. Photoswitching in polymer-dispersed liquid crystals. J. Appl. Phys. 100: 024508. 10. Lackner, A.M., Margerum, J.D., Ramos, E., and Lim, K.C. 1989. Droplet size control in polymer dispersed liquid crystal films. Proc. SPIE 1080: 53–61. 11. Kato, K., Tanaka, K., Tsuru, S., and Sakai, S. 1993. Multipage display using stacked polymer-dispersed liquid crystal films. Jpn. J. Appl. Phys. 32: 4594–4599. 12. Kato, K., Tanaka, K., Tsuru, S., and Sakai, S. 1994. Reflective color display using polymer-dispersed cholesteric liquid-crystal. Jpn. J. Appl. Phys. 33: 2635–2640. 13. Drevensšek-Olenik, I., Cˇopicˇ, M., Sousa, M.E., and Crawford, G.P. 2006. Optical retardation of in-plane switched polymer dispersed liquid crystals. J. Appl. Phys. 100: 033515. 14. Avnir, D., Levy, D., and Reisfeld, R. 1984. The nature of the silica cage as reflected by spectral changes and enhanced photostability of trapped rhodamine-6G. J. Phys. Chem. 88: 5956–5959. 15. Levy, D. 1992. Glasses, ceramic, powders, membranes and composites. 10. Sol-gel glasses for optics and electrooptics. J. Non-Cryst. Solids 147: 508–517. 16. Levy, D., Pena, J.M.S., Serna, C.J., and Otón, J.M. 1992. Glass dispersed liquid-crystals for electrooptical devices. J. Non-Cryst. Solids 147: 646–651. 17. Levy, D., Serrano, A., and Otón, J.M. 1994. Electrooptical properties of gel-glass dispersed liquid crystals. J. Sol-gel Sci. Tech. 2(1–2): 803–807. 18. Levy, D., Serna, C.J., and Otón, J.M. 1991. Preparation of electrooptical active liquid-crystal microdomains by the sol-gel process. Mat. Lett. 10(9–10): 470–476. 19. Otón, J.M., Serrano, A., Serna, C.J., and Levy, D. 1991. Glass dispersed liquid-crystals. Liq. Cryst. 10(5): 733–739. 20. Chang, W.P., Whang, W.T., and Wong, J.C. 1995. Electrooptic characteristics of amino-gel-glassdispersed liquid-crystal and its matrix formation. Jpn. J. Appl. Phys. 34(4A): 1888–1894. 21. Hori, M., and Toki, M. 2000. Electro-optical properties of inorganic oxide/liquid crystal composite film by the sol-gel process. J. Sol-gel Sci. Tech. 19(1–3): 349–352. 22. Levy, D. and Avnir, D. 1988. Effects of the changes in the properties of silica cage along the gel/xerogel transition on the photochromic behavior of trapped spiropyrans. J. Phys. Chem. 92: 4734–4738. 23. Zayat, M. and Levy, D. 2003. Surface organic modifications and the performance of sol-gel derived gelglass dispersed liquid crystals (GDLCs). Chem. Mater. 15(11): 2122–2128. 24. Zayat, M. and Levy, D. 2005. The performance of hybrid organic-active-inorganic GDLC electrooptical devices. J. Mater. Chem. 15: 3769–3775. 25. Otón, J.M., Pena, J.M.S., Serrano, A., and Levy, D. 1995. Light-scattering spectral behavior of liquidcrystal dispersion in silica glasses. Appl. Phys. Lett. 66(8): 929–931.

21

Nano-Emulsion Formation by Low-Energy Methods and Functional Properties Conxita Solans, Isabel Solè, Alejandro Fernández-Arteaga, Jordi Nolla, Núria Azemar, José Gutiérrez, Alicia Maestro, Carmen González, and Carmen M. Pey

CONTENTS 21.1 Introduction ...................................................................................................................... 21.2 Condensation or Low-Energy Emulsification Methods: Relation between Phase Behavior and Nano-Emulsion Properties ................................................ 21.2.1 Self-Emulsification ............................................................................................... 21.2.2 Phase Inversion Temperature Method .................................................................. 21.2.3 Phase Inversion Composition Method .................................................................. 21.3 Functional Characteristics of Nano-Emulsions ................................................................ 21.3.1 Stability of Nano-Emulsions ................................................................................. 21.3.2 Droplet Size-Based Applications .......................................................................... 21.3.3 Solubilization-Based Applications ....................................................................... 21.4 Summary .......................................................................................................................... Acknowledgments ...................................................................................................................... References ..................................................................................................................................

21.1

457 458 459 460 465 473 473 475 478 479 479 479

INTRODUCTION

Nano-emulsions are defined as a class of emulsions with uniform and extremely small droplet size, typically in the range 20–200 nm [1,2]. Due to their characteristic size, nano-emulsions appear transparent or translucent (resembling microemulsions) and possess stability against sedimentation or creaming. The formation of kinetically stable liquid–liquid dispersions of such small sizes is of great interest from fundamental and applied viewpoints. Nano-emulsions are also referred to in the literature as miniemulsions [3–5], submicrometer-sized emulsions [6], finely dispersed emulsions [7], ultrafine emulsions [8], and so on. The term “nano-emulsion” has been increasingly adopted because it gives an idea of the nanoscale size range of the droplets and it is concise. Nano-emulsions, being nonequilibrium systems, require energy input for their formation, either from mechanical devices or from the chemical potential of the components [9]. The methods using mechanical energy are called dispersion or high-energy emulsification methods while those making use of the chemical energy stored in the components are referred to as condensation or low-energy emulsification methods [10]. Nano-emulsions are generally prepared by dispersion or high-energy emulsification 457

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Structure and Functional Properties of Colloidal Systems

methods. However, there is an increasing interest in low-energy methods, which in this chapter are classified as “self-emulsification” [11] and “phase transition” [12–14] methods. In the selfemulsification methods, emulsification is achieved by a dilution process without phase transitions taking place in the system. As it is discussed below, it is not always easy to make a clear distinction between some so-called spontaneous methods and phase transitions because emulsification is often achieved by both mechanisms. The phase transition methods make use of the chemical energy released during the phase transitions that take place during the emulsification process. Among these methods, the phase inversion temperature (PIT) method introduced by Shinoda in 1968 [12] is based on the changes in solubility of polyoxyethylene-type nonionic surfactants with temperature. Other low-energy emulsification methods take advantage of the phase transitions that take place by changing the composition at constant temperature during emulsification [13,14] (i.e., phase inversion composition, PIC, method). Low interfacial tension values and the presence of lamellar liquid crystalline or bicontinuous microemulsion phases (i.e., surfactant aggregates with zero or almost zero curvature) are among the factors that have been shown to be important for the formation of nano-emulsions with minimum droplet size. However, other type of structures, such as micellar cubic liquid crystalline phases and the kinetics of the emulsification process have been reported to play a key role in the properties of the resulting nano-emulsions. Nano-emulsions have found an increasing use in many different applications (chemical, pharmaceutical, cosmetic, etc.). Nano-emulsions are used in the pharmaceutical field as drug delivery systems [6], in cosmetics as personal care formulations [7,15], in agrochemical applications for pesticide delivery [16], in the chemical industry for the preparation of latex particles [3,4,17,18], and so on. The advantages of using nano-emulsions over conventional emulsions (or macroemulsions) are a consequence of their characteristic properties, namely, small droplet size, high kinetic stability, and optical transparency. In addition, nano-emulsions offer the possibility of using microemulsion-like dispersions without the need of high surfactant concentrations. The applications of nano-emulsions depend evidently on their functional characteristics. However, the possibilities of developing applications are mainly limited by stability [2]. In this chapter, nano-emulsion formation by low-energy emulsification methods, with special emphasis on phase inversion methods and their relation to surfactant phase behavior will be discussed first. This will be followed by an analysis of nano-emulsion functional characteristics. The relation with their structure is discussed regarding the applications in which they are relevant. Prior to discussing nano-emulsion functional characteristics, the main nano-emulsion destabilization mechanism, Ostwald ripening, is described.

21.2

CONDENSATION OR LOW-ENERGY EMULSIFICATION METHODS: RELATION BETWEEN PHASE BEHAVIOR AND NANO-EMULSION PROPERTIES

As mentioned in introduction, nano-emulsions being nonequilibrium systems require an energy input for their formation, which can be supplied mechanically (high-energy emulsification), or from the chemical energy of the components (condensation or low-energy emulsification methods). In this section, the different condensation methods, which are classified as self-emulsification and phase inversion methods, will be discussed. In self-emulsification or direct emulsification methods, a solution of the oil and surfactant in an appropriate solvent (that is also soluble in the continuous aqueous phase) is simply added to water to form oil-in-water (O/W) emulsions in one step, under agitation. Similarly, a solution of water and surfactant in an appropriate solvent (also soluble in the oil phase) is added to the oil to form waterin-oil (W/O) emulsions [19]. It is called direct or self-emulsification because the emulsion is just obtained by a dilution process without any phase inversion. This method uses the chemical energy of dissolution in the continuous phase of the solvent present in the initial system (which is going to constitute the dispersed phase). When the intended continuous phase and dispersed phase are mixed,

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459

the solvent present in the latter phase is dissolved into the continuous phase, dragging and dispersing the components of the initial system, thus giving rise to the nano-emulsion droplets. In phase inversion methods, the spontaneous curvature of the surfactant changes during emulsification. As a result, phase transitions occur during the emulsification process that releases chemical energy, used to form the nano-emulsion. For nonionic surfactants, the change of curvature implies a change in the hydration of poly(oxyethylene) chains of the PEO-type nonionic surfactants, induced by a change in temperature (PIT method) [20,1], or by a change in composition (PIC methods). The PIT method can only be applied to surfactants sensitive to changes in temperature, that is, the PEO-type surfactants. In the PIC method, the change in curvature can be achieved with other type of surfactants and can be induced by several mechanisms. Some of the proposed mechanisms that induce phase transitions at constant temperature are the partitioning of alcohol from the oil to the aqueous phase or diffusion of water into the initial droplet [20–23], both producing a shift from lipophilic to hydrophilic conditions; chemical reactions which convert lipophilic surfactants to hydrophilic surfactants through the emulsification path [24,25]; and progressive addition of the intended continuous phase, which may be pure water or oil [26,27] or may contain a dissolved surfactant with affinity to this phase [19]. When phase inversion methods are used, studies on surfactant phase behavior are important, since the phases involved during emulsification are crucial in order to obtain nano-emulsions with small droplet size and low polydispersity. The most used condensation methods, that is, the selfemulsification, PIT, and PIC methods, are discussed next.

21.2.1

SELF-EMULSIFICATION

The so-called self-emulsification is referred to those methods in which the nano-emulsion is just obtained by a dilution process without any inversion of phases. It is also called direct emulsification because the initial emulsion type is that of the intended final emulsion [19]. It should be taken into account that the term “self-emulsification” is frequently used in the literature to describe emulsification mechanisms, in which not only dilution processes but also processes implying changes in the spontaneous curvature of the surfactant film are involved. Therefore, this terminology is often misleading. A common way to prepare O/W nano-emulsions by self-emulsification is to dilute into water an O/W microemulsion. Microemulsions are thermodynamically stable liquid dispersion with droplet diameters generally lower than 20 nm and stabilized by a surfactant–cosurfactant mixture with an appropriate hydrophile–lipophile balance (HLB) [28–30]. The nano-emulsion formation can be explained by considering the thermodynamic aspects involved in this process. Nano-emulsion droplets develop because the droplets would no longer be thermodynamically stable since the surfactant concentration would no longer be sufficiently high to maintain the very low interfacial tension required (g < 104 N m1) for thermodynamic stability. Microemulsions are thermodynamically stable because the entropy of formation (DSf ) of such a large number of very small droplets outweighs the positive interfacial free energy (DHf gA, where A is the interfacial area of the droplets). The latter is minimized by the use of surfactant mixtures that give very low interfacial tensions. The free energy of formation (DG f ) of a microemulsion is given in simple terms by Equation 21.1 [31]: DG f = DHf - T ◊ DSf = g ◊ A - T ◊ DSf

(21.1)

Thus, in a microemulsion T ◊ DSf > g ◊ A, hence DG f < 0 and so the system is thermodynamically stable. However, dilution into water would result in a decrease in the surfactant concentration, resulting in an increase in the interfacial tension g, typically from less than 104 up to around 102 N m1. In this case, g ◊ A > T ◊ DSf and DG f > 0, and the system would now be unstable and the droplets would show a tendency to grow, thus reducing their now large interfacial free energy. The growth process could proceed via either coalescence or Ostwald ripening [20].

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Structure and Functional Properties of Colloidal Systems

Taylor and Ottewill [20,21] studied the mechanisms of destabilization of nano-emulsions prepared in this way using microemulsions of the water/sodium dodecyl sulfate (SDS)/hexanol/decane system and found that Ostwald ripening was the main mechanism. Latter Pons et al. [22], using the same system and the same emulsification method, reported a direct relation between the amount of water in the initial microemulsion (governing the aggregate structure) and the properties of the final nanoemulsion. For high water content microemulsions, the formation of nano-emulsions by dilution with water implies only the dilution of oil droplets already present (self-emulsification). The structure of the obtained nano-emulsions is apparently independent of the water dilution factor and does not affect significantly the average drop size. However, the dilution of lower water content microemulsions implies the transformation from a bicontinuous structure to the closed oil structures through a region of phase separation which consist of structures that at equilibrium are water continuous and liquid crystalline. Therefore, it cannot be considered a self-emulsification but a PIC method. Nano-emulsions stabilized by ionic and nonionic surfactants have also been prepared by dilution of microemulsions with excess water. Wang et al. [32] have recently reported the generation of nano-emulsions with an industrially relevant system composed of water/poly(oxyethylene) 7-lauryl ether (AEO-7)/methyl decanoate by rapid dilution with water, without spontaneous curvature transition. The formation of bluish transparent O/W nano-emulsions with a narrow size distribution was attributed to homogeneous nucleation of oil from the microemulsion phase upon dilution. The droplet sizes of nano-emulsions were determined by both dynamic light scattering (DLS) and small angle neutron scattering (SANS) and the two techniques showed no striking differences in the nanoemulsion droplet radii [32]. Self-emulsification of oil is of interest in various applications such as agrochemical [33,34], machining [35], personal care, drug delivery [36,37], and so on. Self-emulsification of water in an oil phase is also possible although it has not been widely used [11]. Dicharry et al. reports selfemulsification processes related to microemulsions, used in the metal-processing industry, in which dilution is carried out with hard water [38,35,39]. In this application, for easiness of use, a cutting fluid concentrate has to be a stable, not very viscous and “water-soluble” one-phase system. The authors propose a mechanism responsible for the formation of these finely dispersed emulsions and evaluate the influence of the composition of microemulsions upon the result of their dispersion into hard water. Stability of the emulsions is also reported, indicating that their high stability is to a great extent the result of the small size of the droplets and also to the fact that droplets are coated with a layer of charged surfactant complexes. The dilution is also made with a brine solution in order to modulate the affinity of the surfactant film for water and favor the spontaneous emulsification [39]. The so-called Ouzo effect is a self-emulsification method that occurs when water is added to a dilute binary solution of a solute whose solubility in water is very small (e.g., anise oil) in a watermiscible solvent (e.g., ethanol). Most of the solute rapidly come out of solution and a dispersion of small droplets in a surrounding liquid phase is formed without the use of surfactants, dispersing agents, or mechanical agitation [40–43]. For certain solute, solvent, and water ratios, relatively stable dispersions of very small solute droplets are formed, showing typically a log-normal size distribution.

