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

113 downloads 626 Views 10MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

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

xi

xii

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

xiii

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.

xv

xvi

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.

xvii

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

xxi

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

xxii

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


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


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


_______

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


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


·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


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


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


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


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


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


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


[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


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,


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


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.


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.


27

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

REFERENCES 1. Hess, W. and Klein, R. 1983. Generalized hydrodynamics of systems of Browninan particles. Adv. Phys. 32: 173. 2. Pusey, P. N. 1991. Colloidal suspensions. In: J. P. Hansen, D. Levesque, and J. Zinn-Justin (Eds), Liquids, Freezing and Glass Transition. Amsterdam: Elsevier, Chapter 10. 3. Nägele, G. 1996. On the dynamics and structure of charge-stabilized suspensions. Phys. Rep. 272: 215. 4. Ackerson, B. J. 1976. Correlations for interacting Brownian particles. J. Chem. Phys. 64: 242. 5. Ackerson, B. J. 1978. Correlations for interacting Brownian particles. II. J. Chem. Phys. 69: 684. 6. Pusey, P. N. and Tough, R. J. A. 1982. Langevin approach to the dynamics of interacting Brownian Particles. J. Phys. A 15: 1291. 7. Cichocki, B. 1988. Linear kinetic theory of a suspension of interacting Brownian particles. I. Enskog renormalization. Physica A 148: 165–191. 8. Leegwater, J. A. and Szamel, G. 1992. Dynamical properties of hard-sphere suspensions. Phys. Rev. A 46: 4999. 9. Szamel, G. and Leegwater, J. A. 1992. Long-time self-diffusion coefficients of suspensions. Phys. Rev. A 46: 5012. 10. Nägele, G., Bergenholtz, J., and Dhont, J. K. G. 1999. Cooperative diffusion in colloidal mixtures. J. Chem. Phys. 110: 7037. 11. Nägele, G. and Bergenholtz, J. 1998. Linear viscoelasticity of colloidal mixtures. J. Chem. Phys. 108: 9893. 12. Nägele, G. and Dhont, J. K. G. 1998. Tracer-diffusion in colloidal mixtures: A mode-coupling scheme with hydrodynamic interactions. J. Chem. Phys. 108: 9566. 13. Nägele, G. and Baur, P. 1997. Long-time dynamics of charged colloidal suspensions: Hydrodynamic interaction effects. Physica A 245: 297. 14. Banchio, A. J., Bergenholtz, J., and Nägele, G. 1999. Rheology and dynamics of colloidal suspensions. Phys. Rev. Lett. 82: 1792. 15. Banchio, A. J., Bergenholtz, J., and Nägele, G. 2000. Collective diffusion, self-diffusion and freezing criteria of colloidal suspensions. J. Chem. Phys. 113: 3381. 16. Yeomans-Reyna, L., Acuña-Campa, H., and Medina-Noyola, M. 2000. Vineyard-like approximations for colloid dynamics. Phys. Rev. E 62: 3395. 17. Yeomans-Reyna, L. and Medina-Noyola, M. 2000. Overdamped van Hove function of colloidal suspensions. Phys. Rev. E 62: 3382. 18. Yeomans-Reyna, L. and Medina-Noyola, M. 2001. Self-consistent generalized Langevin equation for colloid dynamics. Phys. Rev. E 64: 066114. 19. Yeomans-Reyna, L., Acuña-Campa, H., Guevara-Rodríguez, F., and Medina-Noyola, M. 2003. Selfconsistent theory of collective Brownian dynamics: Theory versus simulation. Phys. Rev. E 67: 021108. 20. Chávez-Rojo, M. A. and Medina-Noyola, M. 2006. Van Hove function of colloidal mixtures: Exact results. Physica A 366: 55. 21. Chávez-Rojo, M. A. and Medina-Noyola, M. 2005. Self-consistent generalized Langevin equation for colloidal mixtures. Phys. Rev. E 72: 031107; Erratum. 2007. Phys. Rev. E 76: 039902. 22. Medina-Noyola, M. and del Río-Correa, J. L. 1987. The fluctuation dissipation theorem for non-Markov process and their contractions: The role of the stationary condition. Physica A 146: 483. 23. Medina-Noyola, M. 1987. The generalized Langevin equation as a contraction of the description. Faraday Discuss. Chem. Soc. 83: 21. 24. Angell, C. A. 1995. Formation of glasses from liquids and biopolymers. Science 267: 1924. 25. Debenedetti, P. G. and Stillinger, F. H. 2001. Super-cooled liquids and the glass transition. Nature 410: 359. 26. Götze, W. 1991. Aspects of structural glass transitions. In: J. P. Hansen, D. Levesque, and J. Zinn-Justin (Eds), Liquids, Freezing and Glass Transition. Amsterdam: Elsevier, Chapter 5. 27. Götze, W. and Sjögren, L. 1992. Relaxation processes in supercooled liquids. Rep. Prog. Phys. 55: 241. 28. Götze, W., Leutheusser, E., and Yip, S. 1981. Dynamical theory of diffusion and localization in a random, static field. Phys. Rev. A 23: 2634.

28


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.


29

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


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


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


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


(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)


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


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


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


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)


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 ~ 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


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 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.

255


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)

256


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


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.

258


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).

REFERENCES 1. Batchelor, G.K. 2007. An Introduction to Fluid Dynamics. UK: Cambridge University Press. 2. Morrison, F.A. 2001. Understanding Rheology. USA: Oxford University Press. 3. Bird, R.B., Armstrong, R.C., and Hassager, O. 1987. Dynamics of Polymeric Liquids: Volume 1, Fluid Mechanics. USA: Wiley. 4. Bird, R.B., Curtiss, C.F., Armstrong, R.C., and Hassager, O. 1987. Dynamics of Polymeric Liquids: Volume 2, Kinetic Theory. USA: Wiley. 5. Larson, R.G. 1999. The Structure and Rheology of Complex Fluids. USA: Oxford University Press. 6. Pal, R. 2007. Rheology of Particulate Dispersions and Composites. Canada: Taylor & Francis Group. 7. Macosko, C.W. 1994. Rheology: Principles, Measurements and Applications. USA: Wiley-VCH. 8. Mezger, T.G. 2006. The Rheology Handbook. Germany: Vincentz Network GmbH & Co. 9. Lodge, A.S. 1964. Elastic Liquids. USA: Academic Press. 10. Oldroyd, J.G. 1950. On the formulation of rheological equations of state. Proc. Roy. Soc. A 200: 523–541. 11. White, J.L. and Metzner, A.B. 1963. Development of constitutive equations for polymeric melts and solutions. J. Appl. Polym. Sci. 7: 1867–1889. 12. Larson, R.G. 1988. Constitutive Equations for Polymer Melts and Solutions. USA: Butterworths. 13. Kaye, A. 1962. College of Aeronautics, Cranfield, Note 134. 14. Bernstein, B., Kearsley, E., and Zapas, L. 1963. A study of stress relaxation with finite strain. Trans. Soc. Rheol. 7: 391–410. 15. Wall, F.T. 1942. Statistical thermodynamics of rubber II. J. Chem. Phys. 10: 485–488. 16. Flory, P.J. and Rehner, J. 1943. Statistical mechanics of cross-linked polymer networks I. Rubberlike elasticity. J. Chem. Phys. 11: 512–520. 17. James, H.M. and Guth, E. 1943. Theory of the elastic properties of rubber. J. Chem. Phys. 11: 455–481. 18. Treloar, L.R.G. 1943. The elasticity of a network of longchain molecules II. Trans. Faraday Soc. 39: 241–246. 19. Green, M.S. and Tobolsky, A.V. 1946. A new approach to the theory of relaxing polymeric media. J. Chem. Phys. 14: 80–92. 20. Klein, J. 1978. Evidence for reptation in an entangled polymer melt. Nature 271: 143–145. 21. de Gennes, P.G. 1979. Scaling Concepts in Polymer Physics. USA: Cornell University Press. 22. Quemada, D. and Berli, C. 2002. Energy of interaction in colloids and its implications in rheological modeling. Adv. Colloid Interface Sci. 98: 51–85. 23. Liu, S.J. and Masliyah, J.H. 1996. Rheology of suspensions. In: L.L. Schramm (Ed.), Suspensions: Fundamentals and Applications in the Petroleum Industry. Washington, DC: American Chemical Society Books. 24. Leal, L.G. and Hinch, E.J. 1971. Effect of weak Brownian rotations on particles in shear flow. J. Fluid Mech. 46: 685–703. 25. Booth, F. 1950. The electroviscous effect for suspensions of solid spherical particles. Proc. Roy. Soc. 203: 533–551. 26. Russel, W.B. 1978. Bulk stresses due to deformation of electrical double layer around a charged sphere. J. Fluid Mech. 85: 673–683. 27. Russel, W.B. 1978. Rheology of suspensions of charged rigid spheres. J. Fluid Mech. 85: 209–232. 28. Goddard, J.D. and Miller, C. 1967. Nonlinear effects in rheology of dilute suspensions. J. Fluid Mech. 28: 657–673. 29. Roscoe, R. 1967. On rheology of a suspension of viscoelastic spheres in a viscous liquid. J. Fluid Mech. 28: 273–293. 30. Taylor, G.I. 1932. The viscosity of a fluid containing small drops of another fluid. Proc. Roy. Soc. A 138: 41–48. 31. Frankel, N.A. and Acrivos, A. 1970. Constitutive equation for a dilute emulsion. J. Fluid Mech. 44: 65–78. 32. Bentley, B.J. and Leal, L.G. 1986. An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows. J. Fluid Mech. 167: 241–283.


259

33. Oldroyd, J.G. 1955. The effect of interfacial stabilizing films on the elastic and viscous properties of emulsions. Proc. Roy. Soc. A 232: 567–577. 34. Danov, K.D. 2001. On the viscosity of dilute emulsions. J. Colloid Interface Sci. 235: 144–149. 35. Drochon, A., Barthes-Biesel, D., Lacombe, C., and Lelievre, J.C. 1990. Determination of the red-bloodcell apparent membrane elastic-modulus from viscometric measurements. J. Biomech. Eng. Trans. ASME 112: 241–249. 36. Drochon, A. 2003. Rheology of dilute emulsions of red blood cells: Experimental and theoretical approaches. Eur. Phys. J. Appl. Phys. 22: 155–162. 37. Zackrisson, M. and Bergenholtz, J. 2003. Intrinsic viscosity of dispersions of core-shell particles. Colloids Surf. A 225: 119–127. 38. Davis, A.M.J. and Brenner, H. 1981. Emulsions containing a 3RD solid internal phase. J. Eng. Mech. 107: 609–621. 39. Stone, H.A. and Leal, L.G. 1990. Breakup of concentric double emulsion droplets in linear flows. J. Fluid Mech. 211: 123–156. 40. Walters, K. 1975. Rheometry. USA: Chapman & Hall. 41. Binnington, R.J. and Boger, D.V. 1985. Die swell-empirical techniques and theoretical predictions. J. Rheol. 29: 111. 42. Khan, S.A. and Larson, R.G. 1987. Comparison of simple constitutive-equations for polymer melts in shear and biaxial and uniaxial extensions. J. Rheol. 31: 207–234. 43. Laun, H.M. 1978. Description of nonlinear shear behavior of a low-density polyethylene melt by means of an experimentally determined strain dependent memory function. Rheol. Acta 17: 1–15. 44. Sofou, S., Muliawan, E.B., Hatzikiriakos, S.G., and Mitsoulis, E. 2008. Rheological characterization and constitutive modeling of bread dough. Rheol. Acta 47: 369–381. 45. Vijay, R., Deshpande, A.P., and Varughese, S. 2008. Nonlinear rheological modeling of asphalt using White–Metzner model with structural parameter variation based asphaltene structural build-up and breakage. Appl. Rheol. 18: 23214–23228. 46. Meissner, J. 1971. Dehnungsverhalten van polyiithylen-schmelzen. Rheol. Acta 10: 230–242. 47. Pearson, D.S. 1987. Recent advances in the molecular aspects of polymer viscoelasticity. Rubber Chem. Technol., Rubber Reviews 60: 439–496. 48. Manley, R.S. and Mason, S.G. 1954. The viscosity of suspensions of spheres: A note on the particle interaction coefficient. Can. J. Chem. 32: 763–767. 49. Saunders, F.L. 1961. Rheological properties of monodisperse latex systems 1. Concentration dependence of relative viscosity. J. Colloid Sci. 16: 13–22. 50. Krieger, I.M. 1972. Rheology of monodisperse lattices. Adv. Colloid Interface Sci.3: 111–136. 51. Krieger, I.M. and Dougherty, T.J. 1959. A mechanism for non-Newtonian flow in suspensions of rigid spheres. Trans. Soc. Rheol. 3: 137–152. 52. Quemada, D. 1977. Rheology of concentrated disperse systems and minimum energy dissipation principle 1. Viscosity-concentration relationship. Rheol. Acta 16: 82–94. 53. Brady, J.F. 1993. The rheological behaviour of concentrated colloidal dispersions. J. Chem. Phys. 99: 567–581. 54. Whitcomb, P. and Macosko, C.W. 1978. Rheology of xanthan gum. J. Rheol. 22: 493–505. 55. Phan-Thien, N. and Graham, A.L. 1991. A new constitutive model for fiber suspensions. Flow past a sphere. Rheol. Acta 30: 44–57. 56. Chan, F.S. and Goring, D.A.I. 1966. Primary electroviscous effect in a sulfonated polystyrene latex. J. Colloid Interface Sci. 22: 371–381. 57. Krieger, I.M. and Eguiluz, M. 1976. 2ND electroviscous effect in polymer lattices. Trans. Soc. Rheol. 20: 29–45. 58. Fagan, M.E. and Zukoski, C.F. 1997. The rheology of charge stabilized silica suspensions. J. Rheol. 41: 373–397. 59. Wolfe, M.S. and Scopazzi, C. 1989. Rheology of swellable microgel dispersions: Influence of crosslink density. J. Colloid Interface Sci. 133: 265–277. 60. Evans, I.D. and Lips, A. 1990. Concentration dependence of the linear elastic behavior of model microgel dispersions. J. Chem. Soc., Faraday Trans. 86: 3413–3417. 61. Wolfe, M.S. 1992. Dispersions and solution rheology control with swellable microgels. Prog. Org. Coat. 20: 487–500 62. Kiminta, D.M.O., Luckham, P.F., and Lenon, S. 1995. The rheology of deformable and thermoresponsive microgel particles. Polymer 36: 4827–4831. 63. Raquois, C., Tassin, J.F., Rezaiguia, S., and Gindre, A.V. 1995. Microgels in coating technology: Structure and rheological properties. Prog. Org. Coat. 26: 239–250.