21.2.2 PHASE INVERSION TEMPERATURE METHOD The PIT method, introduced by Shinoda and Saito [44], is probably the best-known phase inversion process and the most widely used in industry [45]. This method makes use of the changes in lipophilic–hydrophilic balance of some nonionic surfactants with temperature. The PIT method can be used when polyoxyethylene-type nonionic surfactants are present in the system, as their solubility changes with temperature [44]. At low temperatures, these surfactants are hydrophilic; their polyoxyethylene chains are hydrated acquiring a relatively big volume, as compared to the lipophilic part of the molecule. Consequently, the surfactant monolayer has a large positive spontaneous curvature, forming O/W microemulsions (Wm) at equilibrium, which may coexist

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with an excess of oil (Wm + O). With increasing temperature, dehydration of the polyoxyethylene chains occurs and these surfactants become more and more lipophilic. As a result, at high temperatures the polyoxyethylene chains are dehydrated and the spontaneous curvature becomes negative. Then, W/O microemulsions (Om) appear, coexisting with excess water phase (Om + W). At an intermediate temperature, the HLB temperature, surfactant affinity for water and oil phases is balanced. Then, the spontaneous curvature becomes close to zero and a thermodynamically stable planar structure with zero curvature, containing comparable amounts of water and oil, appears which can be a bicontinuous microemulsion (D), or a lamellar liquid crystal (L a), depending on the concentration of the surfactant and the system [46]. This phase can coexist with excess water and/or excess oil (W + D + O) or (W + L a + O). The PIT emulsification method takes advantage of the extremely low interfacial tensions achieved at the HLB temperature, around 102–105 mNm1 [47], to promote emulsification. In such conditions, very small droplet sizes can be obtained. However, due to the fact that the curvature of small droplets is very high and around the HLB temperature the spontaneous curvature is near to zero, the barriers that oppose coalescence processes are low and the coalescence rate is extremely high [48,49]. Consequently, at the HLB temperature, although emulsification in very small droplets is favored, the emulsions are very unstable. Therefore, to produce kinetically stable nano-emulsions, the temperature has to be quickly moved away from the HLB temperature by a rapid cooling or heating (obtaining O/W or W/O emulsions, respectively). If the cooling or heating process is not fast, coalescence predominates and polydisperse coarse emulsions are formed. As a summary, a schematic representation of the PIT method can be observed in Figure 21.1, which shows the paths

3 W+D +O

3

3

W + Om Om

2 D

La + Om

2’

2

La

Wm + La

Path (b)

Path (a)

Path (c)

T(∞C)

2

Wm Wm + O 1

1

1 [S]

FIGURE 21.1 Schematic representation of a typical phase diagram of a ternary water/nonionic surfactant/ oil system as a function of temperature and surfactant content at constant O/W weight ratio (Row). O/W nanoemulsion formation paths by the PIT method are indicated by arrows. The process starts from a bicontinuous microemulsion (D), path (a); an inverse microemulsion (Om), path (b, point 2); a lamellar liquid crystal (L a), path (b, point 2), and a three-phases region (W D O), path (c). (1) indicates compositions below the HLB temperature where all components are mixed and where O/W nano-emulsions are formed), (2) indicates compositions at the HLB temperature, and (3) indicates compositions above the HLB temperature.

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Structure and Functional Properties of Colloidal Systems

to form O/W nano-emulsions. First of all, all the components that will form the O/W final nanoemulsion are mixed at room temperature, where the equilibrium phases are a Wm microemulsion and excess oil (Wm + O) (points 1 of paths a, b, and c). Then, the emulsion formed is heated up to the HLB temperature (points 2 of paths a, b, and c) [sometimes, e.g., for cosmetic purposes, higher temperatures (point 3) are required to melt some components, but, in essence, for the PIT method it is enough to heat just up to the HLB temperature]. Depending on the composition of the system, a bicontinuous microemulsion (path a) or a lamellar liquid crystal (path b) will be present around the HLB temperature, at relatively low and high concentrations of surfactant, respectively, as shown in Figure 21.1. At lower concentrations of surfactant, the bicontinuous microemulsion can coexist with an excess of oil and/or water (path c). Here, nano-emulsions can also be obtained, but, as it will be shown later, bigger sizes and higher polydispersities are obtained when separate oil is present [50]. As discussed below, if a structure with zero curvature is not formed at the HLB temperature, big and polydisperse droplets are obtained by the PIT method. Once this temperature is reached, the two immiscible liquids are intimately dispersed in a single phase. Then, to obtain O/W nano-emulsions, the system is rapidly cooled, inducing a sudden change in the spontaneous curvature of surfactant monolayers. The equilibrium phases present at the end temperature are again a Wm microemulsion and oil (Wm + O). However, by this method the surfactant layer can be “trapped” in a curvature lower than the spontaneous one, yet retaining all the oil inside the “swollen” nanodroplets. Of course the resulting dispersions are not thermodynamically stable, since due to the excess of oil present in the droplets the surfactant films are not at their equilibrium curvature, and will tend to expel the excess water, separating into the two phases at equilibrium. However, these emulsions can be kinetically stable due to their small droplet size and narrow size distribution. The formation of W/O nano-emulsions is analogous, but they form at temperatures higher than the HLB temperature. Then, emulsification paths are in the opposite direction than those presented in Figure 21.1: the components are mixed at a temperature higher than the HLB temperature, then cooled at least up to the HLB temperature, and quickly heated again. Wadle et al. [51] investigated some years ago the water/C16–18E12/glycerol monostearate/cetearyl isononanoate system and reported that in order to obtain fine emulsions (of the order of 100 nm), the transition through a bicontinuous microemulsion (D) and/or lamellar liquid crystalline phase was necessary. In more recent studies, Izquierdo et al. prepared nano-emulsions using the PIT method with different technical grade nonionic surfactants and hexadecane [52] or isohexadecane [53,54] as the oil component. The water content of the final nano-emulsions was 80%. They found that the droplet radius of nano-emulsions decreased from around 130 to around 30 nm when the concentration of surfactants was increased from 4% to 8%, accompanied by a reduction in polydispersity. This decrease was in part attributed to the increase in interfacial area and the decrease in interfacial tension produced by the increasing concentration of these technical grade surfactants. However, for some formulations this effect was not progressive, but a discontinuity was observed above a critical surfactant concentration, obtaining smaller droplets and polydispersities than that expected for the tendency followed up to that concentration of surfactant. When the phase behavior along the emulsification paths was studied [53,54], it was observed that below the critical concentration of surfactant, a region with three phases (W + D + O) was present at the HLB temperature, where emulsification took place, while over this surfactant concentration, a single D microemulsion phase or a two-phase region consisting of D microemulsion and lamellar liquid crystalline L a phases was present. These findings confirmed results reported earlier [46,55,51], which showed that the bicontinuous and/or lamellar structures are determinant for the emulsification mechanism. Morales et al. [50] systematically studied the relation between the emulsion droplet sizes and the phase behavior at the HLB temperatures for different compositions in the water/C16E6/mineral oil system. Figure 21.2 shows the fish-shaped phase diagram obtained as a function of temperature and surfactant concentration for a fixed oil/(water + oil) weight ratio, Row = 0.2. Several nano-emulsions were prepared at different concentrations of the C16E6 surfactant, from 1 to 9 wt%, observing a sharp decrease in emulsion droplet sizes when the C16E6 concentration was around 3–4 wt%

Nano-Emulsion Formation by Low-Energy Methods and Functional Properties

463

110

Temperature (∞C)

90 W+D+O D

70

50

30 0

2

4

6

8

10

12

C16E6 (wt%)

FIGURE 21.2 Phase behavior of the water/C16E6/mineral oil system as a function of temperature and surfactant content. The O/W weight ratio (Row) is fixed at 0.2. A single-phase shear birefringent (D) microemulsion and a three-phase region of water (W), D, and oil (O) are shown. The surrounding regions of the diagram consist of multiphase equilibria. (From Morales, D. et al., Langmuir, 2003, 19, 7196–7200. With permission.)

(the critical surfactant concentration in this case, Figure 21.3). At lower concentrations of surfactant, free oil phase coexisted with water (W) and D phases at the HLB temperature (see Figure 21.2), and at concentrations above 3–4 wt%, oil and water were incorporated into a single bicontinuous D phase microemulsion. By fixing the water content at 95 wt% and changing the oil/(oil + surfactant) ratio, Ros, a dramatic increase in droplet size and polydispersity was observed at Ros values above 0.67 wt%, where a separated oil phase in equilibrium with D was present at the HLB temperature. In all these experiments, an aqueous phase, W, was present at the emulsification temperature. At Ros lower than 0.6 wt%, a slight increase in droplet size was observed with the decrease in Ros. This was related to the appearance of a lamellar liquid crystalline phase, L a , where volume fraction increased with the decrease in Ros. Although an L a phase implies low interfacial tensions that favor nanoemulsification, the relatively high viscosity of this phase probably hinders the homogenization of the system. Another approach was to keep the Ros value constant to 0.67 and the water content changed

Droplet Diameter (¥ 103 nm)

2

0.2

0.1

0 0

2

4

6

8

10

C16E6 (wt%)

FIGURE 21.3 Droplet size, at 40°C, as a function of surfactant concentration for samples with an O/W weight ratio (Row) fixed at 0.2, after emulsification by the PIT method. (From Morales, D. et al., Langmuir, 2003, 19, 7196–7200. With permission.)

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Structure and Functional Properties of Colloidal Systems HLB Temperatures, ∞C 57

57

56

56

56 D

W 95

D

D

D

D

75

80

W 85

W 90

Water, wt%

FIGURE 21.4 Phase equilibria for samples with an O/S weight ratio (Ros) equal to 0.67, and different water concentrations at the corresponding HLB temperature. (From Morales, D. et al., Langmuir, 2003, 19, 7196– 7200. With permission.)

from 75 to 95 wt%. As shown in Figure 21.4, at this Ros no excess oil, O, phase was observed at the HLB temperature at any of the concentratrions studied. A single D microemulsion phase was observed up to around 80 wt% water, and at higher water concentrations D and W phases coexisted (W + D). The droplet diameter, which was found to be independent of the water content, was around 40 nm and, the nano-emulsion droplets showed narrow size distributions. These results indicate that droplet size is mainly controlled by the aggregates with zero curvature structure found at the HLB temperature, and that the key factor to obtain small droplets with low polydispersity is the total incorporation of all the oil into the phase with zero curvature. The oil is trapped into the small droplets during emulsification, and the excess water acts just as a dilution medium for the dispersed oil droplets, and does not influence the final droplet size. The slight increase in droplet sizes when L a appears at the emulsification temperature can be related, as mentioned above, to a more difficult emulsification due to the higher viscosity of this phase compared to the viscosity of the D microemulsion phase. Probably, this drawback could be solved by an appropriate stirring of the system. In order to deepen on the emulsification mechanism, Morales et al. [56] compared the experimental droplet size of nano-emulsions with a theoretical minimum droplet size. The model used to calculate the theoretical size considers an O/W emulsion as being formed by a compact surfactant monolayer. Surfactant solubility in the oil component is also taken into account. Results showed that the model fits quite well droplet sizes of nano-emulsions obtained when D or D + W phases are present at the HLB temperature. On the contrary, when free oil, O, or L a phases were present, the experimental droplet sizes were higher than those predicted by the model, and they were more polydispersed, indicating a less effective emulsification. When O phase is present, it seems more difficult to incorporate it into the droplets when the inversion takes place. On the other hand, when L a is present all the oil is intimately incorporated and should be easy to emulsify it by a sudden change in curvature. The difficulty may be, as mentioned above, the homogenization problems due to high viscosities. These results confirmed the conclusions of the first studies of the authors [50]. Based on these results, the proposed mechanism of nano-emulsion formation from a single D microemulsion phase and from a two-phase system, a D microemulsion plus excess water (D W), consists of a thermal disruption of the D microemulsion phase due to the dramatic hydration of the ethoxylated groups by reducing the temperature, thereby promoting a curvature change of the surfactant monolayer and consequently the oil droplet formation, the excess of water acting just as a dilution medium for the oil droplets. The contribution of Wadle et al. [51] is interesting to be discussed, as they use, apart from the PIT method, a modification of the PIT method that is, in fact, a combination of the two phase inversion methods, PIT, and PIC (which will be discussed more deeply in next section). They describe this emulsification method as a multistep emulsification process. This method consists in the preparation

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of a concentrate containing the oil and emulsifier components as well as part of the water. This concentrate is emulsified around the HLB temperature and then diluted with cold water to obtain the final composition. The emulsion moves away the HLB temperature during the dilution with cold water due to two factors: the cooling of the system and the change in composition. This method saves energy, as not all the sample but just a concentrate has to be heated, and facilitates the cooling, as it is produced by the addition of cold water. Their results show that, with the water/C16–18E12/ glycerol monostearate/cetearyl isononanoate system, nano-emulsions with a droplet size around 100 nm are obtained if during emulsification a bicontinuous or lamellar liquid crystalline phase without free oil is passed through at the HLB temperature, according to the conclusions derived from the other works already discussed. A general conclusion of this subsection is that in order to obtain small droplet-sized nano-emulsions with low polydispersity through the PIT method, the mean feature to take into account is to choose an adequate composition in which, at the HLB temperature, flat structures (bicontinuous microemulsion and/or lamellar liquid crystal) are formed without an excess of oil. The excess of water does not influence the nano-emulsions properties, but acts as a simple dilution medium, probably because when enough water is contained that assures the complete hydration of the PEO chains at the final temperature of emulsions, the water in excess does not interact with surfactants, nor with oil.

21.2.3

PHASE INVERSION COMPOSITION METHOD

At constant temperature, phase transitions (similar to those observed in the PIT method) may take place by changing composition during the emulsification process [23,14], that is, a phase inversion can be induced by a change in composition. Although PEO-type surfactants are not required in the PIC method, numerous investigations have used these type of surfactants for the preparation of direct (O/W) nano-emulsions [57,26,58,19,59,32] and reverse (W/O) nano-emulsions, reported for the first time by Uson et al. [27]. The simplest PIC method to form O/W nano-emulsions consists in the progressive addition of pure water to the oil phase, which contains a surfactant or a mixture of surfactants. The whole emulsification takes place at constant temperature. The mechanism is in essence analogous to that of the PIT method. When water is progressively poured into the oil phase, the initial system is a W/O microemulsion (Om). Upon increasing the volume fraction of water, the hydration grade of the PEO chains of the surfactant progressively increases, changing therefore the spontaneous curvature from inverse (negative) to a planar (zero) curvature. Around this transition composition, surfactant affinity for water and oil phases is balanced, as it happens at the HLB temperature for the PIT method. Consequently, bicontinuous or lamellar structures are formed. With further addition of water, a biphasic region is usually found where direct Wm microemulsion and O phases coexist at equilibrium (Wm + O). When the transition composition is exceeded, the structures with zero curvature separate into metastable small direct (O/W) droplets which still contain all the oil, and have a very small diameter, which implies a very high positive curvature of surfactant layer. The droplet diameter of these emulsions has been related to the bilayer thickness of the lamellar structure [57]. The decomposition into very finely dispersed droplets is facilitated by the low interfacial tension at the transition composition. Further dilution with water does not change the droplet size at this stage of droplet formation, but can be necessary in order to go far away from the compositions with low interfacial tension, where coalescence is favored. However, in many cases the rate at which the system moves away the PIC does not seem to be very crucial in terms of avoiding coalescence, since usually surfactants adequate to be used in PIC method have long PEO chains and, as a result, a steric stabilization can prevent coalescence for some time. In fact, if liquid crystal phases are involved, a slow addition of water may be required to avoid inhomogeneities due to the relatively high viscosity of the system [58]. In order to obtain small and monodisperse droplets, at the inversion point all the oil must be incorporated into the bicontinuous or lamellar liquid crystal structure (as it happens in the PIT

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Structure and Functional Properties of Colloidal Systems

method), which usually implies relatively high surfactant concentrations (but yet low compared to the required to dissolve the same percentage of oil through Wm microemulsions). Low addition rates and an adequate stirring of the system may be necessary to assure homogenization. If at the PIC some oil remains out of the bicontinuous or lamellar structure, this fraction of oil will be emulsified only by mechanical energy, producing big and polydisperse droplets, the more free oil fraction present, the higher mean size, and polydispersity. In some systems, when the limit of stability of the phase with zero curvature is exceeded, a direct single phase (direct microemulsion, Wm [22,59], or direct cubic liquid crystal [25,60]) instead of a two-phase region, is formed. Then, further addition of water leads the system to the region with two-phase equilibria (Wm + O), where kinetically (but not thermodynamically) stable dispersions (i.e., nano-emulsions) can form. This emulsification method could be classified as a self-emulsification method since once the system is in one-phase region with direct-type structure, no inversion takes place. The use of the PIC method to obtain O/W nano-emulsions has been studied extensively. Fernández et al. [57] prepared emulsions with droplet diameters of the order of 300 nm at 10 wt% surfactant in the system formed by water/Cremophor®A6-Cremophor®A25/paraffin oil. Cremophor®A6 is a nonionic surfactant consisting of a polyethylene glycol alkyl ether, C16-18E6, and stearyl alcohol at a weight ratio of 3:1. Cremophor®A25 is analogous to Cremophor®A6 but with C16–18E25. Sadurní et al. [26] reported droplet diameters around 30 nm when working in the water/Cremophor®EL/Miglyol 812 with oil/surfactant (O/S) ratios lower than 40/60 [Cremophor®EL is a polyethylene glycol (35) castor oil, and its HLB lies between 14 and 16. Miglyol 812 is a medium-chain triglyceride]. Pey et al. [58] obtained droplet sizes between 50 and 200 nm in the water/Span 20-Tween 20/paraffin system (Span 20 being sorbitan monolaurate with NHLB = 8.6 and Tween 20 being polyoxyethylene sorbitan monolaurate with NHLB = 16.7). They observed lamellar liquid crystal before phase inversion, but mixed with other phases, including free oil, which can explain the relatively high size of droplets compared with that obtained in other systems. Wang et al. [59] worked with the system methyl decanoate/C12E7 (technical grade)/water and reported droplet diameters ~30 nm around O/S ratios of 1. The formation of W/O nano-emulsion by the PIC method was described for the first time by Uson et al. [27], through the addition of oil into water/surfactant mixtures in the system water/ Cremophor®EL-Cremophor®WO7/isopropyl myristate, obtaining droplet sizes in the range from 60 to 160 nm [Cremophor®WO7 is a polyethylene glycol (7) hydrogenated castor oil]. In all the mentioned studies, lamellar or bicontinuous structures, either as single phases or coexisting with other phases including an oil phase, were found along the emulsification path. The influence of formulation variables (composition), but also formation variables (stirring and addition rate), may be crucial in order to obtain proper nano-emulsions if highly viscous phases are found along the emulsification paths. A systematic approach to the optimization of formulation and formation variables through experimental design tools has been reported recently [58,25]. As an example, the surface response obtained, when addition and stirring rates and their interaction at a fixed composition were analyzed by Pey et al. [58] through experimental design in the water/Span 20-Tween 20/paraffin system, is shown in Figure 21.5. The model predicts smaller droplet sizes when slower addition rates are used. This is attributed to the fact that the high viscosity of liquid crystals present around the composition where inversion takes place produces inhomogeneities, which can be prevented by remaining enough time inside this region (related to slow addition rates) to assure homogenization of the system. However, an optimal stirring rate is observed, because on the one hand a high stirring rate minimizes the mentioned inhomogeneities, but on the other, it can favor coalescence once the border between the viscous multiphasic region and the (Wm + O) region is crossed. It should be noted that it is not easy to encounter systems where the change in the hydration of PEO with the sole change in the water content allows the proper change in curvature from inverse-to planar-to direct. Most of the above-referred authors use a mixture of two surfactants, one lipophilic, with a short hydrophilic chain, and one hydrophilic, with a quite long PEO chain. If only a surfactant

Droplet diameter (nm)

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120 100 80 800 60

600 0

5 10 15 Addition rate (mI/min)

400 20

200

ng

ixi

m)

rp

e( rat

M

FIGURE 21.5 Response surface: Droplet diameter as a function of preparation variables. (From Pey, C.M. et al., Colloids Surf. A: Physicochem. Eng. Aspects, 2006, 288, 144–150. With permission.)

is used with a not very long PEO chain, the increase in hydrophilic volume produced by the hydration of polyoxyethylene chains may not be enough to promote the change to direct curvature, and planar curvatures are preferable. On the other hand, if a mixture of surfactants is used, one highly hydrophilic (with long PEO chains) and one highly hydrophobic, at the PIC, where planar structures are present, the lipophilic surfactant can be located between two hydrophilic surfactant molecules without practically occupying volume. Then, when the PEO chains of the hydrophilic surfactant are hydrated and, as a result, the hydrophilic surfactant molecules are separated, the lipophilic chains of the surfactant with the low HLB number yet interact with the oil located in the bilayer, dissolving it and preventing its segregation into droplets of free oil. However, also in the cases where mixtures of surfactants are used, the mixtures have to be adequately balanced in order to obtain an HLB average number in a narrow range, without a clear preference either for oil or for water, to be able to obtain a sufficient change in the spontaneous curvature by the sole addition of water. All the authors report a decrease in the nano-emulsion droplet size when the surfactant concentration increases, since there is more surfactant available to stabilize more interfaces, and therefore smaller droplets can be obtained. In general, it is observed in the phase diagrams that the region with phases with planar structure, that precedes the region where nano-emulsions are formed, extends to higher water concentrations when the surfactant/oil ratio increases, because when more surfactant is present, it can dissolve more water inside the bicontinuous or liquid crystal structure (as an example, see Figure 21.6). Alternatively to the slow addition of water to the O/S mixture, a concentrate can be directly prepared in the bicontinuous or lamellar region by simple mixing of all the components, in conditions that assure the equilibrium in this phase to be reached, and then the rest of the water can be added in one step. This is the case for Wang et al. [59], who compared nano-emulsion sizes obtained through three different emulsification methods: method A: a two-step process where a concentrate with 50 wt% water was prepared and equilibrated, into the bicontinuous region, and the rest of the water was added in a second step; method B: water was added dropwise to the oil–surfactant mixture; method C: all the components were directly mixed at the final proportions. Figure 21.7a shows the phase diagram of the system. Figure 21.7b shows the droplet size of emulsions obtained through the three methods. It can be seen that the smallest droplet sizes are obtained through the methods A and B, when a bicontinuous D phase or a direct microemulsion, Wm, is present respectively along the emulsification path, immediately before the biphasic zone where nano-emulsions form, with all the oil dissolved into this phase. Method B can be considered as self-emulsification method since dilution of a direct-type, Wm, microemulsion is produced in the last emulsification step. Nevertheless, the sizes obtained with methods A and B were found to be coincident. The ways in which the single phase regions are reached are not relevant if the equilibration of these phases is assured, because

468

Structure and Functional Properties of Colloidal Systems Tween20/Span20 (5.123/4.877) HLB = 12, 75

O:S = 1:1 O:S = 1,5:1 70% water

Multiphasic Om region

O:S = 2:1 O:S = 3:1

Wm + O Biphasic region

Water

Liquid paraffin

FIGURE 21.6 Phase diagram of the W/Tween20 : Span20 (5.123 : 4.877)/liquid paraffin system. Emulsification paths from O/S mixtures by addition of water (until 70 wt%). (From Pey, C.M. et al., Colloids Surf. A: Physicochem. Eng. Aspects, 2006, 288, 144–150. With permission.)

these structures are thermodynamically stable and, as a consequence, they are independent of the way of preparation. As already mentioned, it is not easy to find systems where the mixture of surfactants has the proper HLB to promote the change in curvature by the sole addition of water. Sajjadi [19] proposes a PIC method where the water containing the hydrophilic surfactant is added to the oil containing the lipophilic surfactant. Then, the change in curvature is more easily reached, as the HLB of the surfactant mixture increases continuously by the addition of the aqueous solution containing the hydrophilic surfactant, and the formulation of the system is not so critical. However, they obtain droplet sizes of the order of 0.5–1 mm, which are quite big droplets compared with the other works already discussed. The change in curvature through the PIC method can also be induced by partitioning of alcohol from the oil to the aqueous phase, producing a shift from lipophilic to hydrophilic conditions when water is progressively added to the system. However, most of the reported nano-emulsions are obtained due to a dilution effect (i.e., spontaneous emulsification) rather than a phase inversion process. Examples of this are the contributions of Taylor and Otewill [20] and Pons et al. [22]. They studied the water/SDS/dodecane system using pentanol or hexanol as cosurfactants, and observed that nano-emulsions of a droplet size around 20 nm could be formed by simple dilution of microemulsions with water. A migration of the alcohol from the interface to the water bulk was induced by progressive addition of water, increasing the spontaneous curvature of the interface. Solè [61] studied these systems again in order to deepen on the nano-emulsion formation mechanism. As an example, the pseudoternary phase diagram for the water/SDS-hexanol/dodecane system at a SDS : hexanol ratio of 1 : 1.76 obtained by Solè [61] is shown in Figure 21.8. The diagram is nearly identical to that obtained earlier by Clausse [62], with slight differences due to the different purities of the components used. Solè [61] prepared nano-emulsions departing from the different compositions indicated in Figure 21.8, that is, she departed from several concentrates located in the inverse microemulsion

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Surfactant 0.00 1.00

(a)

0.25

0.75

0.50

Om

La + Om

0.50

La 0.75

0.25

Wm or D

Wm

II

M

1.00

0.00 0.25

0.00 Water

0.50

0.75

1.00 Oil

D or Wm

(b) 600

Droplet size (nm)

500 400 300 200 100 0 0.5

1.5

1.0

2.0

Ros

FIGURE 21.7 (a) Phase behavior of the water/poly(oxyethylene) nonionic surfactant/methyl decanoate system at 25ºC. II, two-liquid isotropic phases; Om, isotropic liquid phase (inverse micellar solution or W/O microemulsion); Wm, isotropic liquid colorless phase (micellar solution of O/W microemulsion); L a , optically anisotropic phase; D, isotropic liquid phase (bicontinuous microemulsion); and M, multiphase region (phases not determined). (b) Droplet sizes at 25ºC as a function of Ros by method A (squares), method B (circles), and method C (triangles). The dotted lines indicate the range of D or Wm phase for the concentrate used in method A. (From Wang, L. et al., J. Colloid Interface Sci., 2007, 314, 230–235. With permission.)

region, Om, and from several concentrates located in the direct microemulsion region, Wm, at five O/S ratios. She used four different emulsification methods: (a) addition of the microemulsion into water in one step; (b) gradual addition of the microemulsion into water; (c) addition of water into the microemulsion in one step; and (d) gradual addition of water into the microemulsion. It was observed that departing from Om1 a turbid emulsion that rapidly suffers creaming is obtained, whatever emulsification mechanism (a, b, c, or d) is selected. It needs to be pointed out that along the emulsification path 1 the direct microemulsion Wm is not crossed. Monodisperse nano-emulsions with droplet sizes within the range 20–40 nm are obtained departing from the Wm concentrates Wm2, Wm3, Wm4, and

470

Structure and Functional Properties of Colloidal Systems SDS / hexanol (1/1.76)

Om1 Om2 Om3 Om Wm

Wm2

Om4 Om5

Wm3

Wm4

Wm5

Multiphasic region Water

Dodecane

FIGURE 21.8 Phase diagram in the pseudoternary system water/SDS:hexanol (1 : 1.76)/dodecane. Emulsification paths for the preparation of nano-emulsions at a final water content of 98 wt%.

Wm5, independent of the emulsification method. These droplet sizes practically coincide with those obtained through the homologue paths Om2, Om3, Om4, and Om5, only if the (d) method (gradual addition of water into the microemulsion) is used. As expected, the droplet size decreases with the O/S ratio. Departing from the Om region, bimodal coarse droplet-sized emulsions are obtained when emulsification methods a, b, and c are used. These results indicate that the key to obtain small and monodisperse nano-emulsions is to reach the equilibrium at the direct Wm zone, with all the oil contained inside the discrete aqueous domains or “swollen micelles.” This requirement is reached, of course, when the system departs directly from the Wm zone, but also departing from the Om zone if the addition of water to the system is slow enough to allow the reaching of the equilibrium in the Wm zone. Then, when further water is added to the Wm, part of the hexanol dissolves into water, leaving the interface and, then, the “swollen micelles” are no more thermodynamically stable, and the microemulsion is transformed into a nano-emulsion. Therefore, as indicated above, this emulsification method falls in the category of self-emulsification. There are few reports on the preparation of nano-emulsions stabilized with ionic surfactants by condensation or low-energy methods. Obviously, the PIT method cannot be used, as temperature does not change the behavior of ionic surfactants. Therefore, the required change in spontaneous curvature has to be reached by a change in the composition, that is, by the PIC method or some modification of the PIC method. Solè et al. [25] proposed to change the curvature (or the lipophylic– hydrophilic affinity) of the surfactant by varying the degree of ionization of the surfactant. They used the water/potassium oleate-oleic acid-C12E10/hexadecane ionic system. Nano-emulsions were prepared by adding proper potassium hydroxide solutions to hexadecane/oleic acid-C12E10 mixtures under controlled mixing and addition rates. Hence, the potassium oleate was formed along the emulsification path. The final concentration of water was 80 wt% in all cases. In order to deepen the nano-emulsion formation mechanism, partial pseudoternary phase diagrams were determined at several oleic acid/C12E10 ratios [25], as shown in Figure 21.9. Along the different emulsification paths, the system crosses an Om region, followed by a biphasic (or a multiphasic, depending on the O/S ratio) region where L a is present. Then, a single L a zone

Nano-Emulsion Formation by Low-Energy Methods and Functional Properties

471

Oleic acid/C12E10 = 20/80 Oleic acid/C12E10 = 30/70 Oleic acid/C12E10 = 40/60 Oleic acid/C12E10 = 50/50

Water + KOH Water + KOH Water + KOH Water + KOH

Hexadecane

FIGURE 21.9 Pseudoternary phase diagrams of the system water/oleic acid-potassium oleate-C12E10/hexadecane at several oleic acid/C12E10 ratios at 25ºC. The concentration of the alkaline aqueous solutions in the vertexes corresponds to a stoichiometric ratio between the oleic acid and the KOH at the final composition (80 wt% water). As the oleic acid is a weak acid, at the final composition both oleic acid and oleate species are present. Black circles correspond to W/O microemulsion, white circles to a W/O microemulsion L a , black squares to L a , white squares to L a Pm3n, black triangles to Pm3n, white triangles to O/W microemulsion oil, and rhombus to a multiphasic phase. (From Sole, I. et al., Langmuir, 2006, 22(20), 8326–8332. With permission.)