260


64. Snabre, P. and Mills, P. 1996. Rheology of weakly flocculated suspensions of viscoelastic particles. 2. J. Phys. III France 6: 1835–1855. 65. Borrega, R., Cloitre, M., Betremieux, I., Ernst, B., and Leibler, L. 1999. Concentration dependence of the low-shear viscosity of polyelectrolite micro-networks: From hard spheres to soft microgels. Europhys. Lett. 47: 729–735. 66. Tan, B.H., Tam, K.C., Lam, Y.C., and Tan, C.B. 2005. Microstructure and rheology of stimuli-responsive microgel systems: Effect of cross-linked density. Adv. Colloid Interface Sci. 113: 111–120. 67. Omari, A., Tabary, R., Rousseau, D., Leal-Calderon, F., Monteil, J., and Chauveteau, G. 2006. Soft watersoluble microgel dispersions: Structure and rheology. J. Colloid Interface Sci. 302: 537–546. 68. Hoare, T. and Pelton, R. 2007. Functionalized microgel swelling: Comparing theory and experiment. J. Phys. Chem. B 111: 11895–11906. 69. Nawab, M.A. and Mason, S.G. 1958. The viscosity of dilute emulsions. Trans. Faraday Soc. 54: 1712–1723. 70. Vinckier, I., Minale, M., Mewis, J., and Moldenaers, P. 1999. Rheology of semi-dilute emulsions: Viscoelastic effects caused by the interfacial tension. Colloids Surf. A 150: 217–228. 71. Choi, S.J. and Schowalter, W.R. 1975. Rheological properties of non-dilute suspensions of deformable particles. Phys. Fluids 18: 420–427. 72. Pal, R. 2008. Viscosity models for multiple emulsions. Food Hydrocolloid 22: 428–438. 73. Saiki, Y., Horn, R.G., and Prestidge, C.A. 2008. Droplet structure instability in concentrated emulsions. J. Colloid Interface Sci. 320: 569–574. 74. Aggarwal, N. and Sarkar, K. 2008. Rheology of an emulsion of viscoelastic drops in steady shear. J. Non-Newtonian Fluid Mech. 150: 19–31. 75. Fischer, P. and Erni, P. 2007. Emulsion drops in external flow fields: The role of liquid interfaces. Curr. Opin. Colloid Interface Sci. 12: 196–205. 76. Rust, A.C. and Manga, M. 2002. Effects of bubble deformation on the viscosity of dilute emulsions. J. Non-Newtonian Fluid Mech.104: 53–63. 77. Lim, Y.M., Seo, D., and Youn, J.R. 2004. Rheological behavior of dilute bubble suspensions in polyol. Korea-Aust. Rheol. J. 16: 47–54. 78. Ichihara, M., Ohkunitani, H., Ida, Y., and Kameda, M. 2004. Dynamics of bubble oscillation and wave propagation in viscoelastic liquids. J. Volcanol. Geoth. Res. 129: 37–60. 79. Muller, F., Peggau, J., Meyer, J., and Scheuermann, R. 2006. New vesicle gels in household applications. Tenside Surfact. Det. 43: 238–241. 80. Oldroyd, J.G. 1953. The elastic and viscous properties of emulsions and suspensions. Proc. R. Soc. London, Ser. A 218: 122–132. 81. Kattige, A. and Rowley, G. 2006. Influence of rheological behavior of particulate/polymer dispersions on liquid-filling characteristics for hard gelatin capsules. Int. J. Pharm. 316: 74–85. 82. Kattige, A. and Rowley, G. 2006. The effect of poloxamer viscosity on liquid-filling of solid dispersions in hard gelatin capsules. Drug Dev. Ind. Pharm. 32: 981–990. 83. Zhang, J., Johnson, P.C., and Popel, A.S. 2007. An immersed boundary lattice Boltzmann approach to simulate deformable liquid capsules and its applications to microscopic blood flows. Phys. Biol. 4: 285–295. 84. Frith, W.J., Strivens, T.A., and Mewis, J. 1990. Dynamic mechanical properties of polymerically stabilized dispersions. J. Colloid Interface Sci. 139: 55–62. 85. Nommensen, P.A., Duits, M.H.G., van den Ende, D., and Mellema, J. 2000. Elastic modulus at high frequency of polymerically stabilized suspensions. Langmuir 16: 1902–1909. 86. Deike, I. and Ballauff, M. 2001. Rheology of thermosensitive latex particles including the high frequency limit. J. Rheol. 45: 709–720. 87. Mewis, J. and Biebaut, G. 2001. Shear thickening in steady and superposition flows effect of particle interaction forces. J. Rheol. 45: 799–813. 88. Tadros, T. 2003. Interaction forces between particles containing grafted or adsorbed polymer layers. Adv. Colloid Interface Sci. 104: 191–226. 89. Zackrisson, M., Stradner, A., Schurtenberger, P., and Bergenholtz, J. 2006. Structure, dynamics, and rheology of concentrated dispersions of poly(ethylene glycol)-grafted colloids. Phys. Rev. E 73: 011408. 90. Nakamura, H. and Tachi, K. 2007. Dynamics of shear-thinning suspensions of core-shell structured latex particles. J. Colloid Interface Sci. 297: 312–316. 91. You, J.L., Park, B.J., Choi, H.J., Choi, S.B., and Jhon, M.S. 2007. Preparation and magnetorheological characterization of CI/PVB core/shell particle suspended MR fluids. Int. J. Mod. Phys. B 21: 4996–5002.

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

264


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).

Surface Functionalization of Latex Particles

265

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

266


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

267

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,

268


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.


269

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.

270


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.


271

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.

272


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


273

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.

274


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


275

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.

276


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.


12.4

277

“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

278


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.


279

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).

280


REFERENCES 1. Pichot, C. 2004. Surface-functionalized latex for biotechnological applications. Curr. Opin. Colloid Interface Sci. 9: 213–21. 2. Ortega-Vinuesa, J.L., Hidalgo-Alvarez, R., de las Nieves, F.J., Davey, C.L., Newman, D.J., and Price, D.P. 1998. Characterization of immunoglobulin G bound to latex particles using surface plasmon resonance and electrophoretic mobility. J. Colloid. Interface Sci. 204(2): 300–11. 3. Borque, L., Rus, A., Bellod, L., and Seco, M.L. 1999. Development of an automated immunoturbidimetric ferritin assay. Clin. Chem. Lab. Med. 37(9): 899–905. 4. Borque, L. 2001. Inmunoanálisis de partículas basado en la tecnología avidina-biotina. Aplicación a la medida de ferritina en suero. PhD dissertation, Zaragoza University. 5. Zhu, B.Y. and Gu, T. 1991. Surfactant adsorption at solid–liquid interfaces. Adv. Colloid Interface Sci. 37: 1–32. 6. Paxton, T.R. 1969. Adsorption of emulsifier on polystyrene and poly(methyl methacrylate) latex particles. J. Colloid Interface Sci. 31: 19–30. 7. Vale, H.M. and Mckenna, T.F. 2005. Adsorption of sodium dodecyl sulfate and sodium dodecyl benzenesulfonate on poly(vinyl chloride) latexes. Colloids Surf. A: Physicochem. Eng. Aspects 268: 68–72. 8. Turner, S.F., Clarke, S.M., Rennie, A.R., et al. 1999. Adsorption of sodium dodecyl sulfate to a polystyrene/water interface studied by neutron reflection and attenuated total reflection infrared spectroscopy. Langmuir 15: 1017–23. 9. Sefcik, J., Verduyn, M., Storti, G., and Morbidelli, M. 2003. Charging of latex particles stabilized by sulfate surfactant. Langmuir 19: 4778–83. 10. Romero-Cano, M.S., Martín-Rodríguez, A., and de las Nieves, F.J. 2000. Adsorption and desorption of Triton X-100 in polystyrene particles with different functionality. I. Adsorption study. J. Colloid Interface Sci. 227: 322–8. 11. Peula, J.M., Puig, J., Serra, J., de las Nieves, F.J., and Hidalgo-Alvarez, R. 1994. Electrokinetic characterization and colloidal stability of polystyrene latex particles partially covered by IgG/a-CRP and m-BSA proteins. Colloids Surf. A 92: 127–36. 12. Galisteo Gonzalez, F., Cabrerizo Vilchez, M.A., and Hidalgo-Alvarez, R. 1991. Adsorption of anionic surfactants on positively charged polystyrene particles. II. Colloid Polym. Sci. 269: 406–11. 13. Zhao, J. and Brown, W. 1996. Dynamic light scattering study of adsorption of a non-ionic surfactant (C12E7) on polystyrene latex particles: Effects of aromatic amino groups and the surface polymer layer. J. Colloid Interface Sci. 179: 281–9. 14. Rutland, M.W. and Senden, T.J. 1993. Adsorption of the poly(oxyethylene) nonionic surfactant C12E5 to silica: A study using atomic force microscopy. Langmuir 9: 412–18. 15. Lee, E.M., Thomas, R.K., Cummins, P.G., Staples, E.J., Penfold, J., and Rennie, A.R. 1989. Determination of the structure of a surfactant layer adsorbed at the silica/water interface by neutron reflection. Chem. Phys. Lett. 162: 196–202. 16. Andersson, M., Fromell, K., Gullberg, E., Artursoon, P., and Caldwell, K.D. 2005. Characterization of surface-modified nanoparticles for in vivo biointeraction. A sedimentation field flow fractionation study. Anal. Chem. 77: 5488–93. 17. Zhao, J. and Brown, W. 1995. Adsorption of a nonionic surfactant (C12E7) on carboxylated styrenebutadiene copolymer latex particles. J. Colloid Interface Sci. 169: 39–47. 18. Cosgrove, T., Mears, S.J., Obey, T., Thompson, L., and Wesley, R.D. 1999. Polymer, particle, surfactant interactions. Colloids Surf. A: Physicochem. Eng. Aspects 149: 329–38. 19. Porcel, R., Jódar, A.B., Cabrerizo, M.A., and Hidalgo-Álvarez, R., and Martín-Rodríguez, A. 2001. Sequential adsorption of Triton X-100 and sodium dodecyl sulfate onto positively and negatively charged polystyrene latexes. J. Colloid Interface Sci. 239: 568–76. 20. Jódar-Reyes, A.B., Ortega-Vinuesa, J.L., and Martín-Rodríguez, A. 2005. Adsorption of different amphiphilic molecules onto polystyrene latices. J. Colloid Interface Sci. 282: 439–47. 21. Lee, S.-Y. and Harris, M.T. 2006. Surface modification of magnetic nanoparticles capped by oleic acids: Characterization and colloidal stability in polar solvents. J. Colloid Interface Sci. 293: 401–8. 22. Klapper, M., Clark Jr, C.G., and Müllen, K. 2008. Application-directed syntheses of surface-functionalized organic and inorganic nanoparticles. Polym. Int. 57:181–202. 23. Tadros, T.F., Vandamme, A., Booten, K., Levecke, B., and Stevens, C.V. 2004. Stabilisation of emulsions using hydrophobically modified inulin (polyfructose). Colloids Surf. A: Physicochem. Eng. Aspects 240: 133–40. 24. Mura, J.-L. and Riess, G. 1995. Polymeric surfactants in latex technology: Polystyrene-poly(ethylene oxide) block copolymers as stabilizers in emulsion polymerization. Polym. Adv. Technol. 6: 497–508.