appears followed by a biphasic region with L a and direct cubic liquid crystal and, with further addition of water, a single direct cubic liquid crystal phase appears (for some O/S ratios, a multiphasic region extends along almost all the emulsification path). After that, a biphasic region with Wm + O appears, which is the region where the nano-emulsions are formed. The change in curvature along the emulsification paths is clear. Small droplet sizes in the range 20–40 nm are obtained departing from O/S ratios at which a direct cubic liquid crystal is formed during the emulsification process, with all the oil contained in the spheres of the cubic structure. As cubic liquid crystals are highly viscous phases, special care is necessary in the mixing and addition rates in order to assure a proper homogenization and equilibration of the cubic phase. If an emulsification path (i.e., an O/S ratio) is chosen that does not form a single liquid crystal phase, but the cubic liquid crystal coexists with an oil phase, the droplet sizes obtained are in the range 120–200 nm and quite polydisperse. This is due to the fact that all the oil is not incorporated into the nano-emulsion and the oil in excess forms big droplets, with a size that depends only on the mechanical energy input during emulsification (mixing rate). The emulsification mechanism in this case is not the break up of bicontinuous or lamellar phases into nanodroplets, trapping the oil included in the bilayer, since the key phase is a direct cubic liquid crystal. The size of the discrete “spheres” or “micelles” packed into the cubic geometry is compared with the size of the resulting nano-emulsions in Figure 21.10 for several formulations. It seems clear that the droplet size of nano-emulsions is related with the size of the discrete “micelles” of the cubic liquid crystal. For an oleic acid/C12E10 ratio of 40/60, both sizes practically coincide. For an oleic acid/C12E10 ratio of 30/70, the nanodroplets are slightly bigger than the corresponding “micelles” of the cubic crystal, increasing the difference when the O/S ratio increases. The important feature for

472

Structure and Functional Properties of Colloidal Systems 20 18 16 Radius (nm)

14 12

Cubic liquid crystal micelles radius

10

Nano-emulsion droplets radius

8 6 4 2 0 O/S O/S O/S 30/70 40/60 50/50

O/S O/S O/S 30/70 40/60 50/50

Oleic acid/C12E10 = 30/70

Oleic acid/C12E10 = 40/60

FIGURE 21.10 Comparison of the radius of the micelles that form the direct cubic liquid crystal with the radius of the nano-emulsion droplets obtained for several emulsification paths for the W/potassium oleateoleic acid-C12E10/hexadecane system.

this system is to obtain a cubic liquid crystal phase with all the oil dissolved into the discrete domains of the liquid crystal during emulsification. Similarly to some of the other studies discussed above [59,61], the path followed for the formation of this phase is not important, because this is a thermodynamically stable phase, although special care is required to avoid inhomogeneities due to the high viscosity of cubic liquid crystals. When further water (with KOH) is added, these “micelles” progressively separate, and the cubic structure is disrupted. During the dilution process, part of the surfactant that was at the interface migrates to the water. Moreover, as KOH is being added with water, at the region where nano-emulsions are formed, the spontaneous curvature is very high, and as a result, “swollen micelles” smaller than the spheres packed in the cubic structure would tend to form at equilibrium (an O/W microemulsion, Wm), expelling the excess of oil as a separated phase. However, through the PIC method, the oil can be trapped in these “swollen micelles” or droplets, forming a nano-emulsion that, of course, is not thermodynamically stable, as the droplets have a curvature lower than the spontaneous one corresponding to Wm, but it can show a reasonable kinetic stability. For an oleic acid/C12E10 ratio of 40/60, the cubic phase is nearer to the final nano-emulsion composition than for the ratio 30/70. As a result, the dilution process suffered is “shorter,” and the size of the nano-emulsion droplets coincide with that of the “micelles” of the cubic crystal. For the oleic acid/C12E10 ratio 30/70, the cubic phase suffers a “longer” dilution process and a stronger change in the spontaneous curvature. The higher the O/S ratio, the longer the dilution process, as the distance between the border of the cubic liquid crystal region and the nano-emulsion composition is higher. Then, a difference is observed between the nano-emulsion droplet size and the size of micelles of the cubic crystal, which becomes more important when O/S ratio is increased. Analogous results were obtained when the W/oleylammonium chloride-oleylamine-C12E10/hexadecane cationic system was used instead the anionic system described above. The conditions for obtaining O/W nano-emulsions with a minimum droplet size and consequently low polydispersity by phase inversion emulsification methods (PIT and PIC) can be summarized as follows: A bicontinuous microemulsion or a lamellar liquid crystalline phase (D or L a , respectively), with all the oil dissolved, must be formed immediately before reaching the final twophase region where the nano-emulsions form. These are composition conditions necessary but not sufficient, because the kinetics of incorporation of oil to these phases or the coalescence of droplets can make the nano-emulsion droplet size also dependent on preparation variables such as mixing rate and aqueous phase addition rate for the PIC method, or cooling rate for the PIT method.

Nano-Emulsion Formation by Low-Energy Methods and Functional Properties

21.3

473

FUNCTIONAL CHARACTERISTICS OF NANO-EMULSIONS

The majority of publications on nano-emulsions emphasize their potential applications, comparing them with conventional emulsions (macroemulsions) and microemulsions. With respect to macroemulsions, the main advantage of nano-emulsions would be the smaller droplet size, whereas with respect to microemulsions the main advantage would be the lower amount of surfactant needed for their stabilization. The possibilities of developing applications are mainly limited by stability [2], which is of course related to the structure of nano-emulsions, and can be considered a functional characteristic. On the other hand, the potential applications of nano-emulsions depend on their functional characteristics. In this section, the functional characteristics of nano-emulsions and the relation with their structure are analyzed and discussed in relation to the applications in which these functional characteristics can be important. The section is divided according to the main functional characteristics determining the applications of nano-emulsions: stability, droplet size, and solubilizing capacity.

21.3.1 STABILITY OF NANO-EMULSIONS Nano-emulsions, as nonequilibrium systems, tend to phase separation by some of the four mechanisms of disperse systems destabilization: sedimentation or creaming and flocculation, as reversible mechanisms, and coalescence and Ostwald ripening, as irreversible ones. Nano-emulsions, due to the small characteristic size are stable against sedimentation or creaming. An adequate selection of surfactant molecules can protect nano-emulsions from flocculation and coalescence. In addition, the greater curvature as a result of the smaller droplet size does not favor flocculation or coalescence phenomena. These considerations leave the Ostwald ripening as the main destabilization mechanism of nano-emulsions. This fact has been experimentally confirmed in numerous studies and discussed in several reviews [63,1]. Ostwald ripening results in the enlargement of the greater droplets by incorporating disperse component from the smaller ones. Solubility of the components inside a droplet depends on the curvature radius of the droplet, following the Kelvin equation (Equation 21.2): C(r) = Ce(2gV /RT)/(1/r), d

(21.2)

where C(r) is the solubility as a function of radius, C is the solubility at infinity radius, g is the surface tension, Vd is the molar volume of dispersed component, R is the gas constant, T is the temperature, and r is the radius of a spherical droplet. The concentration gradient between the exterior of the small droplets and that of big droplets causes a net transport toward the big droplets. A model representing this transport for a distribution of droplet size was developed independently by Lifshitz–Slyozov and Wagner in 1961 [64,65], and discussed and used by many authors since then [66,67], although the resulting equation was not always well reproduced. Through this model, a stationary droplet size distribution is predicted (Figure 21.11). In that distribution, an intermediate droplet radius, critical radius, rc, is identified at each moment. Droplets with this radius gain from smaller droplets the same amount of dispersed component that they lose in favor of the big ones. Stationary droplet size distribution described as a function of an adimensional radius r/rc should be invariant with time according to the model. Droplets with radius greater than the critical radius are continuously growing, and the smallest droplets are disappearing when all the dispersed phase is gone. As a result, all distributions are continuously moving toward greater radii with time, and therefore, the critical radius and average radius are also moving to greater values with time. According to the model, the number averaged radius increases with time following the equation 3

8gDCV2 9RT

d(r ) _______d ____ = , dt

(21.3)

474

Structure and Functional Properties of Colloidal Systems 0.10

g(u)

0.08 0.06 0.04 0.02 0.00 0

FIGURE 21.11

0.2

0.4

0.6

0.8 u = r/rc

1

1.2

1.4

1.6

Droplet size distribution g(u) as a function of u r/rc.

where r is the number averaged radius, g is the interfacial tension, D is the dispersed component diffusivity in the continuous phase, C is the solubility at infinity curvature radius, Vd is the molar volume of dispersed component, R is the gas constant and T is the temperature. From this equation, it can be deduced that the variation of size with time decreases inversely to the size squared: dr = __ dr 3 . 1 ___ __ 2 dt

r dt

(21.4)

This fact explains why Oswald ripening is not generally important for macroemulsions, but it is the most important mechanism for destabilizing nano-emulsions and a serious hindrance in the development of nano-emulsion applications. Nano-emulsions with a dispersed phase with an extremely low solubility in the continuous phase can show a good stability with respect to Ostwald ripening [67], while only the relatively great solubility in water of toluene or hexane can make important Ostwald ripening for macroemulsions. A typical destabilizing behavior of nano-emulsions is to show Ostwald ripening as the main destabilization mechanism at the initial stages and coalescence at latter stages (when the droplet size has grown enough), as described by Porras et al. [68]. To arrive at this equation, some restricted assumptions must be made, as for example, – – – –

Solubility follows the Kelvin equation. Stationary droplet size distribution has been reached. The transport is controlled by the diffusion through the continuous phase. Number averaged radius grows equal than the critical radius.

These restricted conditions normally cannot be all met, but there are many occasions in which the main conclusion of the model can be experimentally proved, and the cube of diameter grows linearly with time: d3 = d3t = 0 wt.

(21.5)

The theoretical prediction for the Ostwald ripening rate, w, according to the model would be 64gDCV 2 w = ________d . 9RT

(21.6)

Nevertheless, the theoretical prediction for w is not fulfilled in the majority of emulsions, because there are factors not taken into account in the model. Some of these factors are, the presence of

Nano-Emulsion Formation by Low-Energy Methods and Functional Properties

475

micelles which facilitate transport that favors destabilization, and the barrier effect of the surfactant at the interface that favors stabilization. It is generally admitted that the presence of micelles facilitates the transport between droplets of different size. Although this transport implies that the dispersed component should be transferred from small droplets to micelles, the micelles should move close to the big droplets, and the dispersed component should be transferred from the micelles to the big droplets, consequently the global resistance can be important. The transport by this mechanism takes place in parallel to diffusion through the continuous phase and causes in any case an increased total transport. There are several publications explaining an experimental increase in average droplet size with time faster than that predicted by the LSW model through micelle transport. Experiments carefully carried out by Taylor [21] proved this fact for systems with SDS and pentanol. However, for a different system, Wang et al. [32] have described exactly the contrary effect: the increase in stability is explained by the presence of micelles. According to these authors, the more surfactant the higher the nanoemulsion stability, because the presence of micelles decrease the concentration of diffusing molecules in the continuous phase, C, and according to the LSW model this results in a decrease in w. This reasoning would be feasible if micelle transportation would have such a high resistance that contribution of micelle transportation would become negligible. Concerning the barrier effect, Capek [69], in a review where the destabilization processes of emulsions is extensively analyzed, explains that the presence of a fatty alcohol can form with an ionic surfactant a denser interfacial layer that can increase the resistance to transfer of the dispersed component from small droplets to the continuous phase and from the continuous phase to the big droplets. Stabilization of oil in water emulsions can be increased by the addition of some components of very low water solubility, hydrophobes. When relatively soluble components are transferred from small droplets, their concentration decrease and so does the activity in the small droplets, whereas concentration and activity of soluble components increase in the big droplets. Activities in small and big droplets are becoming equal and consequently Ostwald ripening decreases toward zero values. Hexadecane is extensively used as hydrophobe in order to stabilize emulsions of more soluble hydrocarbons. On the other hand, another mechanism that has been described by Taisne and Cabane [48] is that the transfer of the dispersed component from small to big droplets takes place directly during collision or aggregation of droplets through their interfacial layers, without diffusing through the continuous phase. This would be the predominant mechanism in case that the frequency of collisions or the life time of aggregates would make it easier than the transport through the continuous phase. Avoiding or limiting collision and aggregation would stabilize emulsions with respect to this mechanism. A small amount of ionic surfactant or a nonionic surfactant with longer hydrophilic head can keep apart the droplets avoiding the transfer of dispersed components through the interfacial layers. It may seem that due to the different mechanisms and factors influencing the stability of nanoemulsions it is not possible to know a priori which will be the behavior of a defined system. However, the growing knowledge about a great variety of systems and a wise use of this knowledge would allow the formulation of nano-emulsions with enough stability for the applications intended.

21.3.2 DROPLET SIZE-BASED APPLICATIONS From the earlier studies on nano-emulsion, their potential application for the preparation of nanoparticles with similar sizes as those of the nano-emulsion droplets has been claimed. In an early work about the preparation of nanoparticles from nano-emulsions (referred as miniemulsions), Ugelstadt [3] proposed a polymerization mechanism with nucleation in the miniemulsion droplet, as a different mechanism to that of macroemulsion polymerization, where nucleation takes place mainly in micelles. This mechanism can result in particles as one-to-one copy from nano-emulsion droplets, and this possibility is deeply discussed in a review by Asua [18] (Figure 21.12).

476

Structure and Functional Properties of Colloidal Systems

FIGURE 21.12 Schematic representation of heterophase polymerization processes: (a) Emulsion polymerization, (b) nano-emulsion polymerization, and (c) microemulsion polymerization. (From Antonietti, M. and Landfester, K., Prog. Polym. Sci., 2002, 27, 689–757. With permission.)

The question where the nucleation takes place is basically probabilistic. If there are much more micelles than droplets, the nucleation will take place predominantly in micelles swelled by monomer, where the radicals have more probability to enter than in the monomer droplets. This situation can occur when nano-emulsions are obtained with a limited amount of surfactant: The coverage of the high surface area present in the nano-emulsions does not leave remaining surfactant molecules to form micelles. In contrast, the macroemulsion polymerization with relatively large droplets can imply an excess of surfactant forming numerous swelled micelles, and the nucleation can take place predominantly in the micelles. Once nucleation occur in a droplet, if there is no monomer transport among the droplets or fusion of droplets, the final particle size is determined and limited by the amount of monomer contained in the droplets, and a one-to-one copy can take place. As it has been mentioned above, there exist numerous evidences that the main destabilization mechanism of nano-emulsions is Ostwald ripening, at least at the first stages with the smaller droplets [52,69,63,1]. The transport through the continuous media is determined by the solubility and the activity differences caused by droplet size differences, and this transport can be facilitated by the presence of micelles. The means for avoiding the transport among the droplets are related to the stability of nano-emulsions, and will be described below.

Nano-Emulsion Formation by Low-Energy Methods and Functional Properties

477

(a)

Transdermal delivery (%)

As a conclusion concerning the possibilities of obtaining polymeric nanoparticles with sizes similar to monomer nano-emulsion droplets, two conditions are required: Nucleation in the monomer droplets is desirable and transport among droplets should be minimized. The presence of excess of surfactant forming micelles is detrimental for the fulfillment of the two previous conditions, so the initial nano-emulsion should be formulated with the minimum excess of surfactant. In earlier studies, where nano-emulsions were obtained by high shear, and surfactant concentrations similar to those in macroemulsion polymerization were used, the low excess of surfactant allowed easily the fulfillment of the two conditions. Consequently, polymeric and small-sized nanoparticles were obtained. The use of nano-emulsions obtained by low-energy methods, where higher excess of surfactant is, generally, required, seems unfavorable for this purpose. However, nano-emulsions avoiding a significant transport by micelles can be obtained. Preparation of inorganic nanoparticles using W/O nano-emulsions is in principle easier, given that transfer of inorganic components among the droplets can be easily avoided, especially if inorganic components are ionic. For example, Porras et al. [70] demonstrated that sizes of ceramic particles prepared by hydrolysis of alcoxides in aqueous droplets of a W/O nano-emulsion correspond to the content of hydrolizable alcoxide in each droplet. However, few processes of synthesis of nanoparticles in W/O nano-emulsions are economically feasible due to the high concentrations of surfactants and organic components required, and due to the fact that these components cannot easily be reused. Another type of application of nano-emulsions where the droplet size could be a key factor is their application as delivery systems, provided that delivery of active components would be more efficient than in conventional emulsions, or macroemulsions. It has been reported that droplet size may influence the delivery of active ingredients [71,15], however, in a recent work Izquierdo et al. [72] have shown that the delivery rate of an active ingredient (a local anesthesic) through the skin is not significantly affected by droplet size (Figure 21.13). It should be noted that contrary to most of 100 80 T F

60 40 20 10

100

1000

10000

(b)

Dermal delivery (mg/mg)

r (nm) 0.40 0.30 T F 0.20 0.10 10

100

1000

10000

r (nm)

FIGURE 21.13 Comparison of the variation of (a) transdermal delivery and (b) dermal delivery, with emulsion droplet radius for emulsions with constant overall surfactant concentration (T) and constant surfactant concentration in the aqueous phase (F). (From Izquierdo, P. et al., Skin Pharmacol. Physiol., 2007, 20, 263–270. With permission.)

478

Structure and Functional Properties of Colloidal Systems

the previous studies, in this work emulsions only differed in droplet size as the system components and compositions were the same. The results obtained were interpreted by considering that penetration through the skin is a molecular diffusion-controlled process and the transfer from the droplets to the skin process is not a controlling step of the process. Evidently, there may be other different factors that can have a significant influence if transfer from the droplets is the controlling step of the process. In these cases, droplet size might have an important influence because this transfer takes place through the droplet surface with their area depending on the square diameter. As a partial conclusion of this subsection, influence of droplet size of nano-emulsions on their functional applications would depend on the very specific characteristics of application and on the system used. Consequently, each system should be specifically studied. It may seem that extensive experimental study should be needed for each new application based on droplet size. However, the accumulated knowledge that can be extracted from experimental published work, especially in the recent years, should allow theoretical predictions about the influence of droplet size and savings in experimental work.

21.3.3

SOLUBILIZATION-BASED APPLICATIONS

The capacity of solubilizing compounds in the droplets of a nano-emulsion gives the possibility of developing applications, implying the transport of these compounds through a continuous phase in which they are insoluble. Pharmaceutical and cosmetic applications for which nonpolar compounds are solubilized in oil droplets dispersed in aqueous media would be typical, and other possible applications in food or agrochemical fields have also been described [1,2]. The majority of active ingredients to be solubilized are amphiphilic and relatively big molecules, so their solubility must be studied for developing the corresponding nano-emulsion applications. In reference [26], lidocaine solubility in Cremophor EL and Mygliol 812 systems was determined. Solubilities in nano-emulsions resulted to be higher than those predicted from individual solubility in the surfactant Cremophor EL or in the oil Mygliol 812. While solubility in water is practically negligible, in Cremophor EL it is 24.5% and in Mygliol 812 30.5%. The global solubilities in nano-emulsions of 90%or 95% are doubled when O/S ratio goes from 0 : 100 to 40 : 60 (Figure 21.14), showing a synergistic effect between surfactant and oil and a favorable effect of the presence of small droplets.

Maximum drug solubilized

4.0

90 wt% water 95 wt% water

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0/100

10/90

20/80

30/70

40/60

Oil/surfactant ratio

FIGURE 21.14 Maximum concentration of lidocaine solubilized in nano-emulsions as a function of O/S ratio at 25ºC. (From Sadurní, N. et al., Eur. J. Pharm. Sci., 2005, 26, 438–445. With permission.)

Nano-Emulsion Formation by Low-Energy Methods and Functional Properties

479

If only the solubility in individual components is taken into account, the possible application of 90% or 95% nano-emulsions would be discarded because it would seem that therapeutic concentrations could not be reached. However, the experimental results show that even in diluted nanoemulsion such as those with 95% water, a concentration of 2.1% active ingredient, within the therapeutic range, can be reached. In a different system, Shakeel et al. [73] showed that the solubilizing capacity is a very critical factor because the active ingredient, aceclofenac, is a very insoluble compound. In this work, studies of solubility in a wide variety of oils and surfactants are described, and the best system was a mixture of oils. Oils and surfactants are selected with the solubility as main criteria. As a main conclusion of this subsection, and from the described examples, the convenience of specific study of the solubility behavior for each system or possible application is pointed out. The experimental work to be made seems enormous given the variety of oils, surfactants, and experimental conditions. However, one approach would be to first focus on systems for which previous information can be obtained, and if selection is made taking into account wise criteria the amount of work could be considerably reduced.

21.4

SUMMARY

Nano-emulsions with minimum droplet size (in the range 20–200 nm) and low polydispersity can be obtained by condensation or low-energy methods. Nano-emulsion formation by the PIT and the PIC methods have been related to the phase behavior of the corresponding emulsion systems. Phase transitions through bicontinuous microemulsion or lamellar liquid crystalline phases (i.e., surfactant aggregates with zero curvature) favor the formation of nano-emulsions with minimum droplet size. Spontaneous emulsification methods, by which emulsification is achieved by a dilution process without phase transitions taking place in the system, have also been described. The main destabilization mechanism for nano-emulsions, Ostwald ripening, and the key factors to prevent nano-emulsion breakdown, for example, addition of a more insoluble oil, and an appropriate interfacial barrier, have been discussed. An analysis of nano-emulsion functional characteristics and the relation with their structure have been discussed with regard to applications.

ACKNOWLEDGMENTS The authors acknowledge financial support by the Spanish Ministry of Science and Education (projects CTQ2005-09063-C03-01 and CTQ2005-09063-C03-02) and Generalitat de Catalunya (Grant 2005-SGR-00812).

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Index A

B

Active targeting, 398 controlled drug delivery, 397 Additive and the multiplicative Vineyard-like approximation, 12 Adsorbed polymer configuration, 267 Adsorption isotherms, 296 Aggregation rate constant, 301 vs. degree of surface coverage, 306 AIBN. See Azobis isobutyronitrile (AIBN) Air-liquid interface, 215 Alpha-relaxation, 101, 103 Amphiphilic molecules, 221 Amplitude sweep curves, 252 Anionic core-shell nanogels, 375 AO. See Asakura-Oosawa (AO) mixture Aqueous phase surface coverage and pH, 298 Arrested phase separation, 150 Arrhenius equation, 131 Artificial swimmers, 118 Asakura-Oosawa (AO) mixture, 25, 144 arrested phase separation, 149, 150 attractive glass, 156 density correlation function, 160 glass transition phase diagram, 26 MSD, 152, 153 squared displacement, 156 Atom transfer radical polymerization (ATRP), 265, 374 latex particles surface functionalization, 278 ATRP. See Atom transfer radical polymerization (ATRP) Attraction-driven glasses theoretical predictions, 141–147 Attractive glasses, 148–158 attraction-driven vitrification, 149–150 attractive glasses and gels simulations, 158, 159 attractive glasses at low density, 158 attractive glasses simulations, 151–154 dynamical heterogeneities, 155 extension to low densities, 159 gelation, 158 gels and attractive glasses, 159 high-order singularities, 156–157 liquid-gas separation, 149–150 low density, 158 MCT, 154 simulations, 151–154 SW system, 152 Average diffusion coefficient, 309 Azidothymidine-triphosphate (AZT-TP) drug loading, 375 Azobis isobutyronitrile (AIBN), 272–273 AZT-TP. See Azidothymidine-triphosphate (AZT-TP) drug loading

Balance condition, 37 BARC. See Bead Array Counter (BARC) Bare particles aggregating at electrolytes, 296 average diffusion coefficient, 309 electrolyte concentrations, 297 fractal dimensions, 296 structural and kinetic parameters, 309 BDMB. See Benzyl-2-dimethylamino-4 morpholinobutyrophenone (BDMB) Bead Array Counter (BARC), 329 tabletop prototype, 330 Benzyl-2-dimethylamino-4 morpholinobutyrophenone (BDMB), 421 Berens and Hopfenberg model, 407 Best fits parameters, 310 BET. See Brunauer–Emmett–Teller (BET) isotherms Beta-casein, 223 dilatational elasticity, 231 foams, 223, 229–231 foams with interfacial studies, 226 interfacial properties, 232 interfacial rheology, 231 rheological properties, 224 surface tensions, 223, 224 tetradecane emulsions, 230 thin film thickness, 224 Biaxial steady elongation, 239 Biaxial stretching flow, 238 Bicomponent model, 25 Bidisperse suspension permittivity, 95–101 bimodal systems permeability experimental data, 98–101 dielectric relaxation, 97 Bidisperse systems bimodal suspensions dynamic mobility, 103–104 SCGLE, 17 Bimodal magnetic fluids colloidal stability, 110–111 Bimodal suspensions dynamic mobility, 102–106 bidisperse systems, 103–104 particle size, 102 Bimodal suspensions magnetic properties, 109 Bimodal suspensions structure, 93–113 bidisperse suspension permittivity, 95–101 bidisperse systems, 103–104 bimodal MF colloidal stability, 110–111 bimodal suspensions dynamic mobility, 102–106 bimodal suspensions magnetic properties, 109 bimodal systems permeability experimental data, 98–101 dielectric relaxation, 97 magnetic particles, 107–111 particle size, 102

483

484 Bimodal systems permeability experimental data, 98–101 Binary hard sphere mixtures dynamic arrest phase diagram, 24 nonadditive diameters, 25 phase diagram, 167 Biochemical detection, 326 Biocompatibility of micro- and nanogels, 373 Biodegradable micro- and nanogels, 374–375 Biological compounds adsorption to hydrophobic surfaces, 268 Biomedical diagnostic applications, 324 Bionanotechnology, 325 Biotin-avidin or DNA-peptide nucleic acid interactions, 369 Bis(2-ethylhexyl) sulfosuccinate, 345 Bjerrum length, 188 Blobs, 203 Body-centered structure, 185–186 Boltzmann constant, 185, 243 Boltzmann exponential expression, 66 Bovine serum albumin (BSA), 294, 369 adsorption isotherms, 296 aggregation rate constant vs. degree of surface coverage, 306 fractal dimension vs. protein surface coverage, 303 particle clusters, 299 Bovine spinal cord phosphatidylserine, 78 Bragg reflection, 417, 418 Bridges, 215 Brownian dynamics, 20 simulations, 13 Brownian motion, 165 and diffusion, 12 individual particles, 13 isolated colloidal particle, 6 vs. magnetic dipole-diploe interaction, 123 Brunauer–Emmett–Teller (BET) isotherms, 265 BSA. See Bovine serum albumin (BSA) Bulk shaken-emulsion, 440

C Cable dampers, 165 Calculated interface tension, 206 Calculated interface width, 207 Capillary charge, 37 Capillary flows, 249 microchannels transport, 432 Capillary forces, 31–62 between colloidal particles at fluid interfaces, 31–62 curved interface small deformation, 50–53 2D colloids at fluid interfaces, 33 flat interface small deformations, 36–49 fluid interface deformation and capillary forces, 34–53 interface under pressure field effect, 46–49 measurement, 36 monopole, 42, 51 pressure-free interface, 38–47 quadrupole, 51–52 spherical charged particle, 53 Capillary polarization, 39 Capillary potential, 37 Capillary rheometer, 251 Capillary torque, 39, 47

Index Cationic polyethylenimine nanogels, 369 Cauchy-Maxwell equation, 240 Cauchy’s equation, 236 CC. See Colloidal crystals (CC) CCD. See Charge coupled device (CCD) spectrometer CC-L. See Colloidal crystals laser (CC-L) device Cell-penetrating peptides (CPP), 399 Cesium chlorine structure, 181 Cetyltrimethylammonium bromide (CTAB), 345 Chain diffusion, 124 Chain transfer agents (CTA), 279 Charge coupled device (CCD) spectrometer, 422 Charged particles binary mixture, 16–17 dispersions, 23 Charged vesicles in aqueous suspensions effective interactions, 77–91 dynamics, 86–88 effective interactions potentials, 82–85 energy content, 89 experimental system and methods, 78 link between static and dynamic properties, 89 results, 79–90 structural hallmarks, 79–81 Charge inversion, 64 Charge reversal, 64 Chemical cross-linking, 369 Chemical potential, 195 Chitosan ethylene diamine tetraacetic acid nanogel, 381 Cholesteryl-pullulan (CHP), 369 CHP. See Cholesteryl-pullulan (CHP) Closure relations, 12 Cluster growth monitoring, 291–293 longtime behavior, 292 short-time behavior, 293 Coated particles, 309 Coions, 96 Collective diffusion, 11 Colloidal crystals (CC), 416 devices fabrication, 420 devices properties, 421–424 fabrication, 416 film fabrication procedure, 418 film reflection spectra, 419 film SEM images, 419 laser applications, 416 laser device, 421 PDMS elastomer, 418, 419 photonic crystals, 416 properties, 417–419 PS, 419 structures manipulation by confinement, 190–195 structures manipulation by electric field, 184–189 Colloidal crystals laser (CC-L) device, 420, 421 laser action, 424 reflection spectrum, 422 Colloidal electric double layers ionic structures, 63–73 divalent counterion, 69 electrolytes mixture, 72–73 multicomponent system HNC/MSA equations, 65 results, 69–72 theoretical background, 65–68 theory-experiment relationship, 66–68 trivalent counterion, 70–71

Index Colloidal fluids diffusion in random porous media, 18–19 Colloidal length scales, 45 Colloidal micro- and nanogels differentiation from other drug carriers, 367–368 Colloidal particles effective attractive interactions, 25 thin liquid film, 34 Colloidal photonic crystals devices fabrication, 420 devices properties, 421–424 fabrication, 416 and laser applications, 415–425 properties, 417–419 Colloidal properties, 93–113 bidisperse suspension permittivity, 95–101 bimodal suspensions dynamic mobility, 102–106 magnetic particles, 107–111 Colloidal suspensions in bulk, 165–196 colloidal crystal structures manipulation by confinement, 190–195 colloidal crystal structures manipulation by electric field, 184–189 confinement and external fields phase behavior and structure, 165–196 DLVO theory, 168–169 external fields colloids, 184–195 hard-core Yukawa particles phase behavior, 170–175 large and small hard spheres binary mixtures, 167 oppositely charged colloids with size ratio 1, 180–183 oppositely charged colloids with size ratio .32 crystal structures, 175–179 primitive model, 168 steric and charge-stabilized colloids, 166–183 Colloidal systems glasses, 135–161 attractive glasses and gels simulations, 148–158 attractive glasses at low density, 158 attractive glasses simulations, 151–154 dynamical heterogeneities, 155 equilibrium structure factor, 142–143 gelation, 158 gels and attractive glasses, 159 high-order singularities, 156–157 input to MCT, 142–143 liquid-gas separation competition, 149–150 MCT of bond formation, 147 MCT of repulsion- and attraction-driven vitrification, 144–146 model and simulation details, 140 repulsion- and attraction-driven glasses theoretical predictions, 141–147 results, 141–159 theory, 138–139 Colloid dynamics and transitions to dynamically arrested states, 3–30 approximate self-consistent theory elements, 12 charged particles binary mixture, 16–17 charged particles dispersions, 23 closure relations, 12 collective diffusion, 11 colloidal fluids diffusion in random porous media, 18–19 colloid-polymer mixtures, 25 exact expression, 8–9 exact memory-function expressions, 11