281

25. Karlberg, M., Thuresson, K., and Lindman, B. 2005. Hydrophobically modified ethyl(hydroxyethyl)cellulose as stabilizer and emulsifying agent in macroemulsions. Colloids Surf. A: Physicochem. Eng. Aspects 262: 158–67. 26. Akiyama, E., Kashimoto, A., Fukuda, K., Hotta, H., Suzuki, T., and Kitsuki, T. 2005. Thickening properties and emulsification mechanisms of new derivatives of polysaccharides in aqueous solution. J. Colloid Interface Sci. 282: 448–57. 27. Torchilin, V.P. and Trubetskoy, V.S. 1995. Which polymers can make nanoparticulate drug carriers longcirculating? Adv. Drug Delivery Rev. 16: 141–55. 28. Roach, P., Farrar, D., and Perry, C.C. 2005. Interpretation of protein adsorption: Surface-induced conformational changes. J. Am. Chem. Soc. 127: 8168–73. 29. Anselmo, K. 2000. Osteoblast adhesion on biomaterials. Biomaterials 21: 667–81. 30. Bookbinder, D.C., Fewkers, E.J., Griffin, J.A., and Smith, F.M. 2000. Producing low binding hydrophobic surfaces by treating with a low HLB number non-ionic surfactant. US Patent 6093559. 31. Simon, S., Picton, L., Le Cerf, D., and Muller, G. 2005. Adsorption of amphiphilic polysaccharides onto polystyrene latex particles. Polymer 46: 3700–7. 32. Blunk, T., Hochstrasser, D., Sanchez, J.C., Mueller, B.W., and Mueller, R.H. 1993. Colloidal carriers for intravenous drug targeting: Plasma protein adsorption patterns on surface-modified latex particles evaluated by two-dimensional polyacrylamide gel electrophoresis. Electrophoresis 14: 1382–7. 33. Esmaeili, F., Ghahremani, M.H., Esmaeili, B., Khoshayand, M.R., Atyabi, F., and Dinarvand, R. 2008. PLGA nanoparticles of different surface properties: Preparation and evaluation of their body distribution. Int. J. Pharm. 349: 249–55. 34. Rouzes, C., Durand, A., Leonard, M., and Dellacherie, E. 2002. Surface activity and emulsification properties of hydrophobically modified dextrans. J. Colloid Interface Sci. 253: 217–23. 35. Rouzes, C., Gref, R., Leonard, M., De Sousa Delgado, A., and Dellacherie, E. 2000. Surface modification of poly(lactic acid) nanospheres using hydrophobically modified dextrans as stabilizers in an o/w emulsion/evaporation technique. J. Biomed. Mater. Res. 50: 557–65. 36. Brannon-Peppas, L. and Blanchette, J.O. 2004. Nanoparticle and targeted systems for cancer therapy. Adv. Drug Delivery. Rev. 56: 1649–59. 37. Wong, J.E., Gaharwar, A.K., Müller-Schulte, D., Bahadur, D., and Richtering, W. 2008. Dual-stimuli responsive PNIPAM microgel achieved via layer-by-layer assembly: Magnetic and thermoresponsive. J. Colloid Interface Sci. 324: 47–54. 38. Decher, G. and Schmitt, J. 1992. Fine-tuning of the film thickness of ultrathin multilayer films composed of consecutively altering layers of anionic and cationic polyelectrolytes. Prog. Colloid Polym. Sci. 89: 160–4. 39. Shafir, A. and Andelman, D. 2006. Polyelectrolyte multilayer formation: Electrostatic and short-range interactions. Eur. Phys. J. E 19: 155–62. 40. Goldenberg, L.M., Jung, B.D., Wagner, J., Stumpe, J., Paulke, B.R., and Goernitz, E. 2003. Preparation of ordered arrays of layer-by-layer modified latex particles. Langmuir 19: 205–07. 41. Trimaille, T., Pichot, C., and Delair, T. 2003. Surface functionalization of poly(D,L-lactic acid) nanoparticles with poly(ethylenimine) and plasmid DNA by the layer-by-layer approach. Colloids Surf. A: Physicochem. Eng. Aspects 221: 39–48. 42. Reb, P., Margarit-Puri, M., Klapper, M., and Müllen, K. 2000. Polymerizable and nonpolymerizable isophthalic acid derivatives as surfactants in emulsion polymerization. Macromolecules 33: 7718–23. 43. Aramendia, E., Barandiaran, M.J., Grade, J., Blease, T., and Asua, J.M. 2005. Improving water sensitivity in acrylic films using surfmers. Langmuir 21: 1428–35. 44. Aramendia, E., Mallegol, J., Jeynes, C., Barandiaran, M.J., Keddie, J.L., and Asua, J.M. 2003. Distribution of surfactants near acrylic latex film surfaces: A comparison of conventional and reactive surfactants (surfmers). Langmuir 19: 3212–21. 45. Schild, R.L., El-Aasser, M.S., Poehlein, G.W., and Vanderhoff, J.W. 1978. Preparation and characterization of polystyrene latexes with the ionic comonomer 2-acrylamido-2-methyl propane sulfonic acid. In: P. Becher and M.N. Yudenfreund (eds), Emulsions, Latices, Dispersions, pp. 99–128. New York: Dekker. 46. El-Aasser, M.S., Makgawinata, T., Vanderhoff, J.W., and Pichot, C. 1983. Batch and semicontinuous emulsion copolymerization of vinyl acetate-butyl acrylate. I. Bulk, surface, and colloidal properties of copolymer latexes. J. Polym. Sci. Polym. Chem. Ed. 21: 2363–82. 47. Polatajko-Lobos, E. and Xanthopoulo, V.G. 1978. Latex stability characteristics related to surface chemistry. Polym. Latex, Int. Conf. 6, 11. 48. Eliseeva, V.I. 1979. The control of the properties of polymers obtained by emulsion polymerization. Acta Polym. 30: 273–82.

282


49. Orsini, A.J., Ingenito, A.C., Leedle, M.A., and DeBari, V.A. 1987. The neutrophil response to polystyrene microspheres bearing defined surface functional groups. Cell Biophys. 10: 33–43. 50. Elimelech, M. and O’Melia, C.R. 1990. Effect of electrolyte type on the electrophoretic mobility of polystyrene latex colloids. Colloids Surf. 44: 165–78. 51. Paulke, B.R., Haertig, W., and Brueckner, G. 1992. Synthesis of nanoparticles for brain cell labelling in vivo. Acta Polym. 43: 288–91. 52. Paulke, B.R., Moeglich, P.M., Knippel, E., Budde, A., Nitzsche, R., and Mueller, R.H. 1995. Electrophoretic 3D-mobility profiles of latex particles with different surface groups. Langmuir 11: 70–4. 53. Schulz, S.F., Gisler, T., Borkovec, M., and Sticher, H. 1994. Surface charge on functionalized latex spheres in aqueous colloidal suspensions. J. Colloid Interface Sci. 164: 88–98. 54. Antonietti, M., Kaspar, H., and Tauer, K. 1996. Swelling equilibrium of small polymer colloids: Influence of surface structure and size-dependent depletion correction. Langmuir 12: 6211–17. 55. Tuncel, A. and Serpen, E. 2001. Emulsion copolymerization of styrene and methacrylic acid in the presence of a polyethylene oxide based-polymerizable stabilizer with a shorter chain length. Colloid Polym. Sci. 279: 240–51. 56. Menshikova, A.Y., Shabsels, B.M., and Shevchenko, N.N., et al. 2007. Surface modified latex particles: Synthesis and self-assembling into photonic crystals. Colloids Surf. A: Physicochem. Eng. Aspects 298: 27–33. 57. Reese, C.E. and Asher, S.A. 2002. Emulsifier-free emulsion polymerization produces highly charged, monodisperse particles for near infrared photonic crystals. J. Colloid Interface Sci. 248: 41–6. 58. Gessner, A., Lieske, A., Paulke, B.R., and Müller, R.H. 2003. Functional groups on polystyrene model nanoparticles: Influence on protein adsorption. Biomed. Mater. Res. 65A: 319–26. 59. Dahman, Y., Puskas, J.E., Margaritis, A., Merali, Z., and Cunningham, M. 2003. Novel thyminefunctionalized polystyrenes for applications in biotechnology. Polymer synthesis and characterization. Macromolecules 36: 2198–205. 60. D’Agosto, F., Charreyre, M.T., Pichot, C., and Gilbert, R.G. 2003. Latex particles bearing hydrophilic grafted hairs with controlled chain length and functionality synthesized by reversible additionfragmentation chain transfer. J. Polym. Sci. Part A: Polym. Chem. 41: 1188–95. 61. Tuncel, A., Kahraman, R., and Piskin, E. 1994. Monosize polystyrene latexes carrying functional groups on their surfaces. J. Appl. Polym. Sci. 51: 1485–98. 62. Antonietti, M., Lohman, S., and Bremser, W. 1992. Polymerization in microemulsion-size and surface control of ultrafine latex particles. Prog. Colloid Polym. Sci. 89: 62–5. 63. Antonietti, M., Basten, R., and Lohman, S. 1995. Polymerization in microemulsions—a new approach to ultrafine, highly functionalized polymer dispersions. Macromol. Chem. Phys. 196: 441–66. 64. Pich, A., Datta, S., Musyanovych, A., Adler, H.J.P., and Engelbrecht, L. 2005. Polymeric particles prepared with fluorinated surfmer. Polymer 46: 1323–30. 65. Holzapfel, V., Musyanovych, A., Landfester, K., Lorenz, M.R., and Mailänder, V. 2005. Preparation of fluorescent carboxyl and amino functionalized polystyrene particles by miniemulsion polymerization as markers for cells. Macromol. Chem. Phys. 206: 2440–9. 66. Zhang, S.W., Zhou, S.X., Weng, Y.M., and Wu, L.M. 2006. Synthesis of silanol-functionalized latex nanoparticles through miniemulsion copolymerization of styrene and gamma-methacryloxypropyltrimethoxysilane. Langmuir 22: 4674–9. 67. Musyanovych, A., Rossmanith, R., Tontsch, C., and Landfester, K. 2007. Effect of hydrophilic comonomer and surfactant type on the colloidal stability and size distribution of carboxyl- and aminofunctionalized polystyrene particles prepared by miniemulsion polymerization. Langmuir 23: 5367–76. 68. Ferguson, C.J., Hughes, R.J., Pham, B.T.T., Hawkett, B.S., Gilbert, R.G., Serelis, A.K., and Such, C.H. 2002. Effective ab initio emulsion polymerization under RAFT control. Macromolecules 35: 9243–5. 69. Pham, B.T.T., Nguyen, D., Ferguson, C.J., Hawkett, B.S., Serelis, A.K., and Such, C.H. 2003. Miniemulsion polymerization stabilized by amphipathic macro RAFT agents. Macromolecules 36: 8907–9. 70. Ferguson, C.J., Hughes, R.J., Nguyen, D., Pham, B.T.T., Gilbert, R.G., Serelis, A.K., Such, C.H., and Hawkett, B.S. 2005. Ab initio emulsion polymerization by RAFT-controlled self-assembly. Macromolecules 38: 2191–204. 71. dos Santos, A.M., Pohn, J., Lansalot, M., and D’Agosto, F. 2007. Combining steric and electrostatic stabilization using hydrophilic macroRAFT agents in an ab initio emulsion polymerization of styrene. Macromol. Rapid Commun. 28: 1325–32. 72. Rieger, J., Stoffelbach, F., Bui, C., Alaimo, D., Jérôme, C., and Charleux, B. 2008. Amphiphilic poly(ethylene oxide) macromolecular RAFT agent as a stabilizer and control agent in ab initio batch emulsion polymerization. Macromolecules 41: 4065–8.