485 extension to colloidal mixtures, 15–17 generalized Langevin equation, 7 GLE formalism, 5–12 GLE for tracer diffusion, 8–10 hard- and soft-sphere systems, 21–22 hard spheres mixtures, 24 model mono-disperse suspensions, 13–14 multicomponent systems, 23–25 nonergodicity parameters, 20 ordinary and generalized Langevin equation, 5–7 ordinary Langevin equation, 6 Ornstein-Uhlenbeck processes, 6 perspectives, 26 SCGLE theory of dynamic arrest transitions, 20–25 self-consistent theory and illustrative applications, 13–19 simplified SCGLE theory, 18 simplifying approximations for DzT (t), 10 systems comparison with experimental data, 21–22 three-dimensional soft sphere systems, 14 time-dependent friction function, 8–9 two-dimensional model system with dipole-dipole (r -3) interaction, 14 vineyard-like approximations, 12 Colloid-polymer mixtures bicomponent model, 25 experimental phase diagrams, 137 SCGLE theory of dynamic arrest transitions, 25 Colloids computer simulations, 166 examples, 165 optical properties, 165 Colloid vibration current (CVI), 94 Colloid vibration potential (CVP), 94 Composite micro- and nanogels, 376–377 Con A. See Concanavalin A (Con A) Concanavalin A (Con A), 331 Concentric cylinders, 251 Condensation or low energy emulsification methods, 458–472 phase inversion composition method, 465–472 phase inversion temperature method, 460–465 self-emulsification, 459 Cone-plate, 251 Confocal microscopy, 183 Contact line, 39 Continuity equation, 236 Continuous wave (CW) excited laser devices, 424 Continuum mechanics, 238 Contrast agents, 119 Controlled drug delivery, 395–398 intracellular delivery and subcellular distribution, 398 passive and active targeting, 397 Controlled radical polymerizations (CRP), 265 latex particles surface functionalization, 277–278 Controlled release systems, 388 Copper gold crystallites, 183 structure, 181 Core-shell and multilayer micro- and nanogels, 378–379 Core-shell nanogels, 379 Coulomb interactions, 147 Coulomb particles, 181 Coulomb phase diagram, 182, 183

486 Counterions (cations), 97 fluxes, 96 Covalent drug binding, 375 CPP. See Cell-penetrating peptides (CPP) CRP. See Controlled radical polymerizations (CRP) CTA. See Chain transfer agents (CTA) CTAB. See Cetyltrimethylammonium bromide (CTAB) Curved interface, 51 small deformation, 50–53 CVI. See Colloid vibration current (CVI) CVP. See Colloid vibration potential (CVP) CW. See Continuous wave (CW) excited laser devices

D DAP. See Diallyl phthalate (DAP) DDS. See Drug delivery systems (DDS) Debye–Hückel parameter, 67 Debye screening length, 185 Debye-Waller factor, 90 Degree of desorption, 207 Density correlation functions logarithmic decay, 158 self part, 150 Deoxyribonucleic acid (DNA) sequencing, 430 Derjaguin–Landau–Verwey–Overbeek (DLVO), 167 interaction energy, 300 particle-particle interaction potential, 304 potential, 147 steric and charge-stabilized colloids, 168–169 theory, 77, 175, 303 Dermal delivery, 477 Dextran and pork skin gelatin, 202 Diallyl phthalate (DAP), 391 microgels, 392 Dielectric increment, 97 calculated values at zero frequency, 101 frequency, 98 zero frequency, 101 Dielectric relaxation, 97 Diethylphosphono-dimethylpropyl nitroxide, 278 Diffusion coefficient, 309 Diffusion-limited cluster aggregation (DLCA), 290, 292, 297, 305 Diffusive filament motion, 122–124 Dilatational elasticity, 231 Dilatational elastic modulus, 220 Dimethylacrylamide (DMA), 278 Dimethylamino ethyl acrylate (DMAEA), 277 Dimethylaminoethyl methacrylate (DMAE-MAA), 378 Dipolar hard-sphere particles, 187 Dipolar interaction, 117–118, 119 Dipolar soft-sphere particles, 188 Direct emulsification, 458 Dispersed systems structured fluids mechanical properties theories, 242–248 structured fluids rheological models, 255–257 Dispersion pH, 392 Divalent counterion, 69 Divinylbenzene (DVB), 320, 390 DLCA. See Diffusion-limited cluster aggregation (DLCA) DLS. See Dynamic light scattering (DLS)

Index DLVO. See Derjaguin–Landau–Verwey–Overbeek (DLVO) DMA. See Dimethylacrylamide (DMA) DMAEA. See Dimethylamino ethyl acrylate (DMAEA) DMAE-MAA. See Dimethylaminoethyl methacrylate (DMAE-MAA) DNA. See Deoxyribonucleic acid (DNA) sequencing Donnan potentials, 202 Double emulsions, 248–249 core-shell structure, 247 DOX. See Doxorubicin (DOX) Doxorubicin (DOX), 371 Droplet based microfluidics, 429–443, 435 capillary flows, 432 lab-on-a-chip, 437 mesoporous particle synthesis, 438–440 microchannels droplet formation, 434–436 microchannels electrokinetic transport, 433 microchannels transport, 431–436 microfluidic device fabrication methods, 430 motivation for working with microfluidics, 429 pressure-driven flows, 431 Droplet diameter, 467 Droplet formation, 436 Droplet size distribution, 474 function of voltage, 438 Drug concentration vs. time profile, 396 Drug delivery applications, 387–408 Drug delivery systems (DDS), 367–368 design criteria, 388 MAA-EA nanogels, 408 Drug delivery vehicles, 400–404 Drug ion selective electrode, 399 Drug loading, 373 micro- and nanogels properties, 372 Drug release, 408 pH, 402 Drug selective electrode, 399, 408 Drug vectorization, 119 DVB. See Divinylbenzene (DVB) Dynamic arrest phase diagram, 24 Dynamic electrophoretic mobility, 103 Dynamic light scattering, 393 Dynamic light scattering (DLS), 123, 128, 294, 372, 460 Dynamic mobility, 105 frequency dependence, 108 spherical particles suspensions, 104 suspensions, 106, 107

E EDL. See Electric double layer (EDL) EDLMP. See Electric double-layered magnetic particles (EDLMP) EDTA. See Ethylene diamine tetraacetic acid (EDTA) nanogel Effective interaction potential, 47 charged vesicles, 82–85 estimated time, 84 Egg phosphatidylcholine, 78 Einstein crystal, 171 Einstein equation, 243 Elastic modules, 90

Index Elastic stresses, 31 Electrically charged two dimensional colloid, 48 Electric double layer (EDL), 63, 95, 433–434 Electric double-layered magnetic particles (EDLMP), 120 Electric field cessation, 455 Electric forces, 31 Electrochemical detection, 327 Electrokinetic sonic amplitude (ESA) technique, 105 Electrolyte concentrations, 297 mixture, 72–73 primitive model (PM), 64 Electromotive force (EMF), 341 Electro-optical coatings, 448 Electrophoresis, 295 Electrophoretic mobility, 69, 71, 73 Electrorheological (ER) fluids, 119, 184 Electrostatic analogy, 37 Elongational creep, 239 Elongational stress growth, 239 EMF. See Electromotive force (EMF) Emulsification, 322 Emulsions copolymerization, 270–272 droplet size, 462 examined with interfacial studies, 222–231 HLB temperature phase behavior, 462 homopolymerization, 270–272 interfacial phenomena, 219–232 polymerization, 390, 476 stabilization, 229–231 Emulsions stability foams and emulsions interfacial phenomena, 221 Energy content, 89 Enhanced permeability and retention (EPR) effect, 397 EPR. See Enhanced permeability and retention (EPR) effect Equation of conservation of linear momentum, 236 Equation of conservation of mass, 236 Equation of motion, 236 Equilibrium phase diagram, 192 ER. See Electrorheological (ER) fluids ESA. See Electrokinetic sonic amplitude (ESA) technique Ethylene diamine tetraacetic acid (EDTA) nanogel, 381 Euler equation, 236 Evaporation, 322 Ewald summation method, 181 Extension to colloidal mixtures, 15–17 charged particles binary mixture, 16–17 External fields colloids, 184–195 colloidal crystal structures manipulation by confinement, 190–195 colloidal crystal structures manipulation by electric field, 184–189 Extremely bimodal suspensions structure and colloidal properties, 93–113

F Factorized K-BKZ model, 242 Factorized Rivlin-Sawyers model, 241 Ferrofluid (FF), 107, 317

487 FF. See Ferrofluid (FF) Field autocorrelation function, 86 half-logarithmic, 87 Filaments aggregates, 131 fractal dimension, 127 magnetic properties, 121 Fish gelatin-rich and dextran-rich solution interface confocal microscopy, 211 water-water interfaces, 215 Fitting parameters, 310 Flat interface small deformations, 36–49 Flow curves, 251 Fluid vertical and horizontal capillaries, 433 Fluid interface deformation and capillary forces, 34–53 curved interface small deformation, 50–53 flat interface small deformations, 36–49 interface under pressure field effect, 46–49 monopole, 51 pressure-free interface, 38–47 quadrupole, 51–52 spherical charged particle, 53 Fluid-solid coexistence chemical potential, 195 transition, 194 Fluorescence resonance energy transfer (FRET) based DNA assay, 442 Fluorescent polystyrene (FPS), 276 Fluxes coions, 96 counterions, 96 Foams, 221 beta-casein, 223, 226, 229–231 and emulsions stabilization, 229–231 examined with interfacial studies, 222–231 film thickness, 228 half-life beta-casein, 230 interfacial rheology, 231 protein stabilizing, 225–228 rheology, 223–224 stability, 223, 225 surface tension, 223–224 surfactant stabilizing, 225–228 thin liquid films, 223–224 Tween 20 system, 226, 228 whole casein, 222–224, 227–228 Foams interfacial phenomena, 219–232 beta-casein foams and emulsions stabilization, 229–231 beta-casein interfacial rheology, 231 beta-casein/Tween 20 system, 226 foams and emulsions examined with interfacial studies, 222–231 foams and emulsions stability, 221 foam stability, 223, 225 interfacial tension, interfacial rheology and thin liquid films, 220 protein and surfactant stabilizing foams, 225–228 surface tension, surface theology and thin liquid films, 223–224 whole casein and beta-casein stabilizing foams, 222–224 whole casein/Tween 20 system, 227–228

488 Fourier transform (FT), 4 FPS. See Fluorescent polystyrene (FPS) Fractal dimensions bare particles aggregating at electrolytes, 296 vs. protein surface coverage, 303 Fractal structures scattering functions, 291 Frenkel–Ladd method, 171, 186 Frequency sweep curves, 252–253 FRET. See Fluorescence resonance energy transfer (FRET) based DNA assay FT. See Fourier transform (FT) Functionalized particles, 309

G GC. See Gouy-Chapman (GC) model GDLC. See Gel–glass dispersed liquid crystals (GDLC) Gelation, 136, 390 attractive glasses, 158 Gel-gel matrices, 452 Gel–glass dispersed liquid crystals (GDLC), 448, 449 devices, 448 devices electro-optical properties, 447–455 devices performance, 451–454 dynamic response, 454 electro-optical coatings, 448 formation, 450 light transmittance, 453 performance, 451–454 phase separation, 449 sol-gel preparation, 449 Gels simulations, 148–158 and attractive glasses, 159 dynamical heterogeneities, 155 gelation, 158 gels and attractive glasses, 159 high-order singularities, 156–157 liquid-gas separation competition with attractiondriven vitrification, 149–150 Generalized Langevin equation (GLE), 4–12 exact expression, 8–9 generalized Langevin equation, 7 ordinary and generalized Langevin equation, 5–7 ordinary Langevin equation, 6 Ornstein-Uhlenbeck processes, 6 simplifying approximations, 10 time-dependent friction function, 8–9 tracer diffusion, 8–10 Genomics, 387 GFP. See Green fluorescent protein (GFP) Giant magnetoresistive (GMR) type multilayer, 328 sensor, 329 Gibbs–Duhem integration method, 171 Gibbs free energy, 167 Gibbs–Marangoni mechanism, 225 Giesekus model, 241, 255 Glass transition lines, 145 AO binary mixture, 26 GLE. See Generalized Langevin equation (GLE) GMR. See Giant magnetoresistive (GMR) type multilayer Gold composite thermosensitive nanogels, 378 crystallites, 183

Index nanogel-based nanoshells, 376 nanorods, 345 structure, 181 Gouy-Chapman (GC) model, 63 Green fluorescent protein (GFP), 442 Ground-state phase diagram, 177, 178

H Hard-core Yukawa model, 172 particles phase behavior, 170–175 phase diagram, 172, 173, 174 Hard spheres (HS) systems, 21–22, 135, 191 binary mixtures, 167 equilibrium phase diagram, 192 mixtures, 24 SCGLE collective correlator, 22 stable solid structures, 193 Helmholtz free energy, 167, 171 Helmholtz–Smoluchowski (HS) equation, 67 HEMA. See Hydroxyethylmethacrylate (HEMA) Heterofunctional polymeric peroxides (HFPP), 275 Heterophase polymerization, 476 HexaDecylPhosphonic acid-MP, 318 HFPP. See Heterofunctional polymeric peroxides (HFPP) HHM-HEC. See Hydrophobically hydrophilically modified hydroxyethylcellulose (HHM-HEC) High-order singularities, 156–157 HLB. See Hydrophile–lipophile balance (HLB) HM-EHEC. See Hydrophobically modified ethyl(hydroxyethyl)cellulose (HM-EHEC) HMEM. See Hydroxy-2-methylpropiophenone ethylene glycol-methacrylate (HMEM) HNC. See Hypernetted-chain (HNC) Homogeneous field, 41 Horizontal capillaries, 433 HS. See Hard spheres (HS) systems; Helmholtz– Smoluchowski (HS) equation Hydrophile–lipophile balance (HLB), 459 emulsion droplet size, 462 phase equilibria, 464 PIT method, 461 temperature, 462, 463, 464–465 W/O nano-emulsion, 462 Hydrophilic micro- and nanogel networks, 369 chemical cross-linking, 369 molecular association, 369 surface grafting and vectorization, 370 Hydrophobically hydrophilically modified hydroxyethylcellulose (HHM-HEC), 268 Hydrophobically modified ethyl(hydroxyethyl)cellulose (HM-EHEC), 268 Hydrophobic and anionic anticancer drugs core-shell nanogels, 379 Hydroxyethylmethacrylate (HEMA), 272, 375 Hydroxy-2-methylpropiophenone ethylene glycol-methacrylate (HMEM), 277 Hypernetted-chain (HNC), 64, 67 colloidal electric double layers ionic structures, 65 Hyperthermia, 119

489

Index I IMI. See Imipramine hydrochloride (IMI) Imipramine hydrochloride (IMI), 401 Immunomagnetic separation, 325 Inherent viscosity, 239 Inorganic matrix embedding, 317 Interaction potential, 141 Interface confocal microscopy, 211 equilibrium shape, 211 permeability, 208–209, 212–213 polymer depleted, 207 under pressure field effect, 46–49 structural information, 215 tension, 202–204, 210–211 total polymer concentration, 206, 209 water-water interfaces, 207, 208–209, 212–213 Interfacial deformation isolated quadrupole, 44 substrate liquid film configuration, 42 Interfacial properties, 232 Interfacial rheology, 220 Interfacial tension, 220 Intermediate scattering function magnetic particles, 14 soft sphere system, 15 Intermediate structures, 194 Intracellular and subcellular distribution, 398 Intracellular delivery and subcellular distribution, 398 Intrinsic viscosity, 239 limiting values, 244 Inverse emulsion, 370 Inversion protocol, 82 Inviscid flow approximation, 237 Ionic clouds, 102 Iron oxide-based magnetic nanoparticles synthesis, 316–322 Irreducible memory functions, 12 SEXP, 18 Isodiffusitivity lines, 151 Isoelectric point, 393–394 Isolated colloidal particle, 6 Isolated quadrupole interfacial deformation, 44 Isononanoate system, 462 Isopropylacrylamide microbeads, 320

J Jeffrey’s model, 240

K K-BKZ model, 255 Kelvin equation, 473, 474 Kelvin-Voigt model, 255 Kinetic viscosity, 239

L Lab-on-a-chip, 437 Lamb–Gromeka equation, 236 La Mer model, 293 Laplace pressure, 208–209, 214

Large hard spheres binary mixtures, 167 Large particles (LP), 94 surrounded by small particles, 102 Laser applications, 415–425 Laser confocal microscopy, 166 Latex particle colloidal stability modification, 265–267 Latex particles physical surface functionalization, 265–268 posttreatment, 268 Latex particles surface functionalization, 263–280 atom transfer radical polymerization, 278 attaching from by conventional RP, 277 attaching from by CRP, 277–278 attaching from surface functionalization, 277–278 attaching to surface functionalization, 270–276 avoiding biological compounds adsorption to hydrophobic surfaces, 268 chemistry for surface functionalization, 276 emulsion homopolymerization, emulsion copolymerization, 270–272 latex particle colloidal stability modification, 265–267 latex particles physical surface functionalization posttreatment, 268 nitroxide-mediated radical polymerization, 277 physical adsorption, 265–269 preformed latexes surface modification, 275 reversible addition-fragmentation chain transfer polymerization, 279 seeded emulsion copolymerization, 273–274 Layer-by-layer (LbL) assembly, 269 Layer by layer (LBL) technique, 403 LBL. See Layer-by-layer (LbL) assembly; Layer by layer (LBL) technique LCD. See Liquid crystal device (LCD) LFDD. See Low-frequency dielectric dispersion (LFDD) Lidocaine nano-emulsions, 478 Light scattering, 125, 165 Light transmittance, 453 Limited valance model, 155 Linear patch, 53 Linear superposition approximation (LSA), 175 Link between static and dynamic properties, 89 Liquid crystal device (LCD), 448 Liquid crystal droplets optical microscopy, 452 Liquid foam stabilization, 221 Living radical polymerizations, 265 Lodge equation, 240 Logarithmic derivative of relative permittivity, 100 London–van der Waals force, 119 London–van der Waals potentials, 303 Low-frequency dielectric dispersion (LFDD), 94 LP. See Large particles (LP) LSA. See Linear superposition approximation (LSA) Lycurgus Chalice, 340

M MAA. See Methacrylic acid (MAA) MAA-EA. See Methacrylic acidethyl acrylate (MAA-EA) Madelung energy, 176, 177, 179 Maeda–Fujime model, 128 Magnetically induced chemical analysis (MICA), 316, 332 immunoassay, 332

490 Magnetic-based particles biomedical applications, 323–332 conventional biomedical diagnostic applications, 324 diagnostics, 323 magnetic gradient as dipstick like approach, 332 magnetic nanoparticles in MRI, 323 magnetic particles as detection labels, 327–331 magnetic particles in biosensing devices, 326 magnetic particles in vitro applications, 324 magnetic separation-based microsystems, 324–325 magnetic transducer, 331 magnetoresistive detectors, 327–330 microfluidic-based systems magnetic particles, 324–326 Magnetic colloids cable dampers, 165 contrast agents, 119 drug vectorization, 119 hyperthermia, 119 utilization, 119 Magnetic colloids biomedical applications preparation advances, 315–333 conventional biomedical diagnostic applications, 324 inorganic matrix embedding, 317 iron oxide-based magnetic nanoparticles synthesis, 316–322 magnetic-based particles biomedical applications, 323–332 magnetic gradient as dipstick like approach, 332 magnetic latex particles from preformed polymers, 321–322 magnetic latex particles via polymerization in dispersed media, 319–320 magnetic nanoparticles in MRI, 323 magnetic nanoparticles modifications advancement, 316 magnetic particles advancement state of the art, 316–322 magnetic particles as detection labels, 327–331 magnetic particles in biosensing devices, 326 magnetic particles in vitro applications, 324 magnetic separation-based microsystems, 324–325 magnetic transducer, 331 magnetoresistive detectors, 327–330 microfluidic-based systems magnetic particles, 324–326 organic matrix embedding, 317–322 in vivo diagnostics, 323 Magnetic dipole-dipole interaction, 119 vs. Brownian motion, 123 Magnetic filament preparation, 120–124 diffusive filament motion, 122–124 filament magnetic properties, 121 Magnetic filament stability, 128–130 Magnetic filament structure, 125–127 Magnetic fluids (MF), 107 bimodal colloidal stability, 110–111 Magnetic forces, 31 Magnetic latex particles emulsification and evaporation, 322 morphologies, 319 polymerization in dispersed media, 319–320 preformed polymers, 322 from preformed polymers, 321–322 TEM, 321 Magnetic nanoparticles modifications advancement, 316 MRI, 323

Index Magnetic particles, 107–111 applications, 324 bimodal MF colloidal stability, 110–111 bimodal suspensions magnetic properties, 109 biosensing devices, 326 detection labels, 327–331 intermediate scattering function, 14 magnetization curves, 109 videomicroscopy, 124 Magnetic particles advancement state of art, 316–322 inorganic matrix embedding, 317 iron oxide-based magnetic nanoparticles synthesis, 316–322 nanoparticles modifications advancement, 316 organic matrix embedding, 317–322 from preformed polymers, 321–322 via polymerization in dispersed media, 319–320 Magnetic polystyrene particles, 118 Magnetic resonance imaging (MRI), 119, 321 Magnetic separation-based microsystems, 324–325 Magnetic transducer, 331 Magnetization curves, 109 Magnetization of suspensions, 110 Magneto-binding assay, 331 Magnetoresistive biosensor, 330 Magnetoresistive detectors, 327–330 Magnetorheological fluid (MRF), 107 Magnetorheological (MR) fluids, 184 Maxwell model, 255 Maxwell’s stress tensor, 37 Maxwell–Wagner effect, 105 Maxwell–Wagner– O’Konski (MWO) relaxation frequency, 97, 103 MC. See Monte Carlo (MC) simulations MCT. See Mode coupling theory (MCT) MEA. See Methoxyethyl acrylamide (MEA) Mean force, 36 Mean-spherical approximation (MSA), 64, 67 Mean-squared displacement (MSD), 142 AO system, 152, 153 experimental intermediate scattering function, 136 time dependence, 88 Memory-function expressions, 11 Mesoporous particle synthesis, 438–440 Messenger ribonucleic acid (mRNA), 325 Metal ions chemical reduction, 341 Metallic nanoparticles chemical synthesis aqueous medium, 340–346 metal ions chemical reduction, 341 nucleation and growth, 342–346 seed nucleation and particle size distribution, 344 soft templates and selective adsorption for shape control, 345–346 Metallic nanoparticles colloidal dispersion formation, 339–352 arrays, 348–351 catalysis, 347 collective properties, 347 colloidal dispersion, 347 functional properties, 347–351 metal ions chemical reduction, 341 metallic NPs chemical synthesis in aqueous medium, 340–346 nucleation and growth, 342–346

491

Index seed nucleation and particle size distribution, 344 sensors, 349–350 SERS, 349 soft templates and selective adsorption for shape control, 345–346 surface plasmon resonance, 347 thermal properties, 351 Metallic nanoparticles formation, 343 Methacrylic acid (MAA), 373 Methacrylic acidethyl acrylate (MAA-EA), 391 drug delivery, 408 nanogels, 402–403 Methoxyethyl acrylamide (MEA), 277 Methyl methacrylate (MMA), 278 MF. See Magnetic fluids (MF) MFFD. See Microfluidic flow focusing device (MFFD) MICA. See Magnetically induced chemical analysis (MICA) Micelle transport, 475 Microchannels droplet formation, 434–436 Microchannels electrokinetic transport, 433 Microchannels transport, 431–436 capillary flows, 432 lab-on-a-chip, 437 microchannels droplet formation, 434–436 microchannels electrokinetic transport, 433 pressure-driven flows, 431 Microemulsion polymerization, 476 Microfluidic-based systems magnetic particles, 324–326 Microfluidic device fabrication methods, 430 optical microscopy of aqueous droplets, 440 Microfluidic flow focusing device (MFFD), 436, 438–439 Microfluidic system, 326 Microfluids, 429 Microgels DAP, 392 drug delivery vehicles, 400–404 solution mobility, 394 Microgels hydrophilic colloidal networks in drug delivery and diagnostics, 367–381 biocompatibility, 373 biodegradable micro- and nanogels, 374–375 chemical cross-linking, 369 colloidal micro- and nanogels differentiation from other drug carriers, 367–368 composite micro- and nanogels, 376–377 core-shell and multilayer micro- and nanogels, 378–379 drug loading, 372 hydrophilic micro- and nanogel networks, 369 micro- and nanogels in drug delivery, 374–380 micro- and nanogels properties, 371–372 molecular association, 369 permeability, 372 stimuli-responsive micro- and nanogels, 380 surface grafting and vectorization, 370 swelling, 371 Microgels in drug delivery, 374–380 biodegradable micro- and nanogels, 374–375 composite micro- and nanogels, 376–377

core-shell and multilayer micro- and nanogels, 378–379 stimuli-responsive micro- and nanogels, 380 Microgels properties, 371–372 biocompatibility, 373 drug loading, 372 permeability, 372 swelling, 371 Microgels structural types, 368 Microgel swelling, 371, 391–395 and deswelling theory, 395 microgel swelling and deswelling theory, 395 stimuli-responsive microgel swelling, 391–394 Micromachined flow-through device, 328 Microsized magnetic particles filament structure and stability, 117–132 diffusive filament motion, 122–124 filament magnetic properties, 121 filaments aggregates, 131 magnetic filament preparation, 120–124 magnetic filament stability, 128–130 magnetic filament structure, 125–127 Micro-total analysis systems (m-TAS), 315–316 Mie theory, 126 MMA. See Methyl methacrylate (MMA) Mode coupling theory (MCT), 4–5, 135, 136, 138–140 attractive glass, 154 mono-component systems, 25 phase diagram, 146 prediction, 148 repulsion- and attraction-driven vitrification, 144–146 Mono-component systems, 25 Mono-disperse colloidal particles suspensions, 25 Mono-disperse suspension model, 13–14 Mononuclear phagocytic system (MPS), 268 Monopole capillary force, 42 fluid interface deformation and capillary forces, 51 Monte Carlo (MC) simulations, 72, 170 MPS. See Mononuclear phagocytic system (MPS) MR. See Magnetorheological (MR) fluids MRF. See Magnetorheological fluid (MRF) MRI. See Magnetic resonance imaging (MRI) MRNA. See Messenger ribonucleic acid (mRNA) MSA. See Mean-spherical approximation (MSA) MSD. See Mean-squared displacement (MSD) m-TAS. See Micro-total analysis systems (m-TAS) Multicomponent systems HNC/MSA equations, 65 SCGLE theory of dynamic arrest transitions, 23–25 MWO. See Maxwell–Wagner– O’Konski (MWO) relaxation frequency

N Nanodrug delivery carriers, 389 Nano-emulsion formation by low energy methods, 457–478 condensation or low energy emulsification methods, 458–472 droplet size-based applications, 475–477 and functional properties, 457–478 nano-emulsions functional characteristics, 473–478 nano-emulsion stability, 473–474

492 Nano-emulsion formation by low energy methods (Continued ) phase inversion composition method, 465–472 phase inversion temperature method, 460–465 self-emulsification, 459 solubilization-based applications, 478 Nano-emulsions defined, 457 lidocaine, 478 polymerization, 476 stability, 473–474 Nano-emulsions functional characteristics, 473–478 droplet size-based applications, 475–477 nano-emulsion stability, 473–474 solubilization-based applications, 478 Nanogels. See also Microgels in drug delivery, 374–380 gold nanoshells, 376 hydrophilic colloidal networks in drug delivery and diagnostics, 367–381 inverse emulsion, 370 properties, 371–372 structural types, 368 Nanoparticles (NP), 339–340 based chemiresistor, 350 embedding in silica colloids, 318 intracellular and subcellular distribution, 398 Nanorod surfactant molecules attachment, 346 NATP. See Triphospates of nucleoside analogs (NATP) Navier-Stokes equation, 237 Nernst equation, 400 Neutron reflection, 215 Newtonian fluid mechanics, 236 NIPAM. See N-isopropylacrylamide (NIPAM) N-isopropylacrylamide (NIPAM), 319 N-isopropylacrylamide microbeads, 320 Nitroxide-mediated radical polymerization (NMRP), 265 latex particles surface functionalization, 277 N-(2-methoxyethyl) acrylamide, 277 NMRP. See Nitroxide-mediated radical polymerization (NMRP) Nonlinear patches, 37 Non-Newtonian fluids, 238 Normalization coincidence, 81 first cumulant time evolution, 300 Nozzle shape channel geometry, 437 NP. See Nanoparticles (NP) N-tert-butyl-N-(1-diethyl phosphono-2,2-dimethylpropyl) nitroxide, 278 Nucleation, 476, 477 and growth, 342–346 Nucleic acid extraction, 325 Nucleoside analogs, 375

O Oil in water (O/W). See also Water in oil (W/O) final nano-emulsion, 462 microemulsion, 472 microemulsions at equilibrium, 460–461 nano-emulsions, 459, 465 self-emulsification, 459 weight ration, 461

Index Oil/surfactant (O/S) ratios, 466, 471 Oldroyd B model, 255 Oleic acid, 318 One-dimensional distribution particle displacements, 157 Oppositely charged colloids with size ratio 0.1, 180–183 with size ratio 0.31 crystal structures, 175–179 Optical microscopy of aqueous droplets, 440 Ordinary and generalized Langevin equation, 5–7 Organic matrix embedding, 317–322 Ormosil matrices, 451 Ornstein-Uhlenbeck processes, 6 Ornstein–Zernike (OZ) equation, 82, 83 Ornstein–Zernike (OZ) formalism, 65 O/S. See Oil/surfactant (O/S) ratios Oscillatory rheology of dispersions, 257 Oscillatory tests, 252 Osmotic swelling, 210 Ostwald ripening, 459, 460, 473, 474, 476 and coalescence, 221 Ouzo effect, 460 Overcharging, 64 O/W. See Oil in water (O/W) OZ. See Ornstein–Zernike (OZ) equation

P PAA. See Polyacrylic acid (PAA) p-aminophenol (PAP), 327 PAP. See p-aminophenol (PAP) Parallel plate comparative analysis, 251 Parameter space, 49 Particle displacements, 157 Passive targeting, 397 PB. See Poisson–Boltzmann (PB) equation PBG. See Photonic band gaps (PBG) PB-GC. See Poisson-Boltzmann-Gouy-Chapman (PB-GC) predictions PC. See Phosphatidylcholine (PC); Photonic crystals (PC) PCL. See Polycaprolactone (PCL); Polyepsilon caprolactone (PCL) PCR. See Polymerase chain reaction (PCR) PDMS. See Polydimethylsiloxane (PDMS) PDVB. See Polydivinylbenzene (PDVB) PE. See Polyelectrolytes (PE) PEDB. See Phenylethyl dithiobenzoate (PEDB) PEG. See Polyethylene glycol (PEG) PEG-DA. See Polyethylene glycol diacrylate (PEG-DA) PEI. See Polyethylenimine (PEI) nanogels Percus-Yevick (PY) approximation, 21 Permanent chains, 127 Permeability, 372 PET. See Polyethylene terephthalate (PET) sheets pH aggregation rate constant, 301 Berens and Hopfenberg model, 407 drug release, 402 responsive microgels mathematical modeling, 405–407 Phase diagram, 189 binary hard spheres, 167 dextran and pork skin gelatin, 202 ternary water/nonionic surfactant/oil system, 461 Phase equilibria, 464