283

73. Delaittre, G., Nicolas, J., Lefay, C., Save, M., and Charleux, B. 2005. Surfactant-free synthesis of amphiphilic diblock copolymer nanoparticles via nitroxide-mediated emulsion polymerization. Chem. Commun. 614–16. 74. Delaittre, G., Nicolas, J., Lefay, C., Save, M., and Charleux, B. 2006. Aqueous suspension of amphiphilic diblock copolymer nanoparticles prepared in situ from a water-soluble poly(sodium acrylate) alkoxyamine macroinitiator. Soft Matter 2: 223–31. 75. Charleux, B. and Nicolas, J. 2007. Water-soluble SG1-based alkoxamines: A breakthrough in controlled/ living free-radical polymerization in aqueous dispersed media. Polymer 48: 5813–33. 76. Delaitre, G. and Charleux, B. 2008. Kinetics of in situ formation of poly(acrylic acid)-b-polystyrene amphiphilic block copolymers via nitroxide-mediated controlled free-radical emulsion polymerization. Discussion on the effect of compartmentalization on the polymerization rate. Macromolecules 41: 2361–7. 77. Bastos, D., Santos, R., Forcada, J., Hidalgo-Alvarez, R., and de las Nieves, F.J. 1994. Surface and electrokinetic characterization of functional aldehyde polymer colloids. Colloids Surf. A: Physicochem. Eng. Aspects 92: 137–46. 78. Peula, J.M., Santos, R., Forcada, J., Hidalgo-Alvarez, R., and de las Nieves, F.J. 1995. Covalent coupling of antibodies to aldehyde groups on polymer carriers. J. Mater. Sci.: Mater. Med. 6: 779–85. 79. Santos, R.M. and Forcada, J. 1996. Synthesis and characterization of latex particles with acetal functionality. Prog. Colloid Polym. Sci. 100: 87–90. 80. Santos, R.M. and Forcada, J. 1997. Acetal-functionalized polymer particles useful for immunoassays. J. Polym. Sci. Part A: Polym. Chem. 35: 1605–10. 81. Peula, J.M., Santos, R., Forcada, J., Hidalgo-Alvarez, R., and de las Nieves, F.J. 1998. Study on the colloidal stability mechanisms of acetal functionalized latexes. Langmuir 14: 6377–84. 82. Santos, R.M. and Forcada, J. 1999. Acetal-functionalized polymer particles useful for immunoassays. II. Surface and colloidal characterization. J. Polym. Sci. Part A: Polym. Chem. 37: 501–11. 83. Santos, R.M. and Forcada, J. 2001. Acetal-functionalized polymer particles useful for immunoassays. III. Preparation of latex–protein complexes and their applications. J. Mater. Sci.: Mater. Med. 12: 173–80. 84. Sarobe, J. and Forcada, J. 1996. Synthesis of core-shell type polystyrene monodisperse particles with chloromethyl groups. Colloid Polym. Sci. 274: 8–13. 85. Sarobe, J., Miraballes, I., Molina, J.A., Forcada, F., and Hidalgo-Alvarez, R. 1996. Nephelometric assay of IgG chemically bound to chloromethyl styrene beads. Polym. Adv. Technol. 7: 749–53. 86. Sarobe, J. and Forcada, J. 1998. Synthesis of monodisperse polymer particles with chloromethyl functionality. Colloids Surf. A: Physicochem. Eng. Aspects 135: 291–5. 87. Sarobe, J., Molina, J.A., Forcada, J., Galisteo, F., and Hidalgo-Alvarez, R. 1998. Functionalized monodisperse particles with chloromethyl groups for the covalent coupling of proteins. Macromolecules 31: 4282–7. 88. Ramos, J., Martín-Molina, A., Sanz-Izquierdo, M.P., Rus, A., Borque, L., Forcada, J., and GalisteoGonzález, F. 2004. Amino, chloromethyl, and acetal-functionalized latex particles for immunoassays: A comparative study. J. Immunol. Methods 287: 159–67. 89. Miraballes-Martinez, I. and Forcada, J. 2000. Synthesis of latex particles with surface amino groups. J. Polym. Sci. Part A: Polym. Chem. 38: 4230–7. 90. Miraballes-Martinez, I., Martín-Molina, A., Galisteo-González, F., and Forcada. J. 2001. Synthesis of amino functionalized latex particles by a multi-step method. J. Polym. Sci. Part A: Polym. Chem. 39: 2929–36. 91. Ramos, J., Martín-Molina, A., Sanz-Izquierdo, M.P., Rus, A., Borque, L., Galisteo-González, F., HidalgoÁlvarez, R., and Forcada, J. 2003. Amino-functionalized latex particles obtained by a multi-step method: Development of a new immuno-reagent. J. Polym. Sci. Part A: Polym. Chem. 41: 2404–11. 92. Ramos, J. and Forcada, J. 2006. Modeling the emulsion polymerization of amino-functionalized latex particles. Polymer 47: 1405–13. 93. Ramos, J. and Forcada, J. 2005. Polymeric and colloidal features of latex particles with surface amino groups obtained by semicontinuous seeded cationic emulsion polymerization. J. Polym. Sci. Part A: Polym. Chem. 43: 3878–86. 94. Forcada, J. and Hidalgo-Álvarez, R. 2005. Functionalized polymer colloids: Synthesis and colloidal stability. Curr. Org. Chem. 9: 1067–84. 95. Forcada, J. 2000. Functionalized latex particles for immunoassays. Recent Res. Dev. Polym. Sci. 4: 107–22. 96. Bucsi, A., Forcada, J., Gibanel, S., Heroguez, V., Fontanille, M., and Gnanou, Y. 1998. Monodisperse polystyrene latex particles functionalized by the macromonomer technique. Macromolecules 31: 2087–97.

284


97. Gibanel, S., Heroguez, V., and Forcada, J. 2001. Surfactants characteristics of PS-PEO macromonomers in aqueous solution and on polystyrene latex particles. Two-step emulsion polymerizations. J. Polym. Sci. Part A: Polym. Chem. 39: 2767–76. 98. Gibanel, S., Forcada, J., Heroguez, V., Schappacher, M., and Gnanou, Y. 2001. Novel gemini-type reactive dispersants based on PS/PEO block copolymers: Synthesis and application. Macromolecules 34: 4451–8. 99. Gibanel, S., Heroguez, V., Gnanou, Y., Aramendia, E., Bucsi, A., and Forcada, J. 2001. Monodisperse polystyrene latex particles functionalized by the macromonomer technique. II. Application in immunodiagnosis. Polym. Adv. Technol. 12: 494–9. 100. Gibanel, S., Heroguez, V., Forcada, J., and Gnanou, Y. 2002. Dispersion polymerization of styrene in ethanol–water mixture using polystyrene-b-poly(ethylene oxide) macromonomers as stabilizers. Macromolecules 35: 2467–73. 101. Charleaux, B., Fanget, P., and Pichot, C. 1992. Radical-initiated copolymers of styrene and p-formylstyrene, 2. Preparation and characterization of emulsifier-free copolymer latexes. Makromol. Chem. 193: 205–20. 102. Charreyre, M.T., Boullanger, P., Delair, T., Mandrand, B., and Pichot, C. 1993. Preparation and characterization of polystyrene latexes bearing disaccharide surface groups. Colloid Polym. Sci. 271: 668–79. 103. Ganachaud, F., Mouterde, G., Delair, T., Elaissari, A., and Pichot, C. 1995. Preparation and characterization of cationic polystyrene latex particles of different aminated surface charges. Polym. Adv. Technol. 6: 480–8. 104. Taniguchi, T., Duracher, D., Delair, T., Elaissari, A., and Pichot, C. 2003. Adsoption/desorption behavior and covalent grafting of an antibody onto cationic amino-functionalized poly(styrene-N-isopropylacrylamide) core–shell latex particles. Colloids Surf. B: Biointerfaces 29: 53–65. 105. Santos, A.M., Elaissari, A., Martinho, J.M.G., and Pichot, C. 2005. Synthesis of cationic poly(methyl methacrylate)-poly(N-isopropyl acrylamide) core–shell latexes via two-stage emulsion copolymerization. Polymer 46: 1181–8. 106. Peula, J.M., Hidalgo-Alvarez, R., and de las Nieves, F.J. 1998. Covalent binding of proteins to acetalfunctionalized latexes. I. Physics and chemical adsoption and electrokinetic characterization. J. Colloid Interface Sci. 201: 132–8. 107. Peula, J.M., Hidalgo-Alvarez, R., and de las Nieves, F.J. 1998. Covalent binding of proteins to acetalfunctionalized latexes. II. Colloidal stability and immunoreactivity. J. Colloid Interface Sci. 201: 139–45. 108. Miraballes-Martinez, I., Martin-Rodriguez, A., Martin, A., and Hidalgo-Alvarez. 1997. Chloroactivated latex particles for covalent coupling of antibodies. Application to immunoassays. J. Biomater. Sci. Polym. Ed. 8: 765–77. 109. Konings, B.L.J.C., Pelssers, E.G.M., Verhoeven, A.J.C.M., and Kamps, K.M.P. 1993. Covalent coupling of antibodies to hydrophilic core–shell particles. Colloids Surf. B: Biointerfaces 1: 69–73. 110. Kim, H.S., Lee, J.Y., Lee, D.Y., and Kim, J.H. 1999. Film properties of reactive latexes modified with epoxy and carboxyl surface functional groups. Polymer (Korea) 23: 852–60. 111. Meifang, Z., Adler, H.J., Yanmo, C., and Zhilian, T. 1999. Study on synthesis and characterization of functional PS microspheres modified by polyperoxide (PPX). J. China Textile. Univ. 16: 11–15. 112. Giacomelli, C.E., Vermeer, A.W.P., and Norde, W. 2000. Adsorption of immunoglobulin G on core–shell latex particles precoated with chaps. J. Colloid Interface Sci. 231: 283–8. 113. Slawinski, M., Meuldijk, J., Van Herk, A.M., and German, A.L. 2000. Seeded emulsion polymerization of styrene: Incorporation of acrylic acid in latex products. J. Appl. Polym. Sci. 78: 875–85. 114. Deppe, O., Subat, M., and Yaacoub, E.J. 2003. Surface functionalization of styrene/butyl acrylate latex with a water-soluble monosaccharide monomer. Synthesis and morphology of the polymer particles. Polym. Adv. Technol. 14: 409–21. 115. Kang, K., Kan, C., Du, Y., and Liu, D. 2005. Synthesis and properties of soap-free poly(methyl methacrylateethyl acrylate-methacrylic acid) latex particles prepared by seeded emulsion polymerization. Eur. Polym. J. 41: 439–45. 116. Cernakova, L., Chrastova, V., and Volfova, P. 2005. Nanosized polystyrene/poly(butyl acrylate) core– shell latex particles functionalized with acrylamides. J. Macromol. Sci. Pure Appl. Chem. A42: 427–39. 117. Kim, H. Daniels, E.S., Dimonie, V.L., and Klein, A. 2006. Preparation and characterization of imidazolefunctionalized microspheres. J. Appl. Polym. Sci. 102: 5753–62. 118. Kawaguchi, H., Hoshino, H., Amagasa, H., and Ohtsuma, Y. 1984. Modifications of a polymer latex. J. Colloid Interface Sci. 97: 465–75. 119. Voronov, S., Tokarev, V., Datsyuk, V., and Kozar, M. 1996. Peroxidation of the interface of colloidal systems as new possibilities for design of compounds. Prog. Colloid Polym. Sci. 101: 189–93.


285

120. Li, P., Liu, J.H., Yiu, H.P., and Chan, K.K. 1997. Surface functionalization of polymer latex particles. I. Catalytic Oxidation of poly(methylstyrene) latex particles in the presence of an anionic surfactant. J. Polym. Sci. Part A: Polym Chem. 35: 1863–72. 121. Li, P., Liu, J.H., Wong, T.K., Yiu, H.P., and Gau, J. 1997. Surface functionalization of polymer latexparticles. 2. Catalylic-oxidation of poly(methylstyrene) latexes in the presence of cetylmethylammonium bromide. J. Polym. Sci. Part A: Polym Chem. 35: 3585–93. 122. Li, P., Xu, J., and Wu, C. 1998. Surface functionalization of polymer latex particles. III. A convenient method of producing ultrafine poly(methylstyrene) latexes with aldehyde groups on the surface. J. Polym. Sci. Part A: Polym Chem. 36: 2103–9. 123. Li, P., Xu, J., Wang, Q., and Wu, C. 2000. Surface functionalization of polymer latex particles: 4. Taylormaking of aldehyde-functional poly(methylstyrene) latexes in an emulsifier-free system. Langmuir 16: 4141–7. 124. Li, P. 2002. A novel surface functionalization method for producing carboxyl-functional poly(methyl styrene) latexes. ACS Symposium Series, 801 (Polymer Colloids): 293–306. 125. Banerjee, P., Irvine, D.J., Mayes, A.M., and Griffith, L.G. 2000. Polymer latexes for cell-resistant and cell-interactive surfaces. J. Biomed. Mater. Res. 50: 331–9. 126. Nakashima, K., Gong, Y., Watanabe, T., Suzuki, N., and Tokunou, K. 2001. Photochemistry of organic dyes on the surface of polystyrene latex particles in aqueous dispersions. 2001. Abstract of papers, 222nd ACS National meeting, Chicago, IL. 127. Bousalem, S., Yassar, A., Basinska, T., Miksa, B., Slomkowski, S., Azioune, A., and Chechini, M.M. 2003. Synthesis, characterization and biomedical applications of functionalized polypyrrole-coated polystyrene latex particles. Polym. Adv. Technol. 14: 820–5. 128. Bousalem, S., Mangeney, C., Chechini, M.M., Basinska, T., Miksa, B., and Slomkowsli, S. 2004. Synthesis, characterization and potential biomedical applications of N-succinimidyl ester functionalized, polypyrrole-coated polystyrene latex particles. Colloid Polym. Sci. 282: 1301–7. 129. Bousalem, S., Benabderrahmane, S., Sang, Y.Y.C., Mangeney, C., and Chechini, M.M. 2005. Covalent immobilization of human serum albumin onto reactive polypyrrole-coated polystyrene latex particles. J. Mater. Chem. 15: 3109–16. 130. Imbert-Laurenceau, E., Berger, M.C., Pavon-Djavid, G., Jouan, A., and Migonney, V. 2005. Surface modification of polystyrene particles for specific antibody adsorption. Polymer 46: 1277–85. 131. Vihola, H., Martila, A.K., Pakkanen, J.S., Andersson, M., Laukkanen, A., Kaukonen, A.M., Tenhu, H., and Hirvonen, J. 2007. Cell-polymer interactions of fluorescent polystyrene latex particles coated with thermosensitive poly(N-isopropylacrylamide) and poly(N-vinylcaprolactam) or grafted with poly(ethylene oxide)-macromonomer. Int. J. Pharm. 343: 238–46. 132. Kolb, H.C., Finn, M.G., and Sharpless, K.B. 2001. Click chemistry: Diverse chemical function from a few good reactions. Angew. Chem. Int. Ed. 40: 2004–21. 133. Huisgen, R. 1963. 1,3-Dipolar cycloadditions. Past and future. Angew. Chem. Int. Ed. 2: 565–98. 134. Hawker, C.J. and Wooley, K.L. 2005. The convergence of synthetic organic and polymer chemistries. Science 309: 1200–4. 135. Johnson, J.A., Finn, M.G., Koberstein, J.T., and Turro, N.J. 2008. Construction of linear polymers, dendrimers, networks, and other polymeric architectures by copper-catalyzed azide–alkyne cycloaddition “click” chemistry. Macromol. Rapid. Commun. 29: 1052–72. 136. Lutz, J.F. 2007. 1,3-Dipolar cycloadditions of azides and alkynes: A universal ligation tool in polymer and materials science. Angew. Chem. Int. Ed. 46: 1018–25. 137. Hawker, C.J., Fokin, V.V., Finn, M.G., and Sharpless, K.B. 2007. Bringing efficiency to materials synthesis: The philosophy of click chemistry. Aust. J. Chem. 60: 381–3. 138. Evans, C. and Lovell, P.A. 2008. Click chemistry as a route to latex particles functionalisation. Abstracts of Communications. UK Polymer Colloids Forum, University of Greenwich, August 28–29. 139. Opsteen, J.A. and van Hest, J.C.M. 2005. Modular synthesis of block copolymers via cycloaddition of terminal azide and alkyne functionalized polymers. Chem. Commun. 1: 57–9. 140. Baek, K.Y., Kamigaito, M., and Sawamoto, M. 2002. Synthesis of end-functionalized poly(methyl methacrylate) by ruthenium-catalyzed living radical polymerization with functionalized initiators. J. Polym. Sci. Part A: Polym. Chem. 40: 1937–44. 141. Opsteen, J.A. and Van Hest, J.C.M. 2007. Modular síntesis of ABC type block copolymers by “click” chemistry. J. Polym. Sci. Part A: Polym. Chem. 45: 2913–24. 142. Agut, W., Taton, D., and Lecommandoux, S. 2007. A versatile synthetic approach to polypeptide based rod-coil block copolymers by click chemistry. Macromolecules 40: 5653–61.

286


143. Thibault, R.J., Takizawa, K., Lowenhein, P., Helms, B., Mynar, J.L., Fréchet, J.M.J., and Hawker, C.J. 2006. A versatile new monomer family: functionalized 4-vinyl-1,2,3-triazoles via click chemistry. J. Am. Chem. Soc. 128: 12084–5. 144. O’Reilly, R.K., Joralemon, M.J., Hawker, C.J., and Wooley, K.L. 2006. Facile syntheses of surfacefunctionalized micelles and shell cross-linked nanoparticles. J. Polym. Sci. Part A: Polym. Chem. 44: 5203–17. 145. O’Reilly, R.K., Joralemon, M.J., Wooley, K.L., and Hawker, C.J. 2005. Functionalization of micelles and shell cross-linked nanoparticles using click chemistry. Chem. Mater. 17: 5976–88. 146. O’Reilly, R.K., Joralemon, M.J., Hawker, C.J., and Wooley, K.L. 2007. Preparation of orthogonallyfunctionalized core click cross-linked nanoparticles. New J. Chem. 31: 718–24. 147. Prucker, O. and Rühe, J. 1998. Synthesis of poly(styrene) monolayers attached to high surface area silica gels through self-assembled monolayers of azo initiators. Macromolecules 31: 592–601. 148. Prucker, O. and Rühe, J. 1998. Mechanism of radical chain polymerization initiated by azo compounds covalently bound to the surface of spherical particles. Macromolecules 31: 602–13. 149. Hritcu, D., Müller, W., and Brooks, D.E. 1999. Poly(styrene) latex carrying cerium(IV)-initiated terminally attached cleavage chains: Analysis of grafted chains and model of the surface layer. Macromolecules 32: 565–73. 150. Guo, X., Weiss, A., and Ballauff, M. 1999. Synthesis of spherical polyelectrolyte brushes by photoemulsion polymerization. Macromolecules 32: 6043–6. 151. Braunecker, W.A. and Matyjaszewski, K. 2007. Controlled/living radical polymerization: Features, developments and perspectives. Prog. Polym. Sci. 32: 93–146. 152. Hodges, J.C., Harikrishnan L.S., and Ault-Justus, S. 2000. Preparation of designer resins via living free radical polymerization of functional monomers on solids support. J. Comb. Chem. 2: 80–8. 153. Bian, K. and Cunningham, M.F. 2005. Synthesis of polymeric microspheres from a Merrifield resin by surface-initiated nitroxide-mediated radical polymerization. J. Polym. Sci. Part A: Polym. Chem. 43: 2145–54. 154. Bian, K. and Cunningham, M.F. 2006. Surface-initiated nitroxide-mediated radical polymerization of 2-(dimethylamino)ethyl acrylate on polymeric microspheres. Polymer 47: 5744–53. 155. Wang, X.S., Jackson, R.A., and Armes, S.P. 2000. Facile synthesis of acidic copolymers via atom transfer radical polymerization in aqueous media at ambient temperature. Macromolecules 33: 255–7. 156. Gaynor, S., Qiu, J., and Matyajaszewski, K. 1998. Controlled/living radical polymerization applied to water-borne system. Macromolecules 31: 5951–7. 157. Mittal, V., Matsko, N.B., Butté, A., and Morbidelli, M. 2007. Functionalized polystyrene latex particles as substrates for ATRP: Surface and colloidal characterization. Polymer 48: 2806–17. 158. Mittal, V., Matsko, N.B., Butté, A., and Morbidelli, M. 2007. Synthesis of temperature responsive polymer brushes from polystyrene latex particles functionalized with ATRP initiator. Eur. Polym. J. 43: 4868–81. 159. Jayachandran, K.N., Takacs-Cox, A., and Brooks, D.E. 2002. Synthesis and characterization of polymer brushes of poly(N,N-dimethylacrylamide) from polystyrene latex by aqueous atom transfer radical polymerization. Macromolecules 35: 4247–57. 160. Kizakkedathu, J.N. and Brooks, D.E. 2003. Synthesis of poly(N,N-dimethylacrylamide) brushes from charged polymeric surfaces by aqueous ATRP: Effect of surface initiator concentration. Macromolecules 36: 591–8. 161. Kizakkedathu, J.N., Norris-Jones, R., and Brooks, D.E. 2004. Synthesis of well-defined environmentally responsive polymer brushes by aqueous ATRP. Macromolecules 37: 734–43. 162. Kizakkedathu, J.N., Kumar, K.R., Goodman, D., and Brooks, D.E. 2004. Synthesis and characterization of well-defined hydrophilic block copolymer brushes by aqueous ATRP. Polymer 45: 7471–89. 163. Guerrini, M.M., Charleux, B., and Vairon J.P. 2000. Functionalized latexes as substrates for atom transfer radical polymerization. Macromol. Rapid Commun. 21: 669–74. 164. Chen, Y., Kang, E.T., Neoh, K.G., and Greiner, A. 2005. Preparation of hollow silica nanospheres by surface-initiated atom transfer radical polymerization on polymer latex templates. Adv. Funct. Mater. 15: 113–17. 165. Zhang, M., Liu, L., Zhao, H., Yang, Y., Fu, G., and He, B. 2006. Double responsive polymer brushes on the surface of colloid particles. J. Colloid Interface Sci. 301: 85–91. 166. Zhang, M., Liu, L., Wu, C., Fu, G., Zhao, H., and He, B. 2007. Synthesis, characterization and application of well-defined environmentally responsive polymer brushes on the surface of colloid particles. Polymer 48: 1989–97.


287

167. Ranganathan, K., Deng, R., Kainthan, R.K., Wu, C., Brooks, D.E, and Kizakkedathu, J.N. 2008. Synthesis of thermoresponsive mixed arm star polymers by combination of RAFT and ATRP from a multifunctional core and its self-assembly in water. Macromolecules 41: 4226–34. 168. Angot, S., Ayres, N., Bon, S.A.F., and Haddleton, D.M. 2001. Living radical polymerization immobilized on Wang resins: Synthesis and harvest of narrow polydispersity poly(methacrylate)s. Macromolecules 34: 768–74. 169. Ayres, N., Haddleton, D.M., Shooter, A.J., and Pears, D.A. 2002. Synthesis of hydrophilic polar supports based on poly(dimethylacrylamide) via copper-mediated radical polymerization from a cross-linked polystyrene surface: Potential resins for oligopeptide solid-phase synthesis. Macromolecules 35: 3849–55. 170. Zheng, G. and Stöver, D.H. 2002. Grafting of polystyrene from narrow disperse polymer particles by surface-initiated atom transfer radical polymerization. Macromolecules 35: 6828–34. 171. Zheng, G. and Stöver, D.H. 2002. Grafting of poly(alkyl (meth)acrylates) from swellable poly(DVB80co-HEMA) microspheres by atom transfer radical polymerization. Macromolecules 35: 7612–19. 172. Min, K., Hu, J., Wang, C., and Elaissari, A. 2002. Surface modification of polystyrene latex particles via atom transfer radical polymerization. J. Polym. Sci. Part A: Polym. Chem. 40: 892–900. 173. Botempo, D., Tirelli, N., Feldman, K., Masci, G., Crescenzi, V., and Hubbell, J.A. 2002. Atom transfer radical polymerization as a tool for surface functionalization. Adv. Mater. 14: 1239–41. 174. Barner, L., Li, C., Hao, X., Stenzel, M.H., Barner-Kowollik, C., and Davis, T.P. 2004. Synthesis of core– shell poly(divinylbenzene) microspheres via reversible addition fragmentation chain transfer graft polymerization of styrene. J. Polym. Sci. Part A: Polym. Chem. 42: 5067–76. 175. Jesberger, M., Barner, L. Stenzel, M.H. Malmstrom, E., and Davis, T.P., Barner-Kowollik, C. 2003. Hyperbranched polymers as scaffolds for multifunctional reversible addition fragmentation chain-transfer agents: A route to polystyrene-core-polyesters and polystyrene-block-poly(butyl acrylate)-corepolyesters. J. Polym. Sci. Part A: Polym. Chem. 41: 3847–61. 176. Perrier, S., Takolpuckdee, P., and Mars, C.A. 2005. Reversible addition–fragmentation chain transfer polymerization mediated by a solid supported chain transfer agent. Macromolecules 38: 6770–4.

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

290


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,

Fractal Structures and Aggregation Kinetics of Protein-Functionalized Colloidal Particles

291

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

292


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


293

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)

294


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


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.

296

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.


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


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


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


[ 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


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


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


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


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,


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


/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


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.