Index Phase inversion composition (PIC) method, 458, 459, 464–465, 468 condensation, 465–472 low energy emulsification methods, 465–472 Phase inversion temperature (PIT) method, 458, 459, 464–465 condensation, 460–465 droplet size, 463 HLB temperature, 461 low energy emulsification methods, 460–465 Phase-separated fish gelation-dextran mixtures, 213 Phase transition, 458 PHEMA. See Polyhydroxyethyl methacrylate (PHEMA) Phenylethyl dithiobenzoate (PEDB), 279 Phenyl-modified alkoxide precursors, 450 Phosphatidylcholine (PC), 78 Phosphatidylserine (PS), 78 Photoinitiator ethylene glycol-methacrylate, 277 Photonic band gaps (PBG), 415 Photonic crystals (PC), 415 Photothermally modulated DDS, 377 Physical adsorption, 264 P.i. See Polydispersity index (p.i.) PIC. See Phase inversion composition (PIC) method Picoliter sized reactors for mesoporous microparticle synthesis, 429–443 PIT. See Phase inversion temperature (PIT) method Planar elongational flow, 238 Planar steady elongation, 239 PLGA. See Polylactide-co-glycolide (PLGA) PLLA. See Polylactide (PLLA) Plugs, 215 PM. See primitive model (PM) PMAA-co-AA. See Polymethacrylic acid-co-acrylic acid (PMAA-co-AA) PNIPA. See PolyN-isopropylacrylamide (PNIPA) Poisson–Boltzmann (PB) equation, 63 Poisson-Boltzmann-Gouy-Chapman (PB-GC) predictions, 71, 73 Poisson-Boltzmann model, 50 Poisson equation of electrostatics, 37 Poisson equation solution electrostatic potential, 66 Polarization charge, 41 Polyacrylamide gel electrophoresis, 295 Polyacrylic acid (PAA), 272 Polycaprolactone (PCL), 319 Polydimethylsiloxane (PDMS) CC film fabrication procedure, 418 CC film reflection spectra, 419 CC film SEM images, 419 channels, 431 coated silica particles, 257 elastomer, 417, 418, 419 gels, 430 Polydispersity silica microspheres, 440 water-water interfaces, 210 Polydispersity index (p.i.), 129 time evolution, 130 Polydivinylbenzene (PDVB), 279 Polyelectrolytes (PE), 393 Polyepsilon caprolactone (PCL), 321 Polyethylene glycol (PEG), 268 Polyethylene glycol diacrylate (PEG-DA), 421

493 Polyethylene terephthalate (PET) sheets, 424 Polyethylenimine (PEI) nanogels, 369 Polyhydroxyethyl methacrylate (PHEMA), 278 Polylactide (PLLA), 321, 322, 323 Polylactide-co-glycolide (PLGA), 322 Polymerase chain reaction (PCR), 325, 442 Polymer concentration interface, 209 vs. interface center total polymer concentration depletion, 206 Polymer depleted interface, 207 Polymeric drug delivery systems, 388, 395 Polymeric liquids mechanical spectrum, 254 structured fluids mechanical properties theories, 238–241 structured fluids rheological models, 253–255 viscosity curve, 254 Polymer solutions interface, 204 Polymethacrylic acid-co-acrylic acid (PMAA-co-AA), 391 Polymethacrylic acid-g-ethylene glycol (PMAA-g-EG), 391 dispersion pH, 392 PolyN-isopropylacrylamide (PNIPA), 257 microgels, 370–371 Polystyrene (PS), 257, 266, 319, 326, 417 CC film reflection spectra, 419 Polyvinyl alcohol (PVA), 321 Polyvinylpyrrolidone (PVP), 317, 346 Porod’s behavior of light scattering, 212 Preformed latexes surface modification, 275 Preformed polymers, 322 Prefreezing, 190 Pressure-free interface, 38–47 PrHy. See Procaine hydrochloride (PrHy) Primitive model (PM) electrolyte, 64 PRISM theory, 147 Procaine hydrochloride (PrHy), 400 in vitro release, 401, 406 Protein-functionalized colloidal particles, 294–301 aggregation mechanisms, 293 cluster growth monitoring, 291–293 fractal aggregates structural coefficient, 306–309 fractal structures and aggregation kinetics, 289–311 fractal structures scattering functions, 291 longtime behavior, 292 protein-functionalized colloidal particles flocculation, 294–301 protein-functionalized colloidal particles fractal aggregates structural coefficient, 306–309 protein net charge influence, 297–301 short-range interactions and protein influence, 302–304 short-time behavior, 293 Protein stabilizing foams, 225–228 Protein surface coverage, 305 Proteomics, 387 PS. See Phosphatidylserine (PS); Polystyrene (PS) Pseudoternary system water oleic acid-potassium oleate, 471 SDS:hexanol phase diagram, 470 PVA. See Polyvinyl alcohol (PVA) PVP. See Polyvinylpyrrolidone (PVP) PY. See Percus-Yevick (PY) approximation

494

Index

Q

S

Quadrupole force, 51–52 Quadrupole-quadrupole force, 45

Sandwich immunoassay, 327 SANS. See Small angle neutron scattering (SANS) Scaling procedure, 89 Scanning electron microscopy (SEM), 417 N-isopropylacrylamide microbeads, 320 PDMS channels, 431 Scattering vector dependence, 81 SCGLE. See Self-consistent generalized Langevin equation (SCGLE) SDBS. See Sodium dodecylbenzenesulfonate (SDBS) SdFFF. See Sedimentation field flow fractionation (SdFFF) SDS. See Sodium dodecylsulfate (SDS) Sedimentation field flow fractionation (SdFFF), 266 Seeded emulsion copolymerization to produce functionalized latexes, 273–274 Seed nucleation and particle size distribution, 344 Self-assembly process, 416 Self-consistent generalized Langevin equation (SCGLE) bidisperse system, 17 charged particles dispersions, 23 collective correlator, 22 colloid-polymer mixtures, 25 hard- and soft-sphere systems, 21–22 hard spheres mixtures, 24 hard-sphere system, 22 multicomponent systems, 23–25 nonergodicity parameters, 20 simplified theory, 4–5 systems comparison with experimental data, 21–22 theoretical predictions, 23 theory, 4–5, 13 theory of dynamic arrest, 20–25 Self-consistent theory, 12 closure relations, 12 illustrative applications, 13–19 model mono-disperse suspensions, 13–14 three-dimensional soft sphere systems, 14 two-dimensional model system with dipole-dipole (r -3) interaction, 14 vineyard-like approximations, 12 Self-diffusion coefficients, 87 Self-emulsification, 458 condensation or low energy emulsification methods, 459 O/W nano-emulsions, 459 SEM. See Scanning electron microscopy (SEM) SERS. See Surface-enhanced Raman scattering (SERS) SEXP. See Single exponential (SEXP) Shear creep, 239 Shear flow, 237 Shear stress decay, 239 Shear stress growth, 239 Shear viscosity, 257 Short-range attraction interaction potential, 141 Short-range interactions and protein influence, 302–304 Silica colloids, 318 Silica matrices, 451 sol-gel reactions, 449 Silica microspheres, 441 Silica particles, 121

R Radical polymerization (RP), 265 RAFT. See Reversible addition-fragmentation transfer (RAFT) Raman spectroscopy, 349 Rayleigh–Gans–Debye (RGD) condition, 124, 126 RBITC. See Rhodamine B isothiocyanate (RBITC) Reaction-limited cluster aggregation (RLCA), 290, 292, 297 Reduced viscosity, 239 Relative permittivity, 100 Relative resonance frequency time dependence, 111, 112 Relative viscosity, 239 Release profiles of procaine hydrochloride, 401 Repulsion-driven glasses theoretical predictions, 141–147 equilibrium structure factor, 142–143 input to MCT, 142–143 MCT of bond formation, 147 MCT of repulsion- and attraction-driven vitrification, 144–146 Repulsive barrier, 85 Repulsive screened Coulomb interactions, 77 RES. See Reticuloendothelial system (RES) Responsive microgels for drug delivery applications, 387–408 controlled drug delivery, 395–398 drug selective electrode, 399 intracellular delivery and subcellular distribution, 398 microgels as drug delivery vehicles, 400–404 microgel swelling, 391–395 microgel swelling and deswelling theory, 395 passive and active targeting, 397 pH-responsive microgels mathematical modeling, 405–407 stimuli-responsive microgel swelling, 391–394 synthesis technique, 390 Restrictive primitive model (RPM), 180, 181 phase diagram, 182 Reticuloendothelial system (RES), 370 Reversible addition-fragmentation transfer (RAFT), 265 chain transfer polymerization, 279 Reynolds number, 431 RGD. See Rayleigh–Gans–Debye (RGD) condition Rheological functions, 237–238 material, 238, 239 standard flows, 237 Rheometers comparative analysis, 251 Rhodamine B isothiocyanate (RBITC), 268 Rigid axisymmetric particles dilute dispersions, 244 RLCA. See Reaction-limited cluster aggregation (RLCA) Root-mean-square displacement, 146 Rotational tests, 251 RP. See Radical polymerization (RP) RPM. See Restrictive primitive model (RPM)

495

Index Silver nanoparticles aggregation chemical sensing, 350 heated, 351 Simple square well, 140 Simplified self-consistent generalized Langevin equation theory, 18 Simulation data mixed potential, 84 Single exponential (SEXP), 18 approximation, 13, 14 irreducible memory functions, 18 SLS. See Static light scattering (SLS) Small-amplitude oscillatory elongation, 239 Small-amplitude oscillatory shear, 239 Small angle neutron scattering (SANS), 460 Small deformation curved interface, 51 Small hard spheres binary mixtures, 167 Small particles (SP), 94 SMCC. See Succinimidy-maleimidomethyl cyclohexanecarboxylate (SMCC) SMSA. See Soft mean spherical approximation (SMSA) Sodium dodecylbenzenesulfonate (SDBS), 266 Sodium dodecylsulfate (SDS), 345 Soft mean spherical approximation (SMSA), 83 Soft sphere systems, 21–22 intermediate scattering function, 15 Soft templates and selective adsorption for shape control, 345–346 Sol-gel reactions phenyl-modified alkoxide precursors, 450 silica matrices, 449 Solid core-hairy shell, 247 Solid core-liquid shell, 247 Solubilization-based applications, 478 Solvent-solvent interfaces, 201 SP. See Small particles (SP) Spatial Fourier transform, 4 Specific viscosity, 239 Spectral distribution, 129 Spherical charged particle, 53 Spherical particle, 33 SPR. See Surface plasmon resonance (SPR) Squared displacement, 156 Squared effective refraction index vs. volume fraction, 79 Square well (SW) system, 140, 142, 144 attractive glass, 152 binary mixture, 151 glass transition lines, 145 structure factor evolution, 151 Stable binary crystal structures, 179 Static light scattering (SLS), 125, 291 Static structure factor, 23 Steady-shear flow, 239 Step elongational strain, 239 Step shear strain, 239 Steric and charge-stabilized colloids, 166–183 DLVO theory, 168–169 hard-core Yukawa particles phase behavior, 170–175 large and small hard spheres binary mixtures, 167 oppositely charged colloids with size ratio 1, 180–183 oppositely charged colloids with size ratio .32 crystal structures, 175–179 primitive model, 168

Stimuli-responsive micro- and nanogels drug delivery, 380 swelling, 391–394 Structural and kinetic parameters of bare particles, 309 Structured fluids mechanical properties theories, 238–248 dispersed systems, 242–248 polymeric liquids, 238–241 Structured fluids rheological models, 235–258 dispersed systems, 242–248, 255–257 experimental techniques, 249–252 material functions, 238 measurement systems, 249–251 Newtonian fluid mechanics, 236 oscillatory tests, 252 polymeric liquids, 238–241, 253–255 rheological functions, 237–238 rotational and oscillatory tests, 251–252 rotational tests, 251 standard flows, 237 structured fluids mechanical properties theories, 238–248 theory vs. experiment, 253–257 Substrate liquid film configuration, 42 Succinimidy-maleimidomethyl cyclohexane-carboxylate (SMCC), 371 Superparamagnetic polystyrene particle suspension, 126 Superposition approximation, 39 Superposition of rotation and oscillation, 253 Surface charge density, 68 parameter space, 49 Surface energy, 342 Surface-enhanced Raman scattering (SERS), 349 Surface functionalization latex particles, 270–278 Surface grafting and vectorization, 370 Surface modification, 270 Surface plasmon resonance (SPR), 347–348 functional properties, 347 metallic nanoparticles colloidal dispersion formation, 347 shape dependence, 348 Surface pressure-concentration isotherms, 226 Surface rheological properties, 223–224 Surface tensions, 223–224 beta-casein, 223 Surfactant molecules attachment, 346 Surfactant stabilizing foams, 225–228 Suspensions dynamic mobility, 106, 107 magnetization, 110 SW. See Square well (SW) system Swelling. See also Microgel swelling maximum swelling vs. cross-link density, 396 micro- and nanogels properties, 371 Synthesis technique, 390

T T cells, 325 TEM. See Transmission electron microscopy (TEM) Temperature-dependent curves, 252, 253 TEOS. See Tetraethylorthosilicate (TEOS) Ternary water/nonionic surfactant/oil system, 461

496 Tetradecane emulsions, 230 Tetraethylorthosilicate (TEOS), 439 Tetragonal structure, 181 Theoretical relative permittivity, 95 Thin liquid films colloidal particle, 34 foams and emulsions interfacial phenomena, 220 Three-dimensional soft sphere systems, 14 Time dependence, 251–252, 253 alpha-relaxation, 101 friction function for tracer diffusion, 8–9 mean-squared displacement, 88 relative resonance frequency, 111, 112 self diffusion coefficient, 19 Time evolution, 300 Time profile vs. drug concentration, 396 Transdermal delivery, 477 Transmission electron microscopy (TEM), 121, 320, 441 gold nanorods, 345 HexaDecylPhosphonic acid-MP, 318 magnetic latex particles, 321 oleic acid-MP, 318 silica microspheres, 441 Triphospates of nucleoside analogs (NATP), 373, 375, 376, 379 Trivalent counterion, 70–71 Tween 20 system foam film thickness, 228 foam half-life, 226 foams and emulsions examined with interfacial studies, 226, 227–228 surface pressure-concentration isotherms, 226 Two-dimensional model system with dipole-dipole interaction, 14 Two-stage emulsion polymerization technique, 274

U Ultra violet-visible light (UV-Vis) transmission spectra, 452 Uniaxial elongational flow, 238 Uniaxial steady elongation, 239 UV-Vis. See Ultra violet-visible light (UV-Vis) transmission spectra

V Van Dongen-Ernst theory, 299 Verlet-Weis (VW) correction, 21 Vertical capillaries, 433 Videomicroscopy magnetic particles, 124 silica particles, 121 Vineyard approximations, 12 Virtual capillary charges, 37 Viscoelastic fluids, 235 Viscosity, 239

Index Volume fraction dependence localization length, 89 repulsive barrier, 85 self-diffusion coefficients, 87 VW. See Verlet-Weis (VW) correction

W Water/C16–18E12/glycerol monostearate/cetearyl isononanoate system, 462 Water/C16E6/mineral oil system, 463 Water in oil (W/O) HLB temperature, 462 microemulsions, 461, 471 nano-emulsions, 462, 465, 477 reverse, 465 Water/polyoxyethylene/methyl decanoate, 460 Water/polyoxyethylene nonionic surfacant/methyl decanoate system, 469 Water/potassium oleate-oleic acid hexadecane system micelles, 472 Water/Tween20:Span20 phase diagram, 468 Water-water interfaces, 210–216 curvatures coexistence, 214 experiments, 210–214 interface excess, 207 interface tension, 202–204, 210–211 interface width, 205–206, 212 permeability, 208–209, 212–213 permeability for low molar mass components, 213 polydispersity, 210 theoretical background, 202–208 Water-water phase droplets, 212 Wave vector static structure factor, 80 structure factors, 143 tail, 144 Wettability, 33 White-Metzner model, 255 Whole casein, 222–223 and beta-casein stabilizing foams, 222–224 foam film thickness, 228 foams and emulsions examined with interfacial studies, 222–224, 227–228 surface tension, 224 Wilhelmy plate method, 202 W/O. See Water in oil (W/O)

Y Yukawa particles, 170 phase diagram, 171 Yukawa potential, 81, 82, 85

Z Zeta-potential, 67