311

REFERENCES 1. Ortega-Vinuesa, J.L., J.A. Molina-Bolívar, and R. Hidalgo-Álvarez. 1996. Particle enhanced immunoaggregation of F(ab¢)2 molecules. Journal of Immunological Methods 190: 29–38. 2. Binks, B.P., D. Chatenay, C. Nicot, W. Urbach, and M. Waks. 1989. Structural parameters of the myelin transmembrane proteolipid in reverse micelles. Biophysical Journal 55 (5): 949–955. 3. Dimitrova, M.N., R. Tsekov, H. Matsumura, and K. Furusawa. 2000. Size dependence of protein-induced flocculation of phosphatidylcholine liposomes. Journal of Colloid and Interface Science 226 (1): 44–50. 4. Tirado-Miranda, M., A. Schmitt, J. Callejas-Fernandez, and A. Fernandez-Barbero. 2003. Aggregation of protein-coated colloidal particles: Interaction energy, cluster morphology, and aggregation kinetics. Journal of Chemical Physics 119 (17): 9251–9259. 5. Fleer, G.J. and J. Lyklema. 1974. Polymer adsorption and its effect on stability of hydrophobic colloids. 2. Flocculation process as studied with silver iodide polyvinyl alcohol system. Journal of Colloid and Interface Science 46 (1): 1–12. 6. Pelssers, E.G.M., M.A.C. Stuart, and G.J. Fleer. 1990. Kinetics of bridging flocculation—role of relaxations in the polymer layer. Journal of the Chemical Society, Faraday Transactions 86 (9): 1355–1361. 7. Stoll, S. and J. Buffle. 1996. Computer simulation of bridging flocculation processes: The role of colloid to polymer concentration ratio on aggregation kinetics. Journal of Colloid and Interface Science 180 (2): 548–563. 8. Adachi, Y. and T. Wada. 2000. Initial stage dynamics of bridging flocculation of polystyrene latex spheres with polyethylene oxide. Journal of Colloid and Interface Science 229 (1): 148–154. 9. Dickinson, E. and S.R. Euston. 1992. Short-range structure of simulated flocs of particles with bridging polymer. Colloids and Surfaces 62 (3): 231–242. 10. Smalley, M.V., H.L.M. Hatharasinghe, I. Osborne, J. Swenson, and S.M. King. 2001. Bridging flocculation in vermiculite-PEO mixtures. Langmuir 17 (13): 3800–3812. 11. Biggs, S., M. Habgood, G.J. Jameson, and Y.D. Yan. 2000. Aggregate structures formed via a bridging flocculation mechanism. Chemical Engineering Journal 80 (1–3):13–22. 12. Glover, S.M., Y.D. Yan, G.J. Jameson, and S. Biggs. 2000. Bridging flocculation studied by light scattering and settling. Chemical Engineering Journal 80 (1–3): 3–12. 13. Swenson, J., M.V. Smalley, and H.L.M. Hatharasinghe. 1998. Mechanism and strength of polymer bridging flocculation. Physical Review Letters 81 (26): 5840–5843. 14. Lin, M.Y., H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein, and P. Meakin. 1989. Universality in colloid aggregation. Nature 339 (6223): 360–362. 15. Broide, M.L. and R.J. Cohen. 1992. Measurements of cluster-size distributions arising in salt-induced aggregation of polystyrene microspheres. Journal of Colloid and Interface Science 153 (2): 493–508. 16. Gonzalez, A.E. 1993. Universality of colloid aggregation in the reaction limit—the computer-simulations. Physical Review Letters 71 (14): 2248–2251. 17. Shih, W.Y., I.A. Aksay, and R. Kikuchi. 1987. Reversible-growth model—cluster-cluster aggregation with finite binding-energies. Physical Review A 36 (10): 5015–5019. 18. Jin, J.M., K. Parbhakar, L.H. Dao, and K.H. Lee. 1996. Gel formation by reversible cluster-cluster aggregation. Physical Review E 54 (1): 997–1000. 19. Haw, M.D., M. Sievwright, W.C.K. Poon, and P.N. Pusey. 1995. Cluster-cluster gelation with finite bondenergy. Advances in Colloid and Interface Science 62 (1): 1–16. 20. Liu, J., W.Y. Shih, M. Sarikaya, and I.A. Aksay. 1990. Fractal colloidal aggregates with finite interparticle interactions—energy-dependence of the fractal dimension. Physical Review A 41 (6): 3206–3213. 21. Burns, J.L., Y.D. Yan, G.J. Jameson, and S. Biggs. 2002. The effect of molecular weight of nonadsorbing polymer on the structure of depletion-induced flocs. Journal of Colloid and Interface Science 247 (1): 24–32. 22. Dhont, J.K.G. 1996. An Introduction to Dynamics of Colloids, Studies in Interface Science, 2. Elsevier, Amsterdam, the Netherlands. 23. Chen, S.H. and J. Teixeira. 1986. Structure and fractal dimension of protein-detergent complexes. Physical Review Letters 57 (20): 2583–2586. 24. Smoluchowski, M. v. 1916. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Zeitschrift für Physikalische Chemie 92: 129–168. 25. Odriozola, G., A. Schmitt, J. Callejas-Fernandez, R. Martinez-Garcia, and R. Hidalgo-Álvarez. 1999. Dynamic scaling concepts applied to numerical solutions of Smoluchowski’s rate equation. Journal of Chemical Physics 111 (16): 7657–7667.

312


26. Van Dongen, P.G.J. and M.H. Ernst. 1985. Dynamic scaling in the kinetics of clustering. Physical Review Letters 54 (13): 1396–1399. 27. Asnaghi, D., M. Carpineti, M. Giglio, and M. Sozzi. 1992. Coagulation kinetics and aggregate morphology in the intermediate regimes between diffusion-limited and reaction-limited cluster aggregation. Physical Review A 45 (2): 1018–1023. 28. Koppel, D.E. 1972. Analysis of macromolecular polydispersity in intensity correlation spectroscopy— method of cumulants. Journal of Chemical Physics 57 (11): 4814–4820. 29. Olivier, B.J. and C.M. Sorensen. 1990. Evolution of the cluster size distribution during slow colloid aggregation. Journal of Colloid and Interface Science 134 (1): 139–146. 30. Tirado-Miranda, M., A. Schmitt, J. Callejas-Fernandez, and A. Fernandez-Barbero. 2000. Test of the physical interpretation of the structural coefficient for colloidal clusters. Langmuir 16 (19): 7541–7544. 31. Tirado-Miranda, M., M.A. Rodriguez-Valverde, A. Schmitt, J. Callejas-Fernandez, and A. FernandezBarbero. 2005. Structural coefficients in aggregates of protein-coated colloidal particles. Colloids and Surfaces A: Physicochemical and Engineering Aspects 270: 309–316. 32. La Mer, V.K. 1966. Filtration of colloidal dispersions flocculated by anionic and cationic polyelectrolytes. Discussions of the Faraday Society 42: 248–254. 33. Hogg, R. 1984. Collision efficiency factors for polymer flocculation. Journal of Colloid and Interface Science 102 (1): 232–236. 34. Moudgil, B.M., B.D. Shah, and H.S. Soto. 1987. Collision efficiency factors in polymer flocculation of fine particles. Journal of Colloid and Interface Science 119 (2): 466–473. 35. Molski, A. 1989. On the collision efficiency approach to flocculation. Colloid and Polymer Science 267 (4): 371–375. 36. Schmitt, A., M.A. Cabrerizo-Vílchez, R. Hidalgo-Álvarez, and A. Fernández-Barbero. 1998. On the identification of bridging flocculation: An extended collision efficiency model. Progress in Colloid and Polymer Science 110: 105–109. 37. Lee, W.K., J.S. Ko, and H.M. Kim. 2002. Effect of electrostatic interaction on the adsorption of globular proteins on octacalcium phosphate crystal film. Journal of Colloid and Interface Science 246 (1): 70–77. 38. Norde, W. and T. Zoungrana. 1998. Surface-induced changes in the structure and activity of enzymes physically immobilized at solid/liquid interfaces. Biotechnology and Applied Biochemistry 28: 133–143. 39. Peula, J.M. and F.J. de las Nieves. 1993. Adsorption of monomeric bovine serum-albumin on sulfonated polystyrene model colloids. 1. Adsorption-isotherms and effect of the surface-charge density. Colloids and Surfaces A: Physicochemical and Engineering Aspects 77 (3): 199–208. 40. Vinuesa, J.L.O., M.J.G. Ruiz, and R. Hidalgo-Álvarez. 1996. F(ab¢)(2)-coated polymer carriers: Electrokinetic behavior and colloidal stability. Langmuir 12 (13): 3211–3220. 41. Norde, W., F. Macritchie, G. Nowicka, and J. Lyklema. 1986. Protein adsorption at solid liquid interfaces—reversibility and conformation aspects. Journal of Colloid and Interface Science 112 (2): 447–456. 42. Schüler, J., J. Frank, W. Saenger, and Y. Georgalis. 1999. Thermally induced aggregation of human transferrin receptor studied by light-scattering techniques. Biophysical Journal 77 (2): 1117–1125. 43. Tirado-Miranda, M., A. Schmitt, J. Callejas-Fernandez, and A. Fernandez-Barbero. 1999. Colloidal clusters with finite binding energies: Fractal structure and growth mechanism. Langmuir 15 (10): 3437–3444. 44. Delgado Calvo-Flores, J.M. 1999. Desarrollo de un inmuno-ensayo óptico de aglutinación coloidal. Estudio cinetico y morfologico, Universidad de Granada. 45. Elgersma, A.V., R.L.J. Zsom, W. Norde, and J. Lyklema. 1990. The adsorption of bovine serum-albumin on positively and negatively charged polystyrene lattices. Journal of Colloid and Interface Science 138 (1): 145–156. 46. Vincent, B., J. Edwards, S. Emmett, and A. Jones. 1986. Depletion flocculation in dispersions of stericallystabilized particles (soft spheres). Colloids and Surfaces 18 (2–4): 261–281. 47. Witten, T.A. and L.M. Sander. 1981. Diffusion-limited aggregation, a kinetic critical phenomenon. Physical Review Letters 47 (19): 1400–1403. 48. Adachi, Y. 1995. Dynamic aspects of coagulation and flocculation. Advances in Colloid and Interface Science 56: 1–31. 49. Oh, C. and C.M. Sorensen. 1997. The effect of overlap between monomers on the determination of fractal cluster morphology. Journal of Colloid and Interface Science 193 (1): 17–25. 50. Samson, R.J., G.W. Mulholland, and J.W. Gentry. 1987. Structural-analysis of soot agglomerates. Langmuir 3 (2): 272–281. 51. Dietler, G., C. Aubert, D.S. Cannell, and P. Wiltzius. 1986. Gelation of colloidal silica. Physical Review Letters 57 (24): 3117–3120.


313

52. Martin, J.E. and B.J. Ackerson. 1985. Static and dynamic scattering from fractals. Physical Review A 31 (2): 1180–1182. 53. Lin, M.Y., R. Klein, H.M. Lindsay, D.A. Weitz, R.C. Ball, and P. Meakin. 1990. The structure of fractal colloidal aggregates of finite extent. Journal of Colloid and Interface Science 137 (1): 263–280. 54. Moncho-Jorda, A., G. Odriozola, M. Tirado-Miranda, A. Schmitt, and R. Hidalgo-Álvarez. 2003. Modeling the aggregation of partially covered particles: Theory and simulation. Physical Review E 68 (1): 011404-1–011404-12. 55. Meakin, P. and M.H. Ernst. 1988. Scaling in aggregation with breakup simulations and mean-field theory. Physical Review Letters 60 (24): 2503–2506. 56. Cai, J., N.L. Lu, and C.M. Sorensen. 1995. Analysis of fractal cluster morphology parameters—structural coefficient and density autocorrelation function cutoff. Journal of Colloid and Interface Science 171 (2): 470–473. 57. Sorensen, C.M. and G.C. Roberts. 1997. The prefactor of fractal aggregates. Journal of Colloid and Interface Science 186 (2): 447–452. 58. Köylü, U.O., Y.C. Xing, and D.E. Rosner. 1995. Fractal morphology analysis of combustion-generated aggregates using angular light scattering and electron microscope images. Langmuir 11 (12): 4848–4854. 59. Tirado-Miranda, M., Schmitt, A., Callejas-Fernandez, J., and Fernández-Barbero, A. 2003. The aggregation behaviour of protein-coated particles: a light scattering study. European Biophysics Journal 32: 128–136.

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


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

2

4 6 8 10 Particle diameter (nm)

12

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.)


319

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.

320


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.

Advances in the Preparation and Biomedical Applications of Magnetic Colloids (a)

(o/w) magnetic emulsion

+ oo

Monomers initiator

Organic phase

321

–c

Iron oxide

coo–

Non ionic surfactant coo–

Ionic surfactant Aqueous phase

+c

oo –

(b)

200 nm

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

322


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.


323

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

324


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


325

(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)

326


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.)

Advances in the Preparation and Biomedical Applications of Magnetic Colloids Label (enzyme)

327

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

328

Structure and Functional Properties of Colloidal Systems 5 mm

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

Advances in the Preparation and Biomedical Applications of Magnetic Colloids Magnetic particle

Z

329

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

330

Structure and Functional Properties of Colloidal Systems (a) Probe DNA Polymer GMR-sensor (b) Biotin

SiO2

Analyte DNA Hybridized analyte DNA Polymer

GMR-sensor

SiO2

Streptavidin

(c)

Magnetic marker

Polymer GMR-sensor

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.)


331

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 > 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.)

332


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

Sample outlet

F-IgG compartment

(a)

D

Sample inlet

Particle surface:

Sample compartment IgG coated magnetic particles

Detection compartment

Sample flow

(b) D

-IgG

Sample flow No flow

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.)


333

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.

REFERENCES 1. Manz, A., Fettinger, J. C., Verpoore, E., Lüdi, H., Widmer, H. M., and Harrison, D. J. 1991. Micromachining of crystalline silicon and glass for chemical analysis systems: A look into next century’s technology or just a fashionable craze? Tr. Anal. Chem. 10: 144–149. 2. Verpoorte, E. 2003. Beads and chips: New recipes for analysis. Lab Chip 3: 60–68. 3. Andersson, H., Van der Wijngaart, W., Enkosson, P., and Stemme, G. 2000. Micromachined flow-through filter-chamber for chemical reactions on beads. Sens. Actuat. B 67: 203–208. 4. Elaissari, A. 2008. Nanocolloids in Biotechnology. Hoboken, NY: Wiley. 5. Rishton, A., Lu. Y., Altman, R. A., Marley, A. C., Bian Hahnes, C., Viswanathan, R., Xiao, G., Gallagher, W. J., and Parkin, S. S. P. 1997. Magnetic tunnel junctions fabricated at tenth-micron dimensions by electron beam lithography. Microelectron. Eng. 35: 249–252. 6. Lee, C. S., Lee, H., and Westervelt, R. M. 2001. Microelectromagnets for the control of magnetic nanoparticles. Appl. Phys. Lett. 79: 3308–3310. 7. Charles, S. W. 1992. Magnetic Fluids (Ferrofluids). North-Holland: Elsevier. 8. Gupta, A. K. and Gurtis, A. S. G. 2004. Lactoferrin and ceruloplasmin derivatized superparamagnetic iron oxide nanoparticles for targeting cell surface receptors. Biomaterials 25: 3029–3040.

334


9. Yang, A., Park, S. B., Yoon, H. G., Huh, Y. M., and Haam, S. 2006. Preparation of poly-caprolactone nanoparticles containing magnetite for magnetic drug carrier. Int. J. Pharm. 324: 185–190. 10. Massart, R. 1981. Preparation of aqueous magnetic liquids in alkaline and acidic media. IEEE Trans. Magn. 17: 1247–1248. 11. Bacri, J. C., Perzynski, R., Salin, D., Cabuil, V., and Massart, R. 1990. Ionic ferrofluids: A crossing of chemistry and physics. J. Magn. Magn. Mater. 85: 27–32. 12. Khalafalla, S. E. 1975. Magnetic fluids. ChemTech 5: 540–546. 13. Khalafalla, S. E. and Reimers, G. W. 1980. Preparation of dilution-stable aqueous magnetic fluids. IEEE. Trans. Magn. 16: 178–183. 14. Bailey, R. L. 1983. Lesser known applications of ferrofluides. J. Magn. Magn. Mater. 39: 178–182. 15. Gupta, A. K. and Gupta, M. 2005. Synthesis and surface engineering of iron oxide nanoparticles for biomedical applications. Biomaterials 26: 3995–4021. 16. Holm, C. and Weis, J. J. 2005. The structure of ferrofluids: A status report. Curr. Opin. Colloid Interface Sci. 10: 133–140. 17. Graf, G., Dembski, S., Hofmannn, A., and Rühl, E. 2006. A general method for the controlled embedding of nanoparticles in silica colloids. Langmuir 22: 5604–5610. 18. De Palma, R., Trekker, J., Peeters, S., Van Bael, M. J., Bonroy, K., Wirix-Speetjens, R., Reekmans, G., Laureyn, W., Borghs, G., and Maes, G. 2007. Surface modification of gamma-Fe2O3SiO2 magnetic nanoparticles for the controlled interaction with biomolecules. J. Nanosci. Nanotechnol. 7: 4626–4641. 19. Núñez, N. O., Tartaj, P., Morales, M. P., Bonville, P., and Serna Yttria, C. J. 2004. Coated FeCo magnetic nanoneedles. Chem. Mater. 16: 3119–3124. 20. Lu, H., Yi, G., Zhao, C., Chen, D., Liang-Hong, G., and Cheng, J. 2004. Synthesis and characterization of multi-functional nanoparticles possessing magnetic, up-conversion fluorescence and bio-affinity properties. J. Mater. Chem. 14: 1336–1341. 21. Mandal, M., Kundu, S., Ghosh, S. K., Panigrahi, S., Sau, T. K., Yusuf, S. M., and Pal, T. 2005. Magnetite nanoparticles with tunable gold or silver shell. J. Colloid Interface Sci. 286: 187–194. 22. Pita, M., Abad, J. M., Vaz-Dominguez, C., Briones, C., Mateo-Marti, E., Martin-Gago, J. A., Puerto Morales, M., and Fernandez, V. M. 2008. Synthesis of cobalt ferrite core/metallic shell nanoparticles for the development of a specific PNA/DNA biosensor. J. Colloid Interface Sci. 321: 484–492. 23. Ulman, A. 1996. Formation and structure of self-assembled monolayers. Chem. Rev. 96: 1533–1554. 24. Rocchiccioli-Deltcheff, C., Franck, R., Cabuil, V., and Massart, R. 1987. Surfacted ferrofluids: Interactions at the surfactant-magnetic iron oxide interface. J. Chem. Res. 5: 126–127. 25. Barnakov, Y. A., Yu, M. H., and Rosenzweig, Z. 2005. Manipulation of the magnetic properties of magnetite-silica nanocomposite materials by controlled Stober synthesis. Langmuir 21: 7524–7527. 26. Sahoo, Y., Pizem, H., Fried, T., Golodnitsky, D., Burstein, L., Sukenik, C. N., and Markovich, G. 2001. Alkyl phosphonate/phosphate coating on magnetite nanoparticles a comparison with fatty acids. Langmuir 17: 7907–7911. 27. Hamoudeh, M., Al Faraj, A., Canet-Soulas, E., Bessueille, F., Léonard, D., and Fessi, H. 2007. Elaboration of PLLA-based superparamagnetic nanoparticles: Characterization, magnetic behaviour study and in vitro relaxivity evaluation. Int. J. Pharm. 338: 248–257. 28. Hamoudeh, M. and Fessi, H. 1996. Preparation, characterization and surface study of poly-epsilon caprolactone magnetic microparticles. J. Colloid Interface Sci. 300: 584–590. 29. Butterworth, M. D., Bell, S. A., Armes, S. P., and Simpson, A. W. 1996. Synthesis and characterization of pyrrole-magnetite-silica particles. J. Colloids Interface Sci. 183: 91–99. 30. Arias, J. L., Gallardo, V., Adolfina Ruiz, M., and Delgado, A. V. 2008. Magnetite/poly(alkylcyanoacrylate) (core/shell) nanoparticles as 5-fluorouracil delivery systems for active targeting. Eur. J. Pharm. Biopharm. 69: 54–63. 31. Arias, J. L., Gallardo, V., Gômez-Lopera, S. A., Plaza, R. C., and Delgado, A. V. 2001. Synthesis and characterization of poly(ethyl-2-cyanoacrylate) nanoparticles with a magnetic core. J. Controlled Release 77: 309–321. 32. Elaissari, A. 2005. Magnetic colloids: Preparation and biomedical applications. e-Polymer 28. 33. Elaissari, A., Sauzedde, F., Montagne, F., and Pichot, C. 2003. Preparation of magnetic latices. In: A. Elaissari (Ed.), Colloidal Polymers, Synthesis and Characterization, Vol. 115, pp. 285–318, Surfactant Science Series. Marcel Dekker. 34. Arshady, R. 2001. Microspheres Microcapsules & Liposomes: Radiolabeled and Magnetic Particles in Medicine & Biology, Vol. 3. Citus Books.


335

35. Ugelstad, J., Ellingsen, T., Berge, A., and Helgee, B. 1986. Process for the production of magnetic polymer particles. European Patent 0 106 873. 36. Furusawa, K., Nagashima, K., and Anzai, C. 1994. Synthetic process to control the total size and component distribution of multilayer magnetic composite particles. Colloid. Polym. Sci. 272: 1104–1110. 37. Sauzedde, F., Elaïssari, A., and Pichot, C. 1999. Hydrophilic magnetic polymer latexes. 1- Adsorption of magnetic iron oxide nanoparticles onto various cationic latexes. Colloid. Polym. Sci. 277: 846–859. 38. Sauzedde, F., Elaïssari, A., and Pichot, C. 1999. Hydrophilic magnetic polymer latexes. 2- Encapsulation of adsorbed iron oxide nanoparticles. Colloid. Polym. Sci. 277: 1041–1050. 39. Daniel, J. C., Schuppiser, J. L., and Tricot, M. 1981. Magnetic polymer latex and preparation process, US Patent 4 358 388. 40. Ramirez, L. P. and Landfester, K. 2003. Magnetic polystyrene nanoparticles with a high magnetic content obtained by miniemulsion processes. Macromol. Chem. Phys. 204: 22–31. 41. Joumaa, N., Toussay, P., Lansalot, M., and Elaissari, A. 2008. Surface modification of iron oxide nanoparticles by a phosphate-based macromonomer and further encapsulation into submicrometer polystyrene particles by miniemulsion polymerization. J. Polym. Sci. Part A 46, 327–340. 42. Muller-Schulte, D. and Schmitz-Rode, T. 2006. Thermosensitive magnetic particles as contacless controllable drug carriers. J. Magn. Magn. Mater. 302: 267–271. 43. Montagne, F., Mondain-Monval, O., Pichot, C., and Elaissari, A. 2006. Highly magnetic latexes from submicrometer oil in water ferrofluid emulsion. J. Polym. Sci. Part A 44: 2642–2656. 44. Grüttner, C., Teller, J., Schütt, W., Westphal, F., Schümichen, C., and Paulke, B. R. 1997. Preparation and characterization of magnetic nanospheres for in vivo application. In: U. Hafeli, W. Schutt, J. Teller, and M. Zborowski (Eds), Scientific and Clinical Applications of Magnetic Carriers, pp. 53–67. New York: Plenum Press. 45. Schütt, W., Grüttner, C., Häfeli, U., Zborowski, M., Teller, J., Putzar, H., and Schümichen, C. 1997. Applications of magnetic targeting in diagnosis and therapy—possibilities and limitations: a mini-review. Hybridoma 16: 109–117. 46. Wang, Y., Wang, X., Luo, G., and Dai, Y. 2008. Adsorption of bovin serum albumin (BSA) onto the magnetic chitosan nanoparticles prepared by a microemulsion system. Bioresour. Technol. 99: 3881–3884. 47. Molday, R. S. and Mackenzie, D. 1982. Immunospecific ferromagnetic iron-dextran reagents for the labeling and magnetic separation of cells. J. Immunol. Methods 52: 353–367. 48. Lee, S. J., Jeong, J. R., Shin, S. C., Kim, J. C., Chang, Y. H., Lee, K. H., and Kim, J. D. 2005. Magnetic enhancement of iron oxide nanoparticles encapsulated with poly(d,l-lactide-co-glycolide). Colloids Surf. A: Physicochem. Eng. Aspects 255: 19–25. 49. Chatterjee, J., Haik, Y., and Chen, C. J. 2001. Synthesis and characterization of heat-stabilized albumin magnetic microspheres. Colloid. Polym. Sci. 279: 1073–1081. 50. Chatterjee, J., Haik, Y., and Chen, C. J. 2004. Biocompatible magnetic film: Synthesis and characterization. Biomagn. Res. Technol. Feb 4; 2(1): 2. 51. Lee, J., Isobe, T., and Senna, M. 1996. Preparation of ultrafine Fe3O4 particles by precipitation in the presence of PVA at high pH. J. Colloid Interface Sci. 177: 490–494. 52. Pollert, E., Knížek, K., Maryško, M., Lancˇok, A., Bohácˇek, J., Horák, D., and Babicˇ, M. 2006. Magnetic poly(glycidyl methacrylate) microspheres containing maghemite prepared by emulsion polymerization. J. Magn. Magn. Mater. 306: 241–247. 53. Autenshlyus, A. I., Brusenstov, N. A., and Lockshim, A. 1993. Magnetic-sensitive dextran-ferrite immunosorbents (for diagnostic and therapy). J. Magn. Magn. Mater. 122: 360–363. 54. Couvreur, P. and Vauthier, C. 2006. Nanotechnology: Intelligent design to treat complex disease. Pharm. Res. 23: 1417–1450. 55. Tan, W. B., Huang, N., and Zhang, Y. 2007. Ultrafine biocompatible chitosan nanoparticles encapsulating multi-coloured quantum dots for bioapplications. J. Colloid Interface Sci. 310: 464–470. 56. Pisani, E., Tsapis, N., Paris, J., Nicolas, V., Cattel, L., and Fattal, E. 2006. Polymeric nano/microcapsules of liquid perfluorocarbons for ultrasonic imaging: Physical characterization. Langmuir 22: 4397–4402. 57. Bridot, J., et al. 2007. Hybrid gadolinium oxide nanoparticles: Multimodal contrast agents for in vivo imaging. J. Am. Chem. Soc. 129: 5076–5084. 58. Jordan, A. and Maier-Hauff, K. 2007. Magnetic nanoparticles for intracranial thermotherapy. J. Nanosci. Nanotechnol. 12: 4604–4606. 59. Chiang, C. L., Sung, C. S., Wu, C. Y., and Hsu, C. Y. 2005. Application of superparamagnetic nanoparticles in purification of plasmid DNA from bacterial cells. J. Chromatogr. B 822: 54–60.

336


60. Lin, B. L., Shen, X. D., and Cui, S. 2007. Application of nanosized Fe3O4 in anticancer drug carriers with target-orientation and sustained-release properties. Biomed. Mater. 2: 132–134. 61. Safaiikova, M., Roy, I., Gupta, M. N., and Safaiik, I. 2003. Magnetic alginate microparticles for purification of a-amylases. J. Biotechnol. 105: 255–260. 62. Chouly, C., Pouliquen, D., Lucet, I., Jeune, J. J., and Jallet, P. 1996. Development of superparamagnetic nanoparticles for MRI: Effect of particle size, charge and surface nature on biodistribution. J. Microencapsulation 13: 245–255. 63. Schulze, K., et al. 2006. Uptake and biocompatibility of functionalized poly(vinyl alcohol) coated superparamagnetic maghemite nanoparticles by synoviocytes in vitro. J. Nanosci. Nanotechnol. 6: 2829–2840. 64. Zheng, W., Gao, F., and Gu, H. Carboxylique magnetic polymer nanolatexes: Preparation, characterization and biomedical applications. J. Magn. Magn. Mater. 293: 199–205. 65. Maensiri, S., Laokul, P., and Klinkaewnaroung, J. 2006. A simple synthesis and room-temperature magnetic behavior of Co-doped anatase TiO2 nanoparticles. J. Magn. Magn. Mater. 302: 448–453. 66. Kryszewski, M. and Jeska, J. K. 1998. Nanostructured conducting polymer composites—superparamagnetic particles in conducting polymers. Synthet. Metal. 94: 99–104. 67. Pich, A., Bhattacharyaa, S., Ghoshb, A., and Adler, H. J. P. 2005. Composite magnetic particles: Encapsulation of iron oxide by surfactant-free emulsion polymerization. Polymer 46: 4596–4603. 68. Xie, X., Zhang, X., Zhang, H., Chen, D., and Fei, W. Y. 2004. Preparation and application of surface-coated superparamagnetic nanobeads in the isolation of genomic DNA. J. Magn. Magn. Mater. 277: 16–23. 69. Ushakova, M. A., Chernyshev, A. V., Taraban, M. B., and Petrov, A. K. 2003. Observation of magnetic field effect on polymer yield in photoinduced dispersion polymerization of styrene. Eur. Polym. J. 39: 2301–2306. 70. Deng, Y., Wang, L., Yang, W., Fu, S., and Elaissari, A. 2003. Preparation of magnetic polymeric particles via inverse microemulsion polymerization process. J. Magn. Magn. Mater. 257: 69–78. 71. Gomez-Lopera, S. A. P., Plaza, R. C., and Delgado, A. V. 2001. Synthesis and characterization of spherical magnetite/biodegradable polymer composite particles. J. Colloid Interface Sci. 240: 40–47. 72. Go, K. G., Bulte, J. W., De Ley, L., The, T. H., Kamman, R. I., Hulstaert, C. E., Blaauw, E. H., and Ma, L. D. 1993. Our approach towards developing a specific tumor-targeted MRI contrast agent for the brain. Eur. J. Radiol. 16: 171–175. 73. Zielhuis, S. W., Nijsen, J. F., Seppenwoolde, J. H., Zonnenberg, B. A., Bakker, C. J., Hennink, W. E., Van Rijik, P. P., and Van Schip, A. D. 2005. Lanthanide bearing microparticulate systems for multimodality imaging and targeted therapy of cancer. Curr. Med. Chem. Anticancer Agents 5: 303–313. 74. Pauser, S., Reszka, R., Wagner, S., Wolf, K. J., Buhr, H. I., and Berger, G. 1997. Liposome-encapsulated superparamagnetic iron oxide particles as markers in an MRI-guided search for tumor-specific drug carriers. Anticancer Drug Des. 12: 125–135. 75. Elaissari, A., Veyret, R., Mandrand, B., and Chatterjee, J. 2003. Biomedical application for magnetic latexes. In: A. Elaissari (Ed.), Colloidal Biomolecules, Biomaterials, and Biomedical Applications, Vol. 116, pp. 1–26, Surfactant Science Series. Marcel Dekker. 76. Kawaguchi, H., Sakamoto, K., Ohtsuka, Y., Ohtake, T., Sekiguchi, H., and Iri, H. 1989. Fundamental study on latex reagents for agglutination tests. Biomaterials 10: 225–229. 77. Stoll, S., Lanet, V., and Pefferkorn, E. 1993. Kinetics and modes of destabilization of antibody-coated polystyrene latices in the presence of antigen: Reactivity of the system IgG–IgM. J. Colloid Interface Sci. 157: 302–311. 78. Veyret, R. 2003. Elaboration de particules magnétiques pour la capture specifique et non specifique d’acides nucleiques et pour la capture generique de virus. These Université Claude bernard-Lyon 1: 157. 79. Rembaum, A., Yen, R. C. K., Kempner, D. H., and Ugelstad, J. 1982. Cell labeling and magnetic separation by means of immunoreagents based on polyacrolein microspheres. J. Immunol. Methods 52: 341–351. 80. Veyret, R., Elaissari, A., Marianneau, P., Sall, A. A., and Delair, T. 2005. Magnetic colloids for the generic capture of viruses. Anal. Biochem. 346: 59–68. 81. Charles, M. H., Charreyre, M. T., Delair, T., Elaissari, A., and Pichot, C. 2001. Oligonucleotide-polymer nanoparticle conjugates: Diagnostic applications. STP Pharma. Sci. 11: 251–263. 82. Meza, M. 1997. Application of magnetic particles in immunoassays. In: U. Häfeli, W. Schütt, J. Teller, and M. Zborowski (Eds), Scientific and Clinical Applications of Magnetic Carriers, pp. 303–309. New York: Plenum Press. 83. Furdui, V. I. and Harrison, D. J. 2004. Immunomagnetic T cell capture from blood for PCR analysis using microfluidic system. Lab Chip 4: 614–618.


337

84. Xie, X., Zhang, X., Yu, B., Gao, H., Zhang, H., and Fei, W. 2004. Rapid extraction of genomic DNA from saliva for HLA typing on microarray based on magnetic nanobeads. J. Magn. Magn. Mater. 280: 164–168. 85. Gijs, M. 2004. Magnetic beads handling on-chip: New opportunities for analytical applications. Microfluid. Nanofluid. 1: 22–40. 86. Jiang, G. and Harrison, D. J. 2000. mRNA isolation in a microfluidic device for eventual integration of cDNA library construction. Analyst 125: 2176–2179. 87. Ostergaard, S., Blankenstein, G., Dirac, H., and Leistiko, O. 1999. A novel approach to the automation of clinical chemistry by controlled manipulation of magnetic particles. J. Magn. Magn. Mater. 194: 156–162. 88. Fan, Z. H., Mangru, S., Granzow, R., Heaney, P., Ho, W., Dong, Q., and Kumar, R. 1999. Dynamic DNA hybridization on a chip using paramagnetic beads. Anal. Chem. 71: 4851–4859. 89. Choi, J., Oh, K. W., Thomas, J., Heineman, W., Haslsall, H., Nevin, J., Helmicki, A., Henderson, H., and Ahn, C. 2002. An integrated microfluidic biochemical detection system for protein analysis magnetic bead-based sampling capabilities. Lab Chip 2: 27–30. 90. Choi, J. W., Liakopoulos, T. M., and Ahn, C. H. 2001. An on-chip magnetic bead separator using spiral electromagnets with semi-encapsulated permalloy. Biosens. Bioelectron. 16: 409–416. 91. Schotter, J., Kamp, P. B., Pühler, A., Reiss, G., and Brückl, H. 2004. Comparison of a prototype magnetoresistive biosensor to standard fluorescent DNA detection. Biosens. Bioelectron. 19: 1149–1156. 92. Baselt, D. R., Lee, G. U., and Colton, R. J. 1996. A biosensor based on force microscope technology. J. Vac. Sci. Technol. B 14: 789–794. 93. Brzeska, M., Panhorst, M., Kamp, P. B., Schotter, J., Reiss, G., Pühler, A., and Brückl, H. 2004. Detection and manipulation of biomolecules by magnetic carriers. J. Biotechnol. 112: 25–33. 94. Rife, J. C., Miller, M. M., Sheehan, P. E., Tamanaha, C. R., and Whitman, L. J. 2003. Design and performance of GMR sensors for the detection of magnetic microbeads in biosensors. Sens. Actuat. A 107: 209–218. 95. Miller, M. M., Sheehan, P. E., Edelstein, R. L., Tamanaha, C. R., Zhong, L., Bounnak, S., Whitman, L. J., and Coton, R. J. 2001. A DNA array sensor utilizing magnetic microbeads and magnetoelectronic detection. J. Magn. Magn. Mater. 225: 138–144. 96. Edelstein, R. L., Tamanaha, C. R., Sheehan, P. E., Miller, C. A., Baselt, D. R., Whitman, L. J., and Coton, R. J. 2000. The BARC biosensor applied to the detection of biological warfare agents. Biosens. Bioelectron. 14: 805–813. 97. Kriz, K., Gehrke, J., and Kriz, D. 1998. Advancement toward magneto immunoassays. Biosens. Bioelectron. 13: 817–823. 98. Berggren, K. C., Radevik, K., and Kriz, D. 1996. Magnetic permeability measurements in bioanalysis and biosensors. Anal. Chem. 68: 1966–1970.

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

340


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

Colloidal Dispersion of Metallic Nanoparticles

341

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).

342


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


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

344


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


345

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

346


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 ( “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

348


[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 (

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

Molecular and Colloidal Electro-optics (Surfactant Science)

Molecular and Colloidal Electro-optics (Surfactant Science)

Colloidal Biomolecules, Biomaterials, and Biomedical Applications (Surfactant Science)

Colloidal Biomolecules, Biomaterials, and Biomedical Applications (Surfactant Science Series)

Reactions and Synthesis in Surfactant Systems (Surfactant Science Series)

Thermal Behavior of Dispersed Systems (Surfactant Science)

Surfactant Science and Technology

Surfactant science and technology

Structure and Properties of Polymers

Structure and Properties of Composites

The Properties of Water and their Role in Colloidal and Biological Systems, Volume 16 (Interface Science and Technology)

Structure and Properties of Ceramics

Handbook of Detergents, Part A - Properties (Surfactant Science Series)

Microporous Media: Synthesis, Properties, and Modeling (Surfactant Science)

Applications of functional analysis and operator theory (Mathematics in Science and Engineering, Volume 146)

Mixed Surfactant Systems

Polymer-surfactant systems

Solubilization in Surfactant Aggregates (Surfactant Science)

Organized Solutions (Surfactant Science)

Cationic Surfactants (Surfactant Science)

Clathrochelates.. Synthesis, Structure and Properties

Quasicrystals: structure and physical properties

Chemical and Functional Properties of Food Saccharides

Clathrochelates: Synthesis, Structure and Properties

Foldamers: Structure, Properties, and Applications

Perovskites: Structure, Properties and Uses

Structure and Properties of Liquid Crystals (Bookmarked)

Chemical and Functional Properties of Food Components

Structure and properties of engineering alloys

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

Molecular and Colloidal Electro-optics (Surfactant Science)

Molecular and Colloidal Electro-optics (Surfactant Science)

Colloidal Biomolecules, Biomaterials, and Biomedical Applications (Surfactant Science)

Colloidal Biomolecules, Biomaterials, and Biomedical Applications (Surfactant Science Series)

Reactions and Synthesis in Surfactant Systems (Surfactant Science Series)

Thermal Behavior of Dispersed Systems (Surfactant Science)

Surfactant Science and Technology

Surfactant science and technology

Structure and Properties of Polymers

Structure and Properties of Composites

The Properties of Water and their Role in Colloidal and Biological Systems, Volume 16 (Interface Science and Technology)

Structure and Properties of Ceramics

Handbook of Detergents, Part A - Properties (Surfactant Science Series)

Microporous Media: Synthesis, Properties, and Modeling (Surfactant Science)

Applications of functional analysis and operator theory (Mathematics in Science and Engineering, Volume 146)

Mixed Surfactant Systems

Polymer-surfactant systems

Solubilization in Surfactant Aggregates (Surfactant Science)

Organized Solutions (Surfactant Science)

Cationic Surfactants (Surfactant Science)

Clathrochelates.. Synthesis, Structure and Properties

Quasicrystals: structure and physical properties

Chemical and Functional Properties of Food Saccharides

Clathrochelates: Synthesis, Structure and Properties

Foldamers: Structure, Properties, and Applications

Perovskites: Structure, Properties and Uses

Structure and Properties of Liquid Crystals (Bookmarked)

Chemical and Functional Properties of Food Components

Structure and properties of engineering alloys

Recommend Documents